{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0001","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Piotr Dworczak; Alex Smolin.\n> Canonical citation: Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"033e78f202e563bc87bd1fb5748500c15cc4c997f5f57eb26d59efb0cc74d85a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0002","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Robust Trust","text":"# Robust Trust\n\n**Authors:** Piotr Dworczak; Alex Smolin\n\n**Manuscript date:** 2026-03-19\n\n#### Abstract\n\nAn agent chooses an action based on her private information and a recommendation from an informed but potentially misaligned adviser. With a known probability, the adviser truthfully reports his signal; with the remaining probability, he can send any message. We characterize optimal robust decision rules that maximize the agent's worst-case expected payoff. Every optimal rule is equivalent to a trust-region policy in belief space: the adviser's reported beliefs are taken at face value if they fall within the trust region but are otherwise clipped to the trust region's boundary. We derive alignment thresholds above which advice is strictly valuable and fully characterize the solution in both binary-state and binary-action environments.\n\nKeywords: robustness, information design, misalignment, human-AI interactions.\nJEL Codes: C72, D81, D83\n\n[^0]","text_sha256":"b444abb5933f8bdf391805db81c01965456137270e84d2a20bee6ac37c9af3dd"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0003","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nModern AI systems increasingly influence decisions with large and sometimes irreversible consequences, including autonomous driving, medical triage, hiring, and credit or security screening (see, e.g., Maslej et al. (2025)). Their appeal is straightforward: they can synthesize information at scale and provide recommendations that exceed unaided human performance in many tasks. The central risk is also well-recognized: when a system is opaque, complex, and trained or deployed under imperfect objectives, users may not be able to tell whether a recommendation is merely noisy, systematically biased, or actively harmful. The misalignment problem is a particularly serious concern in high-stakes environments, and its mitigation is key to ensuring the safe adoption of AI-aided decision making (see, e.g., Russell (2019)).\n\nIn this paper, we study how a decision-maker should use AI when the system may be misaligned. Taking the AI's information structure and an exogenous alignment probability as given, we characterize the optimal robust decision rule that maximizes the decision-maker's payoff under the assumption of worst-case AI behavior in case of misalignment.\n\nOur model features an agent who chooses an action under uncertainty about the state of the world. The agent has access to a private signal reflecting her expertise or contextual information, but she can additionally rely on reports from an adviser. The adviser observes complementary information about the state and sends a message to the agent.\n\nCrucially, the adviser is aligned and reports his information truthfully only with some known alignment probability. With the remaining probability, the adviser is misaligned and can send an arbitrary message. In practice, AI misalignment could take several distinct forms with ambiguous implications for the agent's decision (see, e.g., Amodei et al. (2016)). We therefore adopt a robust approach: we assume that the agent is not willing to make any assumptions about the behavior of the misaligned adviser and hence chooses a policy that maximizes her expected payoff guarantee across all possible forms of misalignment. In the model, this is conceptualized as the misaligned adviser attempting to minimize the agent's payoff. We emphasize, however, that the robust optimality criterion reflects the inability to rule out any output of the misaligned AI, rather than a direct concern that misaligned AI is actively adversarial.\n\nOur main structural result shows that the optimal robust policy can be summarized by a single, interpretable object that we call the \"trust region.\" The trust region is a connected set of reported beliefs about the state that the agent takes at face value. When the adviser's reported belief falls inside this region, the agent behaves as if the adviser were truthful: she combines the reported belief with her private information using Bayes' rule and chooses the corresponding Bayes-optimal action. When the reported belief lies outside the trust region, the agent replaces it with the \"closest safe interpretation\" (formalized by the notion of Bregman distance), which is a belief lying on the boundary of the trust region; she then behaves as if that belief had been reported. Operationally, this is an endogenous form of clipping: moderate recommendations are followed while extreme recommendations are discounted and converted into boundary recommendations that the agent is still willing to accept.\n\nIntuitively, if the agent reacted sharply to extreme reports, the misaligned adviser could exploit that sensitivity to induce large losses. The robust policy responds by limiting how far any recommendation can push behavior. The trust region identifies exactly which recommendations are safe to act upon without additional skepticism, and the boundary mapping formalizes how skepticism should be applied outside that set. On one extreme, a trust region equal to the entire belief simplex corresponds to applying the Bayes-optimal response to all reports; on the other extreme, a trust region only containing the prior belief corresponds to ignoring the adviser's reports. Thus, the shape and size of the trust region yield a disciplined answer to a practical design question: when an AI system outputs highly confident or highly unusual recommendations, optimal robust use requires treating those outputs as \"too informative to be trusted\" and translating them into safer boundary inputs before acting.\n\nAn implication of the characterization is that the optimal robust action rule used by the agent must be defensible as optimal for some coherent set of beliefs about the state of the world-the agent never benefits from distorted use of her own private information. An optimal robust rule simply restricts the set of Bayes-optimal action rules that the agent uses. A further consequence is that implementing the optimal trust region policy does not require commitment. Under mild technical assumptions, we prove a minimax theorem which\nimplies existence of a trust region equilibrium in the zero-sum game between the agent and the misaligned adviser. In a trust region equilibrium, the agent's policy and the misaligned adviser's strategy form a saddle point: after every on-path report, the agent's response is Bayes-optimal given the belief induced by the adviser's strategy, and the misaligned adviser's strategy minimizes the agent's expected payoff. Substantively, this means that robust optimal behavior provides the same payoff guarantee that the agent could obtain had she perfectly known the misaligned adviser's strategy. Practically, this result provides a certification tool: to verify that a proposed policy is optimal, it suffices to exhibit a corresponding adversarial reporting strategy that makes that policy a best response at every recommendation.","text_sha256":"93ac73564b24cfb322127708c332d6b5184b4267b83e5eb426153963d4013982"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0004","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"We then ask when consulting a potentially misaligned adviser is worthwhile for the agent. We formalize this question by examining \"minimal viable alignment\", defined as the threshold alignment probability above which the agent can guarantee a strictly higher payoff than the one she could achieve by only relying on her own information. We derive sharp bounds on this threshold that depend only on the richness of the state space and the adviser's signal distribution. As long as information is useful to the agent, alignment probability exceeding half is sufficient for the agent to benefit from the presence of the adviser. This bound is tight in binary-state problems. In multidimensional settings, however, minimal viable alignment can be much lower-in some cases as low as the reciprocal of the number of states. Thus, when the state space is rich, AI advice can be robustly valuable even when alignment is very unlikely.\n\nOur general characterization becomes particularly sharp when the state space is binary, so that the ground truth is whether a given statement is true or false. In this setting, an adviser's message can be summarized by the implied probability of the statement being true. The trust region is an interval containing the prior belief. Recommendations inside the interval are trusted and acted upon as reported. Recommendations outside the interval are mapped into the nearest endpoint. The misaligned adviser sends messages that push the induced belief to the endpoint that is most harmful for the agent. This structure delivers a sharp phase transition. If the alignment probability is below one half, the optimal interval collapses to the prior belief and the agent ignores the adviser. If the alignment probability is above one half and the agent's decision problem is sufficiently rich in the sense that every\npiece of information is valuable, there is a unique trust interval. This interval expands monotonically with alignment, approaching full trust as the alignment probability approaches one. In addition, we show that the location of the trust region-whether it is skewed towards high or low beliefs-depends on the relative curvature of the agent's indirect utility function. That curvature can be interpreted as a measure of sensitivity of the agent's optimal action to information.\n\nOur characterization also yields a closed-form solution when the agent's downstream choice is binary (e.g., to accept or reject an application) and the agent has no private information. In such problems, the optimal robust use of advice is generically all-or-nothing: either the trust region is the entire belief simplex or it collapses to the prior belief. Which regime obtains is determined by an alignment threshold that depends only on the relative value of the adviser's information across the two actions. In particular, if the alignment probability is below one half, the agent cannot robustly benefit from the presence of the adviser.\n\nFinally, we examine environments where uncertainty concerns many possible states and actions. Here, the geometry of the trust region plays a major role: some directions of belief change are far more consequential than others because they trigger actions whose payoffs are highly sensitive to the true state. In a robust solution, the misaligned adviser chooses recommendations that are farthest from the truth within the trusted set in an incentive-based sense, again formalized by Bregman distance. In general, the trust region may have a complex shape, for example, it need not be convex. In symmetric environments, however, the trust region inherits the symmetry of incentives and information and can be tractably characterized.\n\nThe primary application of our framework is the AI alignment problem; in Section 4.1, we develop a parametric example to illustrate how recommendations from an AI system should be combined with human expertise in applications such as medical triage. However, our model is general, and could be applied in other contexts where the information source is not fully trusted. For example, our framework can be seen as a theory of behavioral belief updating, in which the decision-maker wants to ensure some degree of protection against potential misspecification; from this perspective, our main result provides a robust-optimality foundation for the phenomenon known as \"extreme-belief aversion\" (see Benjamin (2019) for a review of the experimental evidence and Whitmeyer (2026) for a related theoretical\nframework). The assumption that the misaligned adviser attempts to minimize the agent's payoff could also be interpreted literally in some contexts, such as when messages exchanged between two allies can be intercepted and manipulated by an adversary.","text_sha256":"d9e8dfdaebaa336920970fa04e4c13ab8d0e9a75f4b3704f0804c8ee70592c17"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0005","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1.1 Literature review","text":"### 1.1 Literature review\n\nOur model is closely related to two foundational models in information economics: the cheap talk model (Crawford and Sobel (1982)) and the Bayesian-persuasion model (Kamenica and Gentzkow (2011)). Relative to the cheap-talk model, our framework effectively assumes that the Sender maximizes the Receiver's utility with some probability $\\alpha$ and minimizes the Receiver's utility with the complementary probability $1-\\alpha .{ }^{1}$ Our characterization of trust region equilibria shows that equilibrium behavior is very different from that arising in the more standard constant-bias case; in particular, information transmission is perfect for intermediate beliefs of the Sender but completely blocked for extreme beliefs. Relative to the Bayesian-persuasion model, due to the minimax theorem which we prove in our setting, our framework is equivalent to the case in which the Sender tries to minimize the Receiver's payoff but is committed to revealing his signal truthfully with probability $\\alpha$. One of our contributions is to provide a characterization of threshold levels of $\\alpha$ above which the adversarial Sender cannot prevent the Receiver from learning some information.\n\nMore recently, several papers in information economics have studied models in which the Sender is truthful-or committed to an information structure-with some probability, but may otherwise send fake or manipulated messages. To the best of our knowledge, this literature did not consider the case that is central to our AI-alignment motivation: a misaligned Sender who is adversarial and seeks to minimize the Receiver's payoff. Instead, the misaligned Sender is modeled as having known and often state-independent preferences. Lipnowski, Ravid, and Shishkin (2022) and Min (2021) study settings in which the Sender is committed to an information structure with some probability and sends a cheap-talk message otherwise. Glazer, Herrera, and Perry (2020) and Lahr and Winkelmann (2019) analyze communication games where some senders are truthful and others have state-independent preferences, such as\n\n[^1]pushing the Receiver's beliefs upward. Alonso and Padró i Miquel (2025) model competitive capture of public opinion by assuming that informative signals about a binary state may be manipulated (i.e., replaced by an arbitrary message) by two opposed \"interested parties,\" one of which wants the induced beliefs to be as high as possible and the other one as low as possible. They characterize a communication equilibrium in which citizens correctly update beliefs given the equilibrium strategies of the interested parties. Interestingly, the structure of their communication equilibria shares similarities with our trust region equilibria in the special case of a binary state: messages in some intermediate interval are interpreted at face value, while messages outside of that interval induce beliefs at the endpoints of the interval.\n\nOur modeling of uncertainty about the behavior of the adviser is inspired by the classical Hurwicz criterion, also known as the alpha-max-min approach (Hurwicz (1951)), under which the decision-maker maximizes a weighted sum of her best-case and worst-case payoffs. We interpret $\\alpha$ as the probability of alignment. A similar criterion has recently been applied in the context of information design by Dworczak and Pavan (2022).\n\nThe version of our model in which the agent does not have private information is related to the delegation literature in that the agent effectively chooses which decisions to delegate to an informed adviser. Within that literature, the closest paper is Frankel (2014) who adopts a worst-case approach with respect to the adviser's preferences, assumed to lie in a known set. More recently, Alonso, Gan, and Hu (2026) show optimality of convex delegation sets under max-min preferences when the principal only knows the agent's preferred action in every state, but not the exact shape of the agent's quasi-concave utility function. Our setting differs both in primitives and in methods: we require robustness to the adviser's behavior in case of misalignment and the resulting optimization problem has a different structure.\n\nRegarding the AI alignment problem, a few approaches have recently been proposed in the microeconomic theory literature. Chen, Ghersengorin, and Petersen (2024) develop a model of screening for alignment in an environment in which the decision-maker can simulate the task and impose imperfect recall on AI, obscuring whether the task is real or part of a test. ${ }^{2}$ Closer to our approach, Fudenberg and Liang (2025) assume that the AI system is aligned with some known probability and that it performs adversarially in case of misalignment.\n\n[^2]Unlike us, Fudenberg and Liang (2025) assume that the decision-maker can impose the true unconditional distribution of optimal actions but faces non-Bayesian uncertainty about the correlation of the optimal action with a set of covariates she controls. Correspondingly, their main research question is which covariates should be revealed to AI. In our framework, the decision-maker knows the distribution of AI's signals-uncertainty is only about AI's behavior in case of misalignment-and thus she would never optimally disclose her private information to AI. Overall, these papers focus on different (and complementary) aspects of the misalignment problem: Chen et al. (2024) ask how to test AI's alignment; Fudenberg and Liang (2025) ask how to provide information to a misaligned AI; and we ask how the decision-maker should combine advice from a misaligned AI with her private information.\n\nMore broadly, our framework is part of a rapidly growing literature trying to understand optimal human-AI interactions. Closest to our paper are Dreyfuss and Hoong (2025) and Agarwal, Moehring, and Wolitzky (2026) who also adopt an information-design approach; the latter ask how AI advice interacts with human decision-making in the presence of potential biases and when the decision-maker's effort in acquiring information is endogenous.","text_sha256":"2d01283b7a2dd292f4254ae8031610fa17e7977082f4ed98ff59bbf7aa7fd2d6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0006","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"## 2 Model\n\nA state $\\omega$ is drawn from a finite state space $\\Omega$, with $|\\Omega|=N$, according to a full-support prior distribution $\\mu_{0} \\in \\Delta(\\Omega)$. An adviser observes partial information about $\\omega$, captured by a signal $s$ whose distribution is pinned down by a signal function $\\pi: \\Omega \\rightarrow \\Delta(S) .^{3}$ We will identify the adviser's information with the posterior belief about the state that a signal realization induces; let $S=\\Delta(\\Omega)$ and renormalize so that $s$ is equal to the posterior belief about $\\omega$ induced by $s$. Let $\\tau$ denote the unconditional distribution of the adviser's posteriors $s$, with $M=\\operatorname{supp}(\\tau)$.\n\nAn agent takes an action $a \\in A$, where $A$ is a compact metric set. The agent observes a private type $\\theta \\in \\Theta$, where $\\Theta$ is a compact metric set, that captures the agent's own\n\n[^3]information about $\\omega$ and her preferences. The distribution of the type $\\theta$ is determined by a signal function $f: \\Omega \\rightarrow \\Delta(\\Theta)$. We assume that, conditional on the state, $s$ and $\\theta$ are distributed independently. The agent's ex-post payoff is given by a utility function $u(a, \\omega, \\theta)$, assumed continuous in $a$.\n\nThe adviser sends a message $m \\in \\Delta(\\Omega)$ to the agent, where, without loss of generality, we take the message space to be the space of beliefs about the state. The agent chooses a strategy $\\sigma: \\Delta(\\Omega) \\times \\Theta \\rightarrow \\Delta(A)$ that assigns a distribution over actions to each message-type pair. Let $\\Sigma$ denote the set of all such strategies.\n\nThe adviser's strategy maps his beliefs into distributions over messages sent to the agent. With probability $\\alpha$, the adviser is aligned and non-strategically reports his belief according to the identity function $\\mathrm{id}: M \\rightarrow M$ such that $\\mathrm{id}(m)=m$ for all $m \\in M .{ }^{4}$ With probability $1-\\alpha$, the adviser is misaligned and sends a message according to some strategy $\\beta: M \\rightarrow \\Delta(\\Delta(\\Omega))$. Let $\\mathcal{B}$ denote the set of all such strategies.\n\nFaced with non-Bayesian uncertainty about the form of misalignment, the agent adopts a cautious posture and aims to maximize her guaranteed payoff. Concretely, she evaluates each possible strategy $\\sigma$ according to its worst-case payoff\n\n$$\nV(\\sigma) \\triangleq \\alpha \\mathbb{E}_{\\mathrm{id}, \\sigma}[u(a, \\omega, \\theta)]+(1-\\alpha) \\inf _{\\beta \\in \\mathcal{B}} \\mathbb{E}_{\\beta, \\sigma}[u(a, \\omega, \\theta)],\n$$\n\nwhere the expectations are taken with respect to the underlying distributions of the primitive variables $\\omega, s$, and $\\theta$, as well as the respective adviser's and agent's strategies. ${ }^{5}$ We will call any misaligned adviser's strategy $\\beta$ that attains the infimum in expression (1) for a fixed strategy $\\sigma$ of the agent an adversarial strategy against $\\sigma$.\n\nOur main goal is to characterize the agent's optimal strategy $\\sigma^{*}$ that attains:\n\n$$\nV^{*} \\triangleq \\sup _{\\sigma \\in \\Sigma} V(\\sigma) .\n$$\n\n[^4]","text_sha256":"739438c11d0f324bae0117aeb316713bd75868854cc0b4f1a23819e160e35cad"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0007","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Main Results","text":"## 3 Main Results","text_sha256":"841e606e1a29998dd91af5d2b4b2d76fd76657c7b4b80600f11120ccfffd6be4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0008","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.1 Trust Region Strategies","text":"### 3.1 Trust Region Strategies\n\nIn what follows, it will be convenient to separate the dependence of the agent's strategy on the adviser's message and the agent's private information. To this end, we call a private strategy $\\hat{\\sigma}$ the mapping from types to actions $\\hat{\\sigma}: \\Theta \\rightarrow \\Delta(A)$ that specifies how the agent uses her private information. We will refer to the agent's belief about the state prior to updating based on her private type $\\theta$ as the interim belief. If the agent has an interim belief $\\mu$ and uses a private strategy $\\hat{\\sigma}$, her expected payoff is:\n\n$$\nU(\\hat{\\sigma}, \\mu) \\triangleq \\mathbb{E}_{\\omega \\sim \\mu, \\hat{\\sigma}}[u(a, \\omega, \\theta)],\n$$\n\nwhere the expectation is taken with respect to the conditional distribution of $\\theta$ and the distribution of agent's actions induced by $\\hat{\\sigma} .^{6}$ A private strategy $\\hat{\\sigma}$ is called Bayes-optimal for belief $\\mu \\in \\Delta(\\Omega)$ if it maximizes the agent's expected payoff when she holds an interim belief $\\mu: \\hat{\\sigma} \\in \\arg \\max _{\\hat{\\sigma}} U(\\hat{\\sigma}, \\mu)$. The agent's strategy can be viewed as a specification of a private strategy for each possible message received from the adviser, $\\sigma \\sim(\\hat{\\sigma}(m))_{m \\in \\Delta(\\Omega)}$.\n\nDefinition 1. $\\sigma \\sim(\\hat{\\sigma}(m))_{m \\in \\Delta(\\Omega)}$ is a trust region strategy (TRS) if there exists a compact set $T \\subset \\Delta(\\Omega)$ such that\n\n1. if $m \\in T, \\hat{\\sigma}(m)$ is Bayes-optimal for $m$,\n2. if $m \\notin T, \\hat{\\sigma}(m)$ is Bayes-optimal for $P(m)$, where $P(m) \\in \\arg \\max _{m^{\\prime} \\in T} U\\left(\\hat{\\sigma}\\left(m^{\\prime}\\right), m\\right)$.\n\nIntuitively, under a TRS, the agent treats messages $m$ reported within the trust region $T$ \"at face value,\" i.e., she takes an optimal action treating $m$ as her correct interim belief about the state. If a message $m$ does not belong to the trust region $T$, the agent maps $m$ to the trust region by acting as if her interim belief were $P(m) \\in T$. The point $P(m)$ is chosen to maximize, over all beliefs in the trust region, the agent's expected payoff under distribution $m$ when the action is taken to be optimal for $P(m)$.\n\n[^5]To provide further intuition, with slight abuse of notation, let\n\n$$\nU(\\mu) \\triangleq \\max _{\\hat{\\sigma}} U(\\hat{\\sigma}, \\mu)\n$$\n\nbe the payoff to the agent when she uses the Bayes-optimal strategy at belief $\\mu$. Note that $U(\\mu)$ is a convex function on $\\Delta(\\Omega)$; moreover, it is differentiable on the interior of the belief simplex if there exists a unique Bayes-optimal private strategy $\\hat{\\sigma}_{0}(\\mu)$ at every belief $\\mu$. In that case, we can define $\\nabla U(\\mu)$ as the gradient of the indirect payoff function, viewed as a function on $\\mathbb{R}^{N}$. It maps each belief $\\mu$ into the $N$-dimensional vector of state-contingent payoffs associated with the Bayes-optimal strategy. ${ }^{7}$ In particular, $U(\\mu)=\\nabla U(\\mu) \\cdot \\mu$, where • denotes the standard dot product in $\\mathbb{R}^{N}$. Moreover, for a TRS $\\sigma$ and any $m^{\\prime} \\in T$, $U\\left(\\hat{\\sigma}\\left(m^{\\prime}\\right), m\\right)=\\nabla U\\left(m^{\\prime}\\right) \\cdot m$. Thus,\n\n$$\n\\arg \\max _{m^{\\prime} \\in T} U\\left(\\hat{\\sigma}\\left(m^{\\prime}\\right), m\\right)=\\arg \\min _{m^{\\prime} \\in T} \\underbrace{U(m)-U\\left(m^{\\prime}\\right)-\\nabla U\\left(m^{\\prime}\\right) \\cdot\\left(m-m^{\\prime}\\right)}_{D_{U}\\left(m, m^{\\prime}\\right)} .\n$$\n\nThe expression $D_{U}\\left(m, m^{\\prime}\\right)$ is called the Bregman distance (associated with function $U$ ) between beliefs $m$ and $m^{\\prime}$. Thus, under a TRS, messages outside of the trust region $T$ are mapped into the \"closest safe interpretation\"-the belief in the trust region $T$ that is closest in the Bregman distance. In particular, $P(m)$ always lies on the visible part of the boundary of $T$ from the perspective of point $m .^{8}$","text_sha256":"480c4c12a4176f67031d5bf6c5167a100a454a97d6865d41ee28e817fba503b4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0009","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.2 Optimality of Trust Region Strategies","text":"### 3.2 Optimality of Trust Region Strategies\n\nWe call two strategies of the agent equivalent if, together with some corresponding adviser's adversarial strategies, they induce the same joint distribution over states, types, messages, and actions. The importance of TRSs stems from the following key result.\n\n[^6]Theorem 1 (Trust Region Solution). Any optimal strategy $\\sigma^{*}$ is equivalent to a trust region strategy with a connected trust region $T$.\n\nProof. See Appendix A.1. $\\square$\n\nTheorem 1 states that any optimal strategy can be interpreted as a TRS for some connected trust region $T$. This result provides a sharp characterization of optimal robust behavior under misalignment risk. Messages in the trust region are taken at face value while messages outside the trust region are mapped into the closest safe interpretation within the trust region. Thus, the agent's problem reduces to choosing the trust region $T$.\n\nIf the adviser is always aligned, a TRS with the trust region equal to the entire belief space is trivially optimal. On the other extreme, if the adviser is always misaligned, the optimal TRS has a trust region equal to the prior belief-the agent always ignores the message of the adviser. In Section 3.4, we explore conditions under which the trust region is guaranteed to be non-trivial. In general, however, it is difficult to pin down the exact shape of the optimal trust region. A trade-off is created by two opposing forces: when the trust region expands, the expected payoff of the agent weakly increases conditional on the adviser being aligned but weakly decreases conditional on the adviser being misaligned. In Section 4, we study the binary-state case, in which the trust region is an interval; in Section 5, we look at the case of multiple states but binary actions, in which the trust region is either the prior belief or the entire simplex.\n\nTheorem 1 shows that the trust region may be chosen to be connected. It need not, however, be convex in belief space. The reason is that convexifying the trust region by adding the line segment between two trusted beliefs may lead some types of the misaligned adviser to use those newly trusted intermediate reports; the resulting losses may outweigh the gains from additional truthful reports by the aligned adviser. Our proof instead establishes convexity in dual coordinates, that is, in the space of state-contingent payoff vectors. Each belief can be associated with the state-contingent payoff induced by a Bayes-optimal private strategy at that belief; when the optimum is unique, this payoff vector is given by the gradient $\\nabla U(\\mu)$ of the indirect utility function. Although the misaligned adviser's payoff is generally nonlinear in the reported belief, it is linear in the induced state-contingent payoff vector. In particular,\nan adviser with belief $\\mu$ chooses among trusted reports so as to minimize $\\mu \\cdot w$, where $w$ ranges over the state-contingent payoff vectors induced by beliefs in the trust region. Thus, the object that can be convexified is not the trust region itself, but the associated set of induced payoff vectors. Convexifying this dual set, in turn, connects the corresponding set of beliefs in the trust region.\n\nTo further understand the geometry of the optimal trust region, note that the misaligned adviser with belief $\\mu$ will choose a message $m \\in T$ to minimize $U(\\hat{\\sigma}(m), \\mu)$, or equivalently, to maximize the Bregman distance between $\\mu$ and $m$; in particular, the chosen message $m$ must lie on the boundary of the trust region (see Section 5.2 for an extended discussion). As a consequence, adding non-boundary points to a trust region can only weakly increase the agent's payoff. Formally, we say that a set $A \\subset \\mathbb{R}^{N}$ is non-hollow if it contains all points $x \\in \\mathbb{R}^{N}$ with the property that every line going through $x$ intersects $A$ on both sides of $x$.\n\nCorollary 1. Theorem 1 remains true with the additional requirement that $T$ is non-hollow.\nNote that being non-hollow is not implied by connectedness, although it is weaker than convexity. An example of a connected but hollow set is a sphere. If the trust region of some TRS is a sphere, then we can expand the trust region to the corresponding ball, since the misaligned adviser will never send messages in the interior of the ball.\n\nThe trust region is typically not unique and our results in this section emphasized that it can be taken to be a relatively large set. However, when the support $M$ of the adviser's beliefs is finite, it is also possible to construct an optimal discrete trust region $T$ with $|T| \\leq|M|$. Intuitively, at most one belief in the trust region is needed for every possible belief of the aligned adviser. ${ }^{9}$ In such cases, a connected trust region can still be constructed but most beliefs in the trust region are never reported by the adviser. Uniqueness of the trust region can sometimes be established if the adviser's beliefs have full support, $M=\\Delta(\\Omega)$ (see Section 4).","text_sha256":"b327143a1811ec61a3916a807aa2f7d964f02c348e3a04de800bebbdc230b533"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0010","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.3 Robust Rationalizability","text":"### 3.3 Robust Rationalizability\n\nOur model assumes that the agent commits to a strategy at the outset of the game, not knowing the strategy adopted by the misaligned adviser. As we show next, neither the\n\n[^7]commitment assumption nor the timing of moves matter for the value that the agent can achieve. This is because we can construct an optimal solution that is a saddle point of the zero-sum game between the agent and the misaligned adviser. For any strategy $\\beta \\in \\mathcal{B}$ of the adversarial adviser, we let $\\mathbb{P}_{\\beta}(\\cdot \\mid m)$ denote the agent's interim belief induced by message $m$ given the adviser's strategy. ${ }^{10}$\n\nDefinition 2 (Robustly Rationalizable Strategy). A strategy $\\sigma \\sim(\\hat{\\sigma}(m))_{m \\in \\Delta(\\Omega)}$ is robustly rationalizable if there exists an adversarial strategy $\\beta^{*}$ of the misaligned adviser against $\\sigma$ such that for all $m \\in M, \\hat{\\sigma}(m) \\in \\arg \\max _{\\hat{\\sigma}^{\\prime}} U\\left(\\hat{\\sigma}^{\\prime}, \\mathbb{P}_{\\beta^{*}}(\\cdot \\mid m)\\right)$.\n\nThe rationalizability condition means that the agent does not need commitment to follow the strategy. She can view the misaligned adviser as choosing an adversarial reporting strategy such that, after every message, the prescribed private strategy is myopically optimal.\n\nTheorem 2 (Robust Rationalizability). Any robustly rationalizable strategy is optimal. If $M$ and $\\Theta$ are finite, a robustly rationalizable strategy exists.\n\nProof. See Appendix A.2. $\\square$\n\nAssuming finite support of beliefs, ${ }^{11}$ Theorem 2 implies that there exists an adversarial strategy $\\beta^{*}$ for the misaligned adviser such that the agent's optimal strategy is sequentially rational: the agent can simply observe the adviser's message, update her beliefs using Bayes' rule given $\\beta^{*}$, and then use the Bayes-optimal private strategy for the resulting interim belief. In particular, implementing the optimal strategy does not require commitment by the agent.\n\nIn light of Theorem 1, the agent's equilibrium strategy can still be taken to be a TRS. Treating the problem as a zero-sum game between the agent and the misaligned adviser, we will call $\\left(\\sigma^{*}, \\beta^{*}\\right)$ a trust region equilibrium (TRE) if $\\sigma^{*}$ is a TRS that is robustly rationalizable against the adversarial strategy $\\beta^{*}$.\n\nIn a TRE, messages $m \\in M$ in the trust region are taken at face value because they are only reported by the aligned adviser (thus, Bayes' rule implies that $\\mathbb{P}_{\\beta^{*}}(\\cdot \\mid m)=m$ ). Messages\n\n[^8]$m \\in M$ outside of the trust region are reported by both types of the adviser with probabilities such that $\\mathbb{P}_{\\beta^{*}}(\\cdot \\mid m)=P(m)$, where $P(m)$ is the mapping to the boundary of the trust region defined in Theorem 1. Messages $m \\notin M$ are sent with probability zero. In other words, the mappings from messages to beliefs induced by (i) Bayes' rule and (ii) minimizing Bregman distance to the trust region, coincide on the equilibrium path of a TRE. In Section 4, we use this structural property to characterize the trust region in a binary-state setting.\n\nFrom a technical perspective, Theorem 2 provides a practical way of certifying the optimality of solutions in applications, even with infinite belief and message spaces. To construct an optimal solution, it is sufficient to construct a saddle point of the zero-sum game between the agent and the misaligned adviser-verifying the mutual best-response property is often easier than evaluating the agent's objective for every possible strategy.\n\nFinally, Theorem 2 implies that our problem is equivalent to a constrained persuasion problem for the misaligned adviser. When the misaligned adviser moves first, he is effectively choosing a Bayes-plausible distribution of the agent's interim beliefs subject to the constraint that the signal must be truthful in every state with probability at least $\\alpha$; the constraint reflects the presence of the aligned adviser. Thus, the misaligned adviser is effectively attempting to \"jam\" the signal sent by the aligned adviser. We exploit this perspective in the next section to derive thresholds on $\\alpha$ below which no TRE can sustain informative communication.","text_sha256":"8c7e7108546c461eaca21f03191244accc95fa4db8e0c9272517e781dfde9a67"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0011","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.4 Minimal Viable Alignment","text":"### 3.4 Minimal Viable Alignment\n\nIn this section, we derive bounds on the alignment probability $\\alpha$ above which the agent finds it worthwhile to consult the adviser. Equivalently, we characterize the threshold at which the agent's trust region becomes nontrivial.\n\nFormally, define the value of an adviser as\n\n$$\n\\Delta V \\triangleq V^{*}-V_{0},\n$$\n\nwhere $V_{0} \\triangleq \\sup _{\\sigma \\in \\Sigma} \\mathbb{E}_{\\sigma}[u(a, \\omega, \\theta)]$ is the agent's optimal payoff in the absence of the adviser. Since the agent can always ignore the adviser's messages, this value is non-negative, $\\Delta V \\geq 0$.\n\nWe ask when this value is strictly positive, $\\Delta V>0$ (cf. the value of information by Blackwell (1951)).\n\nTo answer this question, we assume that the adviser's beliefs are finitely supported, $|M|=K<\\infty$, and derive a bound on $\\alpha$ that is independent of the agent's problem. If $\\alpha$ is small enough, the misaligned adviser can use a strategy $\\beta$ that \"jams\" the signal created by truthful reporting of the aligned adviser. In such a case, the distribution of interim beliefs $\\mathbb{P}_{\\beta}(\\cdot \\mid m)$ held by the agent is degenerate: the trust region contains only the prior belief. However, if $\\alpha$ is large enough, there exists no strategy for the misaligned adviser that makes the equilibrium message uninformative. In such cases, as long as information is useful to the agent $(U(\\mu)$ is strictly convex in the relevant range), $\\Delta V$ must be strictly positive.\n\nDefinition 3 (Minimal Viable Alignment). The minimal viable alignment $\\operatorname{MVA}(\\tau)$ is the smallest upper bound on $\\alpha$ for which there exists a strategy $\\beta$ of the misaligned adviser such that the induced interim belief satisfies $\\mathbb{P}_{\\beta}(\\cdot \\mid m)=\\mu_{0}$ for every $m \\in M$.\n\nMVA depends on the adviser's information $\\tau \\in \\Delta(\\Delta(\\Omega))$. Define the rank of the matrix of the adviser's posteriors $\\mu_{1}, \\ldots, \\mu_{K} \\in \\operatorname{supp} \\tau$ :\n\n$$\nR(\\tau) \\triangleq \\operatorname{rank}\\left(\\left[\\begin{array}{llll}\n\\mu_{1} & \\mu_{2} & \\cdots & \\mu_{K}\n\\end{array}\\right]\\right) .\n$$\n\nRoughly, $R(\\tau)$ captures the richness of the adviser's information: the adviser's beliefs are located in an $(R(\\tau)-1)$-dimensional subspace of the $(N-1)$-dimensional belief simplex $\\Delta(\\Omega)$. For any $\\tau, R(\\tau) \\leq \\min \\{K, N\\}$. The rank weakly decreases when the adviser's information is garbled. In what follows, we assume that the adviser has some information, $K \\geq 2$, so $R(\\tau) \\geq 2$.\n\nTheorem 3 (Minimal Viable Alignment). The agent strictly benefits from the presence of the adviser, $\\Delta V>0$, in some decision problem (equivalently, in any decision problem with strictly convex $U$ ) if and only if $\\alpha>\\operatorname{MVA}(\\tau)$. For any $\\tau, \\operatorname{MVA}(\\tau) \\in[1 / N, 1 / 2]$. Moreover, for any $\\alpha \\in[1 / N, 1 / 2]$, there exists $\\tau$ such that $\\operatorname{MVA}(\\tau)=\\alpha$. If $R(\\tau)=K$, then $\\operatorname{MVA}(\\tau)=1 / K$.\n\nProof. See Appendix A.3. $\\square$\n\nThe proof of Theorem 3 shows that, for any given $\\tau, \\operatorname{MVA}(\\tau)$ can be computed as the solution to a finite-dimensional linear program. We establish bounds on this solution and then show that these bounds are tight by explicitly constructing adviser information structures that attain every MVA within the admissible range. In fact, the proof yields the stronger statement that, for any $\\tau, \\operatorname{MVA}(\\tau) \\in[1 / R(\\tau), 1 / 2]$.\n\nBy Theorem 3, if the alignment $\\alpha$ exceeds 1/2 (and information is strictly useful everywhere), then the agent always benefits from the presence of the adviser. Conversely, if the state is binary and $\\alpha<1 / 2$, the agent cannot benefit from the adviser. In higher-dimensional problems, the adviser can be valuable at much lower alignment levels. In particular, if $R(\\tau)=K=N$, then it suffices that $\\alpha>1 / N$. Thus, when the state space is very rich, even a small amount of trust is enough for the agent to benefit from the advice of a misaligned adviser.","text_sha256":"621a3074eca7d874cb0162fb837d2da2b98165387c6670d254c5fdec9945f8df"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0012","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Binary State","text":"## 4 Binary State\n\nConsider the case of a binary state, $\\Omega=\\{0,1\\}$ (we can intuitively think of the state as capturing whether a given statement is false or true). The belief is effectively one-dimensional: with slight abuse of notation, let $\\mu \\in[0,1]$ denote the probability of state $\\omega=1$. For expositional clarity, we further assume that the agent's indirect payoff function $U(\\mu)$ is strictly convex and twice differentiable, and the adviser's posterior is distributed over $M=[0,1]$ with a strictly positive probability density $\\tau(\\mu) .{ }^{12}$ In this case, each $\\mu \\in[0,1]$ can be associated with a unique Bayes-optimal private strategy $\\hat{\\sigma}_{0}(\\mu)$.\n\nSince any connected one-dimensional compact set is a closed interval, a straightforward corollary of Theorem 1 is:\n\nCorollary 2. If $|\\Omega|=2$, any optimal strategy $\\sigma^{*}$ is characterized by a trust region $T=[\\underline{\\mu}, \\bar{\\mu}]$. If $m \\in[\\underline{\\mu}, \\bar{\\mu}], \\hat{\\sigma}(m)=\\hat{\\sigma}_{0}(m)$; if $m<\\underline{\\mu}, \\hat{\\sigma}(m)=\\hat{\\sigma}_{0}(\\underline{\\mu})$; if $m>\\bar{\\mu}, \\hat{\\sigma}(m)=\\hat{\\sigma}_{0}(\\bar{\\mu})$.\n\nIf $\\underline{\\mu}=\\bar{\\mu}$, the agent effectively ignores the adviser, implying $\\underline{\\mu}=\\bar{\\mu}=\\mu_{0}$. If $\\underline{\\mu}<\\bar{\\mu}$, the agent plays according to the Bayes-optimal strategy $\\hat{\\sigma}_{0}(\\underline{\\mu})$ if $m \\leq \\underline{\\mu}$, and according to the\n\n[^9]Bayes-optimal strategy $\\hat{\\sigma}_{0}(\\bar{\\mu})$ if $m \\geq \\bar{\\mu}$.\nRecall that the adversarial strategy of the misaligned adviser induces a belief from the trust region that maximizes the Bregman distance from his true posterior belief; when the trust region is an interval, its boundary consists of the two endpoints, and the adversarial strategy admits a simple threshold characterization:\n\nLemma 1. When the agent commits to a TRS with the trust region $T=[\\underline{\\mu}, \\bar{\\mu}]$, the misaligned adviser with belief $\\mu$ finds it optimal to send any message $m \\geq \\bar{\\mu}$ if $\\mu \\leq b(\\underline{\\mu}, \\bar{\\mu})$ and any message $m \\leq \\underline{\\mu}$ if $\\mu \\geq b(\\underline{\\mu}, \\bar{\\mu})$, where\n\n$$\nb(\\underline{\\mu}, \\bar{\\mu})=\\frac{\\int_{\\underline{\\mu}}^{\\bar{\\mu}} \\mu U^{\\prime \\prime}(\\mu) d \\mu}{\\int_{\\underline{\\mu}}^{\\bar{\\mu}} U^{\\prime \\prime}(\\mu) d \\mu}\n$$\n\nProof. See Appendix A.5. $\\square$\n\nLemma 1 states that the misaligned adviser with high enough beliefs $\\mu$ will induce the private strategy Bayes-optimal at the lowest belief in the trust region, $\\underline{\\mu}$, by reporting some message $m$ lower than $\\mu$; similarly, the misaligned adviser with low enough beliefs $\\mu$ will induce the private strategy Bayes-optimal at the highest belief in the trust region, $\\bar{\\mu}$, by reporting some message $m$ higher than $\\bar{\\mu}$. The threshold belief is given by the conditional expectation of a random variable whose distribution is determined by the curvature of the indirect utility function: $b(\\underline{\\mu}, \\bar{\\mu})=\\mathbb{E}[\\nu \\mid \\nu \\in[\\underline{\\mu}, \\bar{\\mu}]]$, where $\\nu$ is distributed with full support over $[0,1]$ according to probability density $U^{\\prime \\prime}(\\cdot) / \\int_{0}^{1} U^{\\prime \\prime}(\\mu) d \\mu$.\n\nTo characterize the trust region's boundaries, we will use the observation from Theorem 2 that it is sufficient to construct mutual best responses for the agent and the misaligned adviser. Lemma 1 characterizes the best response of the misaligned adviser. A best response of the agent must use the Bayes-optimal strategies at each interim belief induced by the adviser's strategy. A necessary condition is that the average interim belief induced by messages $m \\leq \\underline{\\mu}$\nis exactly $\\underline{\\mu}$, and the average interim belief induced by messages $m \\geq \\bar{\\mu}$ is exactly $\\bar{\\mu}$ : ${ }^{13}$\n\n$$\n\\begin{gathered}\n\\frac{\\alpha \\int_{0}^{\\underline{\\mu}} \\mu \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{b(\\underline{\\mu}, \\bar{\\mu})}^{1} \\mu \\tau(\\mu) d \\mu}{\\alpha \\int_{0}^{\\underline{\\mu}} \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{b(\\underline{\\mu}, \\bar{\\mu})}^{1} \\tau(\\mu) d \\mu}=\\underline{\\mu}, \\\\\n\\frac{\\alpha \\int_{\\bar{\\mu}}^{1} \\mu \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{0}^{b(\\underline{\\mu}, \\bar{\\mu})} \\mu \\tau(\\mu) d \\mu}{\\alpha \\int_{\\bar{\\mu}}^{1} \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{0}^{b(\\underline{\\mu}, \\bar{\\mu})} \\tau(\\mu) d \\mu}=\\bar{\\mu} .\n\\end{gathered}\n$$\n\nAs it turns out, these conditions are also sufficient for a TRE:\n\nProposition 1 (Binary-State Characterization). An optimal strategy exists; it is unique and robustly rationalizable. Its trust region, $T=[\\underline{\\mu}, \\bar{\\mu}]$, is equal to the prior belief $\\left\\{\\mu_{0}\\right\\}$ when $\\alpha \\leq 1 / 2$; otherwise, it is defined by the unique solution to the system (6)-(7) that satisfies $\\underline{\\mu} \\leq \\mu_{0} \\leq \\bar{\\mu}$.\n\nProof. See Appendix A.6. $\\square$\n\nNote that while the structure of the trust region characterized by Proposition 1 is simple, the underlying strategy of the misaligned adviser is quite complex in a TRE. By Lemma 1, the misaligned adviser with belief $\\mu \\geq b(\\underline{\\mu}, \\bar{\\mu})$ is indifferent between sending all messages $m \\leq \\underline{\\mu}$ since they all result in the same Bayes-optimal strategy $\\hat{\\sigma}_{0}(\\underline{\\mu})$. In a commitment solution, the misaligned adviser can send any of these messages; for example, he can always send $m=\\underline{\\mu}$. But in a TRE, the strategy $\\beta^{*}$ of the misaligned adviser must be such that every message $m \\leq \\underline{\\mu}$ induces the interim belief $\\underline{\\mu}$ via Bayes' rule. Since all messages $m \\leq \\underline{\\mu}$ are sent on equilibrium path (the aligned adviser simply reports his belief truthfully), $\\beta^{*}$ must rely on the misaligned adviser's indifference to put just enough probability mass on each of these messages to induce $\\underline{\\mu}$.\n\nProposition 1 fully characterizes the optimal trust region. A natural next question is how the trust region depends on the problem's parameters. We offer two comparative statics results, one related to the size of the trust region, and one related to its location.\n\nFirst, higher alignment results in more trust:","text_sha256":"c0e63fa3816afec11bbb46e69dd7d5067e3a081538e65b5e6c56d5978024ed5a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0013","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Binary State","text":"[^10]Proposition 2 (Change in Alignment). When $\\alpha \\geq 1 / 2, \\underline{\\mu}(\\alpha)$ is strictly and continuously decreasing in $\\alpha$ and $\\bar{\\mu}(\\alpha)$ is strictly and continuously increasing in $\\alpha$. At $\\alpha=1 / 2,[\\underline{\\mu}, \\bar{\\mu}]=$ $\\left[\\mu_{0}, \\mu_{0}\\right]$. At $\\alpha=1,[\\underline{\\mu}, \\bar{\\mu}]=[0,1]$.\n\nProof. See Appendix A.7. $\\square$\n\nProposition 2 shows that the trust region gradually and monotonically expands from the prior belief, at $\\alpha \\leq 1 / 2$, to the entire belief simplex. For any $\\alpha<1$, the trust region excludes the most extreme beliefs. Intuitively, the aligned adviser is unlikely to hold such extreme beliefs, whereas the misaligned adviser would be relatively likely to report them if they were included in the trust region.\n\nSecond, we show that the trust region tends to include beliefs at which the decision problem of the agent is less \"information-sensitive.\" In other words, the agent will avoid expanding the trust region to beliefs where small changes in information lead to large changes in the optimal action. We formalize this notion via the indirect utility function $U(\\mu)$, noting that its curvature reflects the sensitivity of the agent's optimal private strategy to her interim beliefs.\n\nDefinition 4 (Information Sensitivity). We say that the indirect utility function $U_{1}(\\mu)$ is less information-sensitive at higher beliefs than the indirect utility function $U_{2}(\\mu)$ if\n\n$$\n\\frac{U_{1}^{\\prime \\prime}(\\mu)}{U_{2}^{\\prime \\prime}(\\mu)} \\text { is decreasing in } \\mu \\text {. }\n$$\n\nThe definition states that the convexity of the indirect utility function $U_{1}$ relative to the convexity of $U_{2}$ is smaller at higher beliefs $\\mu$. Intuitively, under $U_{1}$, the decision of the agent is less sensitive to new information at higher beliefs. It turns out that in this case the trust region will be skewed towards higher beliefs.\n\nProposition 3 (Change in Information Sensitivity). Suppose that $U_{1}(\\mu)$ is less informationsensitive at higher beliefs than $U_{2}(\\mu)$. Then, the trust region $T_{1}$ corresponding to $U_{1}$ is higher in the strong set order than the trust region $T_{2}$ corresponding to $U_{2}$.\n\nProof. See Appendix A.8. $\\square$\n\nProposition 3 shows that the trust region skews towards beliefs at which the agent's optimal action is less sensitive to new information. We illustrate the usefulness of the result in the next subsection, where we consider an application.","text_sha256":"54e676c729a79a6656f9fd9ff7202fd35481fce80713af9f0ced07d4341f4358"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0014","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.1 Application: Medical Triage","text":"### 4.1 Application: Medical Triage\n\nThe agent is a doctor deciding whether a patient should undergo additional testing, $a=1$, or not, $a=0$, based on an interview and a preliminary test result that can be analyzed by AI (e.g., an x-ray image; cf. Agarwal et al. (2025)). We model this by assuming a binary state, where $\\omega=1$ means that the patient is sick and $\\omega=0$ means that the patient is healthy, and with conditionally independent signals for the doctor and AI, both inducing a full-support uniform distribution of posterior beliefs. We let $\\theta \\in[0,1]$ denote the doctor's private belief and $\\mu \\in[0,1]$ denote AI's belief. The doctor's payoff is\n\n$$\nu(a, \\omega)= \\begin{cases}r \\geq 1 & \\text { if } a=1 \\text { and } \\omega=1, \\\\ 1 & \\text { if } a=0 \\text { and } \\omega=0, \\\\ 0 & \\text { if } a \\neq \\omega .\\end{cases}\n$$\n\nThus, the doctor would like to match the action to the state. When $r>1$, payoffs are more sensitive to taking the correct action when the patient is sick, that is, when $\\omega=1$.\n\nFirst best. As a benchmark, consider the case in which the doctor has direct access to $\\mu$. By Bayes' rule, the posterior belief that the state is 1 after observing the realization $(\\mu, \\theta)$ is $p(\\mu, \\theta) \\triangleq \\mu \\theta /(\\mu \\theta+(1-\\mu)(1-\\theta))$. The doctor chooses additional testing, $a=1$, when $p(\\mu, \\theta) \\geq 1 /(1+r)$. The doctor's indirect payoff from interim belief $\\mu$ is\n\n$$\nU(\\mu)=1-\\frac{(1-\\mu)^{2}}{r \\mu+1-\\mu} .\n$$\n\nNote that $U$ is strictly convex: the doctor's rich but imperfect private information makes additional information locally valuable at all beliefs.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: Evolution of the optimal trust region as a function of alignment probability $\\alpha$.\n\nExplicit solution when $r=1$. Suppose that the signal $\\mu$ is reported by an AI system that is aligned with probability $\\alpha$. In the symmetric case $r=1$, we can solve the system of equations (5)-(7) and obtain a symmetric trust region $T_{\\alpha}=[\\underline{\\mu}(\\alpha), 1-\\underline{\\mu}(\\alpha)]$, where\n\n$$\n\\underline{\\mu}(\\alpha)= \\begin{cases}\\frac{1}{2} & \\alpha \\leq \\frac{1}{2}, \\\\ \\frac{\\sqrt{(1-\\alpha)(1+2 \\alpha)}-(1-\\alpha)}{2 \\alpha} & \\alpha>\\frac{1}{2} .\\end{cases}\n$$\n\nFigure 1 depicts the optimal trust region, and Figure 2 illustrates the resulting decision rule for the doctor as a function of the realized signals. In line with Proposition 1 and Proposition 2, the trust region is equal to the prior belief when the alignment probability is below 1/2. In that case, the doctor should not use AI and instead rely exclusively on her own signal, as shown in the left panel of Figure 2. When the alignment probability is above $1 / 2$, the doctor trusts moderate AI reports. Extreme reports, namely those with $m<\\underline{\\mu}(\\alpha)$ or $m>1-\\underline{\\mu}(\\alpha)$, are clipped at the endpoints of the trust region. The optimal decision rule is therefore sensitive to AI's recommendations only in the intermediate range of beliefs. In particular, the AI's signal alone is never sufficient to induce testing without corroboration from the doctor's information, as shown in the middle panel of Figure 2. Finally, as $\\alpha$ approaches 1 , the trust region converges to the entire simplex (Figure 1), and the optimal decision rule converges to the first-best decision rule, as shown in the right panel of Figure 2.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 2: Optimal decision rule in three cases: $\\alpha \\leq 1 / 2, \\alpha=3 / 4$, and $\\alpha=1$.\n\nComparative statics with respect to $r$. When $r>1$, the system of equations (5)-(7) no longer admits a closed-form solution. However, it is easy to verify that, as $r$ increases, $U(\\mu)$ becomes less information-sensitive at higher beliefs. Intuitively, when $r>1$, the objective function makes the doctor effectively place more weight on state 1, in which the patient is sick. As a result, she reacts less strongly to new information when she believes that state 1 is likely. By Proposition 3, increasing $r$ shifts the optimal trust region upward. The doctor therefore trusts higher reports more than before and becomes more skeptical of low reports.\n\nTakeaways. The example resonates with how imaging AI is often deployed in practice. Many real-world systems are used as second readers or prioritization aids rather than as autonomous decision-makers. ${ }^{14}$ The same caution is also consistent with growing evidence on automation bias: when clinicians are shown incorrect AI suggestions, their decisions can deteriorate, including among experienced readers. ${ }^{15}$\n\nOur analysis provides a simple economic rationale for such guardrails. Under the trust-\n\n[^11]region protocol, intermediate probabilities should be used as reported, whereas extremely low or extremely high probabilities should be replaced by the corresponding boundary values of the trusted range. This caps the operational leverage of near-certain predictions: an output presented as almost conclusive is treated as strong evidence, but not as decisive on its own. A conclusion approaching certainty requires corroboration from other sources, such as clinical context and human judgment.","text_sha256":"6bc231210f80247a9c23dd5cf55d08cb9fb1269ba21d76d85c04e4bde4716ad4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0015","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Multiple States","text":"## 5 Multiple States\n\nIn this section, we consider the general case $|\\Omega| \\geq 2$. First, we provide a full characterization of the robustly rationalizable solution in the case of binary private strategies. Second, we analyze the case of rich private strategies and develop the robustly rationalizable solution in a symmetric example.","text_sha256":"6600b7fa590cebbf1eaeb79caa2a5332fe91a491f0ba3314a3c17cb23e8c38e6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0016","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.1 Binary Action","text":"### 5.1 Binary Action\n\nConsider the setting in which the agent has only two pure private strategies, i.e., $A=\\left\\{a_{1}, a_{2}\\right\\}$ and $|\\Theta|=1$. Thus, the agent has no private information and we drop the type throughout.\n\nWithout loss of generality, we can normalize the agent's payoff from action $a_{1}$ to zero, $u\\left(a_{1}, \\omega\\right) \\equiv 0$, and denote the expected payoff from action $a_{2}$ when the adviser's posterior is $\\mu$ by $v(\\mu) \\triangleq \\mathbb{E}_{\\mu}\\left[u\\left(a_{2}, \\omega\\right)\\right]$. Denote by $\\hat{\\tau} \\in \\Delta(\\mathbb{R})$ the distribution of $v$ when $\\mu$ is distributed according to $\\tau \\in \\Delta(\\Delta(\\Omega))$.\n\nDefine the absolute losses and gains from taking the second action relative to the first one:\n\n$$\nL(\\hat{\\tau})=\\int_{-\\infty}^{0}(-v) \\hat{\\tau}(d v), \\quad G(\\hat{\\tau})=\\int_{0}^{+\\infty} v \\hat{\\tau}(d v)\n$$\n\nAlso, define the following threshold:\n\n$$\n\\hat{\\alpha}(\\hat{\\tau})=\\frac{\\max \\{L(\\hat{\\tau}), G(\\hat{\\tau})\\}}{L(\\hat{\\tau})+G(\\hat{\\tau})} .\n$$\n\nTo rule out trivial cases and to simplify the exposition of the optimal strategy, we make the\nfollowing assumption that holds in generic environments.\nAssumption 1 (Genericity). $L(\\hat{\\tau})>0, G(\\hat{\\tau})>0, L(\\hat{\\tau}) \\neq G(\\hat{\\tau}), \\tau(\\{\\mu: v(\\mu)=0\\})=0$.\nProposition 4 (Binary Action Solution). Suppose that Assumption 1 holds. If $\\alpha \\neq \\hat{\\alpha}(\\hat{\\tau})$, then the optimal solution exists, is unique, and is robustly rationalizable. In particular, if $\\alpha>\\hat{\\alpha}(\\hat{\\tau})$, then all messages are trusted, $T=\\Delta(\\Omega)$; if $\\alpha<\\hat{\\alpha}(\\hat{\\tau})$, then no messages are trusted, $T=\\left\\{\\mu_{0}\\right\\}$. If $\\alpha=\\hat{\\alpha}(\\hat{\\tau})$, both full trust and no trust are optimal and robustly rationalizable.\n\nBy Proposition 4, generically, the optimal solution is stark: either all or none of the adviser's messages are trusted. This is in contrast to the binary-state case with a rich strategy space, where the trust region expanded continuously with the alignment probability (Proposition 2).\n\nNotably, only the aggregate quantities $L(\\hat{\\tau})$ and $G(\\hat{\\tau})$ matter for the determination of the trust region; the detailed distribution of relative payoffs $\\hat{\\tau}$ is irrelevant. The threshold $\\hat{\\alpha}(\\hat{\\tau})$ is minimized at $L(\\hat{\\tau})=G(\\hat{\\tau})$, in which case $\\hat{\\alpha}(\\hat{\\tau})=1 / 2$. Hence, if $\\alpha<1 / 2$, the agent never trusts the adviser, regardless of $\\hat{\\tau} .{ }^{16}$ By contrast, for a given $\\alpha>1 / 2$, the trust condition $\\hat{\\alpha}(\\hat{\\tau})<\\alpha$ is equivalent to $(L(\\hat{\\tau}), G(\\hat{\\tau}))$ lying in the cone defined by the two linear inequalities\n\n$$\n(1-\\alpha) L(\\hat{\\tau}) \\leq \\alpha G(\\hat{\\tau}), \\quad(1-\\alpha) G(\\hat{\\tau}) \\leq \\alpha L(\\hat{\\tau}) .\n$$\n\nThus, in binary decision problems, the adviser is beneficial only when the expected gains and losses of one action relative to the other are not too far apart.","text_sha256":"304daa7db94fb46ca9db6445ebbaa7bc553dae7ad4ce5245e399b92bd7c748f2"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0017","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.2 Rich Private Strategies","text":"### 5.2 Rich Private Strategies\n\nWe now assume that the agent's indirect utility $U(\\mu)$ is twice differentiable and strictly convex everywhere. Denote by $h\\left(\\mu \\mid \\mu^{\\prime}\\right)$ the value of the supporting hyperplane to the graph of $U$ at $\\mu^{\\prime}$ evaluated at $\\mu$. Fixing the agent's TRS with trust region $T$, the set of messages\n\n[^12]that the misaligned adviser might send at belief $\\mu$ is given by\n$$\nM^{*}(\\mu)=\\arg \\min _{\\mu^{\\prime} \\in T} h\\left(\\mu \\mid \\mu^{\\prime}\\right) .\n$$\nAs we have argued earlier, a simple transformation establishes the following fact:\n\nCorollary 3 (Bregman distance). $M^{*}(\\mu)$ is the set of maximizers of the Bregman distance $D_{U}\\left(\\mu, \\mu^{\\prime}\\right)$ between the adviser's true belief $\\mu$ and a report $\\mu^{\\prime}$ in the trust region.\n\nBy Corollary 3, $M^{*}(\\mu)$ are the furthest points from $\\mu$ in $T$ with respect to Bregman distance. Bregman distance always strictly increases along each ray from $\\mu$ and thus the misaligned adviser always chooses points on the \"opposite\" boundary of $T$. Therefore, $U$ determines the geometry of the trust region. For example, if $U(\\mu)=\\|\\mu-b\\|^{2}$ for Euclidean norm and some vector $b$, then the Bregman distance between $\\mu$ and $\\mu^{\\prime}$ coincides with the squared Euclidean distance between $\\mu$ and $\\mu^{\\prime}$. In such cases, the trust region can be taken to be convex. ${ }^{17}$ However, in general, Bregman distance is not a (square of a) metric-it may not satisfy the triangle inequality or symmetry. Thus, the geometry of $T$ may be quite complex, and we do not expect a characterization of the trust region in full generality to be tractable.\n\nThe trust region can sometimes be found explicitly in symmetric environments-we illustrate this with an example.\n\nExample 1 (Spherical Environment). Let $U(\\mu)=\\tilde{U}(\\|\\mu-b\\|)$ for some $b$ and $\\tilde{U}$. Let the adviser's belief be symmetrically distributed over a ball $C=\\left\\{\\mu:\\|\\mu-b\\| \\leq r_{0}\\right\\}$ with the radial density $\\tau(r)$. Then, there exists a robustly rationalizable solution in which the trust region $T$ is a ball centered at $b: T=\\left\\{\\mu:\\|\\mu-b\\| \\leq r^{*}(\\alpha)\\right\\}$.\n\nWe will show this result via Theorem 2 by explicitly constructing the corresponding TRE. The key observation, that we formalize and prove in Lemma 9 in Appendix A.10, is that the misaligned adviser with belief $\\mu$ induces an antipodal belief on the boundary of $T$. This fact combined with the symmetry of the problem implies that the adversarial strategy is the\n\n[^13]same on each ray going through the center of the ball, and hence analogous to the strategy constructed in Section 4.\n\nThe radius $r^{*}(\\alpha)$ can be found via the balancing condition applied to any ray going through the center. Indeed, consider any line passing through $b$. Consider a coordinate system on that line such that $b$ is located at $r=0$, and the points on the boundary of $C$ are located at coordinates $-r_{0}$ and $r_{0}$. Then, the belief at $r=r^{*}(\\alpha)$ will be induced by the misaligned adviser only when his belief is at negative coordinates, and by the aligned adviser only when his belief is at positive coordinates. Thus, the agent's interim beliefs satisfy the TRE property if and only if:\n\n$$\nr^{*}=\\frac{\\alpha \\int_{r^{*}}^{r_{0}} r \\tau(r) d r-(1-\\alpha) \\int_{0}^{r_{0}} r \\tau(r) d r}{\\alpha \\int_{r^{*}}^{r_{0}} \\tau(r) d r+(1-\\alpha) \\int_{0}^{r_{0}} \\tau(r) d r} .\n$$\n\nRearranging, we obtain:\n\n$$\n(2 \\alpha-1) \\int_{r^{*}}^{r_{0}}\\left(r-r^{*}\\right) \\tau(r) d r=(1-\\alpha)\\left(\\int_{0}^{r^{*}}\\left(r+r^{*}\\right) \\tau(r) d r+\\int_{r^{*}}^{r_{0}} 2 r^{*} \\tau(r) d r\\right)\n$$\n\nFor $\\alpha<1 / 2$, the equation does not admit a solution. At $\\alpha=1 / 2, r^{*}=0$ is a solution. For $\\alpha \\in(1 / 2,1)$, the left-hand side is continuously and strictly decreasing in $r^{*}$, and the right-hand side is continuously and strictly increasing in $r^{*}$, with a derivative with respect to $r^{*}$ that is strictly positive. Therefore, the equation admits a unique solution $r^{*}(\\alpha)$. As the left-hand side strictly increases in $\\alpha$ and the right-hand side strictly decreases in $\\alpha, r^{*}(\\alpha)$ strictly increases in $\\alpha$. Furthermore, $r^{*}(1)=r_{0}$.\n\nThis example features two notable properties. First, the MVA does not depend on $U$ or the number of states; the threshold alignment probability is always 1/2. Second, the shape of $\\tilde{U}$, which captures the details of the decision problem, does not matter for the trust region $T$; the trust region is uniquely pinned down by $\\tau(r)$. For example, if $\\tau$ is uniform, $\\tau \\sim U\\left[0, r_{0}\\right]$, then $r^{*}(\\alpha)=\\frac{1-\\sqrt{1+\\alpha-2 \\alpha^{2}}}{\\alpha} r_{0}$. $\\square$","text_sha256":"55280e67ce5b17f22c01f3795f66c55d5b72843c15a5073e13ee67314bedf07a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0018","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Concluding Remarks","text":"## 6 Concluding Remarks\n\nSummary. We studied robust decision-making when an agent relies on an informed adviser who may be misaligned. We characterized the decision rule that maximizes the agent's expected payoff guarantee over all possible forms of misalignment. We showed that every optimal policy is equivalent to a trust region policy in belief space: the agent limits exposure to manipulation while preserving value from moderately informative advice. We proved that the optimal solution can be implemented as an equilibrium of a zero-sum game between the agent and the misaligned adviser and derived minimal alignment probabilities required for advice to be robustly valuable.\n\nImplications for AI use. Our results support a cautiously optimistic view about deploying AI in high-stakes settings. Even if misalignment is serious and plausibly frequent, there are provably effective ways to limit the resulting harm while deriving value-provided that the human decision-maker retains final authority over actions. At the same time, our analysis makes clear that safe deployment requires concrete, pre-specified decision protocols rather than informal, case-by-case trust judgments.\n\nThe trust-region characterization translates into a simple design rule for AI-assisted choice under misalignment risk. The decision-maker should specify in advance a rule that maps model outputs into actions, separating a set of outputs that will be used directly from those that will be treated more conservatively. In some contexts, this can be implemented as a delegationstyle guardrail. The AI can effectively control decisions within an approved operating range, but recommendations that push toward unusually aggressive actions are automatically clipped to the nearest admissible recommendation or escalated into a higher-friction path (additional tests, second reads, or explicit human sign-off).\n\nMore broadly, if the adviser is one component inside a larger AI system, the same idea suggests an architectural and training choice: include an interpretable interface layer that enforces the trust-region mapping between modules. This limits the chance that rare errors or adversarial behavior upstream translate into extreme downstream actions, and it provides a well-defined target for auditing and stress-testing the system as a whole.\n\nFuture research directions. Our analysis points to at least two natural next steps. First, it would be useful to obtain comparative statics of the trust region with respect to the agent's decision problem, the adviser's informativeness, and alignment probability beyond the binary-state case, where the geometry of the trust region starts playing a central role. Second, with an eye toward applications, it is important to develop tractable computational methods for finding the trust region. Such methods would need to confront the fact that the value function mapping candidate trust regions into the agent's payoff is a convex combination of a supermodular and a submodular function, making many standard algorithms inappropriate. We leave these directions for future research.","text_sha256":"fe4a46a4188145429d60b0d2a93de036fdef4090e1c48b0020db8951fc40ea36"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0019","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAgarwal, N., A. Moehring, P. Rajpurkar, and T. Salz (2025): \"Combining Human Expertise with Artificial Intelligence: Experimental Evidence from Radiology,\" Working paper.\nAgarwal, N., A. Moehring, and A. Wolitzky (2026): \"Designing Human-AI Collaboration: A Sufficient-Statistic Approach,\" Working paper.\nAlonso, R., T. Gan, and J. Hu (2026): \"Robust Delegation,\" Working paper.\nAlonso, R. and G. Padró i Miquel (2025): \"Competitive Capture of Public Opinion,\" Econometrica, 93, 1265-1297.\nAmodei, D., C. Olah, J. Steinhardt, P. Christiano, J. Schulman, and D. Mané (2016): \"Concrete Problems in AI Safety,\" Working paper.\nBenjamin, D. J. (2019): \"Errors in Probabilistic Reasoning and Judgment Biases,\" Handbook of Behavioral Economics: Applications and Foundations 1, 2, 69-186.\nBlackwell, D. (1951): \"Comparison of Experiments,\" in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, ed. by J. Neyman, Berkeley and Los Angeles: University of California Press, 93-102.\nChen, E. O., A. Ghersengorin, and S. Petersen (2024): \"Imperfect Recall and AI Delegation,\" Working paper.\n\nCrawford, V. P. and J. Sobel (1982): \"Strategic Information Transmission,\" Econometrica, 50, 1431-51.\nDoval, L. and A. Smolin (2024): \"Persuasion and Welfare,\" Journal of Political Economy, 132, 2451-2487.\nDratsch, T., X. Chen, M. Rezazade Mehrizi, R. Kloeckner, A. Mähringer-Kunz, M. Püsken, B. Baessler, S. Sauer, D. Maintz, and D. Pinto dos Santos (2023): \"Automation Bias in Mammography: The Impact of Artificial Intelligence BI-RADS Suggestions on Reader Performance,\" Radiology, 307, e222176.\nDreyfuss, B. and R. Hoong (2025): \"Calibrated Coarsening: Designing Information for AI-Assisted Decisions,\" Working paper.\nDworczak, P. and A. Pavan (2022): \"Preparing for the Worst but Hoping for the Best: Robust (Bayesian) Persuasion,\" Econometrica, 90, 2017-2051.\nFrankel, A. (2014): \"Aligned Delegation,\" American Economic Review, 104, 66-83.\nFudenberg, D. and A. Liang (2025): \"Friend or Foe: Delegating to an AI whose Alignment is Unknown,\" Working paper.\nGale, D. and H. Nikaido (1965): \"The Jacobian Matrix and Global Univalence of Mappings,\" Mathematische Annalen, 159, 81-93.\nGershkov, A., B. Moldovanu, and X. Shi (2025): \"Order Independence in Sequential, Issue-by-Issue Voting,\" Mathematics of Operations Research, 50, 1635-1653.\nGlazer, J., H. Herrera, and M. Perry (2020): \"Fake Reviews,\" The Economic Journal, 131, 1772-1787.\nHurwicz, L. (1951): \"Optimality Criteria for Decision Making Under Ignorance,\" Cowles Commission Discussion Paper.\nKamenica, E. and M. Gentzkow (2011): \"Bayesian Persuasion,\" American Economic Review, 101, 2590-2615.\nLahr, P. and J. Winkelmann (2019): \"Fake Experts,\" Working paper.\nLevy, M. and B. Szentes (2025): \"Information Design for AI Proxies under Imperfect Recall,\" Working paper.\nLipnowski, E., D. Ravid, and D. Shishkin (2022): \"Persuasion via Weak Institutions,\" Journal of Political Economy, 130, 2705-2730.\n\nMaslej, N., L. Fattorini, R. Perrault, Y. Gil, V. Parli, N. Kariuki, E. Capstick, A. Reuel, E. Brynjolfsson, J. Etchemendy, K. Ligett, T. Lyons, J. Manyika, J. C. Niebles, Y. Shoham, R. Wald, T. Walsh, A. Hamrah, L. Santarlasci, J. B. Lotufo, A. Rome, A. Shi, and S. Oak (2025): \"The AI Index 2025 Annual Report,\" Tech. rep., AI Index Steering Committee, Institute for Human-Centered AI, Stanford University, Stanford, CA.\n\nMin, D. (2021): \"Bayesian Persuasion under Partial Commitment,\" Economic Theory, 72, 743-764.\n\nRussell, S. (2019): Human Compatible: AI and the Problem of Control, Penguin UK.\nSion, M. (1958): \"On General Minimax Theorems,\" Pacific Journal of Mathematics, 8, 171-176.\n\nSobel, J. (2020): \"Lying and Deception in Games,\" Journal of Political Economy, 128, 907-947.\n\nWenderott, K., J. Krups, F. Zaruchas, and M. Weigl (2024): \"Effects of artificial intelligence implementation on efficiency in medical imaging-a systematic literature review and meta-analysis,\" npj Digital Medicine, 7, 265.\n\nWhitmeyer, M. (2026): \"Blackwell-Monotone Updating Rules,\" Journal of Political Economy, forthcoming.\n\nWojtaszczyk, P. (1991): Banach Spaces for Analysts, vol. 25 of Cambridge Studies in Advanced Mathematics, Cambridge, UK: Cambridge University Press.","text_sha256":"a57ba1bca139a1981994b8ee532a7df1459f04f148d5f9e161130a10bfd685ba"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0020","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Proofs","text":"## A Proofs","text_sha256":"ff9a8c951ab04be2fe76aa1991ac6598bd9cdde8807050e79679d91fbe727004"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0021","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 1 Proof of Theorem 1","text":"## A. 1 Proof of Theorem 1\n\nWe begin with a key lemma.\n\nLemma 2. Any optimal solution $\\sigma^{*}$ is equivalent to an optimal solution that uses Bayesoptimal private strategies for all $m \\in \\Delta(\\Omega)$.\n\nProof. Consider the set of state-contingent payoff profiles that are feasible for the agent (cf. Doval and Smolin (2024)):\n\n$$\nW=\\left\\{\\mathrm{w} \\in \\mathbb{R}^{N}: \\exists \\hat{\\sigma}, \\mathrm{w}(\\omega)=\\mathbb{E}_{\\hat{\\sigma}}[u(a, \\omega, \\theta) \\mid \\omega], \\forall \\omega \\in \\Omega\\right\\} .\n$$\n\nSince $\\theta$ and $s$ are conditionally independent, if the adviser has posterior $s$ and the agent plays a private strategy that corresponds to payoff profile w, the resulting agent's expected payoff is $\\mathrm{w} \\cdot s$.\n\nThe set $W$ is convex, because a convex combination of the private strategies delivers a convex combination of their respective payoff profiles. The set $W$ is compact, because for any $\\lambda \\in \\mathbb{R}^{|\\Omega|}, \\max _{\\mathrm{w} \\in W} \\lambda \\cdot \\mathrm{w}$ exists and is attained by some $\\mathrm{w} \\in W$ by the boundedness and continuity of $u$ in $a$ and the measurable maximum theorem.\n\nDenote the (weak) Pareto frontier of $W$ by $W^{P}$ :\n\n$$\nW^{P}=\\left\\{\\mathrm{w} \\in W: \\nexists \\mathrm{w}^{\\prime} \\in W, \\forall \\omega \\in \\Omega, \\mathrm{w}^{\\prime}(\\omega)>\\mathrm{w}(\\omega)\\right\\} .\n$$\n\nSince $W$ is convex and compact, by the supporting hyperplane theorem, a private strategy $\\hat{\\sigma}$ is Bayes-optimal for some belief if and only if it delivers a payoff profile in $W^{P}$. Therefore, if $\\hat{\\sigma}$ is not Bayes-optimal, there exists a dominating $\\hat{\\sigma}^{\\prime}$, which can be taken to be Bayes-optimal itself, such that for all $\\omega \\in \\Omega, \\mathbb{E}_{\\hat{\\sigma}^{\\prime}}[u(a, \\omega, \\theta) \\mid \\omega]>\\mathbb{E}_{\\hat{\\sigma}}[u(a, \\omega, \\theta) \\mid \\omega]$.\n\nTake an optimal solution $\\sigma^{*}$ and, for every message $m \\in \\Delta(\\Omega)$, if $\\hat{\\sigma}^{*}(m)$ is not Bayesoptimal for some belief, replace it with a Bayes-optimal dominating strategy $\\hat{\\sigma}^{\\prime}(m)$. The new strategy, which we call $\\sigma_{0}$, must still be optimal. Indeed, the agent's payoff is\n\n$$\n\\mathbb{E}_{\\mu \\sim \\tau}\\left[\\alpha \\mathrm{w}(\\hat{\\sigma}(\\mu)) \\cdot \\mu+(1-\\alpha) \\inf _{m \\in \\Delta(\\Omega)}\\{\\mathrm{w}(\\hat{\\sigma}(m)) \\cdot \\mu\\}\\right],\n$$\n\nwhich pointwise increases after the change. Moreover, the ex-ante expected payoff must stay the same since $\\sigma^{*}$ was optimal to begin with; in particular, $\\sigma_{0}$ makes changes to the strategy only for messages $m$ that have joint probability zero. Thus, $\\sigma^{*}$ is equivalent to $\\sigma_{0}$. $\\square$\n\nWe can now finish the proof of Theorem 1. Pick any optimal solution $\\sigma^{*}$. By Lemma 2, $\\sigma^{*}$ is equivalent to an optimal strategy that uses only Bayes-optimal private strategies. Denote\nby $\\Sigma_{0}$ the set of those private strategies, and let $T_{0}$ be the closure of the set of beliefs at which those private strategies are Bayes-optimal. By continuity, taking the closure does not affect the expected payoff of the strategy $\\sigma^{*}$ in the worst-case scenario.\n\nObserve that the agent's expected payoff conditional on the adviser being misaligned is pinned down by the set $\\Sigma_{0}$; it does not depend on how individual messages are mapped to different elements of $\\Sigma_{0}$ (because the misaligned adviser can report any message). Thus, the mapping from messages to the private strategies in $\\Sigma_{0}$ must maximize the expected payoff conditional on the adviser being aligned. Since the aligned adviser is non-strategic, maximization can be performed pointwise, message by message (without loss of optimality, also for messages that are sent with probability zero by the aligned adviser). In particular, for $m \\in T_{0}$, we can set $\\hat{\\sigma}^{*}(m)$ to be the Bayes-optimal strategy for $m$; for $m \\notin T_{0}$, we can set $\\hat{\\sigma}^{*}(m)=\\hat{\\sigma}^{*}(P(m))$ where $P(m) \\in \\arg \\max _{m^{\\prime} \\in T_{0}} U\\left(\\hat{\\sigma}^{*}\\left(m^{\\prime}\\right), m\\right)$. This way we have constructed a TRS (with the trust region $T_{0}$ ) that is equivalent to $\\sigma^{*}$-and is hence optimal. ${ }^{18}$\n\nWe now show that for any optimal TRS $\\sigma^{*}$, the trust region $T_{0}$ can be enlarged (while preserving the payoffs) to a connected trust region $T_{1}$. Assume that $T_{0}$ is not connected and take any $m_{1}, m_{2} \\in T_{0}$ that belong to different connected components of $T_{0}: m_{1} \\in T_{0}^{1}$ and $m_{2} \\in T_{0}^{2}$. Consider the welfare profiles $\\mathrm{w}_{1} \\triangleq \\mathrm{w}\\left(\\hat{\\sigma}^{*}\\left(m_{1}\\right)\\right)$ and $\\mathrm{w}_{2} \\triangleq \\mathrm{w}\\left(\\hat{\\sigma}^{*}\\left(m_{2}\\right)\\right)$ induced by those messages in the considered solution. Define the subset of Pareto optimal welfare profiles (as in the proof of Lemma 2) that dominate some weighted average of those profiles $\\mathrm{w}(\\gamma) \\triangleq \\gamma \\mathrm{w}_{1}+(1-\\gamma) \\mathrm{w}_{2}:$\n\n$$\nW^{D}\\left(m_{1}, m_{2}\\right)=\\left\\{\\mathrm{w} \\in W^{P}: \\exists \\gamma \\in[0,1], \\mathrm{w} \\geq \\mathrm{w}(\\gamma)\\right\\} .\n$$\n\nConsider any $\\mathrm{w} \\in W^{D}\\left(m_{1}, m_{2}\\right)$ that dominates $\\mathrm{w}(\\gamma)$ for some $\\gamma$. Since $\\mathrm{w} \\in W^{P}$, w is generated by a private strategy $\\hat{\\sigma}(\\mathrm{w})$ Bayes-optimal at a set of beliefs $M_{1}(\\mathrm{w})$. We enlarge $T_{0}$ by adding to it the messages in $M_{1}(\\mathrm{w}) \\backslash T_{0}$ together with the prescription to play $\\hat{\\sigma}(\\mathrm{w})$ at those messages. Doing so does not decrease the payoff from the misaligned adviser because he could already send messages $m_{1}$ and $m_{2}$, and it weakly increases the payoff from the aligned adviser because the trust region increases (in the sense of set inclusion).","text_sha256":"f029f6a21b50cea1e217f61454f5eb444284f4d631eeb13152fcb6ae931bceed"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0022","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 1 Proof of Theorem 1","text":"[^14]Since $W$ is convex, $W^{D}\\left(m_{1}, m_{2}\\right)$ is connected. Furthermore, $M_{1}(\\mathrm{w})$ is upper-hemicontinuous with connected (convex) values because it is a normal-cone correspondence. Therefore, the union $\\bigcup_{\\mathrm{w} \\in W^{D}\\left(m_{1}, m_{2}\\right)} M_{1}(\\mathrm{w})$ is connected and contains $m_{1}$ and $m_{2}$. Therefore, adding these beliefs to the original trust region connects the components $T_{0}^{1}$ and $T_{0}^{2}$, with the trust region weakly expanding and remaining optimal. Since this modification can be performed for all connected components of $T_{0}$, this modification results in a connected optimal trust region $T_{1}$.\n\nFinally, by continuity of payoffs, we can without loss of generality consider the closure of the set of used private strategies, and hence the trust region can be chosen to be equal to $T=\\operatorname{cl} T_{1}$, which is a compact and connected subset of $\\Delta(\\Omega)$. Call the new strategy constructed this way $\\sigma_{1}$.\n\nBy construction, the strategy $\\sigma_{1}$ is optimal. Moreover, the expected payoff must stay the same since $\\sigma^{*}$ was optimal to begin with; therefore, the new strategy makes changes to the strategy only for messages that have joint probability zero in equilibrium. Thus, $\\sigma^{*}$ is equivalent to $\\sigma_{1}$.","text_sha256":"65cef17adb462e4436e95307f79317f6c8a6d318f84b9e1449479deab1f36251"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0023","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 2 Proof of Theorem 2","text":"## A. 2 Proof of Theorem 2\n\nSuppose $M$ and $\\Theta$ are finite. For any given strategy of the misaligned adviser (which was assumed to only use messages in $M$ ) and the agent, $(\\beta, \\sigma)$, the agent's payoff is, with a slight overload of notation for $V$,\n\n$$\n\\begin{aligned}\nV(\\beta, \\sigma) \\triangleq & \\alpha \\sum_{\\mu \\in M, \\omega \\in \\Omega, \\theta \\in \\Theta} \\tau(\\mu) \\mu(\\omega) f(\\theta \\mid \\omega) \\int_{A} u(a, \\omega, \\theta) \\sigma(d a \\mid \\mu, \\theta)+ \\\\\n& (1-\\alpha) \\sum_{\\mu, m \\in M, \\omega \\in \\Omega, \\theta \\in \\Theta} \\tau(\\mu) \\mu(\\omega) \\beta(m \\mid \\mu) f(\\theta \\mid \\omega) \\int_{A} u(a, \\omega, \\theta) \\sigma(d a \\mid m, \\theta) .\n\\end{aligned}\n$$\n\nClearly, $\\mathcal{B}$ and $\\Sigma$ are convex. Since $M$ is finite, $\\mathcal{B}=\\times_{m \\in M} \\Delta(M)$ is compact. Since $M$ and $\\Theta$ are finite and $\\Delta(A)$ can be equipped with the weak* topology, $\\Sigma=\\times_{m \\in M, \\theta \\in \\Theta} \\Delta(A)$ is compact. $V(\\beta, \\sigma)$ is affine in $\\beta$ and in $\\sigma$; therefore it is concave-convexlike in Sion (1958)'s terminology. For each $\\sigma \\in \\Sigma, V(\\beta, \\sigma)$ is continuous in $\\beta$. For each $\\beta \\in \\mathcal{B}, V(\\beta, \\sigma)$ is continuous in $\\sigma$. Therefore, a minimax theorem applies in its infsup variation (e.g., Theorem 4.2', Sion (1958))\nand\n\n$$\n\\sup _{\\sigma \\in \\Sigma} \\inf _{\\beta \\in \\mathcal{B}} V(\\beta, \\sigma)=\\inf _{\\beta \\in \\mathcal{B}} \\sup _{\\sigma \\in \\Sigma} V(\\beta, \\sigma) .\n$$\n\nFurthermore, for any given $\\beta, \\phi(\\beta) \\triangleq \\sup _{\\sigma \\in \\Sigma} V(\\beta, \\sigma)$ is attained because $\\Sigma$ is compact and $V(\\beta, \\sigma)$ is continuous in $\\sigma$. Similarly, for any given $\\sigma, \\psi(\\sigma) \\triangleq \\inf _{\\beta \\in \\mathcal{B}} V(\\beta, \\sigma)$ is attained because $\\mathcal{B}$ is compact and $V(\\beta, \\sigma)$ is continuous in $\\beta$. Because $V(\\beta, \\sigma)$ is continuous, $\\phi(\\beta)$ is lower-semicontinuous and $\\psi(\\sigma)$ is upper-semicontinuous. Thus, we can choose $\\sigma^{*} \\in \\arg \\max _{\\sigma \\in \\Sigma} \\psi(\\sigma)$ and $\\beta^{*} \\in \\arg \\min _{\\beta \\in \\mathcal{B}} \\phi(\\beta)$. Then, $\\left(\\beta^{*}, \\sigma^{*}\\right)$ form a saddle point:\n\n$$\nV\\left(\\beta^{*}, \\sigma\\right) \\leq V\\left(\\beta^{*}, \\sigma^{*}\\right) \\leq V\\left(\\beta, \\sigma^{*}\\right), \\quad \\forall \\beta \\in \\mathcal{B}, \\sigma \\in \\Sigma .\n$$\n\nTherefore, $\\beta^{*}$ is an adversarial adviser's strategy to $\\sigma^{*}$, whereas $\\sigma^{*}$ is a best-response of the agent to $\\beta^{*}$. Since $\\alpha>0$ and all $m \\in M$ are on-path, the latter implies that after any $m \\in M$, the private strategy $\\hat{\\sigma}^{*}(m)$ is Bayes-optimal given $\\beta^{*}$, and hence $\\sigma^{*}$ is robustly rationalizable.\n\nConversely, for any $M$ and $\\Theta$, consider $\\left(\\beta^{*}, \\sigma^{*}\\right)$ such that $\\sigma^{*}$ is robustly rationalizable and $\\beta^{*}$ is adversarial against $\\sigma^{*}$, i.e., $\\left(\\beta^{*}, \\sigma^{*}\\right)$ form a saddle point with property (12). Then, for any $\\sigma \\in \\Sigma$ :\n\n$$\nV(\\sigma)=\\inf _{\\beta \\in \\mathcal{B}} V(\\beta, \\sigma) \\leq V\\left(\\beta^{*}, \\sigma\\right) \\leq V\\left(\\beta^{*}, \\sigma^{*}\\right)=\\min _{\\beta \\in \\mathcal{B}} V\\left(\\beta, \\sigma^{*}\\right)=V\\left(\\sigma^{*}\\right),\n$$\n\nwhere the third comparison uses the saddle property and the fourth comparison uses the fact that $\\beta^{*}$ is adversarial to $\\sigma^{*}$. Therefore, $\\sigma^{*}$ is an optimal solution.","text_sha256":"7038e0d00a03b84a7394672d1545a5cf0ba64108b2402b138320162697812417"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0024","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 3 Proof of Theorem 3","text":"## A. 3 Proof of Theorem 3\n\nNotation: In this section, we denote by $I_{K}$ a unit matrix of dimension $K$, by $1_{K}$ a vector of ones of dimension $K$, by $0_{N \\times K}$ a matrix of zeros of dimension $N \\times K$, by $e_{K}^{i}$ an $i$ th standard basis vector of dimension $K$, by $x_{i, k}$ a $k$ th element of vector $x_{i}$, by $\\operatorname{diag} x$ a diagonal matrix with vector $x$ on the main diagonal, and by $X^{\\top}$ a transpose of a matrix $X$.\n\nSince $\\mu_{0}$ has full support, we can equivalently identify adviser's information with a (row)\nstochastic $N \\times K$ matrix $\\Pi$, where $\\Pi_{i j}$ is the probability of the $j$ th signal observed by the adviser in the $i$ th state. Moreover, $\\operatorname{rank} \\Pi=R(\\tau) .{ }^{19}$\n\nWe identify the strategy of the misaligned adviser with a stochastic $K \\times K$ matrix $B$. Since the aligned adviser reports truthfully, the overall adviser's strategy can be written as a garbling of his information:\n\n$$\nG(B) \\triangleq \\alpha I_{K}+(1-\\alpha) B .\n$$\n\nBy Blackwell (1951), the MVA is a maximal $\\alpha$ for which there exists a stochastic matrix $B$ such that $\\Pi G(B)$ is Blackwell uninformative; we show below that it is attained. This also implies that MVA depends on $\\tau$ only via $\\Pi$, so we will write MVA( $\\Pi$ ).\n\nWe start with preliminary observations. First, note that $G_{k k} \\geq \\alpha$ and $G 1_{K}=1_{K}$. Second, note that $\\Pi G(B)$ is uninformative if and only if all of its rows are equal to each other, that is if and only if\n\n$$\nD(\\Pi) G(B)=D(\\Pi)\\left(\\alpha I_{K}+(1-\\alpha) B\\right)=0_{(N-1) \\times K},\n$$\n\nwhere $D(\\Pi)$ is the row-difference matrix of $\\Pi$ :\n\n$$\nD(\\Pi) \\triangleq\\left(\\begin{array}{c}\n\\left(\\pi_{2}-\\pi_{1}\\right)^{\\top} \\\\\n\\vdots \\\\\n\\left(\\pi_{N}-\\pi_{1}\\right)^{\\top}\n\\end{array}\\right),\n$$\n\nand $\\pi_{i}^{\\top}$ is the $i$ th row of $\\Pi$. Consider the auxiliary finite linear program:\n\n$$\n\\begin{aligned}\n\\Lambda(\\Pi)=\\max _{G \\in \\mathbb{R}^{K \\times K}, \\alpha \\in \\mathbb{R}} & \\alpha \\\\\n\\text { s.t. } & G \\geq \\alpha I_{K}, G 1_{K}=1_{K}, \\\\\n& D(\\Pi) G=0_{(N-1) \\times K} .\n\\end{aligned}\n$$\n\n[^15]Lemma 3. $\\operatorname{MVA}(\\Pi)=\\Lambda(\\Pi)$.\nProof. We need to show that there exists a stochastic matrix $B$ such that $\\Pi G(B)$ is Blackwell uninformative if and only if $\\alpha \\leq \\Lambda(\\Pi)$.\n\nOnly if: For any given $\\alpha$, if $B$ is such that $\\Pi G(B)$ is Blackwell uninformative, then we showed that $G(B)$ must satisfy conditions (16-17). By the maximization nature of the problem, if $\\alpha>\\Lambda(\\Pi)$, those conditions cannot be satisfied.\n\nIf: If $\\alpha \\leq \\Lambda(\\Pi)$, then there exists $G$ that satisfies conditions (16-17) (e.g., the argmax). If $\\Lambda(\\Pi)=1$, then $B$ can be arbitrary. Otherwise, set $B=\\left(G-\\alpha I_{K}\\right) /(1-\\alpha)$. It is straightforward that the so-defined $B$ is a stochastic matrix and by construction $\\Pi G(B)$ is uninformative. $\\square$\n\nLemma 3 provides a computationally tractable characterization of MVA for any given $\\Pi$ and sets the stage for the rest of the proof, which we split into two lemmas.\n\nLemma 4. $\\operatorname{MVA}(\\Pi) \\in[1 / R(\\Pi), 1 / 2]$. If $R(\\Pi)=K$, then $\\operatorname{MVA}(\\Pi)=1 / K$.\n\nProof. To ease notation, in the proof we omit the dependence of $R$ on $\\Pi$.\n1.) $\\operatorname{MVA}(\\Pi) \\leq 1 / 2$.\n\nIf $\\alpha$ and $G$ satisfy (16-17), then $B=\\left(G-\\alpha I_{K}\\right) /(1-\\alpha)$ is a stochastic matrix and\n\n$$\nD(\\Pi) B=-\\frac{\\alpha}{1-\\alpha} D(\\Pi) .\n$$\n\nIn other words, the rows of $D(\\Pi)$ are left eigenvectors of $B$ associated with eigenvalue $-\\alpha /(1-\\alpha)$. Since $B$ is stochastic, its spectral radius equals 1. Thus, $|-\\alpha /(1-\\alpha)| \\leq 1$ and $\\alpha \\leq 1 / 2$. It follows that MVA $(\\Pi) \\leq 1 / 2$.\n2.) If $R=K$, then $\\operatorname{MVA}(\\Pi)=1 / K$.\n\nIf $R=K$, then $K \\leq N$ and $\\operatorname{rank} D(\\Pi)=K-1$. Thus, rank $\\operatorname{ker} D(\\Pi)=K-(K-1)=1$ and, because $D 1_{K}=1_{N-1}-1_{N-1}=0_{N-1}, \\operatorname{ker} D=\\operatorname{span}\\left\\{1_{K}\\right\\}$. Thus, for $(G, \\alpha)$ to satisfy (17), every column of $G$ must be a multiple of $1_{K}$. But since $G$ is stochastic, it follows that $\\sum_{k=1}^{K} G_{k k}=1$ and $\\min _{k} G_{k k} \\leq 1 / K$. To further satisfy (16), it must be that $\\alpha \\leq 1 / K$. Thus, $\\operatorname{MVA}(\\Pi) \\leq 1 / K$.\n\nAt the same time, if $\\alpha \\leq 1 / K$, then $(G, \\alpha)$ satisfy (16-17) for $G=1 / K 1_{K} 1_{K}^{\\top}$. In this case, $G$ is uninformative, not only $\\Pi G$, so the misaligned adviser can make the signal to be uninformative about his estimate, not only about the state. It follows that MVA $\\geq 1 / K$ and, therefore, $\\operatorname{MVA}(\\Pi)=1 / K$.\n3.) $\\mathrm{MVA} \\geq 1 / R$.\n\nLet $\\alpha=1 / R$ (recall that $R \\geq 2$ ). Consider the normed space ( $\\mathbb{R}^{K},\\|\\cdot\\|_{1}$ ) and its linear $(R-1)$-dimensional subspace $\\mathbb{W}$ spanned by rows of $D(\\Pi)$. By the Auerbach basis theorem, there exist vectors $w_{1}, \\ldots, w_{R-1} \\in \\mathbb{W}$ and $x_{1}, \\ldots, x_{R-1} \\in \\mathbb{R}^{K}$ such that ${ }^{20}$\n\n$$\n\\left\\|w_{i}\\right\\|_{1}=1, \\quad\\left\\|x_{i}\\right\\|_{\\infty}=1, \\quad w_{i}^{\\top} x_{j}=\\delta_{i j}, \\quad 1 \\leq i, j \\leq R-1 .\n$$\n\nDefine the corresponding matrices $W \\triangleq\\left(w_{1}, \\ldots, w_{R-1}\\right), X \\triangleq\\left(x_{1}, \\ldots, x_{R-1}\\right)$. By construction,\n\n$$\nW^{\\top} X=I_{R-1},\n$$\n\nand by properties of $D(\\Pi)$,\n\n$$\nW^{\\top} 1_{K}=0_{R-1} .\n$$\n\nDefine the vector of weights of rows of $W, \\bar{w} \\in \\mathbb{R}^{K}$, as $\\bar{w}_{k} \\triangleq \\sum_{i=1}^{R-1}\\left|w_{i, k}\\right|$. Since $\\left\\|w_{i}\\right\\|_{1}=1$, we have\n\n$$\n\\sum_{k=1}^{K} \\bar{w}_{k}=\\sum_{i=1}^{R-1}\\left\\|w_{i}\\right\\|_{1}=R-1 .\n$$\n\nWe explicitly construct the desired strategy of the misaligned adviser $B$ as:\n\n$$\nB=\\frac{1}{R-1}\\left(1_{K} \\bar{w}^{\\top}-X W^{\\top}\\right) .\n$$\n\n[^16]Nonnegativity. For all $j, k$,\n\n$$\nB_{j k}=\\frac{1}{R-1}\\left(\\sum_{i=1}^{R-1}\\left|w_{i, k}\\right|-\\sum_{i=1}^{R-1} x_{i, j} w_{i, k}\\right) \\geq 0,\n$$\n\nbecause $\\left|x_{i, j}\\right| \\leq\\left\\|x_{i}\\right\\|_{\\infty}=1$.\nStochasticity. By (20) and (21):\n\n$$\nB 1_{K}=\\frac{1}{R-1}\\left(1_{K}\\left(\\bar{w}^{\\top} 1_{K}\\right)-X\\left(W^{\\top} 1_{K}\\right)\\right)=\\frac{1}{R-1}\\left(1_{K}(R-1)-0_{K}\\right)=1_{K} .\n$$\n\nUninformativeness. By (19) and (20):","text_sha256":"595b6e1c0ef0fb8567e3a782d6c77b9e024802ab87452e14d256cf78cbd6b147"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0025","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 3 Proof of Theorem 3","text":"$$\nW^{\\top} B=\\frac{1}{R-1}\\left(\\left(W^{\\top} 1_{K}\\right) \\bar{w}^{\\top}-\\left(W^{\\top} X\\right) W^{\\top}\\right)=\\frac{1}{R-1}\\left(0_{R-1}-W^{\\top}\\right)=-\\frac{1}{R-1} W^{\\top} .\n$$\n\nSince by construction columns of $W$ form a basis in the row space of $D(\\Pi)$, it follows that\n\n$$\nD(\\Pi) B=-\\frac{1}{R-1} D(\\Pi) .\n$$\n\nAs $\\alpha=1 / R$, this corresponds exactly to constraint (17). The result follows. $\\square$\n\nLemma 5. For any $N \\geq 2$ and $\\alpha \\in[1 / N, 1 / 2]$, there exist $K$ and $\\Pi$ such that $\\operatorname{MVA}(\\Pi)=\\alpha$.\nProof. The proof is by direct construction. For $N=2$, the result is trivial. For $N \\geq 3$, consider $K \\in[4, N+1]$ and, for $\\delta \\in[0,1]$, the $N \\times K$ matrix $\\Pi$ such that\n\n$$\n\\begin{aligned}\n\\pi_{i}^{\\top} & =\\frac{1}{K} 1_{K}^{\\top}, \\quad i=1 \\text { or } i=K, K+1, \\ldots, N, \\\\\n\\pi_{i}^{\\top} & =\\frac{1}{K}\\left(1_{K}+e_{K}^{i}-e_{K}^{1}\\right)^{\\top}, \\quad i=2, \\ldots, K-2, \\\\\n\\pi_{i}^{\\top} & =\\frac{1}{K}\\left(1_{K}+e_{K}^{i}-\\delta e_{K}^{1}-(1-\\delta) e_{K}^{K}\\right)^{\\top}, \\quad i=K-1,\n\\end{aligned}\n$$\n\nwhere $e_{K}^{i}$ is the $i$ th basis vector of $\\mathbb{R}^{K}$. By construction, $\\Pi$ is a stochastic matrix. Consider MVA(П) that solves the corresponding problem (15).\n\nThe constraint $D(\\Pi) G=0_{(N-1) \\times K}$ reduces to:\n\n$$\n\\left(e_{K}^{i}-e_{K}^{1}\\right)^{\\top} G=0, \\quad i=2, \\ldots, K-2, \\quad\\left(e_{K}^{K-1}-\\delta e_{K}^{1}-(1-\\delta) e_{K}^{K}\\right)^{\\top} G=0,\n$$\n\nwhich effectively states that the first $K-2$ rows are equal to each other and the $(K-1)$ th row is a convex combination of the 1st and the $K$ th rows with weight $\\delta$. Thus, the effective variables are the 1st and the $K$ th rows of the matrix $G$. The constraints $G \\geq \\alpha I_{K \\times K}$ and $G 1_{K}=1_{K}$ then reduce to those rows being probability vectors, such that\n\n$$\nG_{1 k} \\geq \\alpha, k=1, \\ldots, K-2,\\left(\\delta G_{1, K-1}+(1-\\delta) G_{K, K-1}\\right) \\geq \\alpha, G_{K K} \\geq \\alpha .\n$$\n\nTherefore,\n\n$$\n\\alpha \\leq\\left(\\delta G_{1, K-1}+(1-\\delta) G_{K, K-1}\\right) \\leq \\delta(1-(K-2) \\alpha)+(1-\\delta)(1-\\alpha)=1-\\alpha(1+\\delta(K-3)) .\n$$\n\nRearranging yields\n\n$$\n\\alpha \\leq \\alpha^{\\dagger} \\triangleq \\frac{1}{2+\\delta(K-3)},\n$$\n\nand thus MVA $(\\Pi) \\leq \\alpha^{\\dagger}$. Whenever $\\delta \\geq(K-4) /(K-3), \\alpha^{\\dagger} \\leq 1 /(K-2)$ and the bound $\\alpha^{\\dagger}$ can be attained by $G$ with the 1st and the $K$ th rows being (the rest of $G$ is pinned down by condition (23)):\n\n$$\n\\begin{array}{rlrlrl}\nG_{1 k} & =\\alpha^{\\dagger}, k=1, \\ldots, K-2, & & G_{1, K-1} & =1-(K-2) \\alpha^{\\dagger}, & \\\\\nG_{K k} & =0, k=1, \\ldots, K-2, & & G_{K, K-1} & =1-\\alpha^{\\dagger}, & \\\\\nG_{K K} & =\\alpha^{\\dagger} .\n\\end{array}\n$$\n\nThus, $\\mathrm{MVA}(\\Pi)=\\alpha^{\\dagger}$. At $\\delta=(K-4) /(K-3), \\alpha^{\\dagger}=1 /(K-2)$; at $\\delta=1, \\alpha^{\\dagger}=1 /(K-1)$.\nThis establishes that, for all $K \\in[4, N+1]$, as $\\delta$ spans $[(K-4) /(K-3), 1]$, the proposed $\\Pi$ achieves MVA( $\\Pi$ ) that spans $[1 /(K-1), 1 /(K-2)]$. Spanning $K$ from 4 to $N+1$, we obtain the result. $\\square$","text_sha256":"837794cffd766c2012a9ec9c49ca3a2f0cc1d24f9a2bbd1463c3882a8a21e65b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0026","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 4 On Strictly Convex Indirect Utility","text":"## A. 4 On Strictly Convex Indirect Utility\n\nIn this section, we show that the indirect utility is strictly convex when the agent's private information induces a full-support distribution of beliefs.\n\nSpecifically, we assume that the agent's ex-post payoff is type-independent, $u(a, \\omega)$, and identify $\\theta$ with the belief it induces in the absence of any other information: $\\Theta \\subseteq \\Delta(\\Omega)$, $\\theta(\\omega)=\\operatorname{Pr}(\\omega \\mid \\theta)$. We denote by $\\nu$ the final posterior belief that the agent forms, i.e., conditional on both the adviser's message and the agent's type:\n\n$$\n\\nu_{\\mu, \\theta} \\triangleq \\operatorname{Pr}(\\omega \\mid \\mu, \\theta)=\\frac{\\mu(\\omega) f(\\theta \\mid \\omega)}{\\sum_{\\omega^{\\prime} \\in \\Omega} \\mu\\left(\\omega^{\\prime}\\right) f\\left(\\theta \\mid \\omega^{\\prime}\\right)} .\n$$\n\nA necessary and sufficient condition for a private strategy $\\hat{\\sigma}$ to be Bayes-optimal at any given interim belief $\\mu, \\hat{\\sigma} \\in \\arg \\max _{\\hat{\\sigma}^{\\prime}} U\\left(\\hat{\\sigma}^{\\prime}, \\mu\\right)$, is that $\\hat{\\sigma}(\\cdot \\mid \\theta) \\in \\Delta(A)$ is an optimal best-response with respect to $\\nu_{\\mu, \\theta}$ : for all $a \\in \\operatorname{supp} \\hat{\\sigma}(\\cdot \\mid \\theta)$,\n\n$$\na \\in \\arg \\max _{a^{\\prime} \\in A} \\sum_{\\omega \\in \\Omega} \\nu_{\\mu, \\theta}(\\omega) u\\left(a^{\\prime}, \\omega\\right) .\n$$\n\nAssumption 2. $A$ is finite and there exist $a_{1}, a_{2} \\in A$ and $\\mu \\in \\operatorname{int}(\\Delta(\\Omega))$ such that $\\mathbb{E}_{\\mu}\\left[u\\left(a_{1}, \\omega\\right)\\right]=\\mathbb{E}_{\\mu}\\left[u\\left(a_{2}, \\omega\\right)\\right]>\\mathbb{E}_{\\mu}[u(a, \\omega)]$ for all $a \\notin\\left\\{a_{1}, a_{2}\\right\\}$. In addition, for each $\\omega \\in \\Omega$ either $u\\left(a_{1}, \\omega\\right)>u\\left(a_{2}, \\omega\\right)$ or $u\\left(a_{2}, \\omega\\right)>u\\left(a_{1}, \\omega\\right)$.\n\nLemma 6. Suppose $\\theta$ has full support on $\\Delta(\\Omega)$ and Assumption 2 holds. Then, $U(\\mu)$ is strictly convex in the interior of $\\Delta(\\Omega)$.\n\nProof. A sufficient condition for strict convexity of $U(\\mu)$ in the interior of $\\Delta(\\Omega)$ is that for any $\\mu_{1}, \\mu_{2} \\in \\operatorname{int}(\\Delta(\\Omega)), \\mu_{1} \\neq \\mu_{2}$,\n\n$$\n\\arg \\max _{\\hat{\\sigma}} U\\left(\\hat{\\sigma}, \\mu_{1}\\right) \\cap \\arg \\max _{\\hat{\\sigma}} U\\left(\\hat{\\sigma}, \\mu_{2}\\right)=\\emptyset .\n$$\n\nFix any such $\\mu_{1}, \\mu_{2}$. Let $\\mu \\in \\operatorname{int}(\\Delta(\\Omega))$ be the belief from Assumption 2 and define $d(\\omega) \\triangleq u\\left(a_{1}, \\omega\\right)-u\\left(a_{2}, \\omega\\right), r(\\omega) \\triangleq \\mu_{2}(\\omega) / \\mu_{1}(\\omega)$. By Assumption 2 and continuity of the expected payoff in belief, there exists an open neighborhood $O \\subset \\operatorname{int}(\\Delta(\\Omega))$ of $\\mu$ such that for every $\\nu \\in O$, action $a_{1}$ is uniquely optimal whenever $\\nu \\cdot d>0$, and not optimal whenever\n$\\nu \\cdot d<0$, because it is outperformed by $a_{2}$. Define\n\n$$\nR_{1} \\triangleq\\{\\nu \\in O: \\nu \\cdot d>0\\}, \\quad R_{2} \\triangleq\\{\\nu \\in \\Delta(\\Omega): \\nu \\cdot d<0\\} .\n$$\n\nBayes' rule implies that for every $\\omega$ and $\\theta, \\nu_{\\mu_{2}, \\theta}=\\Gamma\\left(\\nu_{\\mu_{1}, \\theta}\\right)$, where $\\Gamma: \\operatorname{int}(\\Delta(\\Omega)) \\rightarrow \\operatorname{int}(\\Delta(\\Omega))$ is the map defined by\n\n$$\n\\Gamma(\\nu)(\\omega) \\triangleq \\frac{\\nu(\\omega) r(\\omega)}{\\sum_{\\omega^{\\prime}} \\nu\\left(\\omega^{\\prime}\\right) r\\left(\\omega^{\\prime}\\right)}\n$$\n\nSince $\\mu_{1} \\neq \\mu_{2}, r$ is not constant; because $d(\\omega) \\neq 0$ for all $\\omega$, the hyperplanes $\\{\\nu: \\nu \\cdot d=0\\}$ and $\\{\\nu: \\nu \\cdot(r * d)=0\\}$, where $*$ denotes the component-wise product, are distinct. As $\\mu \\in O \\cap\\{\\nu: \\nu \\cdot d=0\\}$, we can choose $\\bar{\\nu} \\in O$ such that $\\bar{\\nu} \\cdot d=0$ and $\\bar{\\nu} \\cdot(r * d) \\neq 0$. Without loss of generality, suppose $\\bar{\\nu} \\cdot(r * d)<0$; otherwise swap the labels of $a_{1}$ and $a_{2}$. By continuity, there exists a nonempty open set $A \\subset R_{1}$ such that $\\nu \\cdot(r * d)<0$ for all $\\nu \\in A$. For every $\\nu \\in A$,\n\n$$\n\\Gamma(\\nu) \\cdot d=\\frac{\\nu \\cdot(r * d)}{\\nu \\cdot r}<0,\n$$\n\nso $\\Gamma(A) \\subset R_{2}$. The map $\\theta \\mapsto \\nu_{\\mu_{1}, \\theta}$ is continuous and onto $\\operatorname{int}(\\Delta(\\Omega))$. Hence $\\Theta_{0} \\triangleq\\{\\theta$ : $\\left.\\nu_{\\mu_{1}, \\theta} \\in A\\right\\}$ is nonempty and open; since $\\theta$ has full support on $\\Delta(\\Omega)$, it has strictly positive probability.\n\nFor every $\\theta \\in \\Theta_{0}$ we have $\\nu_{\\mu_{1}, \\theta} \\in R_{1}$ and $\\nu_{\\mu_{2}, \\theta} \\in R_{2}$. This means that the private strategies optimal at $\\mu_{1}$ and $\\mu_{2}$ must necessarily differ on $\\theta \\in \\Theta_{0}$. The result follows. $\\square$","text_sha256":"85c4f3223eaf28aaaafe42edd6c56e79906386e5b04efa6340643ec00abebc19"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0027","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 5 Proof of Lemma 1","text":"## A. 5 Proof of Lemma 1\n\nThe misaligned adviser with signal realization $\\mu$ minimizes $U\\left(\\hat{\\sigma}\\left(\\mu^{\\prime}\\right), \\mu\\right)$ over $\\mu^{\\prime}$ in the trust region. Recall that the function $U\\left(\\hat{\\sigma}\\left(\\mu^{\\prime}\\right), \\mu\\right)$ is linear in $\\mu$, and we assumed that $U(\\mu)=$ $\\max _{\\hat{\\sigma}} U(\\hat{\\sigma}, \\mu)$ is strictly convex and twice differentiable in $\\mu$. This means that $U\\left(\\hat{\\sigma}\\left(\\mu^{\\prime}\\right), \\mu\\right)$ is the value at $\\mu$ of the hyperplane supporting $U$ at $\\mu^{\\prime}$. Under our convention that $\\mu$ is the\nprobability of state 1, this means that\n\n$$\nU\\left(\\hat{\\sigma}\\left(\\mu^{\\prime}\\right), \\mu\\right)=U\\left(\\mu^{\\prime}\\right)+U^{\\prime}\\left(\\mu^{\\prime}\\right)\\left(\\mu-\\mu^{\\prime}\\right) .\n$$\n\nBy convexity of $U$, this function is quasi-concave in $\\mu^{\\prime}$, and hence for all $\\mu^{\\prime} \\in[\\underline{\\mu}, \\bar{\\mu}]$, $U\\left(\\hat{\\sigma}\\left(\\mu^{\\prime}\\right), \\mu\\right) \\geq \\min \\{U(\\hat{\\sigma}(\\underline{\\mu}), \\mu), U(\\hat{\\sigma}(\\bar{\\mu}), \\mu)\\}$. Thus, the misaligned adviser's strategy takes a threshold form. The threshold $b(\\underline{\\mu}, \\bar{\\mu})$ is the intersection point of the supporting lines to $U$ at points $\\underline{\\mu}$ and $\\bar{\\mu}$ :\n\n$$\nU(\\underline{\\mu})+U^{\\prime}(\\underline{\\mu})(b(\\underline{\\mu}, \\bar{\\mu})-\\underline{\\mu})=U(\\bar{\\mu})+U^{\\prime}(\\bar{\\mu})(b(\\underline{\\mu}, \\bar{\\mu})-\\bar{\\mu}) .\n$$\n\nIf $\\underline{\\mu}=\\bar{\\mu}=\\mu, b(\\underline{\\mu}, \\bar{\\mu})=\\mu$, coinciding with (5) by continuity. Otherwise, rearranging, we obtain:\n\n$$\nb(\\underline{\\mu}, \\bar{\\mu})=\\frac{\\bar{\\mu} U^{\\prime}(\\bar{\\mu})-\\underline{\\mu} U^{\\prime}(\\underline{\\mu})-(U(\\bar{\\mu})-U(\\underline{\\mu}))}{U^{\\prime}(\\bar{\\mu})-U^{\\prime}(\\underline{\\mu})} .\n$$\n\nApplying integration by parts, the numerator equals $\\int_{\\underline{\\mu}}^{\\bar{\\mu}} \\mu U^{\\prime \\prime}(\\mu) d \\mu$ and the denominator equals $\\int_{\\underline{\\mu}}^{\\bar{\\mu}} U^{\\prime \\prime}(\\mu) d \\mu$. The result follows.","text_sha256":"5effe7826544f62f6dff4224dae656b71f092c544103ff5e7b4d8fceda5d9bb5"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0028","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 6 Proof of Proposition 1","text":"## A. 6 Proof of Proposition 1\n\nBy Lemma 1 and Corollary 2, the choice of an optimal strategy for the agent reduces to optimization over the extreme points $\\underline{\\mu}, \\bar{\\mu}$ of the trust interval with the corresponding payoff:\n\n$$\n\\begin{aligned}\n& V(\\underline{\\mu}, \\bar{\\mu}) \\triangleq \\\\\n& \\alpha\\left(\\int_{0}^{\\underline{\\mu}}\\left(U(\\underline{\\mu})+U^{\\prime}(\\underline{\\mu})(\\mu-\\underline{\\mu})\\right) \\tau(\\mu) d \\mu+\\int_{\\underline{\\mu}}^{\\bar{\\mu}} U(\\mu) \\tau(\\mu) d \\mu+\\int_{\\bar{\\mu}}^{1}\\left(U(\\bar{\\mu})+U^{\\prime}(\\bar{\\mu})(\\mu-\\bar{\\mu})\\right) \\tau(\\mu) d \\mu\\right) \\\\\n& \\quad+(1-\\alpha)\\left(\\int_{0}^{b(\\underline{\\mu}, \\bar{\\mu})}\\left(U(\\bar{\\mu})+U^{\\prime}(\\bar{\\mu})(\\mu-\\bar{\\mu})\\right) \\tau(\\mu) d \\mu+\\int_{b(\\underline{\\mu}, \\bar{\\mu})}^{1}\\left(U(\\underline{\\mu})+U^{\\prime}(\\underline{\\mu})(\\mu-\\underline{\\mu})\\right) \\tau(\\mu) d \\mu\\right)\n\\end{aligned}\n$$\n\nThe function $V(\\underline{\\mu}, \\bar{\\mu})$ is continuously differentiable with partial derivatives (whenever\n\n$$\n\\underline{\\mu}<\\bar{\\mu}):\n$$\n\n$$\n\\begin{aligned}\n& \\frac{\\partial V}{\\partial \\underline{\\mu}}=U^{\\prime \\prime}(\\underline{\\mu})\\left(\\alpha \\int_{0}^{\\underline{\\mu}}(\\mu-\\underline{\\mu}) \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{b(\\underline{\\mu}, \\bar{\\mu})}^{1}(\\mu-\\underline{\\mu}) \\tau(\\mu) d \\mu\\right) \\\\\n& \\frac{\\partial V}{\\partial \\bar{\\mu}}=U^{\\prime \\prime}(\\bar{\\mu})\\left(\\alpha \\int_{\\bar{\\mu}}^{1}(\\mu-\\bar{\\mu}) \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{0}^{b(\\underline{\\mu}, \\bar{\\mu})}(\\mu-\\bar{\\mu}) \\tau(\\mu) d \\mu\\right)\n\\end{aligned}\n$$\n\nIntuitively, the first-order impact of a change in the trust boundary equals the change in the action played at that boundary, measured by $U^{\\prime \\prime}(\\cdot)$, integrated over the belief regions in which the aligned and misaligned advisers induce that action, weighted by the alignment parameter. (Terms involving $\\partial b / \\partial \\underline{\\mu}$ and $\\partial b / \\partial \\bar{\\mu}$ vanish because at $\\mu=b(\\underline{\\mu}, \\bar{\\mu})$ the misaligned adviser is indifferent between the two messages.)\n\nWhenever the trust region is non-singleton, $\\underline{\\mu}<\\bar{\\mu}$, at the optimal choice of $\\underline{\\mu}$ and $\\bar{\\mu}$ these partial derivatives must equal zero, $\\partial V / \\partial \\underline{\\mu}=0$ and $\\partial V / \\partial \\bar{\\mu}=0$. Since $U^{\\prime \\prime}(\\cdot)>0$, these first-order conditions can be rearranged as follows. Define functions $\\Psi_{1}$ and $\\Psi_{2}$ as\n\n$$\n\\begin{aligned}\n& \\Psi_{1}(\\underline{\\mu}, \\bar{\\mu}, \\alpha) \\triangleq \\alpha \\int_{0}^{\\underline{\\mu}}(\\mu-\\underline{\\mu}) \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{b(\\underline{\\mu}, \\bar{\\mu})}^{1}(\\mu-\\underline{\\mu}) \\tau(\\mu) d \\mu \\\\\n& \\Psi_{2}(\\underline{\\mu}, \\bar{\\mu}, \\alpha) \\triangleq \\alpha \\int_{\\bar{\\mu}}^{1}(\\mu-\\bar{\\mu}) \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{0}^{b(\\underline{\\mu}, \\bar{\\mu})}(\\mu-\\bar{\\mu}) \\tau(\\mu) d \\mu\n\\end{aligned}\n$$\n\nThen, $\\Psi_{1}(\\underline{\\mu}, \\bar{\\mu}, \\alpha)=\\Psi_{2}(\\underline{\\mu}, \\bar{\\mu}, \\alpha)=0$ is equivalent to conditions (6) and (7).\nFirst, we show that conditions (6) and (7) are incompatible with $\\alpha<1 / 2$. (If $M$ were finite, this would follow directly from Theorem 3.) Indeed, if those conditions hold then (for the rest of the proof, we will often omit the arguments of the function $b$ for brevity):\n\n$$\n\\begin{aligned}\n\\alpha\\left(\\int_{0}^{b}(b-\\mu) \\tau(\\mu) d \\mu+\\int_{b}^{1}(\\mu-b) \\tau(\\mu) d \\mu\\right) & \\geq \\alpha\\left(\\int_{0}^{\\underline{\\mu}}(\\underline{\\mu}-\\mu) \\tau(\\mu) d \\mu+\\int_{\\bar{\\mu}}^{1}(\\mu-\\bar{\\mu}) \\tau(\\mu) d \\mu\\right) \\\\\n& =(1-\\alpha)\\left(\\int_{0}^{b}(\\bar{\\mu}-\\mu) \\tau(\\mu) d \\mu+\\int_{b}^{1}(\\mu-\\underline{\\mu}) \\tau(\\mu) d \\mu\\right) \\\\\n& \\geq(1-\\alpha)\\left(\\int_{0}^{b}(b-\\mu) \\tau(\\mu) d \\mu+\\int_{b}^{1}(\\mu-b) \\tau(\\mu) d \\mu\\right)\n\\end{aligned}\n$$\n\nwhere the inequalities hold because $\\underline{\\mu} \\leq b(\\underline{\\mu}, \\bar{\\mu}) \\leq \\bar{\\mu}$ and the equality is a consequence of (6) and (7). Because $\\tau$ has full support, the multipliers on both sides of the inequality are\nstrictly positive, and thus $\\alpha \\geq 1-\\alpha$, i.e., $\\alpha \\geq 1 / 2$.\nNow we argue that for $\\alpha \\geq 1 / 2$ the solution to (6) and (7) such that $\\underline{\\mu} \\leq \\bar{\\mu}$ exists. Note that at $\\alpha=1 / 2,[\\underline{\\mu}, \\bar{\\mu}]=\\left[\\mu_{0}, \\mu_{0}\\right]$ is a solution. For the rest of this proof, we omit the dependence of $\\Psi_{i}$ on $\\alpha$. By Lemma 1 and direct inspection, $b(\\underline{\\mu}, \\bar{\\mu})$ is strictly and continuously increasing in its arguments, so $\\Psi_{1}(\\underline{\\mu}, \\bar{\\mu})$ is strictly and continuously decreasing in $\\underline{\\mu}$ for each $\\bar{\\mu}$. Furthermore,\n\n$$\n\\begin{aligned}\n& \\Psi_{1}(0, \\bar{\\mu})=(1-\\alpha) \\int_{b(0, \\bar{\\mu})}^{1} \\mu \\tau(\\mu) d \\mu \\geq 0 \\\\\n& \\Psi_{1}(1, \\bar{\\mu})=\\alpha \\int_{0}^{1}(\\mu-1) \\tau(\\mu) d \\mu<0\n\\end{aligned}\n$$\n\nTherefore, for each $\\bar{\\mu}$, a best-response $b_{1}(\\bar{\\mu})$ such that $\\Psi_{1}\\left(b_{1}(\\bar{\\mu}), \\bar{\\mu}\\right)=0$ exists and is unique. Since $\\Psi_{1}(\\underline{\\mu}, \\bar{\\mu})$ strictly decreases in $\\bar{\\mu}, b_{1}(\\bar{\\mu})$ strictly decreases in $\\bar{\\mu}$. Finally, for any $\\bar{\\mu}$,\n\n$$\n\\int_{b_{1}(\\bar{\\mu})}^{1}\\left(\\mu-b_{1}(\\bar{\\mu})\\right) \\tau(\\mu) d \\mu \\geq \\int_{b\\left(b_{1}(\\bar{\\mu}), \\bar{\\mu}\\right)}^{1}\\left(\\mu-b_{1}(\\bar{\\mu})\\right) \\tau(\\mu) d \\mu \\geq \\int_{0}^{b_{1}(\\bar{\\mu})}\\left(b_{1}(\\bar{\\mu})-\\mu\\right) \\tau(\\mu) d \\mu\n$$\n\nwhere the second inequality holds because $\\alpha \\geq 1 / 2$ and $\\Psi_{1}\\left(b_{1}(\\bar{\\mu}), \\bar{\\mu}\\right)=0$. Thus, for any $\\bar{\\mu}$, $b_{1}(\\bar{\\mu}) \\leq \\mu_{0}$.\n\nAnalogously, for each $\\underline{\\mu}$, a best-response $b_{2}(\\underline{\\mu})$ such that $\\Psi_{2}\\left(\\underline{\\mu}, b_{2}\\right)=0$, exists, is unique, strictly decreases in $\\underline{\\mu}$, and is everywhere greater than $\\mu_{0}$.","text_sha256":"cbe5b13784ff6073b78d6a6477f77602038e1edc316c98932119fbb4d9de6a09"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0029","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 6 Proof of Proposition 1","text":"Therefore, a solution to (6) and (7) is any $\\underline{\\mu} \\in\\left[0, \\mu_{0}\\right]$ and $\\bar{\\mu}=b_{2}(\\underline{\\mu}) \\in\\left[\\mu_{0}, 1\\right]$ such that $b_{1}\\left(b_{2}(\\underline{\\mu})\\right)=\\underline{\\mu}$. By the established properties of $b_{1}$ and $b_{2}, b_{1}\\left(b_{2}(\\underline{\\mu})\\right)$ is continuous in $\\underline{\\mu}$ with $b_{1}\\left(b_{2}(\\underline{\\mu})\\right) \\in\\left[0, \\mu_{0}\\right]$ for all $\\underline{\\mu} \\in\\left[0, \\mu_{0}\\right]$; hence, $b_{1}\\left(b_{2}(0)\\right)-0 \\geq 0$ and $b_{1}\\left(b_{2}\\left(\\mu_{0}\\right)\\right)-\\mu_{0} \\leq 0$. By the intermediate value theorem, there exists $\\underline{\\mu} \\in\\left[0, \\mu_{0}\\right]$ such that $b_{1}\\left(b_{2}(\\underline{\\mu})\\right)=\\underline{\\mu}$.\n\nSo far, we showed that for $\\alpha \\geq 1 / 2$, a solution exists and belongs to a closed rectangular set $D=\\left\\{(\\underline{\\mu}, \\bar{\\mu}): \\underline{\\mu} \\in\\left[0, \\mu_{0}\\right], \\bar{\\mu} \\in\\left[\\mu_{0}, 1\\right]\\right\\}$. To establish uniqueness, consider the function $-\\Psi=\\left(-\\Psi_{1},-\\Psi_{2}\\right)$ on $D$. Any solution must satisfy $-\\Psi(\\underline{\\mu}, \\bar{\\mu})=(0,0)$. Observe that for any\n$(\\underline{\\mu}, \\bar{\\mu}) \\in D$,\n\n$$\n\\begin{aligned}\n& \\frac{\\partial\\left[-\\Psi_{1}\\right]}{\\partial \\underline{\\mu}}=\\alpha \\int_{0}^{\\underline{\\mu}} \\tau(\\mu) d \\mu+(1-\\alpha) \\frac{\\partial b}{\\partial \\underline{\\mu}} \\tau(b)(b-\\underline{\\mu})+(1-\\alpha) \\int_{b}^{1} \\tau(\\mu) d \\mu>0, \\\\\n& \\frac{\\partial\\left[-\\Psi_{1}\\right]}{\\partial \\bar{\\mu}}=(1-\\alpha) \\frac{\\partial b}{\\partial \\bar{\\mu}} \\tau(b)(b-\\underline{\\mu}) \\geq 0, \\\\\n& \\frac{\\partial\\left[-\\Psi_{2}\\right]}{\\partial \\underline{\\mu}}=-(1-\\alpha) \\frac{\\partial b}{\\partial \\underline{\\mu}} \\tau(b)(b-\\bar{\\mu}) \\geq 0, \\\\\n& \\frac{\\partial\\left[-\\Psi_{2}\\right]}{\\partial \\bar{\\mu}}=\\alpha \\int_{\\bar{\\mu}}^{1} \\tau(\\mu) d \\mu+(1-\\alpha) \\frac{\\partial b}{\\partial \\bar{\\mu}} \\tau(b)(\\bar{\\mu}-b)+(1-\\alpha) \\int_{0}^{b} \\tau(\\mu) d \\mu>0 .\n\\end{aligned}\n$$\n\nMoreover, for all $(\\underline{\\mu}, \\bar{\\mu}) \\in D$, the Jacobian of $[-\\Psi]$ is a P-matrix, i.e., it has strictly positive principal minors:\n\n$$\n\\frac{\\partial\\left[-\\Psi_{1}\\right]}{\\partial \\underline{\\mu}}>0, \\quad \\frac{\\partial\\left[-\\Psi_{1}\\right]}{\\partial \\underline{\\mu}} \\frac{\\partial\\left[-\\Psi_{2}\\right]}{\\partial \\bar{\\mu}}-\\frac{\\partial\\left[-\\Psi_{1}\\right]}{\\partial \\bar{\\mu}} \\frac{\\partial\\left[-\\Psi_{2}\\right]}{\\partial \\underline{\\mu}}>0 .\n$$\n\nBy the Gale-Nikaido Theorem (Gale and Nikaido (1965), Theorem 4), it follows that $[-\\Psi]$ is injective on $D$, and thus there exists at most one solution to the equation $-\\Psi(\\underline{\\mu}, \\bar{\\mu})=(0,0)$.\n\nFinally, we show that the proposed trust region strategy is robustly rationalizable by explicitly constructing a TRE. For $\\alpha \\geq 1 / 2$, we need to construct a measurable strategy of the misaligned adviser $\\beta:[0, b] \\rightarrow[\\bar{\\mu}, 1]$ such that for every set $X \\subseteq[\\bar{\\mu}, 1]$ with $\\alpha \\tau(X)+(1-$ $\\alpha) \\tau\\left(\\beta^{-1}(X)\\right)>0$,\n\n$$\n\\frac{\\alpha \\int_{X} \\mu \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{\\beta^{-1}(X)} \\mu \\tau(\\mu) d \\mu}{\\alpha \\int_{X} \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{\\beta^{-1}(X)} \\tau(\\mu) d \\mu}=\\bar{\\mu}\n$$\n\n(The construction of $\\beta:(b, 1] \\rightarrow[0, \\underline{\\mu}]$ is analogous.) To this end, define two finite atomless nonnegative measures:\n\n$$\n\\begin{aligned}\n& \\nu(Y) \\triangleq(1-\\alpha) \\int_{Y}(\\bar{\\mu}-\\mu) \\tau(\\mu) d \\mu, \\quad Y \\subseteq[0, b] \\\\\n& \\eta(X) \\triangleq \\alpha \\int_{X}(\\mu-\\bar{\\mu}) \\tau(\\mu) d \\mu, \\quad X \\subseteq[\\bar{\\mu}, 1]\n\\end{aligned}\n$$\n\nObserve that condition (7) is precisely $\\eta([\\bar{\\mu}, 1])=\\nu([0, b])$ whereas condition (27) is the\npushforward identity:\n\n$$\n\\eta(X)=\\nu\\left(\\beta^{-1}(X)\\right), \\quad X \\subseteq[\\bar{\\mu}, 1] .\n$$\n\nIn other words, we need to find $\\beta$ that transports $\\nu$ to $\\eta$. It is always possible. For a canonical quantile construction, define the cumulative mass functions $F_{\\nu}(\\mu) \\triangleq \\nu([0, \\mu])$ for $\\mu \\in[0, b]$ and $F_{\\eta}(\\mu) \\triangleq \\eta([\\bar{\\mu}, \\mu])$ for $\\mu \\in[\\bar{\\mu}, 1]$. The transport map can then be set:\n\n$$\n\\beta(\\mu)=F_{\\eta}^{-1}\\left(F_{\\nu}(\\mu)\\right), \\quad \\mu \\in[0, b],\n$$\n\nwhere $F_{\\eta}^{-1}(\\cdot)$ is the generalized inverse: $F_{\\eta}^{-1}(q)=\\inf \\left\\{\\mu \\in[\\bar{\\mu}, 1]: F_{\\eta}(\\mu) \\geq q\\right\\}$.\nFor $\\alpha<1 / 2, T=\\left\\{\\mu_{0}\\right\\}$, so the misaligned adviser is indifferent between all messages and it suffices to construct a strategy $\\beta:[0,1] \\rightarrow[0,1]$ such that for all $X \\subseteq[0,1]$ with $\\alpha \\int_{X} \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{\\beta^{-1}(X)} \\tau(\\mu) d \\mu>0$,\n\n$$\n\\frac{\\alpha \\int_{X} \\mu \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{\\beta^{-1}(X)} \\mu \\tau(\\mu) d \\mu}{\\alpha \\int_{X} \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{\\beta^{-1}(X)} \\tau(\\mu) d \\mu}=\\mu_{0}\n$$\n\nwhich is equivalent to:\n\n$$\n\\alpha \\int_{X}\\left(\\mu-\\mu_{0}\\right) \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{\\beta^{-1}(X)}\\left(\\mu-\\mu_{0}\\right) \\tau(\\mu) d \\mu=0\n$$\n\nTo do that, observe that $\\int_{0}^{1}\\left(\\mu-\\mu_{0}\\right) \\tau(\\mu) d \\mu=0$ and $\\int_{0}^{\\mu_{0}}\\left(\\mu_{0}-\\mu\\right) \\tau(\\mu) d \\mu=\\int_{\\mu_{0}}^{1}(\\mu-$ $\\left.\\mu_{0}\\right) \\tau(\\mu) d \\mu>0$. Since $\\alpha \\in(0,1 / 2), \\alpha /(1-\\alpha) \\in(0,1)$ and by the intermediate value theorem, there exist $\\mu_{L} \\in\\left(0, \\mu_{0}\\right)$ and $\\mu_{H} \\in\\left(\\mu_{0}, 1\\right)$ such that\n\n$$\n\\begin{aligned}\n& \\int_{0}^{\\mu_{L}}\\left(\\mu_{0}-\\mu\\right) \\tau(\\mu) d \\mu=\\frac{\\alpha}{1-\\alpha} \\int_{0}^{\\mu_{0}}\\left(\\mu_{0}-\\mu\\right) \\tau(\\mu) d \\mu \\\\\n& \\int_{\\mu_{H}}^{1}\\left(\\mu-\\mu_{0}\\right) \\tau(\\mu) d \\mu=\\frac{\\alpha}{1-\\alpha} \\int_{\\mu_{0}}^{1}\\left(\\mu-\\mu_{0}\\right) \\tau(\\mu) d \\mu\n\\end{aligned}\n$$\n\nBy construction,\n\n$$\n\\begin{aligned}\n& \\int_{0}^{\\mu_{L}}\\left(\\mu_{0}-\\mu\\right) \\tau(\\mu) d \\mu=\\frac{\\alpha}{1-\\alpha} \\int_{\\mu_{0}}^{1}\\left(\\mu-\\mu_{0}\\right) \\tau(\\mu) d \\mu \\\\\n& \\int_{\\mu_{H}}^{1}\\left(\\mu-\\mu_{0}\\right) \\tau(\\mu) d \\mu=\\frac{\\alpha}{1-\\alpha} \\int_{0}^{\\mu_{0}}\\left(\\mu_{0}-\\mu\\right) \\tau(\\mu) d \\mu \\\\\n& \\int_{\\mu_{L}}^{\\mu_{H}}\\left(\\mu-\\mu_{0}\\right) \\tau(\\mu) d \\mu=0\n\\end{aligned}\n$$","text_sha256":"adaf89cff3cb2d53eef8b0057d668e4b3810c5fdcec1004597d2c6de3c4b98a2"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0030","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 6 Proof of Proposition 1","text":"We can set $\\beta(\\mu)=\\beta_{L}(\\mu)$ when $\\mu \\in\\left[0, \\mu_{L}\\right], \\beta(\\mu)=\\mu_{0}$, when $\\mu \\in\\left(\\mu_{L}, \\mu_{H}\\right)$, and $\\beta(\\mu)=\\beta_{H}(\\mu)$, when $\\mu \\in\\left[\\mu_{H}, 1\\right]$. Here, $\\beta_{L}$ is a quantile transport map that transports measure $\\nu_{L}(Y)=(1-\\alpha) \\int_{Y}\\left(\\mu_{0}-\\mu\\right) \\tau(\\mu) d \\mu$ on $\\left[0, \\mu_{L}\\right]$ to measure $\\eta_{L}(X)=\\alpha \\int_{X}\\left(\\mu-\\mu_{0}\\right) \\tau(\\mu) d \\mu$ on [ $\\mu_{0}, 1$ ], just like in the case of $\\alpha \\geq 1 / 2$; it ensures that (28) holds for all $X \\subseteq\\left(\\mu_{0}, 1\\right]$. Similarly, $\\beta_{H}$ is a quantile transport map that transports measure $\\nu_{H}(Y)=(1-\\alpha) \\int_{Y}\\left(\\mu-\\mu_{0}\\right) \\tau(\\mu) d \\mu$ on $\\left[\\mu_{H}, 1\\right]$ to measure $\\eta_{H}(X)=\\alpha \\int_{X}\\left(\\mu_{0}-\\mu\\right) \\tau(\\mu) d \\mu$ on $\\left[0, \\mu_{0}\\right]$; it ensures that (28) holds for all $X \\subseteq\\left[0, \\mu_{0}\\right)$. (The transported masses match the targets by equations (29) and (30).) Finally, by equation (31), condition (28) holds for $\\mu=\\mu_{0}$. The result follows.","text_sha256":"3e7f2384f50db0fd2603736fd47c6bbb926fd7718179bcc5dac0f3e7d3cb7254"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0031","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 7 Proof of Proposition 2","text":"## A. 7 Proof of Proposition 2\n\nAt $\\alpha=1 / 2,[\\underline{\\mu}, \\bar{\\mu}]=\\left[\\mu_{0}, \\mu_{0}\\right]$ satisfies conditions (6) and (7). At $\\alpha=1,[\\underline{\\mu}, \\bar{\\mu}]=[0,1]$ satisfies conditions (6) and (7).\n\nFor $\\alpha \\in(1 / 2,1)$, denote by $\\Psi_{i 1}, \\Psi_{i 2}$, and $\\Psi_{i \\alpha}$ the partial derivatives of $\\Psi_{i}$ with respect to $\\underline{\\mu}, \\bar{\\mu}$, and $\\alpha$ respectively. Define the Jacobian:\n\n$$\nJ(\\underline{\\mu}, \\bar{\\mu}, \\alpha) \\triangleq\\left(\\begin{array}{ll}\n\\Psi_{11} & \\Psi_{12} \\\\\n\\Psi_{21} & \\Psi_{22}\n\\end{array}\\right)\n$$\n\nAs we argued in the proof of Proposition 1, for all $\\underline{\\mu}, \\bar{\\mu}, \\alpha>1 / 2, \\operatorname{det} J(\\underline{\\mu}, \\bar{\\mu}, \\alpha)>0$, and therefore, by the implicit function theorem, optimal $\\underline{\\mu}(\\alpha)$ and $\\bar{\\mu}(\\alpha)$ are continuously\ndifferentiable and ${ }^{21}$\n\n$$\n\\binom{d \\underline{\\mu} / d \\alpha}{d \\bar{\\mu} / d \\alpha}=-J(\\underline{\\mu}, \\bar{\\mu}, \\alpha)^{-1}\\binom{\\Psi_{1 \\alpha}}{\\Psi_{2 \\alpha}} .\n$$\n\nConsequently,\n\n$$\n\\begin{aligned}\n& \\frac{d \\underline{\\mu}}{d \\alpha}=-\\frac{\\Psi_{2 \\bar{\\mu}} \\Psi_{1 \\alpha}-\\Psi_{1 \\bar{\\mu}} \\Psi_{2 \\alpha}}{\\Psi_{1 \\underline{\\mu}} \\Psi_{2 \\bar{\\mu}}-\\Psi_{1 \\bar{\\mu}} \\Psi_{2 \\underline{\\mu}}}<0, \\\\\n& \\frac{d \\bar{\\mu}}{d \\alpha}=\\frac{\\Psi_{2 \\underline{\\mu}} \\Psi_{1 \\alpha}-\\Psi_{1 \\underline{\\mu}} \\Psi_{2 \\alpha}}{\\Psi_{1 \\underline{\\mu}} \\Psi_{2 \\bar{\\mu}}-\\Psi_{1 \\bar{\\mu}} \\Psi_{2 \\underline{\\mu}}}>0,\n\\end{aligned}\n$$\n\nwhere the inequalities hold because, as we already showed, $\\Psi_{1 \\underline{\\mu}}<0, \\Psi_{1 \\bar{\\mu}} \\leq 0, \\Psi_{2 \\underline{\\mu}} \\leq 0$, $\\Psi_{2 \\bar{\\mu}}<0$, and\n\n$$\n\\begin{aligned}\n& \\Psi_{1 \\alpha}=\\int_{0}^{\\underline{\\mu}}(\\mu-\\underline{\\mu}) \\tau(\\mu) d \\mu-\\int_{b}^{1}(\\mu-\\underline{\\mu}) \\tau(\\mu) d \\mu<0 \\\\\n& \\Psi_{2 \\alpha}=\\int_{\\bar{\\mu}}^{1}(\\mu-\\bar{\\mu}) \\tau(\\mu) d \\mu-\\int_{0}^{b}(\\mu-\\bar{\\mu}) \\tau(\\mu) d \\mu>0\n\\end{aligned}\n$$\n\nThe result follows.","text_sha256":"166f1c1e1a34590c938d3fea6f0e70451da255806230702b1fa2e59a0dabd591"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0032","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 8 Proof of Proposition 3","text":"## A. 8 Proof of Proposition 3\n\nThroughout, fix $\\alpha>1 / 2$; the proposition is trivially true otherwise. We begin with a simple lemma.\n\nLemma 7. Let $U_{1}, U_{2}$ be twice differentiable and strictly convex on $[0,1]$. Assume that the ratio $U_{1}^{\\prime \\prime}(\\mu) / U_{2}^{\\prime \\prime}(\\mu)$ is decreasing. Then, with $b_{i}$ defined for each $U_{i}, i \\in\\{1,2\\}$, by equation (5), we have that $b_{1}(\\underline{\\mu}, \\bar{\\mu}) \\leq b_{2}(\\underline{\\mu}, \\bar{\\mu})$ for all $\\underline{\\mu} \\leq \\bar{\\mu}$.\n\nProof. By equation (5), for $i \\in\\{1,2\\}, b_{i}(\\underline{\\mu}, \\bar{\\mu})=\\mathbb{E}_{\\mu \\sim f_{i}}[\\mu]$, where $f_{i}(\\mu)$ is a density of a\n\n[^17]probability measure on $[\\underline{\\mu}, \\bar{\\mu}]$ defined as\n$$\nf_{i}(\\mu) \\triangleq \\frac{U_{i}^{\\prime \\prime}(\\mu)}{\\int_{\\underline{\\mu}}^{\\bar{\\mu}} U_{i}^{\\prime \\prime}(\\mu) d \\mu} .\n$$\nBy assumption, $f_{1}(\\mu) / f_{2}(\\mu)$ is decreasing, and hence $f_{2}$ likelihood-ratio dominates $f_{1}$. This implies that the probability distribution $f_{1}$ is first-order stochastically dominated by $f_{2}$; in particular, it has a lower mean. The result follows. $\\square$\n\nWe now take the second step by showing a monotone relationship between the cutoff function $b$ and the trust region.\n\nLemma 8. Consider two decision problems $U_{i}, i \\in\\{1,2\\}$, and let $\\left(\\underline{\\mu}_{i}, \\bar{\\mu}_{i}\\right) \\in D$ denote the unique solution to the system\n\n$$\n\\Psi_{1}^{i}(\\underline{\\mu}, \\bar{\\mu}, \\alpha)=0, \\quad \\Psi_{2}^{i}(\\underline{\\mu}, \\bar{\\mu}, \\alpha)=0,\n$$\n\nwhere $\\Psi_{1}^{i}, \\Psi_{2}^{i}$ and $D$ are defined as in the proof of Proposition 1 for each $U_{i}$. If $b_{1}(\\underline{\\mu}, \\bar{\\mu}) \\leq$ $b_{2}(\\underline{\\mu}, \\bar{\\mu})$ for all $\\underline{\\mu} \\leq \\bar{\\mu}$, then $\\underline{\\mu}_{2} \\leq \\underline{\\mu}_{1}$ and $\\bar{\\mu}_{2} \\leq \\bar{\\mu}_{1}$.\n\nProof. For any $h \\in[0,1]$, define the auxiliary functions\n\n$$\n\\begin{aligned}\n& \\widetilde{\\Psi}_{1}(\\underline{\\mu}, h) \\triangleq \\alpha \\int_{0}^{\\underline{\\mu}}(\\mu-\\underline{\\mu}) \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{h}^{1}(\\mu-\\underline{\\mu}) \\tau(\\mu) d \\mu \\\\\n& \\widetilde{\\Psi}_{2}(\\bar{\\mu}, h) \\triangleq \\alpha \\int_{\\bar{\\mu}}^{1}(\\mu-\\bar{\\mu}) \\tau(\\mu) d \\mu+(1-\\alpha) \\int_{0}^{h}(\\mu-\\bar{\\mu}) \\tau(\\mu) d \\mu\n\\end{aligned}\n$$\n\nFor each $h$, let $\\underline{\\mu}(h)$ be the unique solution to $\\widetilde{\\Psi}_{1}(\\underline{\\mu}, h)=0$ in $\\left[0, \\mu_{0}\\right]$, and let $\\bar{\\mu}(h)$ be the unique solution to $\\widetilde{\\Psi}_{2}(\\bar{\\mu}, h)=0$ in $\\left[\\mu_{0}, 1\\right]$, with the existence and uniqueness following from an argument analogous to that used in the proof of Proposition 1.\n\nFor each $i \\in\\{1,2\\}$ define the scalar map\n\n$$\n\\varphi_{i}(h) \\triangleq b_{i}(\\underline{\\mu}(h), \\bar{\\mu}(h)) .\n$$\n\nLet $h_{i} \\triangleq b_{i}\\left(\\underline{\\mu}_{i}, \\bar{\\mu}_{i}\\right)$ be the cutoff evaluated at the optimal endpoints of problem $i$. Then $\\left(\\underline{\\mu}_{i}, \\bar{\\mu}_{i}\\right)$ solves $\\Psi_{1}^{i}=\\Psi_{2}^{i}=0$ if and only if $\\underline{\\mu}_{i}=\\underline{\\mu}\\left(h_{i}\\right), \\bar{\\mu}_{i}=\\bar{\\mu}\\left(h_{i}\\right)$, and $h_{i}=\\varphi_{i}\\left(h_{i}\\right)$. By the assumption\n$b_{1} \\leq b_{2}$ pointwise, for every $h$,\n\n$$\n\\varphi_{1}(h)=b_{1}(\\underline{\\mu}(h), \\bar{\\mu}(h)) \\leq b_{2}(\\underline{\\mu}(h), \\bar{\\mu}(h))=\\varphi_{2}(h) .\n$$\n\nDefine a region $H \\triangleq\\{h \\in[0,1]: \\underline{\\mu}(h) \\leq h \\leq \\bar{\\mu}(h)\\}$. Note that $H$ is an interval and $h_{1}, h_{2} \\in H$. For $h \\in H$, implicit differentiation yields\n\n$$\n\\underline{\\mu}^{\\prime}(h)=-\\frac{\\partial \\widetilde{\\Psi}_{1} / \\partial h}{\\partial \\widetilde{\\Psi}_{1} / \\partial \\underline{\\mu}} \\leq 0, \\quad \\bar{\\mu}^{\\prime}(h)=-\\frac{\\partial \\widetilde{\\Psi}_{2} / \\partial h}{\\partial \\widetilde{\\Psi}_{2} / \\partial \\bar{\\mu}} \\leq 0 .\n$$\n\nThus, $\\varphi_{i}$ is weakly decreasing in $h$ on $H$ : both $\\underline{\\mu}(h)$ and $\\bar{\\mu}(h)$ are weakly decreasing in $h$, while $b_{i}(\\underline{\\mu}, \\bar{\\mu})$ is weakly increasing in each endpoint, so the composition $h \\mapsto \\varphi_{i}(h)$ is weakly decreasing.\n\nIt follows that $h_{2} \\geq h_{1}$ : if $h_{2}<h_{1}$, then, since $\\varphi_{2}$ is decreasing on $H$,\n\n$$\nh_{2}=\\varphi_{2}\\left(h_{2}\\right) \\geq \\varphi_{2}\\left(h_{1}\\right) \\geq \\varphi_{1}\\left(h_{1}\\right)=h_{1},\n$$\n\nwhich is a contradiction. Since $\\underline{\\mu}(\\cdot)$ and $\\bar{\\mu}(\\cdot)$ are weakly decreasing,\n\n$$\n\\underline{\\mu}_{2}=\\underline{\\mu}\\left(h_{2}\\right) \\leq \\underline{\\mu}\\left(h_{1}\\right)=\\underline{\\mu}_{1}, \\quad \\bar{\\mu}_{2}=\\bar{\\mu}\\left(h_{2}\\right) \\leq \\bar{\\mu}\\left(h_{1}\\right)=\\bar{\\mu}_{1},\n$$\n\ncompleting the proof. $\\square$\n\nBy Lemma 7, $U_{1}^{\\prime \\prime} / U_{2}^{\\prime \\prime}$ decreasing implies $b_{1}(\\underline{\\mu}, \\bar{\\mu}) \\leq b_{2}(\\underline{\\mu}, \\bar{\\mu})$ for all $\\underline{\\mu} \\leq \\bar{\\mu}$. Lemma 8 then yields $\\underline{\\mu}_{2} \\leq \\underline{\\mu}_{1}$ and $\\bar{\\mu}_{2} \\leq \\bar{\\mu}_{1}$. This proves Proposition 3.","text_sha256":"a830b367588766fc907253883766feac4ae9827d8eb61df3942a802e04889429"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0033","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 9 Proof of Proposition 4","text":"## A. 9 Proof of Proposition 4\n\nWith a small abuse of notation, we can parameterize each private strategy by $\\hat{\\sigma}=\\operatorname{Pr}\\left(a=a_{2}\\right)$. We also drop the dependence of $G$ and $L$ on $\\hat{\\tau}$ in the notation. Then, by the arguments behind Theorem 1, if the agent employs the set of private strategies $\\hat{\\Sigma}_{0}=\\{\\hat{\\sigma}(m)\\}_{m \\in \\Delta(\\Omega)}$, the payoffs coming from both aligned and misaligned adviser depend only on $\\hat{\\sigma}_{L} \\triangleq \\inf \\hat{\\Sigma}_{0}$ and $\\hat{\\sigma}_{H} \\triangleq \\sup \\hat{\\Sigma}_{0}$, and the optimal payoffs from using $\\hat{\\Sigma}_{0}$ are the same as if the agent plays\n$\\hat{\\sigma}(m)=\\hat{\\sigma}_{L}$ when $v(m)<0$ and $\\hat{\\sigma}(m)=\\hat{\\sigma}_{H}$ when $v(m) \\geq 0$. This payoff is:\n\n$$\n\\begin{aligned}\n& \\int_{-\\infty}^{0}\\left(\\alpha \\hat{\\sigma}_{L}+(1-\\alpha) \\hat{\\sigma}_{H}\\right) v \\hat{\\tau}(d v)+\\int_{0}^{+\\infty}\\left(\\alpha \\hat{\\sigma}_{H}+(1-\\alpha) \\hat{\\sigma}_{L}\\right) v \\hat{\\tau}(d v) \\\\\n& =\\hat{\\sigma}_{L}((1-\\alpha) G-\\alpha L)+\\hat{\\sigma}_{H}(\\alpha G-(1-\\alpha) L)\n\\end{aligned}\n$$\n\nThe optimal choice of $\\hat{\\sigma}_{L}$ and $\\hat{\\sigma}_{H}$ must maximize (32) subject to $\\hat{\\sigma}_{L}, \\hat{\\sigma}_{H} \\in[0,1]$ and $\\hat{\\sigma}_{L} \\leq \\hat{\\sigma}_{H}$. This is a linear optimization subject to $\\left(\\hat{\\sigma}_{L}, \\hat{\\sigma}_{H}\\right)$ being in a triangle with vertices (0, 0), (0, 1), and $(1,1)$. A straightforward calculation gives the following solution:\n\nIf $G=L:$ if $\\alpha<\\hat{\\alpha} \\triangleq 1 / 2$, then any $\\hat{\\sigma}_{L}=\\hat{\\sigma}_{H}$ is optimal; if $\\alpha>\\hat{\\alpha}$, then $\\hat{\\sigma}_{L}=0$ and $\\hat{\\sigma}_{H}=1$; if $\\alpha=\\hat{\\alpha}$, then any $\\left(\\hat{\\sigma}_{L}, \\hat{\\sigma}_{H}\\right)$ is optimal. If $G>L$ : if $\\alpha<\\hat{\\alpha}$, then $\\hat{\\sigma}_{L}=\\hat{\\sigma}_{H}=1$; if $\\alpha>\\hat{\\alpha}$, then $\\hat{\\sigma}_{L}=0$ and $\\hat{\\sigma}_{H}=1$; if $\\alpha=\\hat{\\alpha}$, then $\\hat{\\sigma}_{H}=1$ and any $\\hat{\\sigma}_{L}$ is optimal. If $G<L$ : if $\\alpha<\\hat{\\alpha}$, then $\\hat{\\sigma}_{L}=\\hat{\\sigma}_{H}=0$; if $\\alpha>\\hat{\\alpha}$, then $\\hat{\\sigma}_{L}=0$ and $\\hat{\\sigma}_{H}=1$; if $\\alpha=\\hat{\\alpha}$, then $\\hat{\\sigma}_{L}=0$ and any $\\hat{\\sigma}_{H}$ is optimal.\n\nThe cases $\\hat{\\sigma}_{L}=\\hat{\\sigma}_{H}$ correspond to not trusting any message and always acting in the same way, optimal at the prior belief, so $T=\\left\\{\\mu_{0}\\right\\}$. The cases $\\hat{\\sigma}_{L}=0$ and $\\hat{\\sigma}_{H}=1$ correspond to trusting all messages, so $T=\\Delta(\\Omega)$. Since we assumed that the probability of $v(\\mu)=0$ is 0 and $G \\neq L$, the corresponding optimal strategy is uniquely determined.\n\nIt is left to show that those strategies are robustly rationalizable. Define $M_{0}=\\{\\mu:$ $v(\\mu)=0\\}, M_{-}=\\{\\mu: v(\\mu)<0\\}$, and $M_{+}=\\{\\mu: v(\\mu)>0\\}$. Define probability measures $q_{+}(X)=\\int_{X} v(\\mu) \\tau(d \\mu) / G$ for $X \\subseteq M_{+}, q_{-}(Y)=\\int_{Y}(-v(\\mu)) \\tau(d \\mu) / L$ for $Y \\subseteq M_{-}$.\n\nFor $\\alpha>\\hat{\\alpha}$, the agent fully trusts the adviser. Consider the following strategy of the misaligned adviser. If $\\mu \\in M_{0}$, then $\\beta(\\mu)=\\mu$. If $\\mu \\in M_{-}$, then $\\beta$ randomizes over messages $m \\in M_{+}$according to $q_{+}$. If $\\mu \\in M_{+}$, then $\\beta$ randomizes over messages $m \\in M_{-}$according to $q_{-}$. This strategy is clearly adversarial. Furthermore, since $\\alpha>\\hat{\\alpha}$, after any message $m \\in M_{+}$, the posterior expected payoff from action $a_{2}$ is strictly positive: for any $X \\subseteq M_{+}$ with $\\tau(X)>0$,\n\n$$\n\\alpha \\int_{X} v(m) \\tau(d m)+(1-\\alpha) \\int_{\\Delta(\\Omega)} v(\\mu) \\beta(X \\mid \\mu) \\tau(d \\mu)=\\alpha G q_{+}(X)-(1-\\alpha) L q_{+}(X)>0 .\n$$\n\nAnalogously, after any message $m \\in M_{-}$, the posterior expected payoff from action $a_{2}$ is\nstrictly negative: for any $Y \\subseteq M_{-}$with $\\tau(Y)>0$,\n\n$$\n\\alpha \\int_{Y} v(m) \\tau(d m)+(1-\\alpha) \\int_{\\Delta(\\Omega)} v(\\mu) \\beta(Y \\mid \\mu) \\tau(d \\mu)=-\\alpha L q_{-}(Y)+(1-\\alpha) G q_{-}(Y)<0 .\n$$\n\nAnd by construction, after any message $m \\in M_{0}$, the posterior expected payoff from action $a_{2}$ is zero.\n\nFor $\\alpha<\\hat{\\alpha}$, the agent doesn't trust the adviser so any adviser's strategy is adversarial. Consider the case $G>L$; the complementary case is analogous. We need to construct the misaligned adviser strategy that makes communication not valuable. To do so, define $\\gamma=\\alpha L /((1-\\alpha) G) \\in[0,1]$. Consider the following strategy of the misaligned adviser. If $\\mu \\in M_{-}$, then $\\beta$ randomizes over messages $m \\in M_{+}$according to $q_{+}$. If $\\mu \\in M_{+}$, then with probability $\\gamma, \\beta$ randomizes over messages $m \\in M_{-}$according to $q_{-}$and with probability $(1-\\gamma), \\beta$ randomizes over messages $m \\in M_{+}$according to $q_{+}$. This strategy makes the posterior expected payoff from action $a_{2}$ zero after every message $m \\in M_{-}$: for any $Y \\subseteq M_{-}$ with $\\tau(Y)>0$,\n\n$$\n\\alpha \\int_{Y} v(m) \\tau(d m)+(1-\\alpha) \\int_{\\Delta(\\Omega)} v(\\mu) \\beta(Y \\mid \\mu) \\tau(d \\mu)=-\\alpha L q_{-}(Y)+(1-\\alpha) \\gamma G q_{-}(Y)=0 .\n$$\n\nAfter any message $m \\in M_{+}$, the posterior expected payoff from action $a_{2}$ is strictly positive: for any $X \\subseteq M_{+}$with $\\tau(X)>0$,\n\n$$\n\\alpha \\int_{X} v(m) \\tau(d m)+(1-\\alpha) \\int_{\\Delta(\\Omega)} v(\\mu) \\beta(X \\mid \\mu) \\tau(d \\mu)=(G-L) q_{+}(X)>0 .\n$$\n\nThus, this $\\beta$ robustly rationalizes the strategy.\nFinally, at $\\alpha=\\hat{\\alpha}$, both full trust and no trust are optimal, along a continuum of other strategies, and robustly rationalizable by the same strategies $\\beta$ as above.","text_sha256":"31cb59b4e115da768f9537cd6015423bd9417486d9a6cde65bb6df6f16cfd0d2"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0034","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 10 Supporting calculations for Example 1","text":"## A. 10 Supporting calculations for Example 1\n\nThe key step in the construction of the optimal trust region in Example 1 comes from the following simple lemma.\n\nLemma 9 (Spherical U). Let $U(\\mu)=\\tilde{U}(\\|\\mu-b\\|)$ for some vector $b$ and function $\\tilde{U}$. Let $\\mu^{\\prime}(r, n)=b+r n$, where $n \\in \\mathbb{R}^{N}$ with $\\|n\\|=1$ and $r \\in \\mathbb{R}$. Then, (i) $D_{U}\\left(\\mu, \\mu^{\\prime}(r, n)\\right)$ is strictly increasing in $n \\cdot(b-\\mu)$ whenever $r \\neq 0$, and (ii) $D_{U}\\left(\\mu, \\mu^{\\prime}(r, n)\\right)$ is a unimodal function in $r$ with a minimum at $r=n \\cdot(\\mu-b)$.\n\nProof. In this case, $\\nabla U\\left(\\mu^{\\prime}(r, n)\\right)=\\tilde{U}^{\\prime}(r) n$, and thus\n\n$$\nD_{U}\\left(\\mu, \\mu^{\\prime}(r, n)\\right)=\\tilde{U}(\\|\\mu-b\\|)-\\tilde{U}(r)-\\tilde{U}^{\\prime}(r)(n \\cdot(\\mu-b)-r) .\n$$\n\nSince $U(\\mu)$ is strictly convex, $\\tilde{U}^{\\prime}(r)>0$ whenever $r \\neq 0$ and $\\tilde{U}^{\\prime \\prime}(r)>0$. Thus $D_{U}\\left(\\mu, \\mu^{\\prime}(r, n)\\right)$ is strictly increasing in $n \\cdot(b-\\mu)$. Furthermore,\n\n$$\n\\frac{\\partial D_{U}\\left(\\mu, \\mu^{\\prime}(r, n)\\right)}{\\partial r}=-\\tilde{U}^{\\prime}(r)-\\tilde{U}^{\\prime \\prime}(r)(n \\cdot(\\mu-b)-r)+\\tilde{U}^{\\prime}(r)=\\tilde{U}^{\\prime \\prime}(r)(r-n \\cdot(\\mu-b)) .\n$$\n\nBecause $\\tilde{U}^{\\prime \\prime}(r)>0$, the unimodality follows. $\\square$\n\nSuppose that the misaligned adviser holds belief $\\mu$. By Corollary 3, the adviser will try to maximize $D_{U}\\left(\\mu, \\mu^{\\prime}\\right)$ over $\\mu^{\\prime}$ on the boundary of the trust region. Parameterizing $\\mu^{\\prime}=b+r n$ and applying Lemma 9 yields that $n$ is optimally chosen to be $(b-\\mu) /\\|b-\\mu\\|$, whereas $r$ is chosen to be maximal within the trust region, $r=r^{*}(\\alpha)$. Thus, the optimal $\\mu^{\\prime}$ is given uniquely by $b+r^{*}(\\alpha)(b-\\mu) /\\|b-\\mu\\|$.\n\n[^0]:    *Dworczak: Department of Economics, Northwestern University and Group for Research in Applied Economics, piotr.dworczak@northwestern.edu. Smolin: Toulouse School of Economics, alexey.v.smolin@gmail.com. We thank Nageeb Ali, Ricardo Alonso, Ben Brooks, Laura Doval, Tan Gan, Alexis Ghersengorin, Marina Halac, Jason Hartline, Nicole Immorlica, Emir Kamenica, David Levine, Annie Liang, Stephen Morris, Jacopo Perego, Balázs Szentes, and Mark Whitmeyer for helpful conversations. Alex Smolin gratefully acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (grant ANR-17-EURE-0010) and the AI Interdisciplinary Institute ANITI (grant ANR-23-IACL-0002). Part of the analysis in this paper was conducted while Alex Smolin was visiting Northwestern University and Columbia Business School, and we thank both institutions for their hospitality.\n\n[^1]:    ${ }^{1}$ Strictly speaking, we assume that with probability $\\alpha$ the Sender reveals his signal truthfully, but this can be shown to be optimal for the Receiver.\n\n[^2]:    ${ }^{2}$ See also Levy and Szentes (2025) for a related model of AI under imperfect recall.\n\n[^3]:    ${ }^{3}$ We allow $S$ to be infinite, which is useful for constructing tractable examples. Whenever we work with an infinite space, we endow it with the Borel $\\sigma$-algebra, and require all sets and functions that we define to be measurable; statements involving \"for all\" should be interpreted as \"for almost all\" with respect to the underlying distributions.\n\n[^4]:    ${ }^{4}$ It can be shown that the assumption of truthful reporting of the belief is equivalent (in terms of equilibrium payoff consequences) to assuming that the aligned adviser is attempting to maximize the agent's expected payoff. However, the assumption of truthful reporting is natural for an aligned AI system and useful, as it provides a natural meaning to each message (see Sobel (2020)).\n    ${ }^{5}$ Formally, $\\mathbb{E}_{\\beta, \\sigma}[u(a, \\omega, \\theta)]=\\sum_{\\omega \\in \\Omega} \\mu_{0}(\\omega) \\int_{S} \\int_{\\Delta(\\Omega)} \\int_{\\Theta} \\int_{A} u(a, \\omega, \\theta) \\sigma(d a \\mid m, \\theta) f(d \\theta \\mid \\omega) \\beta(d m \\mid s) \\pi(d s \\mid \\omega)$.\n\n[^5]:    ${ }^{6}$ Formally, $\\mathbb{E}_{\\omega \\sim \\mu, \\hat{\\sigma}}[u(a, \\omega, \\theta)]=\\sum_{\\omega \\in \\Omega} \\mu(\\omega) \\int_{\\Theta} \\int_{A} u(a, \\omega, \\theta) \\hat{\\sigma}(d a \\mid \\theta) f(d \\theta \\mid \\omega)$.\n\n[^6]:    ${ }^{7}$ Formally, to define the gradient, we extend the function $U$ beyond the probability simplex by assuming that, for any non-negative measure $\\mu, U(\\mu)=\\mu(\\Omega) U(\\mu / \\mu(\\Omega))$.\n    ${ }^{8}$ Point $m^{\\prime} \\in T$ is visible from $m$ if the line segment connecting $m^{\\prime}$ and $m$ does not intersect $T \\backslash\\left\\{m^{\\prime}\\right\\}$.\n\n[^7]:    ${ }^{9}$ However, it is not without loss of generality to assume that $T \\subseteq M$.\n\n[^8]:    ${ }^{10}$ Without loss of generality, we assume that $\\beta$ uses only messages in $M$; any message $m \\notin M$ cannot be sent by an aligned adviser and hence reveals that the adviser is misaligned.\n    ${ }^{11}$ The assumption of finite $M$ and $\\Theta$ is made for technical reasons; verifying the assumptions of Sion (1958)'s minimax theorem (in particular, its continuity requirements) is difficult for a cheap-talk-like game with infinite-dimensional strategy spaces since the impact of messages on payoffs is endogenous.\n\n[^9]:    ${ }^{12}$ The strictly convex indirect payoff function can be a result of the agent having a continuum of actions or, as we show in Appendix A.4, finitely many actions and a continuum of private types.\n\n[^10]:    ${ }^{13}$ One way to see that is to use our observation that in a TRE, the mapping from messages to belief defined by Bayes' rule must agree with the mapping defined by minimizing the Bregman distance to the trust region.\n\n[^11]:    ${ }^{14}$ A recent systematic review of real-world AI deployment in medical imaging finds that AI most commonly serves as a secondary reader or a triage tool rather than as a fully autonomous reader; see Wenderott et al. (2024). For a regulatory example, the FDA-cleared DrAid chest X-ray system is indicated as a triage and prioritization aid and is not intended for stand-alone clinical decision-making; see FDA 510(k) summary K221241, released in 2024.\n    ${ }^{15}$ See Dratsch et al. (2023). An RSNA summary of the study reports that incorrect AI suggestions reduced reader accuracy even for experienced radiologists.","text_sha256":"314ea0832461e3ee2109afeacbc986adcc308185b93a7fe8817008ee3e56308f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0035","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 10 Supporting calculations for Example 1","text":"[^12]:    ${ }^{16}$ This conclusion is driven by the coarseness of the strategy space; we know from Theorem 3 that, for any $\\alpha>0$, the agent would optimally use a non-trivial trust region if the state space and her action space were sufficiently rich.\n\n[^13]:    ${ }^{17}$ To see why, note that any trust region $T$ can be convexified by replacing it with the intersection of sets $T_{\\mu}$ over all $\\mu \\in \\operatorname{supp}(\\tau)$, where $T_{\\mu}$ is the (convex) set of all points that are not further away from $\\mu$ than any point in $T$.\n\n[^14]:    ${ }^{18}$ It is equivalent to $\\sigma^{*}$ because it is weakly better than $\\sigma^{*}$ and $\\sigma^{*}$ was optimal.\n\n[^15]:    ${ }^{19}$ A matrix of adviser's posteriors can be computed by Bayes' rule as $(\\mu(s))_{s \\in S}=$ $\\left(\\operatorname{diag}\\left(\\mu_{0}(\\omega)\\right)_{\\omega \\in \\Omega}\\right) \\Pi\\left(\\operatorname{diag}(\\tau(s))_{s \\in S}\\right)^{-1}$. The diagonal matrices are invertible and the multiplication by an invertible matrix preserves the rank.\n\n[^16]:    ${ }^{20}$ By the Auerbach theorem, there exist $v_{1}, \\ldots, v_{R-1} \\in \\mathbb{W}$ and $\\phi_{1}, \\ldots, \\phi_{R-1} \\in \\mathbb{W}^{*}$ such that $\\left\\|v_{i}\\right\\|_{1}=1$, $\\left\\|\\phi_{i}\\right\\|_{\\mathbb{W}^{*}}=1$, and $\\phi_{i}\\left(v_{j}\\right)=\\delta_{i j}$ (Section II.E, Lemma 11 in Wojtaszczyk (1991); see also Gershkov et al. (2025) for another recent application). By Hahn-Banach theorem, these $\\phi_{i}$, operating on $\\mathbb{W}$, can be extended to $\\tilde{\\phi}_{i}$, operating on $\\mathbb{R}^{K}$, without a change in their norm. By the duality between spaces $l_{1}$ and $l_{\\infty}$, for each $i$, there exists $x_{i} \\in \\mathbb{R}^{K}$ such that $\\tilde{\\phi}_{i}(z)=z^{\\top} x_{i}$ and $\\left\\|x_{i}\\right\\|_{\\infty}=\\left\\|\\tilde{\\phi}_{i}\\right\\|=1$. Then, for $w \\in \\mathbb{W}$, $w_{i}^{\\top} x_{j}=\\tilde{\\phi}_{j}\\left(w_{i}\\right)=\\phi_{j}\\left(w_{i}\\right)=\\delta_{i j}$.\n\n[^17]:    ${ }^{21}$ Differentiating the optimality conditions with respect to $\\alpha$ we obtain $\\Psi_{11} \\frac{d \\underline{\\underline{\\mu}}}{d \\alpha}+\\Psi_{12} \\frac{d \\bar{\\mu}}{d \\alpha}+\\Psi_{1 \\alpha}=0, \\Psi_{21} \\frac{d \\underline{\\underline{\\mu}}}{d \\alpha}+$ $\\Psi_{22} \\frac{d \\bar{\\mu}}{d \\alpha}+\\Psi_{2 \\alpha}=0$.","text_sha256":"a3028d39a76dd771f9937d5cbc9040c46affef4a8eb2e1d668a6b27abd390523"}
{"schema_version":"1.0","chunk_id":"alex-smolin:robust-trust:2026-03-19:0036","work_id":"alex-smolin:robust-trust","paper_id":"alex-smolin:robust-trust:2026-03-19","title":"Robust Trust","authors":[{"name":"Piotr Dworczak","url":"https://sites.northwestern.edu/dworczak/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-19","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/robust-trust.md","source_record":"https://arxiv.org/abs/2602.09490","citation":"Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Piotr Dworczak; Alex Smolin\n\n**Canonical citation:** Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/robust-trust.md\n\n**Source record:** https://arxiv.org/abs/2602.09490\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"9be85ddacb765ebf0f974a544346b3d2488fbdd28c64e1647d55f27150592923"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0001","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Dirk Bergemann; Alessandro Bonatti; Alex Smolin.\n> Canonical citation: Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"8ba5885a6ff07fa7c9efedbf72f196b06cf70729c1fde41b0684cc61339d4d4f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0002","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Menu Pricing of Large Language Models","text":"# Menu Pricing of Large Language Models\n\n**Authors:** Dirk Bergemann; Alessandro Bonatti; Alex Smolin\n\n**Manuscript date:** 2026-03-10\n\n#### Abstract\n\nWe develop a framework for the optimal pricing and product design of LLMs in which a provider sells menus of token budgets to users who differ in their valuations across a continuum of tasks. Under a homogeneous production technology, we show that users' high-dimensional type profiles are summarized by a scalar index, reducing the seller's problem to one-dimensional screening. The optimal mechanism takes the form of committed-spend contracts: buyers pay for a budget that they allocate across token classes priced at marginal cost. We extend the analysis to environments with multiple differentiated models and to competition between a proprietary leader and an open-source fringe, showing that competitive pressure reshapes both the intensive and extensive margins of compute provision. Each element of our theory (token-budget menus, maximum- and minimum-spend plans, multi-model versioning, and linear API pricing) has a direct counterpart in the observed pricing practices of providers such as Anthropic, OpenAI, and GitHub.\n\nKeywords: Large Language Models, Optimal Pricing, Menu Pricing, Fine-Tuning. JEL Codes: D47, D82, D83.\n\n[^0]","text_sha256":"69eacbe16bb45853512054b11921d74a5ec07628b3e4daad9b2306196c1e65b3"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0003","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nAccess to large language models is fast becoming a major input to economic activity. Businesses use LLMs for coding, customer support, legal research, and content generation; individual users rely on them for writing, analysis, and decision-making. The leading providers, Anthropic, OpenAI, and Google, collectively serve hundreds of millions of users and generate billions of dollars in annualized revenue. Yet the pricing of these services remains strikingly ad hoc: subscription tiers, token-based metering, credit systems, and volume commitments coexist with no clear theoretical foundation. Understanding how a profit-maximizing provider should price LLM access, and quantifying the distortions that arise when it does, is the goal of this paper. ${ }^{1}$\n\nPricing access to LLMs is fundamentally a multidimensional screening problem. An LLM user faces a continuum of tasks, each with a different value. The provider offers token budgets-bundles of input, output, and fine-tuning tokens-that the user allocates across tasks at her discretion. The provider can meter total token usage but cannot observe or contract on the user's task-by-task allocation, creating a combined adverse-selection and moral-hazard problem. User types are infinite-dimensional (a function from tasks to values), the allocation space is high-dimensional (tokens of multiple classes across many tasks), and the user's hidden action (token allocation across tasks) further compounds the difficulty. A priori, this problem seems intractable.\n\nOur central result is that it is not. Under a homogeneous production technology-meaning that the gain function is homogeneous, so the optimal composition of inference tokens is scale-invariant-each user's entire type profile collapses to a scalar index, the $a g$ gregate type. Users with the same aggregate type make the same total token demands, receive the same total surplus, and choose the same menu item, regardless of the fine structure of their task valuations. This aggregation property reduces the provider's problem to one-dimensional screening à la Mussa and Rosen (1978), despite the underlying complexity of the environment. Building on this reduction, we develop the analysis in four steps.\n\nFirst, we characterize the efficient allocation (Section 3). Efficiency requires that all tasks employ inference token classes in common proportions; only the scale of token usage varies with a task's marginal value. When the provider faces capacity constraints in each token class, the efficient allocation can be implemented through linear prices equal to inflated shadow costs (Corollary 1), a result that rationalizes the linear, pay-per-token pricing universally observed in developer-facing API markets.\n\n[^1]Second, we characterize the optimal mechanism for a single-model monopolist (Section 4). By the aggregation result, the buyer's indirect utility from a token-budget bundle takes a multiplicative form: aggregate type times aggregate quality. The optimal menu therefore excludes low types and distorts quality downward for (almost) all others, exactly as in the standard one-dimensional framework. Crucially, the optimal mechanism admits intuitive indirect implementations (Proposition 4): it can be realized as a maximum-spend mechanism (a budget of credits allocated across tokens priced at marginal cost), as a minimum-spend mechanism (volume commitments that unlock lower per-token prices), or as a menu of twopart tariffs. These implementations are not merely theoretical curiosities; they correspond precisely to the pricing structures observed at leading platforms.\n\nThird, we extend the framework to multiple differentiated models (Section 5). When models share a common returns-to-scale parameter in inference but differ in returns to finetuning, the buyer's payoff has a constant elasticity of substitution over model qualities. Cost minimization implies that each type uses a single model for all tasks (Proposition 6). This generates a natural versioning structure: higher-tier plans grant access to more capable models, not merely larger usage allocations. Anthropic's pricing illustrates the single-model benchmark of Section 4: all paid tiers access the same models, with differentiation occurring through compute budgets. OpenAI, by contrast, screens jointly on usage and model access, reserving its most compute-intensive reasoning model for the highest tier, consistent with the multi-model menu of Section 5.\n\nFourth, we study competition between a proprietary leader and an open-source fringe that sells tokens at marginal cost (Section 5). The leader designs an optimal menu subject to the buyer's ability to supplement usage from the fringe. Three regions emerge: low types purchase exclusively from the fringe; intermediate types buy from the leader at quantities that exactly deter fringe top-up; and high types are served as under pure monopoly, with standard downward distortions (Proposition 7). Competition thus reshapes both the intensive margin (how much compute is sold to each buyer) and the extensive margin (which buyers adopt the proprietary model at all).\n\nWe conclude by connecting each theoretical construct to observed pricing (Section 6). Consumer subscriptions at Anthropic and OpenAI implement the nonlinear menus of Sections 4 and 5. Model aggregators (e.g., Quora's Poe and GitHub Copilot) implement the committed-spend mechanisms of Section 4.2, differing in whether they enforce a hard budget cap (maximum spend) or allow overages at linear prices (minimum spend). API pricing is linear in tokens with no volume discounts, consistent with providers prioritizing adoption over rent extraction. Thus, pricing practices observed in the industry suggest that these mechanisms capture fundamental economic forces rather than incidental design choices.","text_sha256":"5e5cb9a206517f55525c4eb64a0f278f638a1be428f7655d63738aead4521bc5"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0004","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"Related Literature Our paper contributes to the rapidly growing literature on the economics of AI. Existing work has examined the impact of LLMs on price competition (Fish, Gonczarowski, and Shorrer, 2024), token auctions (Duetting, Mirrokni, Paes Leme, Xu, and Zuo, 2024), and sponsored search (Bergemann, Bojko, Dütting, Paes Leme, Xu, and Zuo, 2024), and Demirer, Fradkin, Tadelis, and Peng (2025) document pricing and market-share patterns. By contrast, our focus is on the provider's own design problem, how to price and version LLM access, which has received surprisingly little theoretical attention. For example, Mahmood (2024) studies competition among producers of generative AI who can specialize in different tasks, but restricts attention to linear token pricing. We go further and characterize the fully optimal nonlinear mechanism, leveraging the sufficient-statistic reduction generated by the homogeneity of the gain function, which renders the joint screening-and-moral-hazard problem solvable and yields sharp, implementable predictions.\n\nAt their core, LLMs are an information technology, connecting our work to the literature on selling information (Babaioff, Kleinberg, and Paes Leme, 2012; Bergemann, Bonatti, and Smolin, 2018; Yang, 2022). However, LLMs possess distinctive features: they are generalpurpose technologies deployed across many tasks, usage is metered in tokens, and precision is improved by combining inference and fine-tuning, which give rise to a structurally different design problem.\n\nMethodologically, our aggregation result relates to the literature on \"1.5-dimensional\" mechanism design (Fiat, Goldner, Karlin, and Koutsoupias, 2016; Devanur, Goldner, Saxena, Schvartzman, and Weinberg, 2020). Unlike most of that literature, however, we must address potential incentive clashes across the continuum of subproblems that arise from the buyer's hidden allocation of tokens, requiring an additional argument based on the production technology. Even in simpler settings, such as one-dimensional screening with moral hazard (Castro-Pires, Chade, and Swinkels, 2024) or multidimensional screening and bundling without moral hazard (Armstrong, 1996; Rochet and Stole, 2003; Daskalakis, Deckelbaum, and Tzamos, 2017), tractability issues arise. Our environment combines both, and our solution exhibits a constrained efficiency property as in Laffont and Tirole (1990) and Doligalski, Dworczak, Krysta, and Tokarski (2025).","text_sha256":"260a364369d207f468a8fdad5a95a33e7496d4bcd333dc41db285047357253b0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0005","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"## 2 Model","text_sha256":"69d3a5f84483f9aafd18fefcbfe33e9bab8536442b3889a76ddff069b3a561b0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0006","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2.1 Task Environment","text":"### 2.1 Task Environment\n\nA buyer of LLM services faces a unit measure of tasks indexed by $i \\in[0,1]$. In our baseline setting, an LLM provider offers one type of model only. The buyer can decide on (i) how\nmany inference tokens to use for each task $i, x_{i} \\in \\mathbb{R}_{+}^{J}$ and (ii) how many fine-tuning tokens to use to improve the model's performance (e.g., precision) on all tasks, $z \\in \\mathbb{R}_{+}^{K}$. In language models, a token is a unit of text, typically a word.\n\nThe model's performance on task $i$ is given by the gain function $g\\left(x_{i}, z\\right)$. The function $g$ is common across tasks and takes the following form:\n\n$$\ng\\left(x_{i}, z\\right)=\\Psi\\left(x_{i}\\right) \\Phi(z) .\n$$\n\nThus, the gain function is multiplicatively separable in inference and fine-tuning tokens.\nDiminishing returns to inference tokens on a single task are a fundamental feature of LLM performance, consistent with empirically observed scaling laws. Thus the function $\\Psi$ is positive, increasing, strictly concave, differentiable, Inada at 0, ${ }^{2}$ and homogeneous of degree $\\sigma \\in(0,1)$, i.e., $\\Psi\\left(r x_{i}\\right)=r^{\\sigma} \\Psi\\left(x_{i}\\right)$ for all $r>0$. This function captures the returns to inference tokens. A canonical example of such a function is a CES function, $\\Psi\\left(x_{i}\\right)=\\left(\\sum_{j=1}^{J} \\alpha_{j} x_{i j}^{\\rho}\\right)^{\\sigma / \\rho}$, where $\\alpha_{j}>0, \\sum_{j} \\alpha_{j}=1, \\rho<1$. The function $\\Phi(z)$ is positive, increasing, such that $g$ is strictly concave (e.g., $\\Phi(z)^{1 /(1-\\sigma)}$ is strictly concave) and $(1-\\sigma)$-Inada at 0. ${ }^{3}$ This function captures the returns to fine-tuning.\n\nThese properties are similar in shape to the scaling laws for training LLMs. Concavity captures the fundamental assumption that time and computing resources spent on a single task exhibit diminishing marginal returns. Homogeneity is the key tractability assumption. It ensures that the optimal mix of token classes is the same for every task; only the scale of token usage varies. This property drives the aggregation result in Section 3, which reduces the seller's infinite-dimensional screening problem to a one-dimensional problem. The parameter $\\sigma$ captures the returns to scale.\n\nThe buyer's marginal value of performance on each task is captured by $w=\\left(w_{i}\\right)_{i \\in[0,1]}$,\n\n$$\nw:[0,1] \\rightarrow[0,1],\n$$\n\nwhich we refer to as the buyer's type. Using a $z$-fine-tuned model with a profile of $\\left(x_{i}\\right)_{i \\in[0,1]}$ inference tokens delivers the following total payoff for buyer type $w$ :\n\n$$\n\\int_{0}^{1} w_{i} g\\left(x_{i}, z\\right) d i .\n$$\n\nThe buyer's type is distributed according to a commonly known distribution $F_{w}$. The buyer\n\n[^2]knows his type, while the provider does not.\nThe provider bears the cost of processing tokens. We assume that the marginal processing costs are constant but can vary across different token classes. The cost of inference tokens is $c_{j}>0, j \\in[J]$, and the cost of fine-tuning tokens is $\\hat{c}_{k}>0, k \\in[K]$. We let $c \\in \\mathbb{R}_{++}^{J}$ and $\\hat{c} \\in \\mathbb{R}_{++}^{K}$ denote the corresponding cost profiles.\n\nNotation Throughout the text, we use the following notation. \" $\\triangleq$ \" indicates a definition. $[K] \\triangleq\\{1, \\ldots, K\\} . \\mathbb{R}_{+}^{K} \\triangleq\\left\\{x \\in \\mathbb{R}^{K}: x_{k} \\geq 0, k \\in[K]\\right\\}$ and $\\mathbb{R}_{++}^{K} \\triangleq\\left\\{x \\in \\mathbb{R}^{K}: x_{k}>\\right.$ $0, k \\in[K]\\}$. All stand-alone qualifiers such as \"positive,\" \"increasing,\" and \"concave\" are understood in the weak sense, that is, as \"non-negative,\" \"non-decreasing,\" and \"weakly concave,\" respectively. For constants, subscripts refer to variables or tasks; for functions, they indicate partial derivatives. All proofs are in Appendix A.","text_sha256":"73066a3f8bf9e54991f8e14e1e2262c643660c7d33df7db1afd4b8a6834ef959"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0007","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2.2 Mechanism Design","text":"### 2.2 Mechanism Design\n\nThe provider contracts on the total number of tokens of each class used by the buyer. In other words, the provider sells budgets of various kinds of tokens. A mechanism consists of an arbitrary menu of the form\n\n$$\n\\left\\{\\left(X_{1}(w), \\ldots, X_{J}(w), Z_{1}(w), \\ldots, Z_{K}(w), t(w)\\right)\\right\\}_{w},\n$$\n\nwhere $X_{j}$ is the total number of inference tokens of class $j$. Upon purchase, the buyer can freely distribute those tokens across tasks. (We relax this restriction in Appendix B, where we allow the provider to contract on the usage of all tokens by the buyer.) $Z_{k}$ is the number of fine-tuning tokens of class $k$, which is used for fine-tuning and cannot be distributed.\n\nThe seller's problem is an optimal mechanism design problem with infinite-dimensional private information and moral hazard. However, we show that the homogeneity of the gain function renders the problem tractable.","text_sha256":"8331f3c5c69946613c07fe1f7b58f8a6f467b110a2b82dba27e3b5350296c210"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0008","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2.3 Mapping the Setting to Practice","text":"### 2.3 Mapping the Setting to Practice\n\nBefore turning to the analysis, we describe how the primitives of our setting relate to the design and operation of contemporary large language models (LLMs).\n\nFine-tuning tokens, $\\boldsymbol{z} \\in \\mathbb{R}_{+}^{\\boldsymbol{K}}$ The case $z=0$ corresponds to a baseline model that has not been fine-tuned. ${ }^{4}$ In this case, the model can be interpreted as a foundation model. Its\n\n[^3]quality depends on its size (i.e., the number of parameters) ${ }^{5}$, its architecture, the quantity and quality of training data, and the details of the training procedure. Scaling up model size typically improves capability and accuracy, but state-of-the-art performance generally requires commensurate scaling of training data as well (Kaplan et al., 2020). In our setting, we take pretraining as given. Accordingly, we focus on inference-time compute rather than training-time compute.\n\nWhen $z>0$, this variable captures the number of tokens used to fine-tune the foundation model. Fine-tuning resembles pretraining in that it updates the model's parameters, without changing its size or architecture, but it is much smaller in scale. It is performed on a dataset directly relevant to a class of tasks (e.g., labeled X-ray scans, code examples, or conversational transcripts) and can be interpreted as injecting task-specific knowledge and behavior into the model.\n\nInference tokens, $\\boldsymbol{x}_{\\boldsymbol{i}} \\in \\mathbb{R}_{+}^{\\boldsymbol{J}}$ These are the tokens used directly to process a given task. The two standard classes of inference tokens are input and output tokens. Input tokens are those provided by the user. Increasing the number of input tokens can improve predictive quality for two reasons. First, richer prompts provide more context, enabling more tailored and appropriate responses. Second, additional input may supply the data on which the model is meant to operate. For example, retrieval-augmented generation (RAG) uses the prompt to retrieve relevant passages from an external corpus and appends them to the prompt.\n\nOutput tokens are those generated by the model. A larger number of output tokens can also improve quality. First, it allows the model to provide more detailed and nuanced answers. Second, it provides opportunities for multi-step reasoning, often described as chainof-thought (CoT) computation. This additional reasoning channel can enable the model to solve more complex tasks, at the cost of generating many intermediate tokens, often hidden from the user. This inference-time reasoning is qualitatively distinct from pretraining, remains only partially understood, and is an active frontier for improving LLM performance. ${ }^{6}$\n\nGain function, $\\boldsymbol{g}$ Tokens of different classes enter the large language model as distinct inputs. They are neither perfect substitutes nor perfect complements, but each can contribute to higher predictive quality. This motivates our gain-function formulation in (1), which is also in line with the empirically observed scaling laws for inference-time computation (e.g., Wu, Sun, Li, Welleck, and Yang, 2025).\n\n[^4]Furthermore, one can envision other contractible token categories, either by refining the distinctions above (e.g., separating reasoning tokens from response tokens) or by introducing new modalities and resources (e.g., media files, external databases, etc.). Accordingly, our model accommodates an arbitrary number of inference and fine-tuning token classes.\n\nCosts, $\\boldsymbol{c}_{\\boldsymbol{j}}, \\hat{\\boldsymbol{c}}_{\\boldsymbol{k}}$ Processing tokens is costly because it requires executing the neural network and, for fine-tuning, computing parameter updates, which in turn requires energy and specialized hardware. Because tokens within a given class are processed symmetrically from a computational perspective, we assume that the marginal cost per token within a class is constant across tasks. ${ }^{7}$ However, because different token classes correspond to different computational routines, we allow marginal costs to differ across classes.","text_sha256":"e2d5d3e879e05c3bb396202abeb48358390981632e334654232d73beef351072"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0009","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Efficient Solution","text":"## 3 Efficient Solution\n\nIn this section, we begin by analyzing the socially efficient allocation of tokens to buyers and tasks. This solution provides a useful benchmark and also coincides with the optimal monopoly solution when the buyer's type is known. We then turn to the problem of a social planner who faces capacity constraints for each token class, and we use the constrainedefficient solution to characterize the buyer's utility from purchasing a fixed token budget.\n\nUnconstrained problem Given a type $w=\\left(w_{i}\\right)_{i \\in[0,1]}$, the efficient allocation solves the following problem:\n\n$$\n\\max _{\\left(x_{i}\\right)_{i \\in[0,1],}, z \\geq 0} \\int_{0}^{1} w_{i} \\Psi\\left(x_{i}\\right) \\Phi(z) d i-\\sum_{j=1}^{J} c_{j} \\int_{0}^{1} x_{i j} d i-\\sum_{k=1}^{K} \\hat{c}_{k} z_{k}\n$$\n\nConsider the allocation of inference tokens first. A key implication of the homogeneity of the production function $\\Psi$ is that inference tokens are optimally employed in the same proportions: the efficient allocation of tokens across tasks differs solely by the scale at which these inputs are employed. To see this, consider the system of $J$ first-order conditions for each $x_{i}=\\left(x_{i 1}, \\ldots, x_{i J}\\right)$ :\n\n$$\nw_{i} \\nabla \\Psi\\left(x_{i}\\right) \\Phi(z)=c .\n$$\n\nBy equation (6), for all $i, \\nabla \\Psi\\left(x_{i}\\right)$ belongs to a ray with a direction $c$. Because $\\Psi$ is homogeneous and strictly concave, the ray in the space of gradients corresponds to a ray in the\n\n[^5]space of tokens. Thus, for each $i$, any optimal $x_{i}$ can be written as:\n$$\nx_{i}=r_{i} d,\n$$\nwhere $d \\in \\mathbb{R}_{+}^{J}$ is the unique vector that solves $\\nabla \\Psi(d)=c$, i.e., the cost-minimizing input shares. Furthermore, because $\\Psi$ is homogeneous of degree $\\sigma, \\Psi_{j}$ is homogeneous of degree $\\sigma-1$, so that the optimal scale is given by\n$$\nr_{i}=w_{i}^{\\frac{1}{1-\\sigma}} \\Phi(z)^{\\frac{1}{1-\\sigma}} .\n$$\nSubstituting back in the objective function, both the total surplus and the total consumption of each token class depend on the buyer's type $w$ only through $\\int_{0}^{1} w_{i}^{1 /(1-\\sigma)} d i$. It is then convenient to define the aggregate type $\\theta(w)$ as\n$$\n\\theta(w) \\triangleq\\left(\\int_{0}^{1} w_{i}^{\\frac{1}{1-\\sigma}} d i\\right)^{1-\\sigma}\n$$\n\nOur first result establishes that the total surplus and the total amount of inference tokens depend only on the aggregate type and not on the finer details of the type profile $w$.\n\nProposition 1 (Efficient Allocation). Under the efficient allocation, all buyer types $w$ with the same aggregate type $\\theta(w)$ consume the same number of fine-tuning tokens, consume the same total number of inference tokens in each class, and obtain the same total payoff. The number of inference tokens allocated to task $i$ is proportional to $w_{i}^{\\frac{1}{1-\\sigma}}$.\n\nProposition 1 has a key implication: two buyers with very different task profiles-one who values a few tasks intensely and many tasks little, and another who values all tasks moderately-consume the same total resources and obtain the same total surplus, provided they share the same aggregate type. The seller therefore cannot distinguish between them on the basis of total token consumption, nor would it want to. This indistinguishability is not an assumption but a consequence of the technology: homogeneity of $\\Psi$ ensures that the efficient mix of token classes is task-independent, so only the scale of usage varies, and the aggregator (7) is the natural summary of how much total scale a buyer demands.\n\nWe now show that a similar logic can be used to characterize the socially efficient solution under capacity constraints in each class of tokens (training, computation, input, output).\n\nConstrained problem We now introduce capacity constraints on each token class. This detour is not merely a generalization: the special case in which all constraints bind is precisely the problem faced by a buyer who has purchased a fixed bundle of token budgets. Thus,\nthe constrained-efficient allocation provides the foundation for the seller's mechanism design problem in Section 4.\n\nFix a type $w$ and capacity constraints $X_{j}>0, Z_{k}>0$ for all $j, k$. The capacityconstrained planner's problem is\n\n$$\n\\begin{aligned}\n& \\max _{\\left(x_{i}\\right)_{i \\in[0,1], z \\geq 0}} \\int_{0}^{1} w_{i} \\Psi\\left(x_{i}\\right) \\Phi(z) d i-\\sum_{j=1}^{J} c_{j} \\int_{0}^{1} x_{i j} d i-\\sum_{k=1}^{K} \\hat{c}_{k} z_{k} \\\\\n& \\text { s.t. } \\quad \\int_{0}^{1} x_{i j} d i \\leq X_{j}, z_{k} \\leq Z_{k} \\text { for } j \\in[J], k \\in[K]\n\\end{aligned}\n$$\n\nOur next result establishes that the capacity-constrained efficient allocation coincides with the (unconstrained) efficient allocation for marginal costs inflated by the shadow costs of the capacity constraints, $\\left(c^{\\prime}, \\hat{c}^{\\prime}\\right) \\geq(c, \\hat{c})$. This observation yields an immediate implementation of the efficient solution.","text_sha256":"26a3011c3cc9d3b57ea538fb5287fdfa90789852abd65897c9abdf59161a3a5f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0010","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Corollary 1 (Constrained Efficient Allocation).","text":"## Corollary 1 (Constrained Efficient Allocation).\n\n1. In the capacity-constrained efficient allocation, all buyer types $w$ with the same aggregate type $\\theta(w)$ consume the same total number of fine-tuning tokens and inference tokens in each class, and they obtain the same total payoff.\n2. The capacity-constrained efficient allocation can be implemented via linear prices equal to inflated marginal costs.\n\nThus, capacity constraints do not change the qualitative properties of the solution, but lead to an inefficiency in the relative allocation of token types, and in the split between those buyer types who choose to fine-tune $(z>0)$ and those who do not $(z=0)$.\n\nThe special case where all capacity constraints bind is of particular interest, because it is instrumental in the characterization of the buyer's demand for token budgets in Section 4. When all constraints bind, the total production cost is pinned down. The planner's and the buyer's solutions then coincide, i.e., the constrained efficient allocation solves the problem of a buyer who has access to inference and fine-tuning token budgets $(X, Z)$.\n\nThe optimal allocation of a fixed budget $(X, Z)$ solves the following problem:\n\n$$\n\\max _{x_{i j} \\geq 0} \\int_{0}^{1} w_{i} \\Psi\\left(x_{i}\\right) \\Phi(Z) d i, \\quad \\text { s.t. } \\int_{0}^{1} x_{i j} d i=X_{j} \\text { for } j \\in[J] .\n$$\n\nApplying the homogeneity argument again, $\\Phi(Z)$ factors out, and the buyer optimally sets $x_{i}=r_{i} d$ with $r_{i} \\propto w_{i}^{1 /(1-\\sigma)}$. Since the budget constraint for each token class $j$ requires $d_{j} \\int_{0}^{1} r_{i} d i=X_{j}$, the common direction is $d=X / \\int_{0}^{1} r_{i} d i$, which yields a simple expression for the buyer's optimal payoff.\n\nThe following result is the key step in our analysis. It shows that the buyer's indirect utility from any token bundle is multiplicatively separable in the aggregate type and an aggregate quality index, placing the seller's problem squarely in the Mussa-Rosen framework.\n\nProposition 2 (Buyer Indirect Utility). For any inference and fine-tuning token budgets $X=\\left(X_{1}, \\ldots, X_{J}\\right) \\geq 0$ and $Z=\\left(Z_{1}, \\ldots, Z_{K}\\right) \\geq 0$, the indirect utility of buyer type $w$ is\n\n$$\nU(w, X, Z)=\\theta(w) \\Psi(X) \\Phi(Z),\n$$\n\nwhere $\\theta(w)$ is the aggregate type defined in (7).\nWe therefore obtain a tractable expression for the buyer's payoff by means of a representative task, whose value is given by the aggregate type $\\theta$, to which the buyer assigns the entire token budget. Heterogeneity across tasks washes out, and only the aggregate matters. A notable implication is that the buyer's unobserved allocation of tokens across tasks generates no additional incentive constraints beyond those of the standard screening problem. Under homogeneity, the cost-minimizing input mix is independent of scale, so the buyer's within-bundle allocation is pinned down regardless of type. This product form is the key input to the seller's problem, which we turn to next.","text_sha256":"0413eb64a512495e92a699af66e5a7020bc1b5aa6acb666f9175cebd9405932d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0011","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Menus of Token Budgets","text":"## 4 Menus of Token Budgets\n\nThe main result of this section is that the seller's optimal mechanism takes the form of a menu of committed-spend contracts: each buyer pays an upfront fee for a spending budget that she allocates freely across token classes priced at the provider's marginal cost. Higher types purchase larger budgets at higher fees, with quantity discounts that are consistent with observed industry pricing. The formal analysis proceeds in two steps: we first characterize the optimal direct mechanism (Section 4.1) and then show that it admits intuitive indirect implementations (Section 4.2).\n\nWe now characterize the provider's profit-maximizing menu of token budgets. By Proposition 2, the buyer's indirect utility from a bundle $(X, Z)$ is $\\theta(w) \\Psi(X) \\Phi(Z)$. Two features of this expression are worth emphasizing. First, private information enters only through the scalar $\\theta$ : the seller's multidimensional screening problem reduces to a one-dimensional problem. Second, the payoff is multiplicatively separable in type and allocation, placing us in the framework of Mussa and Rosen (1978).","text_sha256":"a20d3b6f34c62596475979b6557851a77cbe06d75ca144cc54bf5173898b0fe3"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0012","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.1 Optimal Budget Mechanism","text":"### 4.1 Optimal Budget Mechanism\n\nDefine the effective type as $\\theta$ and the effective allocation as the aggregate quality given by\n\n$$\nQ(X, Z) \\triangleq \\Psi(X) \\Phi(Z) .\n$$\n\nThe effective production costs of producing the aggregate quality $Q$ can be found via a cost-minimization problem:\n\n$$\nC(Q) \\triangleq \\min _{X, Z \\geq 0} \\sum_{j=1}^{J} c_{j} X_{j}+\\sum_{k=1}^{K} \\hat{c}_{k} Z_{k}, \\quad \\text { s.t. } \\Psi(X) \\Phi(Z)=Q .\n$$\n\nThe solution to the cost-minimization problem has the following properties.\nLemma 1 (Cost Function). The cost function $C(Q)$ is strictly increasing and strictly convex and satisfies $C_{+}^{\\prime}(0)=0 .{ }^{8}$\n\nConvexity follows from the strict concavity of the gain function $g$ : producing higher aggregate quality requires disproportionately more tokens. The property $C_{+}^{\\prime}(0)=0$ follows from the Inada conditions. Economically, $C_{+}^{\\prime}(0)=0$ means that the first unit of aggregate quality is arbitrarily cheap to produce. Combined with convexity, this implies that it is never efficient to exclude any buyer type entirely. Thus, in the monopoly problem, exclusion is driven by information rents rather than production costs.\n\nBy Lemma 1, the analysis of Mussa and Rosen (1978) applies. Denote the prior distribution of $\\theta$ by $F$, which is derived from $F_{w}$. The seller chooses an aggregate quality schedule $Q(\\theta)$ and a transfer schedule $t(\\theta)$ to solve\n\n$$\n\\max _{Q(\\cdot), t(\\cdot)} \\mathbb{E}_{F}[t(\\theta)-C(Q(\\theta))],\n$$\n\nsubject to incentive compatibility and individual rationality.\nAs usual, let\n\n$$\n\\varphi(\\theta) \\triangleq \\theta-\\frac{1-F(\\theta)}{f(\\theta)},\n$$\n\ndenote the virtual (aggregate) type. If the virtual value $\\varphi$ is not increasing, let $\\bar{\\varphi}$ denote its Myerson-ironed version. Then, all $\\theta$ with $\\bar{\\varphi}(\\theta) \\leq 0$ are excluded. For all other $\\theta$, the optimal allocation is uniquely pinned down by:\n\n$$\n\\bar{\\varphi}(\\theta)=C^{\\prime}(Q(\\theta)) .\n$$\n\n[^6]Denoting the corresponding optimal transfers by\n\n$$\nt(\\theta)=\\theta Q(\\theta)-\\int_{0}^{\\theta} Q(r) d r\n$$\n\nwe can then fully characterize the optimal mechanism.\nProposition 3 (Optimal Menu). The optimal menu of token budgets is $\\{(X(\\theta), Z(\\theta), t(\\theta))\\}_{\\theta}$, where $(X(\\theta), Z(\\theta))$ are the cost-minimizing token budgets that deliver quality $Q=Q(\\theta)$ at prices $t(\\theta)$ as defined in (10)-(13).\n\nThe optimal menu offers a continuum of plans. All types $w$ with the same aggregate type $\\theta(w)$ choose the same item. Low-type buyers are excluded. All served buyers receive distorted-downward quality (fewer tokens than the efficient allocation would prescribe) except at the very top. Buyers with higher aggregate types purchase strictly larger token budgets and pay strictly higher transfers, but enjoy quantity discounts: the average price per unit of quality falls with $\\theta$.","text_sha256":"bb136f72afb8fac5fe433a1d95dec83052d39c97be0b1811e8a3d01d9aba695e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0013","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.2 Indirect Mechanisms","text":"### 4.2 Indirect Mechanisms\n\nThe optimal direct mechanism specifies token quantities for each type. In practice, providers do not announce menus of token quantities; they post prices and spending limits. We now show that three natural pricing formats implement the optimum.\n\nWe first consider committed-spend mechanisms, where the terms of the contract with a buyer are related to a maximum or minimum total expenditure.\n\nDefinition 1 (Maximum-Spend Mechanism). A maximum-spend mechanism is a collection of token prices $p$ and a menu of monetary budgets and transfers $\\left\\{\\left(B_{n}, T_{n}\\right)\\right\\}_{n}$ such that, upon selecting item $n$, the buyer pays $T_{n}$ for access to budget $B_{n}$, which he can freely spend on tokens priced at $p$.\n\nWhile the optimal budget (direct) mechanism is a cap on quantities, the maximumspend mechanism limits quality by capping expenditures at fixed prices. Those expenditures are virtual in that they do not contribute to payment beyond $T_{n}$, but they govern token allocation. As we will see in Section 6, the maximum-spend mechanism is used in practice, for example, by Quora's Poe (see Section 6), and it is closely related to the Cost-Based tariffs (where $p=c$ ) introduced in Armstrong (1996). Indeed, Proposition 8 in Appendix B derives analogous conditions to those in Armstrong (1996) under which a maximum-spend mechanism is optimal across all mechanisms, including those that specify a task-by-task token allocation.\n\nAn alternative implementation does not impose a cap on spending, but exposes the buyer to variable marginal prices, much like committed-spend mechanisms in cloud computing.\n\nDefinition 2 (Minimum-Spend Mechanism). A minimum-spend mechanism is a menu of monetary budgets and transfers $\\left\\{\\left(p_{n}, T_{n}\\right)\\right\\}_{n}$ such that, upon selecting item $n$, the buyer commits to spending at least $T_{n}$ on tokens priced at $p_{n}$.\n\nA minimum-spend mechanism allows buyers to commit to larger budgets (i.e., total expenditures) to unlock variable-price discounts. In contrast to a maximum-spend mechanism, payment comes from actual token consumption rather than from an upfront payment.\n\nDefinition 3 (Two-Part-Tariff Mechanism). A two-part-tariff mechanism is a menu of prices and transfers $\\left\\{\\left(p_{n}, T_{n}\\right)\\right\\}_{n}$ such that, upon selecting item $n$, the buyer pays $T_{n}$ and can buy any number of tokens priced at $p_{n}$.\n\nA two-part tariff is a classical mechanism that combines upfront and consumption payments but does not impose any token consumption limits.\n\nDefine a type-dependent markup $m(\\theta)$ as:\n\n$$\nm(\\theta) \\triangleq \\frac{\\theta}{\\bar{\\varphi}(\\theta)} .\n$$\n\nThe markup $m(\\theta)$ is the ratio of the true type to the virtual type; it captures the informationrent wedge that the seller imposes. Higher markups on lower types reflect greater information rents extracted from higher types in the menu.\n\nProposition 4 (Indirect Implementation). An optimal menu of token budgets can be implemented via a maximum-spend mechanism. If $m(\\theta)$ is decreasing (e.g., if $F$ has an increasing hazard rate), then an optimal menu of token budgets can be implemented via a minimum-spend mechanism and a two-part tariff mechanism.\n\nImplementability by a maximum-spend mechanism is straightforward and relies on the fact that the optimal token allocation is constrained-efficient, as in Doligalski et al. (2025): it suffices to set prices equal to marginal costs and to choose token expenditures and transfers so as to mimic those under the direct mechanism.\n\nImplementability by minimum-spend and two-part tariff mechanisms requires more care. Because $\\Psi$ is homogeneous, the cost-minimizing input mix is the same for every quality level $Q$; only the scale changes. A two-part tariff that preserves this input mix must therefore set token prices proportional to marginal costs; any other price vector would distort the buyer's allocation away from cost minimization. The proportionality factor is the markup $m(\\theta)=$\n$\\theta / \\bar{\\varphi}(\\theta)$, which inflates all marginal costs equally. The condition that $m(\\theta)$ is decreasing ensures that higher types, who select larger budgets, face lower per-token prices, so the menu is incentive-compatible. In the minimum-spend implementation, the markup equals the ratio between total revenue and total cost for type $\\theta$. Section 6 shows that each of these three implementations corresponds to an observed pricing format at leading platforms.","text_sha256":"fdadc8019529d271470c61cd7e6fc19f2b9c25fd589a41cec6048129ef6697b1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0014","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Multiple Models and Competition","text":"## 5 Multiple Models and Competition\n\nIn practice, every major LLM provider offers multiple models that differ in capability and cost (e.g., Anthropic's Haiku, Sonnet, and Opus, or OpenAI's GPT-4o-mini and o1). A buyer can assign different tasks to different models, and the provider can screen on both usage quantity and model access. This section extends our framework to this richer environment. Two new questions arise: How does a buyer optimally allocate tasks across models? And how does a profit-maximizing provider design menus that screen on both dimensions?\n\nSpecifically, we extend our setting and allow there to be $L$ models, each with a modelspecific gain function:\n\n$$\ng_{l}\\left(x_{i}, z\\right)=\\Psi_{l}\\left(x_{i}\\right) \\Phi_{l}(z),\n$$\n\nwhere $\\Psi_{l}$ and $\\Phi_{l}$ are assumed to have the same properties as in the baseline model, with $\\Psi_{l}$ being homogeneous of degree $\\sigma_{l}$. We order the models so that $\\sigma_{1} \\leq \\sigma_{2} \\leq \\cdots \\leq \\sigma_{L}$. The token costs are model-specific $\\left(c_{l}, \\hat{c}_{l}\\right)$.\n\nThe buyer can process different tasks with different models; however, he cannot use two models for the same task. ${ }^{9}$ If the buyer of type $w$ processes tasks $i \\in I_{l}$ with model $l$, his total payoff ignoring the payment is:\n\n$$\n\\sum_{l=1}^{L} \\int_{I_{l}} w_{i} g_{l}\\left(x_{l i}, z_{l}\\right) d i\n$$\n\nIn this setting, a bundle specifies token budgets for all available models $\\left(X_{l}, Z_{l}\\right)_{l=1}^{L}=$ $\\left(X_{l 1}, \\ldots, X_{l J}, Z_{l 1}, \\ldots, Z_{l K}\\right)_{l=1}^{L}$.\n\nAs a preliminary step for the subsequent analysis, we show that the buyer's value for a bundle $\\left(X_{l}, Z_{l}\\right)_{l=1}^{L}$ depends only on the aggregate quality of each model,\n\n$$\nQ_{l} \\triangleq g_{l}\\left(X_{l}, Z_{l}\\right),\n$$\n\n[^7]and admits a tractable closed-form expression. To this end, we fix $w$ and order the tasks so that $w_{i}$ is increasing. We assume that $Q_{l}>0$ for all $l$ (if $Q_{l}=0$ then the $l$-th model can be ignored). If the buyer allocates tasks in the set $I_{l}$ to model $l$, then his payoff from model $l$ is, by the arguments of Proposition 2,\n$$\nU_{l}\\left(I_{l}, Q_{l}\\right)=\\theta_{l}\\left(I_{l}\\right) Q_{l} \\text {, }\n$$\nwhere $\\theta_{l}\\left(I_{l}\\right) \\triangleq\\left(\\int_{I_{l}} w_{i}^{1 /\\left(1-\\sigma_{l}\\right)} d i\\right)^{1-\\sigma_{l}}$. The total payoff from the task-model allocation $\\left(I_{l}\\right)_{l=1}^{L}$ is\n$$\n\\sum_{l=1}^{L} U_{l}\\left(I_{l}, Q_{l}\\right)=\\sum_{l=1}^{L} \\theta_{l}\\left(I_{l}\\right) Q_{l} .\n$$\nThe optimal payoff $U^{*}\\left(Q_{1}, \\ldots, Q_{L}\\right)=\\sum_{l=1}^{L} U_{l}\\left(I_{l}^{*}, Q_{l}\\right)$ is evaluated at the optimal task-model allocation $\\left(I_{l}^{*}\\right)_{l=1}^{L}$.\n\nLemma 2 (Buyer-Optimal Payoff). If $\\sigma_{l}=\\sigma$ for all $l \\in[L]$, then\n\n$$\nU^{*}\\left(Q_{1}, \\ldots, Q_{L}\\right)=\\theta\\left(\\sum_{l=1}^{L} Q_{l}^{1 / \\sigma}\\right)^{\\sigma} .\n$$\n\nBy Lemma 2, despite the combinatorial complexity of assigning a continuum of tasks to multiple models, the buyer's payoff from a bundle of tokens across different models admits a simple product decomposition into an aggregate type and a CES-style aggregate quality over models. The elasticity of substitution across models is pinned down by the common returns-to-scale parameter $\\sigma$, and the single-model tractability of Sections 3-4 extends to the multi-model environment.","text_sha256":"9187da6f3517f2841c9a83712414f60910f92f474fec881d040b09a47087e9a6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0015","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.1 Efficient Solution","text":"### 5.1 Efficient Solution\n\nWe first compute the efficient allocation in this setting. Since the buyer's value takes a simple form (20) as a function of aggregate qualities of each model, the efficient allocation delivers the efficient amount of these qualities in a cost-efficient way.\n\nAssumption 1 (Homogeneous Fine-Tuning). For each $l=1, \\ldots, L$ :\n\n1. $\\Psi_{l}$ is homogeneous of degree $\\sigma \\in(0,1)$.\n2. $\\Phi_{l}$ is homogeneous of degree $\\hat{\\sigma}_{l}<1-\\sigma$.\n\nAll subsequent analysis holds under Assumption 1, so we omit it from the statements. Assumption 1 allows us to obtain closed-form expressions because of the following result:\n\nLemma 3 (Model-Specific Cost Function). The cost function of delivering aggregate quality $Q_{l}$ through model $l$ is $C_{l}\\left(Q_{l}\\right)=\\kappa_{l} Q_{l}^{1 /\\left(\\sigma+\\hat{\\sigma}_{l}\\right)}$ for some constant $\\kappa_{l}>0$.\n\nBy Lemma 3, the minimal cost of obtaining quality $Q$ from $L$ models is\n\n$$\nC(Q)=\\min _{Q_{1}, \\ldots, Q_{L} \\geq 0} \\sum_{l=1}^{L} \\kappa_{l} Q_{l}^{1 /\\left(\\sigma+\\hat{\\sigma}_{l}\\right)}, \\quad \\text { s.t. } \\sum_{l=1}^{L} Q_{l}^{1 / \\sigma}=Q^{1 / \\sigma} .\n$$\n\nThe optimal solution utilizes a single model, resulting in\n\n$$\nC(Q)=\\min _{l \\in[L]} \\kappa_{l} Q^{1 /\\left(\\sigma+\\hat{\\sigma}_{l}\\right)} .\n$$\n\nThus, the rich functional model heterogeneity (15) reduces to heterogeneity in two parameters $\\left(\\kappa_{l}, \\hat{\\sigma}_{l}\\right)$ that roughly correspond to \"cost-effectiveness\" and \"propensity to fine-tune.\" Heterogeneity in $\\kappa_{l}$ is vertical: higher $\\kappa_{l}$ correspond to overall costlier models. Heterogeneity in $\\hat{\\sigma}_{l}$ is horizontal: models with high $\\hat{\\sigma}_{l}$ are better at generating high $Q$ whereas models with low $\\hat{\\sigma}_{l}$ are better at generating low $Q$. As a result, higher values of $Q$ are optimally produced by models with higher $\\hat{\\sigma}_{l}$. In other words, models with higher returns to fine-tuning have flatter cost curves and are cheaper for producing high qualities, introducing a natural form of horizontal differentiation among models.\n\nNotably, the cost function $C(Q)$ is not convex but piecewise convex, with kinks at the model switches, which translates into unusual properties of the efficient and profitmaximizing allocations. Specifically, the efficient quality allocation for type $\\theta$ when using model $l$ solves:\n\n$$\n\\max _{Q \\geq 0} \\theta Q-\\kappa_{l} Q^{1 /\\left(\\sigma+\\hat{\\sigma}_{l}\\right)},\n$$\n\nleading to the following characterization:\nProposition 5 (Efficient Allocation). In the efficient allocation, each type $\\theta$ processes all tasks with a single model:\n\n$$\nl^{*} \\in \\arg \\max _{l}\\left(1-\\sigma-\\hat{\\sigma}_{l}\\right)\\left(\\frac{\\left(\\sigma+\\hat{\\sigma}_{l}\\right)}{\\kappa_{l}}\\right)^{\\left(\\sigma+\\hat{\\sigma}_{l}\\right) /\\left(1-\\sigma-\\hat{\\sigma}_{l}\\right)} \\theta^{1 /\\left(1-\\sigma-\\hat{\\sigma}_{l}\\right)},\n$$\n\nwith the aggregate quality:\n\n$$\nQ^{*}(\\theta)=\\left(\\frac{\\theta\\left(\\sigma+\\hat{\\sigma}_{l^{*}}\\right)}{\\kappa_{l^{*}}}\\right)^{\\left(\\sigma+\\hat{\\sigma}_{l^{*}}\\right) /\\left(1-\\sigma-\\hat{\\sigma}_{l^{*}}\\right)}\n$$\n\nThe efficient token allocation is given by Lemma 3 evaluated at $Q^{*}$ from (23).\n\nEvery type uses a single model. Higher aggregate types $\\theta$ consume higher aggregate qualities by using models with higher fine-tuning capabilities $\\hat{\\sigma}_{l}$, i.e., $Q^{*}(\\theta)$ is increasing.\n\nBecause $C(Q)$ is not continuously differentiable, the allocation is discontinuous at the model switch. Indeed, it follows from (23) and (22) that if the efficient allocation at type $\\theta$ is indifferent between $Q_{l}$ of model $l$ and $Q_{l^{\\prime}}$ of model $l^{\\prime}$ such that $\\hat{\\sigma}_{l^{\\prime}}>\\hat{\\sigma}_{l}$, then\n\n$$\n\\frac{Q_{l^{\\prime}}}{Q_{l}}=\\frac{1-\\sigma-\\hat{\\sigma}_{l}}{1-\\sigma-\\hat{\\sigma}_{l^{\\prime}}}>1 .\n$$","text_sha256":"5132c214070fd795c7ebb5b6c63c4594c70b9899fd409131f06d2e2832958eec"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0016","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.2 Multi-Model Monopolist","text":"### 5.2 Multi-Model Monopolist\n\nWe now assume that all $L$ models are sold by a monopolist who offers a menu. Each item specifies a transfer and a collection of token budgets for different models $\\left(X_{l}, Z_{l}\\right)_{l=1}^{L}$.\n\nBy Lemma 2, the seller's problem is equivalent to the optimal screening problem in Mussa and Rosen (1978), in which the buyer's type is $\\theta \\in \\mathbb{R}$, the seller chooses a menu of $(Q, t)$, the buyer's payoff is $U(\\theta, Q)=\\theta Q$, and the cost of obtaining a quality $Q$ is equal to $C(Q)$ as in (21). Although $C(Q)$ is not convex, the Envelope theorem applies and this problem is equivalent to a maximization of virtual surplus:\n\n$$\n\\max _{Q(\\cdot): \\text { increasing }} \\int_{0}^{1}(\\varphi(\\theta) Q(\\theta)-C(Q(\\theta))) d F(\\theta)\n$$\n\nwhere $\\varphi(\\theta)=\\theta-(1-F(\\theta)) / f(\\theta)$.\nAssuming $\\varphi(\\theta)$ is increasing, the solution can be obtained pointwise to equal an efficient allocation with respect to the virtual type: $Q^{\\mathrm{m}}(\\theta)=Q^{*}(\\varphi(\\theta))$. By Proposition 5, if $\\varphi(\\theta)<0$, then the buyer is excluded, $Q^{\\mathrm{m}}=0$; if $\\varphi(\\theta) \\geq 0$,\n\n$$\n\\begin{aligned}\n& l^{\\mathrm{m}}(\\theta) \\in \\arg \\max _{l}\\left(1-\\sigma-\\hat{\\sigma}_{l}\\right)\\left(\\frac{\\left(\\sigma+\\hat{\\sigma}_{l}\\right)}{\\kappa_{l}}\\right)^{\\left(\\sigma+\\hat{\\sigma}_{l}\\right) /\\left(1-\\sigma-\\hat{\\sigma}_{l}\\right)} \\varphi(\\theta)^{1 /\\left(1-\\sigma-\\hat{\\sigma}_{l}\\right)}, \\\\\n& Q^{\\mathrm{m}}(\\theta)=\\left(\\frac{\\varphi(\\theta)\\left(\\sigma+\\hat{\\sigma}_{l^{\\mathrm{m}}}\\right)}{\\kappa_{l^{\\mathrm{m}}}}\\right)^{\\left(\\sigma+\\hat{\\sigma}_{l^{\\mathrm{m}}}\\right) /\\left(1-\\sigma-\\hat{\\sigma}_{l^{\\mathrm{m}}}\\right)} .\n\\end{aligned}\n$$\n\nSince $Q^{*}$ is increasing, the pointwise solution solves the problem. Optimal transfers are\n\n$$\nt^{\\mathrm{m}}(\\theta)=\\theta Q^{\\mathrm{m}}(\\theta)-\\int_{0}^{\\theta} Q^{\\mathrm{m}}(r) d r\n$$\n\nAs under the efficient allocation, because $C(Q)$ is not continuously differentiable, the quality and transfer schedules are discontinuous.\n\nProposition 6 (Multi-Model Monopoly). If $\\varphi(\\theta)$ is increasing, then the optimal token-\nbudget menu is given by $\\left\\{\\left(\\left(X_{l}(\\theta), Z_{l}(\\theta)\\right)_{l=1}^{L}, t(\\theta)\\right)\\right\\}_{\\theta}$, where $\\left(X_{l}(\\theta), Z_{l}(\\theta)\\right)_{l=1}^{L}$ are efficient token budgets that deliver quality $Q^{\\mathrm{m}}(\\theta)$ at prices $t(\\theta)$ as defined in (26) and (27).\n\nThe monopolist assigns each buyer type to a single model, with higher types using more capable models. All types $w$ with the same aggregate type $\\theta(w)$ choose the same item and use one model, $l^{\\mathrm{m}}(\\theta)$ as defined in (25), for all tasks. The quality schedule exhibits discrete jumps at model-switching thresholds: a buyer at the margin between two models jumps to a strictly higher quality when upgrading. OpenAI's tier structure, which reserves its most compute-intensive reasoning model for the highest-paying subscribers, provides a direct empirical counterpart; see the discussion in Section 6.","text_sha256":"f782d709ac6723ed96dc59ba74dbb2afc76086f612a8de3861c327ef5e8608c8"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0017","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.3 Leader-Fringe Competition","text":"### 5.3 Leader-Fringe Competition\n\nWe now study competition between a proprietary leader and an open-source competitive fringe. The key economic forces are: (i) the fringe provides an outside option whose value depends on the buyer's type; (ii) for intermediate types, the leader must provide enough tokens to deter fringe top-up; (iii) for high types, the leader acts as an unconstrained monopolist. The resulting allocation has three distinct regions, generating a richer pattern of distortions than either the single-model monopoly or the multi-model monopoly.\n\nSpecifically, we assume that there is a single leader, corresponding to the highest-capability model $(l=L)$ in the previous section, and a continuum of firms in a competitive fringe that sell their tokens at fixed per-token prices equal to marginal costs. We assume that the leader possesses a proprietary model characterized by the aggregate cost parameter $c_{L}>0$, returns to intensity $\\sigma_{L}=\\sigma>0$, and returns to fine-tuning $\\hat{\\sigma}_{L}>0$, such that $\\sigma+\\hat{\\sigma}_{L}<1$. The fringe possesses an open-source model characterized by the lower aggregate cost parameter $c_{F} \\in\\left[0, c_{L}\\right)$, the same returns to intensity $\\sigma_{F}=\\sigma$, and lower returns to fine-tuning $\\hat{\\sigma}_{F} \\in\\left[0, \\hat{\\sigma}_{L}\\right)$.\n\nThe buyer has a private type $w$. By Lemma 2, only the aggregate type $\\theta(w)$ matters, so we take $\\theta$ as a primitive and assume that $\\theta$ is distributed according to $F$ with strictly positive density $f$ everywhere on [0, 1]. The buyer can multi-home and can buy at most one item from the leader.\n\nWe want to solve the leader's problem, which can be viewed as a monopolistic design of an optimal menu $(q, t)$ subject to the competitive pressure from the fringe. For notational simplicity, define the quantities:\n\n$$\n\\begin{aligned}\n& q_{L} \\triangleq Q_{L}^{1 /\\left(\\sigma+\\hat{\\sigma}_{L}\\right)}=g_{L}\\left(X_{L 1}, \\ldots, X_{L J}, Z_{L 1}, \\ldots, Z_{L K}\\right)^{1 /\\left(\\sigma+\\hat{\\sigma}_{L}\\right)} \\\\\n& q_{F} \\triangleq Q_{F}^{1 /\\left(\\sigma+\\hat{\\sigma}_{F}\\right)}=g_{F}\\left(X_{F 1}, \\ldots, X_{F J}, Z_{F 1}, \\ldots, Z_{F K}\\right)^{1 /\\left(\\sigma+\\hat{\\sigma}_{F}\\right)}\n\\end{aligned}\n$$\n\nThe payoff of type $\\theta$ from having purchased quantity $q_{L}$ from the leader is\n\n$$\n\\max _{q_{F} \\geq 0} \\theta\\left(q_{L}^{\\left(\\sigma+\\hat{\\sigma}_{L}\\right) / \\sigma}+q_{F}^{\\left(\\sigma+\\hat{\\sigma}_{F}\\right) / \\sigma}\\right)^{\\sigma}-c_{F} q_{F},\n$$\n\nwith a (type-dependent) outside option corresponding to $q_{L}=0$. The first-order condition for the buyer's problem of purchasing fringe quantity is\n\n$$\n\\theta\\left(\\sigma+\\hat{\\sigma}_{F}\\right)\\left(q_{L}^{\\left(\\sigma+\\hat{\\sigma}_{L}\\right) / \\sigma}+q_{F}^{\\left(\\sigma+\\hat{\\sigma}_{F}\\right) / \\sigma}\\right)^{\\sigma-1} q_{F}^{\\hat{\\sigma}_{F} / \\sigma}=c_{F} .\n$$\n\nIn general, (29) does not admit a closed-form solution. It does so in the case of $\\hat{\\sigma}_{F}=0$. Specifically, define\n\n$$\n\\hat{q}(\\theta) \\triangleq\\left(\\frac{\\theta \\sigma}{c_{F}}\\right)^{\\sigma /\\left((1-\\sigma)\\left(\\sigma+\\hat{\\sigma}_{L}\\right)\\right)}, \\quad \\psi(\\theta) \\triangleq \\theta^{\\frac{1}{1-\\sigma}}(1-\\sigma)\\left(\\frac{\\sigma}{c_{F}}\\right)^{\\sigma /(1-\\sigma)} .\n$$\n\nIf the quantity purchased from the leader is $q_{L}<\\hat{q}(\\theta)$, the buyer will purchase fringe tokens to achieve the optimal total quantity $\\hat{q}(\\theta)$; if $q_{L}>\\hat{q}(\\theta)$, the buyer will single-home with the leader. The buyer's outside option utility corresponding to $q=0$ is $\\psi(\\theta)$. We then obtain the following characterization of the buyer's demand for the leader's quantity.\n\nLemma 4 (Buyer's Payoff). The leader's problem is equivalent to one in which the buyer has a zero outside option and the following payoff:\n\n$$\nu(\\theta, q)= \\begin{cases}c_{F} q^{\\left(\\sigma+\\hat{\\sigma}_{L}\\right) / \\sigma}, & \\text { if } q<\\hat{q}(\\theta), \\\\ \\theta q^{\\sigma+\\hat{\\sigma}_{L}}-\\psi(\\theta), & \\text { if } q \\geq \\hat{q}(\\theta) .\\end{cases}\n$$\n\nThe payoff $u(\\theta, q)$ in Lemma 4 is continuous, differentiable, and increasing in both parameters. In the region $q<\\hat{q}(\\theta)$, it is convex in $q$ and independent of $\\theta$. In the region $q>\\hat{q}(\\theta)$, it is concave in $q$ and supermodular in $q$ and $\\theta$.\n\nThe two-regime structure has a clear economic interpretation. In the first regime ( $q<$ $\\hat{q}(\\theta))$, the buyer supplements leader tokens with fringe tokens to achieve the optimal total quantity, so her marginal value of additional leader tokens is pinned down by the fringe price $c_{F}$; the leader faces a perfectly elastic residual demand. In the second regime $(q \\geq \\hat{q}(\\theta))$, the buyer single-homes with the leader, and her value is the standard concave payoff minus the outside option $\\psi(\\theta)$ she forgoes by not using the fringe. The transition between regimes is where the leader's competitive constraint binds.\n\nThe leader's problem is an instance of screening with a type-dependent outside option (Jullien, 2000); the reformulation (31) absorbs this outside option into the buyer's payoff.\n\nAn optimal menu can then be characterized by standard methods. ${ }^{10}$ Indeed, $u(\\theta, q)$ satisfies the (weak) single-crossing constraints: $u_{\\theta q}(\\theta, q)=0$ if $q<\\hat{q}(\\theta)$ and $u_{\\theta q}(\\theta, q)>0$ if $q \\geq \\hat{q}(\\theta)$. Allocation $q(\\theta)$ is implementable only if it is increasing whenever $q \\geq \\hat{q}(\\theta)$, and the profit associated with allocation $q$ is:\n\n$$\n\\Pi(q)=\\int_{0}^{1}\\left(u(\\theta, q(\\theta))-c_{L} q(\\theta)-\\frac{1-F(\\theta)}{f(\\theta)} u_{\\theta}(\\theta, q(\\theta))\\right) f(\\theta) d \\theta\n$$\n\nIn regular environments, this profit can be maximized pointwise. To this end, denote by $q^{\\mathrm{m}}(\\theta)$ the optimal monopoly quantity in the absence of the competitive fringe\n\n$$\nq^{\\mathrm{m}}(\\theta)=\\left(\\frac{\\left(\\sigma+\\hat{\\sigma}_{L}\\right) \\varphi(\\theta)}{c_{L}}\\right)^{1 /\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right)} .\n$$","text_sha256":"0c1512f6a296f0299f5170badce1c78d2a52c2056e274857a4c0a0643b6014f2"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0018","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.3 Leader-Fringe Competition","text":"Proposition 7 (Leader-Fringe Competition). Let $F$ satisfy a monotone hazard rate and $\\hat{\\sigma}_{F}=0$. Then, there exist $\\underline{\\theta}$ and $\\bar{\\theta}, \\underline{\\theta} \\leq \\bar{\\theta}$, such that (i) for $\\theta \\leq \\underline{\\theta}, q(\\theta)=0$, (ii) for $\\theta \\in(\\underline{\\theta}, \\bar{\\theta})$, $q(\\theta)=\\hat{q}(\\theta)$ given by (30), and (iii) for $\\theta \\geq \\bar{\\theta}, q(\\theta)=q^{\\mathrm{m}}(\\theta)$ given by (33).\n\nUnder the optimal mechanism, low types purchase exclusively from the fringe. All higher types purchase exclusively from the leader. Out of those, the midline types are on the margin of whether to buy from the fringe, whereas the highline types strictly prefer to purchase from the leader. Depending on parameters, the midline region may not exist, $\\underline{\\theta}=\\bar{\\theta}$, or the highline region may not exist, $\\bar{\\theta}>1$.\n\nThe three-region structure reflects three distinct economic regimes. In the fringe-only region $(\\theta \\leq \\underline{\\theta})$, the buyer's willingness to pay for the leader's superior fine-tuning capability is too low to justify adoption. In the deterrence region $(\\underline{\\theta}<\\theta<\\bar{\\theta})$, the leader offers exactly enough tokens to make the buyer indifferent between single-homing with the leader and supplementing with the fringe; this pins the leader's quantity to $\\hat{q}(\\theta)$, which rises with type. In the monopoly region $(\\theta \\geq \\bar{\\theta})$, the buyer's value is high enough that fringe competition no longer constrains the leader, who reverts to the standard downward-distorted monopoly quantity $q^{\\mathrm{m}}(\\theta)$. The novel region is the deterrence region, which has no analog in either the single-model monopoly or the multi-model monopoly.\n\n[^8]Comparison with Efficient Allocation It is instructive to compare the allocation under leader-fringe competition with the efficient benchmark. By Proposition 5, defining\n\n$$\nu_{l}^{*}(\\theta)=\\left(1-\\sigma_{l}-\\hat{\\sigma}_{l}\\right) \\theta^{1 /\\left(1-\\sigma_{l}-\\hat{\\sigma}_{l}\\right)}\\left(\\frac{\\sigma_{l}+\\hat{\\sigma}_{l}}{c_{l}}\\right)^{\\left(\\sigma_{l}+\\hat{\\sigma}_{l}\\right) /\\left(1-\\sigma_{l}-\\hat{\\sigma}_{l}\\right)}\n$$\n\ntype $\\theta$ uses the fringe model if and only if $u_{F}^{*}(\\theta)>u_{L}^{*}(\\theta)$. The indifference type at which $u_{F}^{*}(\\theta)=u_{L}^{*}(\\theta)$ is (recall that $\\sigma_{L}=\\sigma_{F}=\\sigma$ and $\\hat{\\sigma}_{F}=0$ )\n\n$$\n\\hat{\\theta}=\\left(\\frac{1-\\sigma}{1-\\sigma-\\hat{\\sigma}_{L}}\\right)^{\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right)(1-\\sigma) / \\hat{\\sigma}_{L}}\\left(\\frac{\\sigma}{c_{F}}\\right)^{\\sigma\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right) / \\hat{\\sigma}_{L}}\\left(\\frac{c_{L}}{\\sigma+\\hat{\\sigma}_{L}}\\right)^{\\left(\\sigma+\\hat{\\sigma}_{L}\\right)(1-\\sigma) / \\hat{\\sigma}_{L}} .\n$$\n\nIf $\\theta<\\hat{\\theta}$, then the type uses the fringe model: $q_{F}(\\theta)=\\left(\\theta \\sigma / c_{F}\\right)^{1 /(1-\\sigma)}$ and $q_{L}(\\theta)=0$. If $\\theta \\geq \\hat{\\theta}$, then the type uses the leader model: $q_{F}(\\theta)=0$ and $q_{L}(\\theta)=\\left(\\theta\\left(\\sigma+\\hat{\\sigma}_{L}\\right) / c_{L}\\right)^{1 /\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right)}$.\n\nComparing this with the fringe-competition allocation characterized in Proposition 7, we see two main differences: First, we see that the \"highline\" leader quantity $q^{\\mathrm{m}}(\\theta)$ is distorted downward from efficiency because $\\varphi(\\theta)$ is smaller than $\\theta$. This is natural, since in that range the leader is unconstrained by the fringe and acts as a monopolist. Second, there is an additional extensive margin distortion, because $\\hat{\\theta}<\\underline{\\theta} .{ }^{11}$\n\nComparison with Multi-Model Monopoly Another natural benchmark is the case of a monopolist possessing the leader and fringe models. By Proposition 6, the monopolist effectively faces a Mussa-Rosen problem with the buyer's payoff:\n\n$$\nu\\left(\\theta, q_{L}, q_{F}\\right)=\\theta\\left(q_{F}+q_{L}^{\\left(\\sigma+\\hat{\\sigma}_{L}\\right) / \\sigma}\\right)^{\\sigma} .\n$$\n\nand will be supplying an efficient allocation with respect to the virtual type. If $\\theta<\\theta^{\\mathrm{m}}$, where $\\varphi\\left(\\theta^{\\mathrm{m}}\\right) \\triangleq \\hat{\\theta}$ and $\\hat{\\theta}$ is defined at (34), then the type uses the fringe model and $q_{F}(\\theta)=$ $\\left(\\varphi(\\theta) \\sigma / c_{F}\\right)^{1 /(1-\\sigma)}$ and $q_{L}(\\theta)=0$. If $\\theta \\geq \\theta^{\\mathrm{m}}$, then the type uses the leader model with $q_{F}(\\theta)=0$ and $q_{L}(\\theta)=\\left(\\varphi(\\theta)\\left(\\sigma+\\hat{\\sigma}_{L}\\right) / c_{L}\\right)^{1 /\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right)}$.\n\nRelative to the efficiency benchmark, the allocation is distorted downward within each model, except at the very top. In addition, fewer types consume the leader model.\n\nRelative to the leader-fringe environment of Proposition 7, the integrated multi-model monopolist internalizes the fringe technology and therefore does not face a competitive outside option. This has two implications. First, the fringe allocation is no longer pinned down by marginal-cost pricing: types that use the fringe model receive $q_{F}(\\theta)=\\left(\\varphi(\\theta) \\sigma / c_{F}\\right)^{1 /(1-\\sigma)}$\n\n[^9](and are excluded whenever $\\varphi(\\theta)<0$ ), whereas under leader-fringe competition the corresponding types purchase the efficient quantity $\\left(\\theta \\sigma / c_{F}\\right)^{1 /(1-\\sigma)}$ from the fringe. Second, the \"midline\" region $\\theta \\in(\\underline{\\theta}, \\bar{\\theta})$ in which the leader supplies $q=\\hat{q}(\\theta)$ to deter top-up purchases from the fringe disappears. Because the monopolist controls both models, she uses the fringe model directly as the low-quality product in the Mussa-Rosen menu and assigns each type to a single model with a single cutoff $\\theta^{\\mathrm{m}}$. Finally, once the fringe constraint is slack, the intensive-margin provision coincides and equals the monopoly quantity $q^{\\mathrm{m}}(\\theta)$; differences between the two environments are therefore concentrated among low and intermediate types. ${ }^{12}$\n\nFigure 1 illustrates the optimal allocation and distortions in a uniform example. All calculations for this example are in Appendix A.","text_sha256":"e66cce1dcfb17aaa9f4dc06187ff79429f91557440dd8281c9e46c8097318c74"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0019","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.3 Leader-Fringe Competition","text":"The top panel plots the leader quantity $q(\\theta)$, and the bottom panel plots the fringe quantity $q_{F}(\\theta)$. Four cutoffs, ordered $\\hat{\\theta}<\\underline{\\theta}<\\theta^{\\mathrm{m}}<\\bar{\\theta}$, partition the type space. The efficient allocation (solid) features a single switch at $\\hat{\\theta}$ : types below $\\hat{\\theta}$ use only the fringe, types above use only the leader. The leader-fringe allocation (dotted) shifts this switch rightward to $\\underline{\\theta}$, reflecting the extensive-margin distortion. Between $\\underline{\\theta}$ and $\\bar{\\theta}$, the leader supplies the deterrence quantity $\\hat{q}(\\theta)$, which lies strictly below the efficient leader quantity. Above $\\bar{\\theta}$, the fringe constraint is slack and the leader reverts to the monopoly quantity $q^{\\mathrm{m}}(\\theta)$, which coincides with the integrated monopolist's allocation (dashed). In the bottom panel, the efficient and leader-fringe curves for $q_{F}$ coincide over their respective ranges, since the fringe prices at marginal cost in both cases; the difference is that fringe usage persists up to $\\underline{\\theta}$ rather than $\\hat{\\theta}$. The integrated monopoly distorts fringe provision downward and switches to the leader at the later cutoff $\\theta^{\\mathrm{m}}$.","text_sha256":"56ef4b156a982661f8f0ae6d10ca28a58c97a7405669bdb515afb1ffd080925b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0020","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 LLM Pricing in Practice","text":"## 6 LLM Pricing in Practice\n\nA striking feature of current LLM pricing is that different market segments implement each of the optimal mechanisms in our theoretical framework: consumer subscriptions implement the nonlinear menus of Sections 4 and 5; model aggregators implement the committed-spend mechanisms of Section 4.2; and developer APIs implement the constrained-efficient linear pricing of Corollary 1.\n\nOur main findings are as follows. The two largest proprietary providers, Anthropic and OpenAI, offer remarkably similar price points (\\$20 and \\$200) but implement fundamentally different screening mechanisms. Anthropic screens through quantity alone, holding model ac-\n\n[^10]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: Leader-Fringe allocations across different regimes. Example with $\\theta \\sim U[0,1]$, $\\sigma=1 / 2, \\hat{\\sigma}_{L}=1 / 4, c_{F}=1 / 10, c_{L}=1 / 8$.\n\ncess constant across paid tiers, matching the token-budget mechanism of Proposition 3. OpenAI screens through both quantity and model access, reserving its most compute-intensive reasoning model for the highest tier, matching the multi-model menu of Proposition 6. This contrast illustrates the distinction between the single-model screening of Section 4 and the multi-model versioning of Section 5. Model aggregators, which resell access to upstream providers through a single interface, implement the committed-spend mechanisms of Section 4.2: Quora's Poe enforces a hard budget cap (maximum spend), while GitHub Copilot allows overage purchasing at linear prices (minimum spend). Finally, API pricing across all major providers is linear in tokens with no volume discounts, resembling the constrainedefficient benchmark of Corollary 1 rather than any profit-maximizing mechanism, consistent with providers prioritizing market share over rent extraction in the developer segment.","text_sha256":"20a653b55af1958e7c731cf12f583f4d28728d208a7204c8c5eae161cba8c948"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0021","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6.1 Anthropic","text":"### 6.1 Anthropic\n\nAnthropic's consumer pricing for Claude best illustrates the token-budget mechanism of Section 4. Anthropic offers the same model family across all paid tiers, with differentiation occurring through usage allocations. As in the optimal menu characterized in Proposition 3, users with higher aggregate types $\\theta$ select items with larger token budgets while consuming the same underlying technology.\n\nFigure 2 summarizes the structure of paid tiers as of January 2026. All paid tiers, namely Pro (\\$20/month), Max 5x (\\$100/month), and Max 20x (\\$200/month), grant access to Haiku 4.5, Sonnet 4.5, and Opus 4.5. The key distinction is the quantity of compute: Anthropic measures usage limits in computational intensity. For example, Opus queries consume resources approximately $5 \\times$ faster than Sonnet. This aligns with Proposition 3, where the optimal mechanism defines users' budgets in total tokens consumed across classes. Although Anthropic offers multiple models, all paid tiers grant access to the same set, so the relevant screening dimension is quantity rather than model access. The different depletion rates across models (e.g., Opus consuming resources $5 \\times$ faster than Sonnet) function as heterogeneous per-unit costs within a single compute-budget mechanism, analogous to the token-class costs $c_{j}$ in Proposition 3.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 2: Anthropic subscription tiers (January 2026). Model access is constant across paid tiers; differentiation occurs through usage allocations.","text_sha256":"dcad30128d41736d275ff7d7913a75887ea1b98a92f1a15206142e1f3e728bff"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0022","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6.2 OpenAI","text":"### 6.2 OpenAI\n\nUnlike Anthropic's tier structure, which differentiates mostly through constraints on usage, OpenAI's ChatGPT pricing differentiates through both quantity and exclusive model access: higher tiers grant access to more capable models and relax usage constraints. This joint screening aligns with the multi-model monopolist of Section 5.2.\n\nFigure 3 summarizes OpenAI's tier structure as of January 2026. The Free tier provides limited GPT-4o access with automatic downgrade to GPT-4o-mini when capacity is constrained. Plus (\\$20/month) expands to approximately 80 GPT-4o messages per 3-hour window and adds reasoning models (o3, o4-mini) with a separate weekly limit of roughly 100 queries. Pro (\\$200/month) removes most constraints and provides exclusive access to o1pro, OpenAI's most compute-intensive reasoning model. Thus, both Anthropic and OpenAI converge on similar price points (\\$20 standard, \\$200 premium), but implement very different screening strategies.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 3: OpenAI ChatGPT subscription tiers (January 2026). Higher tiers grant access to more capable models and larger usage allocations.\n\nIn Proposition 6, we showed that when models differ in cost curvature, the optimal mechanism assigns higher-capability models to higher buyer types. OpenAI's decision to reserve its o1-pro model for the highest tier is consistent with this prediction. In consumer subscriptions, the relevant cost differences across models may reflect inference-time compute intensity rather than user fine-tuning, but the qualitative implication of a monotone assignment with discrete jumps at upgrade thresholds is the same.","text_sha256":"63f1e6c4dbcff2c7d2bccf06a7eed7edf5523cd344e87f52f1a93e449983b885"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0023","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6.3 Model Aggregators: GitHub and Quora","text":"### 6.3 Model Aggregators: GitHub and Quora\n\nA number of AI platforms, including GitHub Copilot and Quora's Poe, do not develop their own models. Instead, they aggregate models produced by others (OpenAI, Anthropic, Google, Mistral, and others) and sell access to users through a single interface. A common contractual arrangement is one whereby subscribers pay a monthly fee, receive a budget of platform-specific credits, and allocate those credits across the available models. Selecting a more capable model typically depletes the budget faster. We now describe Quora's and GitHub's platforms in detail and then relate their pricing structures to the indirect implementation of the optimal mechanism discussed in Section 4.2.\n\nQuora-Poe Poe offers subscribers access to over 100 AI models through a unified chat interface. A subscriber pays a fixed monthly fee and receives an allocation of \"compute points,\" which serve as the platform's internal currency. Each model is priced in points per message at a rate that reflects its inference cost. ${ }^{13}$\n\nTable 1 summarizes Poe's tier structure. The entry-level Basic plan provides 300,000 points per month for \\$5; the top-tier Premium plan provides 12,500,000 points for \\$250. Above the Basic tier, the effective price per million points is constant at \\$20, so that higher tiers simply offer a proportionally larger budget at a fixed unit rate. Table 2 reports point costs for selected models, illustrating the wide dispersion across model classes.\n\n| Tier | Monthly Price | Points/Month | Eff. \\$/M Points |\n| :--- | :--- | :--- | :--- |\n| Basic | \\$5 | 300,000 | \\$16.67 |\n| Standard | \\$20 | 1,000,000 | \\$20.00 |\n| Pro | \\$50 | 2,500,000 | \\$20.00 |\n| Advanced | \\$100 | 5,000,000 | \\$20.00 |\n| Premium | \\$250 | 12,500,000 | \\$20.00 |\n\nTable 1: Poe subscription tiers (2025).\n\nThe connection to the maximum-spend framework of Section 4.2 is immediate. The monthly subscription fee corresponds to the transfer $T_{n}$; the point allocation corresponds to the budget $B_{n}$; and the per-model point costs correspond to marginal-cost pricing of different token classes. The subscriber chooses how to allocate the budget taking the marginal prices into account. Finally, once the budget is exhausted, access to all models is suspended until the next billing cycle, and points do not roll over. Thus, Poe's menu is best captured by a maximum-spend mechanism.\n\n[^11]| Class | Model | Points/Message |\n| :--- | :--- | :--- |\n| Budget | GPT-4o-mini | 9 |\n|  | Gemini-2.0-Flash | 9 |\n| Mid-tier | GPT-4o | 224 |\n|  | Claude-3.5-Sonnet | 276 |\n| Frontier | Claude-3-Opus | 1,697 |\n|  | Claude-Opus-4 | 4,105 |\n\nTable 2: Poe per-model point costs (2025).\n\nGitHub Copilot GitHub Copilot provides AI-assisted coding within a developer's integrated environment. Like Poe, Copilot offers multiple pricing tiers, each bundling a monthly fee with a budget of \"premium requests\" that the developer allocates across model calls. Table 3 summarizes the plan structure.\n\n| Plan | Monthly Fee | Premium Requests | Notes |\n| :--- | :--- | :--- | :--- |\n| Free | \\$0 | 50 | Limited quota, basic completions |\n| Pro | \\$10 | 300 | Unlimited completions, chat, IDE support |\n| Pro+ | \\$39 | 1,500 | Larger quota, advanced models |\n| Business | \\$19/user | 300/user | Org controls, policy management |\n| Enterprise | \\$39/user | 1,000/user | Enterprise features |\n\nTable 3: GitHub Copilot pricing tiers and premium request budgets (2026).\n\nThe mechanism by which Copilot meters usage differs slightly from Poe's point system but serves the same function. Rather than assigning each model a point cost per message, Copilot assigns each model a multiplier that determines how many premium requests a single interaction consumes. A set of baseline models, including GPT-5 mini, GPT-4.1, and GPT-4o, carry a multiplier of 0 on paid plans: interactions with these models are included at no additional budget cost. Efficient reasoning models such as Claude Haiku 4.5 and o4-mini carry multipliers of 0.25-0.33. Standard models such as Claude Sonnet 4.x and GPT-5.x carry a multiplier of 1. At the frontier, Claude Opus 4.5 carries a multiplier of 3, and GPT-4.5 carries a multiplier of 50.\n\nThis mechanism matches the framework of Section 4.2 quite closely. The plan fee corresponds to $T_{n}$; the monthly premium-request allocation corresponds to $B_{n}$; and the model multipliers play the role of heterogeneous marginal costs. A distinctive feature of Copilot is that it extends this budget mechanism with an overage option. When a developer (i.e., a buyer) exhausts her monthly allocation, Copilot allows continued use at a fixed charge\nof \\$0.04 per premium request. The minimum-spend framework of Section 4.2 matches this feature rather well: each user commits to a certain amount of monthly spending, there are no refunds, but the developer can increase her consumption ex post at linear prices. ${ }^{14}$\n\nThus, both Poe and Copilot implement the core logic of the committed-spend mechanisms: a non-bankable budget, priced up front via a subscription fee, that the user freely allocates across inputs with heterogeneous per-unit costs. The key distinction lies in how each platform handles the overages. Poe enforces a hard cap: once points are exhausted, access is suspended until the next billing cycle. This corresponds exactly to the maximumspend mechanism. Copilot, by contrast, allows the cap to be relaxed at a known marginal price (which varies across models through the multiplier system), effectively implementing a minimum-spend mechanism.","text_sha256":"675bb18864ba64032bdc3bf6d5cf0b575dfcddf5fcca80b0b61ecdf3658a09b7"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0024","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6.4 API Pricing","text":"### 6.4 API Pricing\n\nAlongside consumer subscriptions, every major LLM provider operates a developer-facing API through which applications can programmatically submit prompts and receive completions. A developer who calls the API pays per token, with separate rates for input tokens (the prompt) and output tokens (the model's response). There is no subscription fee, no bundled allocation, and no minimum commitment: the developer pays only for what she uses. Table 4 reports prices for selected models across three providers as of January 2025.\n\n| Provider | Model | Input (\\$/M) | Output (\\$/M) |\n| :--- | :--- | :--- | :--- |\n| OpenAI | GPT-4o-mini | 0.15 | 0.60 |\n| OpenAI | GPT-4o | 2.50 | 10.00 |\n| OpenAI | o1 (reasoning) | 15.00 | 60.00 |\n| Anthropic | Claude 3.5 Haiku | 0.80 | 4.00 |\n| Anthropic | Claude 3.5 Sonnet | 3.00 | 15.00 |\n| Anthropic | Claude 3 Opus | 15.00 | 75.00 |\n| Google | Gemini 1.5 Flash | 0.075 | 0.30 |\n| Google | Gemini 1.5 Pro | 1.25 | 5.00 |\n\nTable 4: API pricing across major providers (January 2025). Prices per million tokens.\n\nThe pricing structure is notably uniform. Output tokens are priced at $3-5 \\times$ input tokens across all providers, reflecting that output generation is sequential and autoregressive (each token requires a forward pass conditioned on all preceding tokens), whereas input encod-\n\n[^12]ing is parallelizable and therefore substantially cheaper per token. Pricing is strictly linear: there are no quantity discounts, volume commitments, or tiered rates in standard API access. Although outside our model, all providers offer batch discounts (typically 50\\%) for asynchronous processing and prompt-caching discounts (up to 90\\%) for repeated inputs.\n\nThis linear, pay-per-token structure stands in sharp contrast to the nonlinear menus that characterize consumer subscriptions. It corresponds instead to the constrained-efficient allocation of Corollary 1: when a provider faces capacity constraints but wishes to maximize total surplus rather than profit (for instance, to lock in consumers in anticipation of future monetization), the constrained-efficient allocation can be implemented via linear prices equal to marginal costs inflated by shadow costs. The absence of nonlinear screening in API pricing is consistent with providers currently prioritizing adoption over rent extraction.\n\nThe uniformity of the pricing format across providers is itself evidence of competitive pressure: if any single provider were to introduce nonlinear screening in the API segment, it would risk losing developers to competitors who maintain simpler, linear pricing.\n\nEvidence from pricing dynamics is consistent with this interpretation. First, API prices have declined rapidly: GPT-4-class capability fell from \\$30/\\$60 per million tokens at launch (March 2023) to \\$2.50/\\$10 by August 2024, a 90\\% reduction in 16 months. Second, providers appear to cut prices in response to competitive launches rather than cost improvements; OpenAI's August 2024 reductions followed Claude 3.5 Sonnet and Llama 3.1 releases. Third, gross margins of 50-75\\% imply prices remain above marginal cost but are being compressed by competition. The entry of DeepSeek in January 2025 with reasoning-model pricing at \\$0.55/\\$2.19, versus OpenAI's \\$15/\\$60 for comparable capability, suggested that further compression is viable. See Demirer et al. (2025) for a comprehensive treatment of market share dynamics in the API segment.","text_sha256":"a5ff46c255113052345add83735e8d2016cdd8de30204852f77fac54cf9f1c0e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0025","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"7 Conclusion","text":"## 7 Conclusion\n\nA priori, the problem of pricing LLM access appears intractable: users have high-dimensional private information, the allocation space is also high-dimensional, and the user's hidden allocation of tokens across tasks introduces moral hazard. In this paper, we have shown that homogeneity of the gain function generates a sufficient-statistic reduction that makes the problem solvable. Users' high-dimensional type profiles collapse to a scalar aggregate type; the seller's problem reduces to one-dimensional screening; and the optimal mechanism admits simple implementations (committed-spend contracts and two-part tariffs) that correspond precisely to the pricing structures observed at leading providers.\n\nWhile we have phrased the analysis in terms of large language models, the framework applies more broadly to any setting in which a provider sells access to a general-purpose technology with multiple input classes and heterogeneous users. Cloud computing, where providers offer a multiplicity of services with both horizontal and vertical differentiation, is a natural example. In both settings, the core pricing problem is how to allocate and monetize costly computing resources, particularly when operating under capacity constraints.\n\nSeveral extensions would enrich the analysis. First, we assumed that all tasks are homogeneous in their use of input, output, and fine-tuning tokens, and differ only vertically in how valuable a given task is to the buyer. Allowing the gain function parameters to vary across tasks would break the aggregation result. Understanding how the optimal mechanism changes when the scalar sufficient statistic no longer exists is a challenging but important open question, especially in the context of competing specialized models.\n\nSecond, we assumed that all buyer types have fine-tuning data readily available. In practice, buyers face constraints on data availability or may be unwilling to share data with the provider because of privacy, compliance, or intellectual-property concerns. A data-poor buyer may then substitute toward inference-time tokens, while a data-rich buyer substitutes toward fine-tuning tokens. When data availability is the user's private information, the provider's problem involves screening on two dimensions (aggregate type and data availability), which may require new techniques.\n\nThird, LLM platforms exhibit network effects and data externalities that our static framework abstracts from. A provider that attracts more users may improve its model through fine-tuning, reinforcement learning from human feedback, or simply from the revenue that funds further training. These dynamic complementarities between adoption and model quality could substantially reshape optimal pricing, potentially justifying the aggressive belowaverage-cost pricing observed in the API segment not just as a means to capture market share but as investment in model improvement.\n\nFourth, our analysis treats each provider as either a monopolist or a leader facing a competitive fringe. A full oligopoly analysis with multiple differentiated proprietary models would capture the strategic interactions among Anthropic, OpenAI, Google, and others that increasingly shape the market. The tractability of our framework through the reduction to one-dimensional types suggests that such an extension may be feasible and would yield further predictions about equilibrium pricing and product differentiation.\n\nThe LLM industry is at an inflection point between growth-oriented pricing and profitmaximizing pricing. Our framework provides a theoretical foundation for understanding the mechanisms through which this transition will unfold, and for evaluating whether the resulting market structure efficiently allocates this critical economic input.","text_sha256":"7884e609966af13d3b7dde61d1368b1c3c80a55d1c5cc8ae93e08cbb0768109c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0026","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Proofs and Derivations","text":"## A Proofs and Derivations\n\nProof of Proposition 1 For any $z$ and $i$, the optimal allocation of inference tokens on task $i$ solves\n\n$$\n\\max _{x_{i j} \\geq 0} w_{i} \\Psi\\left(x_{i}\\right) \\Phi(z)-\\sum_{j=1}^{J} c_{j} x_{i j} .\n$$\n\nIf $w_{i}=0$ or $\\Phi(z)=0$, then $x_{i}=0$. Otherwise, by the Inada assumption, the problem admits an interior solution, which satisfies the system of $J$ first-order conditions:\n\n$$\nw_{i} \\nabla \\Psi\\left(x_{i}\\right) \\Phi(z)=c .\n$$\n\nBy (35), for all $i, \\nabla \\Psi\\left(x_{i}\\right)$ belongs to a ray with a direction $c$. Because $\\Psi$ is homogeneous and strictly concave, the ray in the space of gradients corresponds to a ray in the space of tokens. ${ }^{15}$ Thus, for each $i$, any optimal $x_{i}$ can be written as:\n\n$$\nx_{i}=d y_{i},\n$$\n\nwhere $d \\in \\mathbb{R}_{+}^{J}$ is the unique vector that solves $\\nabla \\Psi(d)=c$.\nTherefore, by the homogeneity of $\\Psi$, we have $\\nabla \\Psi\\left(x_{i}\\right)=y_{i}^{\\sigma-1} \\nabla \\Psi(d)$, and (35) can be rewritten as:\n\n$$\nw_{i} y_{i}^{\\sigma-1} \\nabla \\Psi(d) \\Phi(z)=\\nabla \\Psi(d),\n$$\n\nwhich pins down the optimal scale as:\n\n$$\ny_{i}=w_{i}^{\\frac{1}{1-\\sigma}} \\Phi(z)^{\\frac{1}{1-\\sigma}} .\n$$\n\nThe resulting total surplus ignoring fine-tuning costs is:\n\n$$\n\\begin{aligned}\n& \\int_{0}^{1} w_{i} \\Psi\\left(x_{i}\\right) \\Phi(z) d i-\\sum_{j=1}^{J} c_{j} \\int_{0}^{1} x_{i j} d i=\\int_{0}^{1} w_{i} y_{i}^{\\sigma} \\Psi(d) \\Phi(z)-y_{i}(d \\cdot c) d i \\\\\n& =\\int_{0}^{1} w_{i}^{\\frac{1}{1-\\sigma}} \\Phi(z)^{\\frac{1}{1-\\sigma}}(\\Psi(d)-d \\cdot c) d i=\\theta(w)^{\\frac{1}{1-\\sigma}} \\Phi(z)^{\\frac{1}{1-\\sigma}}(1-\\sigma) \\Psi(d)\n\\end{aligned}\n$$\n\n[^13]where $\\theta(w)$ is the aggregate type defined as\n$$\n\\theta(w) \\triangleq\\left(\\int_{0}^{1} w_{i}^{\\frac{1}{1-\\sigma}} d i\\right)^{1-\\sigma}\n$$\nand the last equality in (36) follows from the definition of $d$ and Euler's theorem for homogeneous functions: $d \\cdot \\nabla \\Psi(d)=\\sigma \\Psi(d)$.\n\nSimilarly, the total amount of inference tokens of class $j$ is:\n\n$$\n\\int_{0}^{1} x_{i j} d i=\\int_{0}^{1} d_{j} y_{i} d i=\\int_{0}^{1} d_{j} w_{i}^{\\frac{1}{1-\\sigma}} \\Phi(z)^{\\frac{1}{1-\\sigma}} d i=\\theta(w)^{\\frac{1}{1-\\sigma}} d_{j} \\Phi(z)^{\\frac{1}{1-\\sigma}}\n$$\n\nThis completes the proof. $\\square$\n\nProof of Corollary 1 The Lagrangian approach applies. Thus, associating Lagrange multipliers $\\lambda_{j} \\geq 0$ and $\\hat{\\lambda}_{k} \\geq 0$ with the capacity constraints for the different token classes, the optimal solution solves\n\n$$\n\\max _{\\left(x_{i}\\right)_{i \\in[0,1], z \\geq 0}} \\int_{0}^{1} w_{i} g\\left(x_{i}, z\\right) d i-\\sum_{j=1}^{J}\\left(c_{j}+\\lambda_{j}\\right) \\int_{0}^{1} x_{i j} d i-\\sum_{k=1}^{K}\\left(\\hat{c}_{k}+\\hat{\\lambda}_{k}\\right) z_{k} .\n$$\n\nAs such, the solution is an efficient allocation given the adjusted costs $c_{j}^{\\prime} \\triangleq c_{j}+\\lambda_{j}$ and $\\hat{c}_{k}^{\\prime} \\triangleq \\hat{c}_{k}+\\hat{\\lambda}_{k}$. The result follows. $\\square$\n\nProof of Proposition 2 Consider the buyer-optimal inference token allocation across tasks for any given token budgets $(X, Z)$ such that $\\Phi(Z)>0$ :\n\n$$\n\\max _{x_{i j} \\geq 0} \\int_{0}^{1} w_{i} \\Psi\\left(x_{i}\\right) \\Phi(Z) d i, \\quad \\text { s.t. } \\int_{0}^{1} x_{i j} d i=X_{j} \\text { for } j \\in[J] .\n$$\n\nBecause $\\Phi(Z)$ is a strictly positive constant, it can be ignored. Assume that for each $j$, $X_{j}>0$; otherwise, $x_{i j}=0$ for all $i$. Lagrangian approach applies. We associate Lagrange multipliers $\\lambda=\\left(\\lambda_{1}, \\ldots, \\lambda_{J}\\right)$ with the budget constraints of (38); the resulting first-order conditions are:\n\n$$\nw_{i} \\nabla \\Psi\\left(x_{i}\\right)=\\lambda .\n$$\n\nBy the same logic as in the efficiency analysis, these conditions imply that the relative ratios of input tokens are the same across all tasks:\n\n$$\nx_{i}=y_{i} d,\n$$\n\nfor some $d \\in \\mathbb{R}_{+}^{J}$. The budget constraints imply that $\\left(\\int_{0}^{1} y_{i} d i\\right) d=X$ and hence $d$ is proportional to $X$ and can be normalized to satisfy $\\sum_{j=1}^{J} d_{j}=1$, leading to\n\n$$\nd=\\frac{1}{\\sum_{j=1}^{J} X_{j}} X\n$$\n\nThe optimal scales $y_{i}$ solve:\n\n$$\n\\max _{y_{i} \\geq 0} \\int_{0}^{1} w_{i} y_{i}^{\\sigma} \\Psi(d) d i, \\quad \\text { s.t. } \\quad \\int_{0}^{1} y_{i} d i=\\sum_{j=1}^{J} X_{j} .\n$$\n\nAt the optimum, the marginal gains $w_{i} \\sigma y_{i}^{\\sigma-1}$ are equalized across tasks, so:\n\n$$\ny_{i}=w_{i}^{\\frac{1}{1-\\sigma}} \\frac{\\sum_{j=1}^{J} X_{j}}{\\int_{0}^{1} w_{r}^{\\frac{1}{1-\\sigma}} d r} .\n$$\n\nThe optimal buyer's payoff is:\n\n$$\n\\begin{aligned}\nU & =\\int_{0}^{1} w_{i}\\left(w_{i}^{\\frac{1}{1-\\sigma}} \\frac{\\sum_{j=1}^{J} X_{j}}{\\int_{0}^{1} w_{r}^{\\frac{1}{1-\\sigma}} d r}\\right)^{\\sigma} \\Psi(d) \\Phi(Z) d i \\\\\n& =\\left(\\int_{0}^{1} w_{i}^{\\frac{1}{1-\\sigma}} d i\\right)^{1-\\sigma}\\left(\\sum_{j=1}^{J} X_{j}\\right)^{\\sigma} \\Psi(d) \\Phi(Z)=\\theta(w) \\Psi(X) \\Phi(Z)\n\\end{aligned}\n$$\n\nwhere in the last equality we used the homogeneity of $\\Psi$ and (39). $\\square$\n\nProof of Lemma 1 The fact that $C(Q)$ is strictly increasing and strictly convex is immediate because $g(X, Z)=\\Psi(X) \\Phi(Z)$ is strictly increasing and strictly concave in its arguments whenever $g(X, Z)>0$.\n\nTo show $C_{+}^{\\prime}(0)=0$, observe that since $C(Q)$ is convex, $C_{+}^{\\prime}(0)=\\inf _{Q>0} C(Q) / Q$. Pick any $X_{0}$ such that $\\Psi\\left(X_{0}\\right)>0$ and $Z_{0}$ as in Footnote 3. For $r>0$, consider $\\left(X_{r}, Z_{r}\\right)=r\\left(X_{0}, Z_{0}\\right)$ and set $Q_{r}=g\\left(X_{r}, Z_{r}\\right)$. We have,\n\n$$\n\\frac{C\\left(Q_{r}\\right)}{Q_{r}} \\leq \\frac{c \\cdot X_{r}+\\hat{c} \\cdot Z_{r}}{Q_{r}}=\\frac{r\\left(c \\cdot X_{0}+\\hat{c} \\cdot Z_{0}\\right)}{r^{\\sigma} \\Psi\\left(X_{0}\\right) \\Phi\\left(r Z_{0}\\right)}=\\frac{c \\cdot X_{0}+\\hat{c} \\cdot Z_{0}}{\\Psi\\left(X_{0}\\right)} \\frac{1}{\\Phi\\left(r Z_{0}\\right) / r^{1-\\sigma}} .\n$$\n\nBy the choice of $Z_{0}, \\Phi\\left(r Z_{0}\\right) / r^{1-\\sigma} \\rightarrow+\\infty$ as $r \\downarrow 0$, hence the right-hand side goes to 0 . Since $Q_{r} \\downarrow 0$, it follows that $\\inf _{Q>0} C(Q) / Q=0$. $\\square$\n\nProof of Proposition 3 The result is an application of Mussa and Rosen (1978) to the buyer's payoff $\\theta Q$ from Proposition 2 and the cost function $C(Q)$ from Lemma 1. The optimal quality schedule (12) and transfers (13) follow from standard arguments. $\\square$","text_sha256":"d98dbc4f79243325fbbd9dd1bdf9c57cadd3307341b90036316d571993339eeb"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0027","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Proofs and Derivations","text":"Proof of Proposition 4 Fix an optimal direct mechanism $\\left(X^{*}(\\theta), Z^{*}(\\theta), T^{*}(\\theta)\\right)$ with the associated aggregate quality $Q^{*}(\\theta)=\\Psi\\left(X^{*}(\\theta)\\right) \\Phi\\left(Z^{*}(\\theta)\\right)$.\n\nMaximum-spend mechanism. Consider a maximum-spend mechanism with $p=$ $(c, \\hat{c}), T(\\theta)=T^{*}(\\theta)$, and $B(\\theta)=C\\left(Q^{*}(\\theta)\\right)$. Under this mechanism, since tokens are priced at marginal cost, each type $w$ purchasing budget $B\\left(\\theta^{\\prime}\\right)$ would optimally allocate tokens in a constrained-efficient way, thus deriving payoff $\\theta(w) Q$ where $C(Q)=B\\left(\\theta^{\\prime}\\right)$, i.e., $Q=Q^{*}\\left(\\theta^{\\prime}\\right)$. Hence, the choice across items in this mechanism is equivalent to the choice in the direct mechanism, and this mechanism implements the allocation of the direct mechanism.\n\nTwo-part-tariff mechanism. When type $w$ faces any two-part tariff, his optimal token allocation problem is equivalent to the efficient allocation problem (5) with prices taking the roles of costs. Therefore, all types $w$ with the same $\\theta(w)$ have the same payoff from any possible two-part tariff and can be treated as a single type.\n\nFaced with a given item, the problem of type $\\theta$ can be written as value-maximizationpayment-minimization:\n\n$$\n\\max _{Q \\geq 0} \\theta Q-P(Q),\n$$\n\nwhere\n\n$$\nP(Q) \\triangleq \\min _{X_{1}, \\ldots, X_{J}, Z_{1}, \\ldots, Z_{K} \\geq 0} \\sum_{j=1}^{J} p_{j} X_{j}+\\sum_{k=1}^{K} \\hat{p}_{k} Z_{k}, \\quad \\text { s.t. } \\Psi(X) \\Phi(Z)=Q .\n$$\n\nConsider the following two-part-tariff mechanism:\n\n$$\n\\begin{aligned}\n& p_{j}(\\theta)=m(\\theta) c_{j}, \\hat{p}_{k}(\\theta)=m(\\theta) \\hat{c}_{k}, j \\in[J], k \\in[K], \\\\\n& p_{0}(\\theta)=t(\\theta)-m(\\theta) C(Q(\\theta)),\n\\end{aligned}\n$$\n\nwhere $C(Q)$ is as defined in (10), $Q(\\theta)$ is as defined in (12), and $t(\\theta)$ is as defined in (13). Under this mechanism, $\\sum_{j=1}^{J} p_{j} X_{j}+\\sum_{k=1}^{K} \\hat{p}_{k} Z_{k}=m(\\theta)\\left(\\sum_{j=1}^{J} c_{j} X_{j}+\\sum_{k=1}^{K} \\hat{c}_{k} Z_{k}\\right)$. Thus, the buyer-optimal allocation is efficient, and $P(Q)=m(\\theta) C(Q)$.\n\nThe rest of the argument is standard (e.g., Tirole (1988, pp. 154-157)). A buyer-optimal $Q(\\theta)$ is determined by the first-order condition:\n\n$$\n\\theta=P^{\\prime}(Q(\\theta))=m(\\theta) C^{\\prime}(Q(\\theta)),\n$$\n\nand thus the buyer-optimal level of quality satisfies the optimality condition:\n\n$$\n\\bar{\\varphi}(\\theta)=C^{\\prime}(Q(\\theta)) .\n$$\n\nIf $m(\\theta)$ is decreasing, then the implied payment schedule in units of $C(Q)$ is concave. Thus, a menu of two-part tariffs with markups $m(\\theta)$, constant across inputs, implements the desired quality schedule $Q(\\theta)$, with each type $\\theta$ consuming the optimal amount of tokens and paying the optimal total transfer.\n\nMinimum-spend mechanism. Consider a minimum-spend mechanism that consists of $p(\\theta)=r(\\theta)(c, \\hat{c})$ and $B(\\theta)=T^{*}(\\theta)$ for all types such that $Q^{*}(\\theta)>0$, where $r(\\theta)=$ $\\frac{T^{*}(\\theta)}{C\\left(Q^{*}(\\theta)\\right)}$. By construction, since prices are proportional to costs, type $w$, when consuming aggregate quality $Q$, would optimally derive value $\\theta(w) Q$. Thus, the buyer's problem of optimal spending can be formulated in terms of $Q$. The restriction of minimum spend when reporting $\\theta^{\\prime}$ can be written as $r\\left(\\theta^{\\prime}\\right) C(Q) \\geq T^{*}\\left(\\theta^{\\prime}\\right)$, which simplifies to $Q \\geq Q^{*}\\left(\\theta^{\\prime}\\right)$. Thus, the optimal payoff of type $\\theta$ when reporting type $\\theta^{\\prime}$ is\n\n$$\nU\\left(\\theta, \\theta^{\\prime}\\right)=\\max _{Q \\geq Q^{*}\\left(\\theta^{\\prime}\\right)} \\theta Q-r\\left(\\theta^{\\prime}\\right) C(Q) .\n$$\n\nThis mechanism implements the allocation of the direct mechanism if each type prefers to report truthfully and choose $Q=Q^{*}(\\theta)$. Since $B(\\theta)$ is increasing, for this to happen, $r(\\theta)$ must be decreasing.\n\nIn fact, $r(\\theta)$ being decreasing is not only necessary but also sufficient for implementability. To see this, note that $Q^{*}(\\theta)$ is continuously increasing and $r(\\theta) \\geq 1$. Thus, the optimal deviation is always weakly below the maximum quality offered in the direct menu:\n\n$$\n\\arg \\max _{Q \\geq Q^{*}\\left(\\theta^{\\prime}\\right)} \\theta Q-r\\left(\\theta^{\\prime}\\right) C(Q) \\leq \\arg \\max _{Q \\geq Q^{*}\\left(\\theta^{\\prime}\\right)} \\bar{\\theta} Q-C(Q)=Q^{*}(\\bar{\\theta}),\n$$\n\nwhere $\\bar{\\theta}$ is the maximum point in $\\operatorname{supp} F$. Moreover, if type $\\theta$ deviates to type $\\theta^{\\prime}$ and consumes quality $Q \\leq Q^{*}(\\bar{\\theta})$, then there exists a type $\\theta^{\\prime \\prime} \\geq \\theta^{\\prime}$ such that $Q=Q^{*}\\left(\\theta^{\\prime \\prime}\\right)$. Since $r$ is decreasing, it is more profitable for type $\\theta$ to deviate to type $\\theta^{\\prime \\prime}$ and consume $Q^{*}\\left(\\theta^{\\prime \\prime}\\right)$. But such a deviation is equivalent to the deviation to $\\theta^{\\prime \\prime}$ in the direct menu and is therefore not profitable.\n\nIt remains to show that decreasing $m$ implies decreasing $r$. Assume that $m$ is decreasing. Since $m$ and $r$ are continuous, it suffices to show that their derivatives are negative at all\ndifferentiable points. At those points, $m^{\\prime}(\\theta) \\leq 0$, which implies $\\bar{\\varphi}(\\theta)-\\bar{\\varphi}^{\\prime}(\\theta) \\theta \\leq 0$, and\n\n$$\nr^{\\prime}(\\theta)=\\frac{T^{* \\prime}(\\theta) C\\left(Q^{*}(\\theta)\\right)-T^{*}(\\theta) C^{\\prime}\\left(Q^{*}(\\theta)\\right) Q^{* \\prime}(\\theta)}{C\\left(Q^{*}(\\theta)\\right)^{2}}=\\frac{Q^{\\prime}(\\theta)}{C\\left(Q^{*}(\\theta)\\right)^{2}}\\left(\\theta C\\left(Q^{*}(\\theta)\\right)-T^{*}(\\theta) \\bar{\\varphi}(\\theta)\\right),\n$$\n\nwhere we used the fact that $T^{* \\prime}(\\theta)=\\theta Q^{* \\prime}(\\theta)$ and $C^{\\prime}\\left(Q^{*}(\\theta)\\right)=\\bar{\\varphi}(\\theta)$. Since $Q$ is increasing, it suffices to show that $\\frac{\\theta}{\\bar{\\varphi}(\\theta)} \\leq \\frac{T^{*}(\\theta)}{C\\left(Q^{*}(\\theta)\\right)}$, i.e., the markup in the two-part tariff implementation is lower than the markup in the minimum-spend implementation. To show this, observe that at the maximal excluded type $\\theta_{0}, \\theta C\\left(Q^{*}(\\theta)\\right)-T^{*}(\\theta) \\bar{\\varphi}(\\theta)=0$, and for all types $\\theta>\\theta_{0}$,","text_sha256":"510ed134ddfb2ddca87893a87daad2b9d2f05dece9bf3cf4b7df933f15e1691f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0028","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Proofs and Derivations","text":"$$\n\\left(\\frac{\\theta C\\left(Q^{*}(\\theta)\\right)-T^{*}(\\theta) \\bar{\\varphi}(\\theta)}{\\theta}\\right)^{\\prime}=\\frac{T(\\theta)}{\\theta^{2}}\\left(\\bar{\\varphi}(\\theta)-\\bar{\\varphi}^{\\prime}(\\theta) \\theta\\right) \\leq 0 .\n$$\n\nTherefore, $\\theta C\\left(Q^{*}(\\theta)\\right)-T^{*}(\\theta) \\bar{\\varphi}(\\theta) \\leq 0$, and $r$ is decreasing. $\\square$\n\nProof of Lemma 2 For general, not necessarily identical, $\\sigma_{l}$, the marginal gain from assigning an infinitesimal task with value $w_{i}$ to model $l$ is, by (18):\n\n$$\n\\delta_{l}\\left(w_{i}\\right)=\\frac{1-\\sigma_{l}}{\\left(\\int_{I_{l}} w_{r}^{1 /\\left(1-\\sigma_{l}\\right)} d r\\right)^{\\sigma_{l}}} Q_{l} w_{i}^{1 /\\left(1-\\sigma_{l}\\right)}\n$$\n\nUnder an optimal task split, if a task with value $w_{i}$ is assigned to a model $l$, then it must be that for all $m, \\delta_{l}\\left(w_{i}\\right) \\geq \\delta_{m}\\left(w_{i}\\right)$. Because $\\delta_{m}\\left(w_{i}\\right) / \\delta_{l}\\left(w_{i}\\right)$ is increasing in $w_{i}$ for all $m>l$, it follows that an optimal task-model allocation exists that forms a monotone partition, with models of higher $l$ (and thus higher $\\sigma_{l}$ ) being allocated the more valuable tasks.\n\nDenoting by $i_{l}$ the delimiting points, with $i_{0}=0$ and $i_{L}=1$, the optimal partition is determined by equalizing the marginal gains, $\\delta_{l}\\left(w_{i_{l}}\\right)=\\delta_{l+1}\\left(w_{i_{l}}\\right)$ for $l \\in[L-1]$ :\n\n$$\n\\frac{\\left(1-\\sigma_{l}\\right) Q_{l} w_{i_{l}}^{1 /\\left(1-\\sigma_{l}\\right)}}{\\left(1-\\sigma_{l+1}\\right) Q_{l+1} w_{i_{l}}^{1 /\\left(1-\\sigma_{l+1}\\right)}}=\\frac{\\left(\\int_{I_{l}} w_{i}^{1 /\\left(1-\\sigma_{l}\\right)} d i\\right)^{\\sigma_{l}}}{\\left(\\int_{I_{l+1}} w_{i}^{1 /\\left(1-\\sigma_{l+1}\\right)} d i\\right)^{\\sigma_{l+1}}} .\n$$\n\nWhen $\\sigma_{l} \\equiv \\sigma$, the optimality condition (42) simplifies to the following expression: ${ }^{16}$\n\n$$\n\\frac{Q_{l}}{Q_{l+1}}=\\frac{\\left(\\int_{I_{l}^{*}} w_{i}^{1 /(1-\\sigma)} d i\\right)^{\\sigma}}{\\left(\\int_{I_{l+1}^{*}} w_{i}^{1 /(1-\\sigma)} d i\\right)^{\\sigma}}\n$$\n\n[^14]Thus,\n\n$$\n\\int_{I_{l}^{*}} w_{i}^{1 /(1-\\sigma)} d i=\\frac{Q_{l}^{1 / \\sigma}}{\\sum_{m} Q_{m}^{1 / \\sigma}} \\int_{0}^{1} w_{i}^{1 /(1-\\sigma)} d i\n$$\n\nor, equivalently,\n\n$$\n\\theta_{l}\\left(I_{l}^{*}\\right)=\\left(\\frac{Q_{l}^{1 / \\sigma}}{\\sum_{m} Q_{m}^{1 / \\sigma}}\\right)^{1-\\sigma} \\theta .\n$$\n\nThe collection $\\left(\\theta_{l}\\left(I_{l}^{*}\\right)\\right)_{l=1}^{L}$ determines an optimal (monotone) partition $\\left(I_{l}^{*}\\right)_{l=1}^{L} .{ }^{17}$ All models with $Q_{l}>0$ are employed at some tasks. The resulting optimal payoff is then (20). $\\square$\n\nProof of Lemma 3 Drop the $l$ index. First, consider the case $\\hat{\\sigma}>0$. Define the unit-cost indices\n\n$$\ne \\triangleq \\min _{x \\geq 0}\\{c \\cdot x: \\Psi(x) \\geq 1\\}, \\quad \\hat{e} \\triangleq \\min _{z \\geq 0}\\{\\hat{c} \\cdot z: \\Phi(z) \\geq 1\\} .\n$$\n\nFor strictly concave, increasing, homogeneous $\\Psi, \\Phi$ these are finite and attained, and the respective minimizers $d, \\hat{d}$ are unique.\n\nBy homogeneity and the definition of $e$, the minimal cost to reach $\\Psi(x)=\\Psi_{0}$ is\n\n$$\n\\min _{x \\geq 0}\\left\\{c \\cdot x: \\Psi(x) \\geq \\Psi_{0}\\right\\}=\\Psi_{0}^{1 / \\sigma} e,\n$$\n\nattained at $x=\\Psi_{0}^{1 / \\sigma} d$. Similarly, $\\min _{z \\geq 0}\\left\\{\\hat{c} \\cdot z: \\Phi(z) \\geq \\Phi_{0}\\right\\}=\\Phi_{0}^{1 / \\hat{\\sigma}} \\hat{e}$, attained at $z=\\Phi_{0}^{1 / \\hat{\\sigma}} \\hat{d}$.\nThe cost minimization reduces to\n\n$$\n\\min _{\\Psi_{0}, \\Phi_{0} \\geq 0} \\Psi_{0}^{1 / \\sigma} e+\\Phi_{0}^{1 / \\hat{\\sigma}} \\hat{e} \\quad \\text { s.t. } \\quad \\Psi_{0} \\Phi_{0}=Q\n$$\n\nStraightforward calculation gives:\n\n$$\n\\Psi_{0}=Q^{\\frac{\\sigma}{\\sigma+\\tilde{\\sigma}}}\\left(\\frac{\\hat{e} \\sigma}{e \\hat{\\sigma}}\\right)^{\\frac{\\sigma \\hat{\\sigma}}{\\sigma+\\tilde{\\sigma}}}, \\quad \\Phi_{0}=Q^{\\frac{\\hat{\\sigma}}{\\sigma+\\hat{\\sigma}}}\\left(\\frac{\\hat{e} \\sigma}{e \\hat{\\sigma}}\\right)^{-\\frac{\\sigma \\hat{\\sigma}}{\\sigma+\\hat{\\sigma}}},\n$$\n\nwhich corresponds to the optimal token allocation:\n\n$$\nx^{*}(Q)=Q^{\\frac{1}{\\sigma+\\hat{\\sigma}}}\\left(\\frac{\\hat{e} \\sigma}{e \\hat{\\sigma}}\\right)^{\\frac{\\hat{\\sigma}}{\\sigma+\\hat{\\sigma}}} d, \\quad z^{*}(Q)=Q^{\\frac{1}{\\sigma+\\hat{\\sigma}}}\\left(\\frac{\\hat{e} \\sigma}{e \\hat{\\sigma}}\\right)^{-\\frac{\\sigma}{\\sigma+\\hat{\\sigma}}} \\hat{d}\n$$\n\nThe resulting cost function is:\n\n$$\nC(Q)=e \\Psi_{0}^{1 / \\sigma}+\\hat{e} \\Phi_{0}^{1 / \\hat{\\sigma}}=e^{\\frac{\\sigma}{\\sigma+\\hat{\\sigma}}} \\hat{e}^{\\frac{\\hat{\\sigma}}{\\sigma+\\hat{\\sigma}}}\\left[\\left(\\frac{\\sigma}{\\hat{\\sigma}}\\right)^{\\frac{\\hat{\\sigma}}{\\sigma+\\hat{\\sigma}}}+\\left(\\frac{\\hat{\\sigma}}{\\sigma}\\right)^{\\frac{\\sigma}{\\sigma+\\hat{\\sigma}}}\\right] Q^{\\frac{1}{\\sigma+\\hat{\\sigma}}}=C_{0} Q^{\\frac{1}{\\sigma+\\hat{\\sigma}}} .\n$$\n\n[^15]The case $\\hat{\\sigma}=0$ is analogous and simpler because in that case $\\Phi(z) \\equiv \\Phi_{0}>0$ and fine-tuning can be ignored. The resulting cost function is\n\n$$\nC(Q)=e\\left(\\frac{Q}{\\Phi_{0}}\\right)^{\\frac{1}{\\sigma}}=C_{0} Q^{\\frac{1}{\\sigma}} .\n$$\n\nThis completes the proof. $\\square$\n\nProof of Proposition 5 By Lemma 3, each model's surplus $\\theta Q-\\kappa_{l} Q^{1 /\\left(\\sigma+\\hat{\\sigma}_{l}\\right)}$ is concave in $Q$ and can be maximized in closed form. The efficient allocation selects the model that yields the highest surplus, yielding (22)-(23). Single-model usage follows because the cost function (21) is concave in $Q^{1 / \\sigma}$. $\\square$\n\nProof of Proposition 6 The result follows by pointwise maximization of virtual surplus with $\\varphi(\\theta)$ in place of $\\theta$ in Proposition 5. Monotonicity of $\\varphi$ ensures that the pointwise solution $Q^{\\mathrm{m}}(\\theta)=Q^{*}(\\varphi(\\theta))$ is increasing and therefore implementable. $\\square$\n\nProof of Lemma 4 When $\\hat{\\sigma}_{F}=0$, (29) simplifies into:\n\n$$\nq_{L}^{\\left(\\sigma+\\hat{\\sigma}_{L}\\right) / \\sigma}+q_{F}=\\left(\\frac{\\theta \\sigma}{c_{F}}\\right)^{1 /(1-\\sigma)}\n$$\n\nThe outside option corresponding to $q_{L}=0$ is\n\n$$\n\\psi(\\theta)=\\theta^{\\frac{1}{1-\\sigma}}(1-\\sigma)\\left(\\frac{\\sigma}{c_{F}}\\right)^{\\sigma /(1-\\sigma)}\n$$\n\nThe formulation (31) follows. $\\square$\n\nProof of Proposition 7 Denote the expression in the brackets of (32) by $\\Pi(\\theta, q)$ and observe that","text_sha256":"0c76f4ff2de2aff96ad45f26aa70b1d2b9eab8f60b890d7394dafbb51bab9342"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0029","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Proofs and Derivations","text":"$$\n\\Pi(\\theta, q(\\theta))= \\begin{cases}c_{F} q(\\theta)^{\\left(\\sigma+\\hat{\\sigma}_{L}\\right) / \\sigma}-c_{L} q(\\theta), & \\text { if } q(\\theta)<\\hat{q}(\\theta), \\\\ \\varphi(\\theta) q(\\theta)^{\\sigma+\\hat{\\sigma}_{L}}-c_{L} q(\\theta)-\\psi(\\theta)+\\frac{1-F(\\theta)}{f(\\theta)} \\psi^{\\prime}(\\theta), & \\text { if } q(\\theta)>\\hat{q}(\\theta) .\\end{cases}\n$$\n\nConsider the pointwise maximization of $\\Pi(\\theta, q(\\theta))$. In the region $q(\\theta) \\in[0, \\hat{q}(\\theta)], \\Pi(\\theta, q(\\theta))$ is convex in $q(\\theta)$ and therefore attains its maximum at a corner, $q(\\theta)=0$ or $q(\\theta)=\\hat{q}(\\theta)$.\n\nDirect calculation shows that $\\Pi(\\theta, 0)>\\Pi(\\theta, \\hat{q}(\\theta))$ if and only if $\\theta<\\theta_{1}$, where\n\n$$\n\\theta_{1} \\triangleq \\frac{c_{F}}{\\sigma}\\left(\\frac{c_{L}}{c_{F}}\\right)^{(1-\\sigma)\\left(\\sigma+\\hat{\\sigma}_{L}\\right) / \\hat{\\sigma}_{L}}\n$$\n\nIn the region $q(\\theta) \\geq \\hat{q}(\\theta)$, if $\\varphi(\\theta) \\leq 0$, then optimally $q(\\theta)=\\hat{q}(\\theta)$. If $\\varphi(\\theta)>0$, then $\\Pi(\\theta, q(\\theta))$ is concave in $q(\\theta)$ and therefore the maximum is either at the corner, $q(\\theta)=\\hat{q}(\\theta)$, or in the interior, in which case it equals the optimal monopoly quantity\n\n$$\nq^{\\mathrm{m}}(\\theta)=\\left(\\frac{\\left(\\sigma+\\hat{\\sigma}_{L}\\right) \\varphi(\\theta)}{c_{L}}\\right)^{1 /\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right)}\n$$\n\nThus, the solution over the region $q(\\theta) \\geq \\hat{q}(\\theta)$ is interior if and only if $\\Pi_{q}(\\theta, \\hat{q}(\\theta))>0$, or\n\n$$\n\\varphi(\\theta)>\\frac{c_{L}}{\\sigma+\\hat{\\sigma}_{L}}\\left(\\frac{\\theta \\sigma}{c_{F}}\\right)^{\\frac{\\sigma\\left(1-\\sigma-\\sigma_{L}\\right)}{(1-\\sigma)\\left(\\sigma+\\tilde{\\sigma}_{L}\\right)}} .\n$$\n\nCondition (48) can hold on disjoint intervals of $\\theta$ even if $\\varphi(\\theta)$ is increasing, because the right-hand side is increasing in $\\theta$. However, the power of $\\theta$ on the right-hand side is strictly less than 1 . Thus, if $F$ satisfies monotone hazard rate, then $\\varphi^{\\prime}(\\theta) \\geq 1$, and condition (48) holds, if at all, for all $\\theta>\\theta_{2}$ where $\\theta_{2}$ solves:\n\n$$\n\\varphi\\left(\\theta_{2}\\right)=\\frac{c_{L}}{\\sigma+\\hat{\\sigma}_{L}}\\left(\\frac{\\theta_{2} \\sigma}{c_{F}}\\right)^{\\frac{\\sigma\\left(1-\\sigma-\\sigma_{L}\\right)}{(1-\\sigma)\\left(\\sigma+\\sigma_{L}\\right)}}\n$$\n\nIf (49) doesn't admit a solution, then the solution is never interior and thus for all $\\theta<\\theta_{1}$, $q(\\theta)=0$ and for all $\\theta>\\theta_{1}, q(\\theta)=\\hat{q}(\\theta)$. If (49) admits a solution at $\\theta_{2}>\\theta_{1}$, then: for all $\\theta<\\theta_{1}, q(\\theta)=0$; for all $\\theta \\in\\left(\\theta_{1}, \\theta_{2}\\right), q(\\theta)=\\hat{q}(\\theta)$; for all $\\theta \\geq \\theta_{2}, q(\\theta)=q^{\\mathrm{m}}(\\theta)$. Finally, if (49) admits a solution at $\\theta_{2}<\\theta_{1}$, then $\\theta_{3} \\in\\left[\\theta_{2}, \\theta_{1}\\right]$ exists such that for all $\\theta<\\theta_{3}, q(\\theta)=0$ and for all $\\theta \\geq \\theta_{3}, q(\\theta)=q^{\\mathrm{m}}(\\theta)$. (The threshold $\\theta_{3}>\\theta_{2}$ is determined by the condition $\\Pi\\left(\\theta_{3}, q^{\\mathrm{m}}\\left(\\theta_{3}\\right)\\right)=0$.)\n\nSince all these (relaxed) solutions are implementable, the result follows. $\\square$","text_sha256":"60cfcd69a532adccf99b0c1ed661b4577e7a8abbc15fa1f6509bb14be5684e3f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0030","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Leader-Fringe Competition with Uniform Types","text":"## Leader-Fringe Competition with Uniform Types\n\nCompetitive Mechanism- With $\\hat{\\sigma}_{F}=0$ the buyer's first-order condition implies\n\n$$\nq_{L}^{\\left(\\sigma+\\hat{\\sigma}_{L}\\right) / \\sigma}+q_{F}=\\left(\\frac{\\theta \\sigma}{c_{F}}\\right)^{1 /(1-\\sigma)}\n$$\n\nOn the fringe-indifference boundary $\\left(q_{F}=0\\right)$ the leader's quantity solves\n\n$$\n\\hat{q}(\\theta)=\\left(\\frac{\\theta \\sigma}{c_{F}}\\right)^{\\frac{\\sigma}{(1-\\sigma)\\left(\\sigma+\\hat{\\sigma}_{L}\\right)}} .\n$$\n\nWhen the buyer single-homes on the leader and the fringe constraint is slack,\n\n$$\nq^{\\mathrm{m}}(\\theta)=\\left(\\frac{\\left(\\sigma+\\hat{\\sigma}_{L}\\right) \\varphi(\\theta)}{c_{L}}\\right)^{\\frac{1}{1-\\sigma-\\hat{\\sigma}_{L}}}=\\left(\\frac{\\left(\\sigma+\\hat{\\sigma}_{L}\\right)(2 \\theta-1)}{c_{L}}\\right)^{\\frac{1}{1-\\sigma-\\hat{\\sigma}_{L}}} .\n$$\n\nWe want the optimal menu to be characterized by $0<\\theta_{1}<\\theta_{2}<1$ such that\n\n$$\nq^{L F}(\\theta)=\\left\\{\\begin{array}{ll}\n0, & \\theta \\leq \\theta_{1}, \\\\\n\\hat{q}(\\theta), & \\theta_{1}<\\theta<\\theta_{2}, \\\\\nq^{\\mathrm{m}}(\\theta), & \\theta \\geq \\theta_{2},\n\\end{array} \\quad q_{F}^{L F}(\\theta)= \\begin{cases}\\left(\\frac{\\theta \\sigma}{c_{F}}\\right)^{\\frac{1}{1-\\sigma}}, & \\theta \\leq \\theta_{1}, \\\\\n0, & \\theta>\\theta_{1} .\\end{cases}\\right.\n$$\n\nThe entry cutoff is\n\n$$\n\\theta_{1}=\\frac{c_{F}}{\\sigma}\\left(\\frac{c_{L}}{c_{F}}\\right)^{\\frac{(1-\\sigma)\\left(\\sigma+\\hat{\\sigma}_{L}\\right)}{\\hat{\\sigma}_{L}}} .\n$$\n\nThe boundary-interior switch $\\theta_{2}$ is the unique solution to\n\n$$\n\\varphi(\\theta)=\\frac{c_{L}}{\\sigma+\\hat{\\sigma}_{L}}\\left(\\frac{\\theta \\sigma}{c_{F}}\\right)^{\\frac{\\sigma\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right)}{(1-\\sigma)\\left(\\sigma+\\hat{\\sigma}_{L}\\right)}},\n$$\n\nwhich exists in (1/2, 1) if and only if $\\frac{c_{L}}{\\sigma+\\hat{\\sigma}_{L}}\\left(\\frac{\\sigma}{c_{F}}\\right)^{\\frac{\\sigma\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right)}{(1-\\sigma)\\left(\\sigma+\\hat{\\sigma}_{L}\\right)}}<1$. Furthermore, $\\theta_{2}>\\theta_{1}$ if and only if $\\theta_{1}<\\left(\\sigma+\\hat{\\sigma}_{L}\\right) /\\left(\\sigma+2 \\hat{\\sigma}_{L}\\right)$. Therefore, for $0<\\theta_{1}<\\theta_{2}<1$ to take place, the necessary and sufficient conditions are:\n\n$$\n\\frac{c_{L}}{\\sigma+\\hat{\\sigma}_{L}}\\left(\\frac{\\sigma}{c_{F}}\\right)^{\\frac{\\sigma\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right)}{(1-\\sigma)\\left(\\sigma+\\hat{\\sigma}_{L}\\right)}}<1, \\quad \\frac{c_{F}}{\\sigma}\\left(\\frac{c_{L}}{c_{F}}\\right)^{\\frac{(1-\\sigma)\\left(\\sigma+\\hat{\\sigma}_{L}\\right)}{\\hat{\\sigma}_{L}}}<\\frac{\\sigma+\\hat{\\sigma}_{L}}{\\sigma+2 \\hat{\\sigma}_{L}} .\n$$\n\nEfficient Allocation - Efficiency features single-homing with a model-switch cutoff $\\hat{\\theta}$ :\n\n$$\n\\left(q_{L}^{*}(\\theta), q_{F}^{*}(\\theta)\\right)= \\begin{cases}\\left(0,\\left(\\frac{\\theta \\sigma}{c_{F}}\\right)^{\\frac{1}{1-\\sigma}}\\right), & \\theta<\\hat{\\theta} \\\\ \\left(\\left(\\frac{\\theta\\left(\\sigma+\\hat{\\sigma}_{L}\\right)}{c_{L}}\\right)^{\\frac{1}{1-\\sigma-\\hat{\\sigma}_{L}}}, 0\\right), & \\theta \\geq \\hat{\\theta}\\end{cases}\n$$\n\nwith\n\n$$\n\\hat{\\theta}=\\left(\\frac{1-\\sigma}{1-\\sigma-\\hat{\\sigma}_{L}}\\right)^{\\frac{\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right)(1-\\sigma)}{\\hat{\\sigma}_{L}}}\\left(\\frac{\\sigma}{c_{F}}\\right)^{\\frac{\\sigma\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right)}{\\hat{\\sigma}_{L}}}\\left(\\frac{c_{L}}{\\sigma+\\hat{\\sigma}_{L}}\\right)^{\\frac{\\left(\\sigma+\\hat{\\sigma}_{L}\\right)(1-\\sigma)}{\\hat{\\sigma}_{L}}} .\n$$\n\nMulti-Model Monopoly- Virtual type is $\\varphi(\\theta)=2 \\theta-1$. Types $\\theta<1 / 2$ are excluded. Among\nserved types the monopolist assigns a single model, switching at $\\theta^{\\mathrm{m}}=(1+\\hat{\\theta}) / 2$. Optimal quantities are\n\n$$\n\\left(q_{L}^{\\mathrm{m}}(\\theta), q_{F}^{\\mathrm{m}}(\\theta)\\right)= \\begin{cases}(0,0), & \\theta<\\frac{1}{2}, \\\\ \\left(0,\\left(\\frac{\\sigma \\varphi(\\theta)}{c_{F}}\\right)^{\\frac{1}{1-\\sigma}}\\right), & \\frac{1}{2} \\leq \\theta<\\theta^{\\mathrm{m}}, \\\\ \\left(\\left(\\frac{\\left(\\sigma+\\hat{\\sigma}_{L}\\right) \\varphi(\\theta)}{c_{L}}\\right)^{\\frac{1}{1-\\sigma-\\hat{\\sigma}_{L}}}, 0\\right), & \\theta \\geq \\theta^{\\mathrm{m}} .\\end{cases}\n$$","text_sha256":"921a7fd405703fec8b30ea39c65e675752befb24d58c3944286fa480396722ec"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0031","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B Contractible Tasks","text":"## B Contractible Tasks\n\nThroughout the manuscript, we focus on the more realistic case in which the LLM provider cannot contract on the allocation of tokens across the buyer's tasks. However, the fullcontracting setting provides a natural benchmark and we study it in this section. Thus, the seller designs a direct menu that specifies a token allocation across different tasks:\n\n$$\n\\left\\{\\left(\\left(x_{i 1}(w), \\ldots, x_{i J}(w)\\right)_{i \\in[0,1]}, z_{1}(w), \\ldots, z_{K}(w), t(w)\\right)\\right\\}_{w} .\n$$\n\nThe allocation being contractible means the buyer has no freedom to reallocate tokens across tasks. This naturally increases the scope for screening. This also makes the problem intractable in full generality, because the reduction to a one-dimensional aggregate type and quality as in Proposition 2 does not apply. Nevertheless, in this section we obtain complete characterizations in two special settings: the case of separable type distributions (Section B.1) and the case of two types (Section B.2). In the former case, the optimal solution can be obtained via a menu of token budgets. In the latter case, the optimal solution can be obtained by identifying the structure of binding incentive constraints.","text_sha256":"4d81ea807c765293d55e1ec7d4d1d0f0a1d404831f0fde8d418e9c0c0ddb768b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0032","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B. 1 Separable Type Distribution and Cost-Based Pricing","text":"## B. 1 Separable Type Distribution and Cost-Based Pricing\n\nIn this section, we provide a sufficient condition on type distribution under which contracting on tasks is not profitable. In fact, under that condition an even less contractually restrictive class of mechanisms, cost-based tariffs, is optimal:\n\nDefinition 4 (Cost-Based Tariff). A cost-based tariff is a menu of monetary budgets and transfers $\\left\\{\\left(B_{n}, T_{n}\\right)\\right\\}_{n}$ such that, upon purchasing item $n$, the buyer pays $T_{n}$ for access to budget $B_{n}$, which he can freely spend on tokens priced at their marginal costs.\n\nWe build on the analysis of Armstrong (1996) and introduce the conditions that are jointly sufficient for the optimality of cost-based tariffs: demand separability and type separability. We show that the former is always satisfied in our setting, whereas the latter is equivalent to the aggregate type not being informative about the relative task weights.\n\nTo this end, consider a problem of optimal token allocation by type $w$ given a monetary budget $B$ and token prices equal to marginal costs:\n\n$$\nV(w, B)=\\max _{\\left\\{\\left(x_{i}\\right)_{i \\in[0,1], z \\geq 0\\}}\\right.} \\int_{0}^{1} w_{i} g\\left(x_{i}, z\\right) d i, \\quad \\text { s.t. } \\int_{0}^{1} \\sum_{j=1}^{J} c_{j} x_{i j} d i+\\sum_{k=1}^{K} \\hat{c}_{k} z_{k}=B,\n$$\n\nDefinition 5 (Demand Separability). Demand is separable if there exist functions $V_{1}(w)$ and $V_{2}(B)$ such that\n\n$$\nV(w, B)=V_{1}(w) V_{2}(B) .\n$$\n\nIf demand is separable, then faced with a cost-based tariff, all types $w$ with the same $V_{1}(w)$ purchase the same item.\n\nDemand separability always holds in our setting. Indeed, by Proposition 2, if type $w$ purchases a total amount of inference tokens $X=\\left(X_{1}, \\ldots, X_{J}\\right)$ and fine-tuning tokens $Z=$ $\\left(Z_{1}, \\ldots, Z_{K}\\right)$, then his optimal payoff is $\\theta(w) g(X, Z)$. Therefore, (51) holds with $V_{1}(w)=$ $\\theta(w)$ and\n\n$$\nV_{2}(B)=\\max _{X \\geq 0, Z \\geq 0} g(X, Z), \\quad \\text { s.t. } \\sum_{j=1}^{J} c_{j} X_{j}+\\sum_{k=1}^{K} \\hat{c}_{k} Z_{k}=B .\n$$\n\nDefinition 6 (Type Separability). Type distribution is separable if $f_{1}, f_{2}$ exist such that $f(w)=f_{1}(\\theta(w)) f_{2}(w)$ for all $w$ and $f_{2}$ is homogeneous of degree zero.\n\nA separable type distribution means that knowing $\\theta(w)$ provides no information about which ray from the origin $w$ lies on, that is, about the relative weights the buyer assigns to different tasks. Equivalently, denoting by $\\|\\cdot\\|_{p}$ a standard $L_{p}$ norm, $\\theta(w)=\\|w\\|_{1 /(1-\\sigma)}$ and the type separability is equivalent to $\\|w\\|_{1 /(1-\\sigma)}$ and $w /\\|w\\|_{1 /(1-\\sigma)}$ to be independent. Thus, any type distribution generated by a draw of a \"total size\" $\\|w\\|_{1 /(1-\\sigma)}$ according to an arbitrary distribution together with an independent draw of \"relative weights\" $w /\\|w\\|_{1 /(1-\\sigma)}$ according to any other distribution, is separable.\n\nProposition 8 (Cost-Based Optimality). If the type distribution is separable, then a cost-based tariff is optimal.\n\nProof. Since in our setting demand is always separable, the result follows from the arguments presented by Armstrong (1996). Specifically, we can relax the IC constraints between types located on different rays from the origin. Then, applying the Envelope Theorem ray by ray, we can find a solution to the relaxed problem as follows:\n\n$$\n\\left(x^{*}, z^{*}\\right) \\in \\arg \\max _{\\left\\{\\left(x_{i}\\right)_{i \\in[0,1],} z \\geq 0\\right\\}}\\left(1-\\frac{\\int_{1}^{\\infty} r^{n-1} f(r w) d r}{f(w)}\\right) u(w, x, z)-c(x, z) .\n$$\n\nGiven the demand separability, this condition can be rewritten as:\n\n$$\n\\left(x^{*}, z^{*}\\right) \\in \\arg \\max _{\\left\\{\\left(x_{i}\\right)_{i \\in[0,1]}, z \\geq 0\\right\\}} u(w, x, z), \\quad \\text { s.t. } c(x, z) \\leq B^{*}(w),\n$$\n\nand\n\n$$\nB^{*}(w) \\in \\arg \\max _{B \\geq 0}\\left(1-\\frac{\\int_{1}^{\\infty} r^{n-1} f(r w) d r}{f(w)}\\right) \\theta(w) V_{2}(B)-B .\n$$\n\nAt the same time, the optimal cost-based tariff implements an allocation\n\n$$\n\\left(x^{*}, z^{*}\\right) \\in \\arg \\max _{\\left\\{\\left(x_{i}\\right)_{i \\in[0,1]}, z \\geq 0\\right\\}} u(w, x, z), \\quad \\text { s.t. } c(x, z) \\leq B^{*}(w),\n$$\n\nwhere\n\n$$\nB^{*}(w) \\in \\arg \\max _{B}\\left(\\theta(w)-\\frac{1-F(\\theta(w))}{f(\\theta(w))}\\right) V_{2}(B)-B .\n$$\n\nIf the type distribution is separable, then\n\n$$\n\\left(1-\\frac{\\int_{1}^{\\infty} r^{n-1} f(r w) d r}{f(w)}\\right) \\theta(w)=\\theta(w)-\\frac{1-F(\\theta(w))}{f(\\theta(w))},\n$$\n\nand the cost-based tariff implements the solution to the relaxed problem. Therefore, it is (indirectly) optimal. $\\square$","text_sha256":"ba7d99e745658521e37ae7b96c507249dec22738cd072132058705e025a28a8b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0033","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B. 2 Binary Types","text":"## B. 2 Binary Types\n\nAlternatively, let there be only two types, $w_{1}$ and $w_{2}$, which occur with prior strictly positive probabilities $f_{1}$ and $f_{2}$, respectively. In this case, the optimal mechanism depends on what happens if the seller attempts to extract the first-best level of surplus, that is, if she offers a menu containing the efficient amounts of tokens for each type with prices equal to their respective added values. If this menu is incentive compatible, then it is clearly optimal, and we call the type with the higher payment \"high\" and associate the label $H$ with it. If this menu is not incentive compatible, then we call \"high\" the type whose incentive constraint is violated. We call the other type \"low\" and associate the label $L$ with it.\n\nTo design an optimal menu, it is important to determine which type out of $w_{1}$ and $w_{2}$ is high and which one is low. If $w_{2}$ dominates $w_{1}$ for every task, then it is clearly high; alternatively, the types are in some sense horizontally differentiated and the distinction is less clear. However, it turns out that the ranking can be derived from the aggregate types, and the optimal menu admits a simple characterization.\n\nProposition 9 (Binary Types). The high type is the one with the higher aggregate type, i.e., $\\theta\\left(w_{H}\\right) \\geq \\theta\\left(w_{L}\\right)$. In the optimal menu, the token allocation of $w_{H}$ is always efficient. If\n\n$$\n\\int_{0}^{1}\\left(w_{H i}-w_{L i}\\right) w_{L i}^{\\frac{\\sigma}{1-\\sigma}} d i \\leq 0\n$$\n\nthen the token allocation of $w_{L}$ is also efficient and the seller extracts full surplus. Otherwise, the token allocation of $w_{L}$ is efficient with respect to a virtual type $w_{L}-\\left(w_{H}-w_{L}\\right) f_{H} / f_{L}$.\n\nProof. With a small abuse of notation relative to previous sections, denote by $q=\\left(q_{i}\\right)_{i \\in[0,1]}$ the profile of qualities delivered to each task, $q_{i} \\triangleq g\\left(x_{i}, z\\right)$, and denote by $C(q)$ the minimal total cost of generating a given profile $q$ :\n\n$$\nC(q) \\triangleq \\min _{x_{i}, z \\geq 0} \\int_{0}^{1} \\sum_{j=1}^{J} c_{j} x_{i j} d i+\\sum_{k=1}^{K} \\hat{c}_{k} z_{k}, \\quad \\text { s.t. } g\\left(x_{i}, z\\right)=q_{i}, \\forall i \\in[0,1] .\n$$\n\nBecause the set of feasible profiles $q$ is convex, it follows from the analysis of Haghpanah and Siegel (2024) on general screening problems with two buyer types that in our setting either (i) the seller extracts full surplus; or (ii) the incentive constraint of type $w_{H}$ and the individual rationality constraint of type $w_{L}$ bind.\n\nIt follows that if the seller cannot extract full surplus, then the seller's problem can be written as\n\n$$\n\\begin{aligned}\n\\max _{q_{L}, q_{H}, t_{L}, t_{H}} & f_{L}\\left(t_{L}-C\\left(q_{L}\\right)\\right)+f_{H}\\left(t_{H}-C\\left(q_{H}\\right)\\right) \\\\\n\\text { s.t. } & \\int_{0}^{1} w_{H i} q_{H i} d i-t_{H}=\\int_{0}^{1} w_{H i} q_{L i} d i-t_{L}, \\int_{0}^{1} w_{L i} q_{L i} d i-t_{L}=0\n\\end{aligned}\n$$\n\nSolving for transfers from the constraints, the problem can be restated as:\n\n$$\n\\max _{q_{L}, q_{H}} f_{L}\\left(\\int_{0}^{1}\\left(w_{L i}-\\frac{f_{H}}{f_{L}}\\left(w_{H i}-w_{L i}\\right)\\right) q_{L i} d i-C\\left(q_{L}\\right)\\right)+f_{H}\\left(\\int_{0}^{1} w_{H i} q_{H i} d i-C\\left(q_{H}\\right)\\right),\n$$\n\nwhich is solved by the allocation efficient relative to the virtual types.\nIt is left to determine which type is high and provide conditions for full surplus extraction. Consider a mechanism that attempts full surplus extraction. By the efficiency analysis behind Proposition 1, under this mechanism type $w$, when reporting type $\\tilde{w}$, obtains added value\n\n$$\nu(w, \\tilde{w})=\\int_{0}^{1} w_{i} \\Psi\\left(x_{i}(\\tilde{w})\\right) \\Phi(z(\\tilde{w})) d i=\\int_{0}^{1} w_{i} \\tilde{w}_{i}^{\\frac{\\sigma}{1-\\sigma}} \\Psi(d) \\Phi(z(\\tilde{w}))^{\\frac{1}{1-\\sigma}} d i\n$$\n\nBecause under truth-telling each type obtains zero rents, the corresponding payment is\n$t(w)=u(w, w)$, and the incentive constraint $u(w, w)-t(w) \\geq u(w, \\tilde{w})-t(\\tilde{w})$ is violated if and only if:\n\n$$\n\\Psi(d) \\Phi(z(\\tilde{w}))^{\\frac{1}{1-\\sigma}} \\int_{0}^{1}\\left(w_{i}-\\tilde{w}_{i}\\right) \\tilde{w}_{i}^{\\frac{\\sigma}{1-\\sigma}} d i>0\n$$\n\nBy Jensen's inequality and the concavity of the logarithm:\n\n$$\nw_{i} \\tilde{w}_{i}^{\\frac{\\sigma}{1-\\sigma}}=w_{i}^{\\frac{1-\\sigma}{1-\\sigma}} \\tilde{w}_{i}^{\\frac{\\sigma}{1-\\sigma}} \\leq(1-\\sigma) w_{i}^{\\frac{1}{1-\\sigma}}+\\sigma \\tilde{w}_{i}^{\\frac{1}{1-\\sigma}} .\n$$\n\nTherefore, inequality (53) implies\n\n$$\n\\int_{0}^{1}\\left(w_{i}^{\\frac{1}{1-\\sigma}}-\\tilde{w}_{i}^{\\frac{1}{1-\\sigma}}\\right) d i>0\n$$\n\nand the incentive violation is possible only if $\\theta(w)>\\theta(\\tilde{w})$, i.e., from high to low aggregate type. Therefore, if full surplus extraction is not incentive compatible, then the high type is the one with the higher aggregate type. If full surplus extraction is incentive compatible, then the high type is the one with the higher aggregate type directly by the analysis behind Proposition 1. The result follows. $\\square$\n\nNote that even though the virtual types in Proposition 9 follow the standard formula of the single-dimensional case, each type there is infinite-dimensional.\n\nOne might wonder whether the correspondence between the incentive order and aggregate types extends beyond the binary-type case. This is not the case. In Section B.3, we study a special case of our model in which the buyer's type can be effectively parameterized by two variables: the number of ex ante homogeneous tasks to which the buyer attaches positive value, and the value he attaches to each of those tasks. We show that the incentive constraints that bind in the optimal mechanism do not admit a one-dimensional structure. Instead, they form an infinite collection of one-dimensional segments. In that setting the optimal mechanisms also admit a natural implementation via a menu of two-part tariffs. However, in contrast to the indirect implementations described in Section 4.2, each item in the menu must be accompanied by task caps, that is, restrictions on the number of tasks the buyer can process.","text_sha256":"c4338dbc2ecb0a9a62acdc5671431a0c8a184ccaa8fd8e13671c348da6556ee9"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0034","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B. 3 Value-Scale Heterogeneity","text":"## B. 3 Value-Scale Heterogeneity\n\nIn this section, we characterize an optimal menu with contractible token allocations across tasks in the case of value-scale heterogeneity. In this case, each multidimensional type $w$ is\ncharacterized (with a small abuse of notation) by two parameters $(w, s)$ such that\n\n$$\nw_{i}=\\left\\{\\begin{array}{l}\nw, \\text { if } i \\leq s \\\\\n0, \\text { if } i>s\n\\end{array}\\right.\n$$\n\nin which $w$ and $s$ are independently distributed according to CDFs $F_{w}$ and $F_{s}$, with $F_{w}$ featuring increasing virtual values. We will argue that in this case, under additional assumptions, the binding incentive constraints are those within each scale, and the seller is able to generate the same profit as if the scale were observable.\n\nFor this section, we drop the product structure and the homogeneity requirements of (1) and instead require that $g\\left(x_{i}, z\\right)$ is (i) positive and continuous on $\\mathbb{R}_{+}^{J+K}$, (ii) strictly monotone, twice continuously differentiable, strictly concave with negative definite Hessian, and with all cross-partial derivatives strictly positive on $\\mathbb{R}_{++}^{J+K}$, (iii) $g(0)=0$, and (iv) Inada at zero, i.e., $\\lim _{y_{m} \\downarrow 0} g_{y_{m}}\\left(y_{m}, y_{-m}\\right)=+\\infty$ for all $y_{-m} \\in \\mathbb{R}_{+}^{J+K-1}$ such that $g\\left(0, y_{-m}\\right)>0 .{ }^{18}$\n\nConsider the problem in which the scale is commonly known to be $s>0$. The buyer's payoff from any given item on the menu (50) is\n\n$$\nw \\int_{i=0}^{s} g\\left(x_{i}, z\\right) d i-t\n$$\n\nFor any reported $w$ the seller should optimize token allocation to deliver a promised level of (total) quality $q$,\n\n$$\nw q-t,\n$$\n\nwith the minimal cost function $C(q, s)$ of delivering a given quality being:\n\n$$\nC(q, s)=\\min _{\\left(x_{i}\\right)_{i \\in[0,1], z \\geq 0}} \\int_{i=0}^{s} \\sum_{j=1}^{J} c_{j} x_{i j} d i+\\sum_{k=1}^{K} \\hat{c}_{k} z_{k}, \\quad \\text { s.t. } \\int_{i=0}^{s} g\\left(x_{i}, z\\right) d i=q \\text {. }\n$$\n\nSince $g$ is strictly concave, the solution to this problem is achieved by allocating the inference tokens uniformly across the $s$ tasks. The problem can be equivalently stated as:\n\n$$\nC(q, s)=\\min _{x, z \\geq 0} s \\sum_{j=1}^{J} c_{j} x_{j}+\\sum_{k=1}^{K} \\hat{c}_{k} z_{k}, \\quad \\text { s.t. } s g(x, z)=q .\n$$\n\nThe resulting cost function $C(q, s)$ satisfies, over the domain of admissible $(q, s)$, the following properties:\n\n[^16]","text_sha256":"7e05b8802c4689932b1e1bfc8dbd3171e5c67dad498f954ac06debe74dcd684a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0035","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Lemma 5 (Cost Function).","text":"## Lemma 5 (Cost Function).\n\n1. $C(q, s)$ is strictly increasing and strictly convex in $q$ with $C_{q}(0, s)=0$.\n2. $C(q, s)$ is strictly decreasing in $s$ for $q>0$.\n3. $C(q, s)$ is submodular, i.e., $C_{q}(q, s)$ is decreasing in $s$ for all $q$.\n\nProof. The first two statements follow directly from our assumptions on $g$. To establish the third property, posit the Lagrangian for the cost minimization problem:\n\n$$\nL=s \\sum_{j=1}^{J} c_{j} x_{j}+\\sum_{k=1}^{K} \\hat{c}_{k} z_{k}+\\lambda(q-s g(x, z)) .\n$$\n\nBy the Envelope Theorem, $C_{q}(q, s)=\\lambda(q, s)$. Thus, it suffices to show that $\\lambda(q, s)$ is decreasing in $s$ for a fixed $q$.\n\nTo this end, for any $q>0$, by the Inada condition, the solution must be interior, $(x, z) \\gg$ 0 . The constraint binds, so $\\lambda>0$. Denoting $(x, z)$ by $y$, the first-order conditions are:\n\n$$\ng_{x_{j}}(y)=c_{j} / \\lambda, \\quad g_{z_{k}}(y)=\\hat{c}_{k} /(\\lambda s) .\n$$\n\nDefine the Hessian function $H(y) \\triangleq \\nabla^{2} g(y)$. Denote by $A, B \\in \\mathbb{R}^{J+K}$ vectors such that $A_{r}=c_{r} / \\lambda^{2}$ if $r \\leq J$ and $=\\hat{c}_{r-J} /\\left(\\lambda^{2} s\\right)$ if $r>J, B_{r}=0$ if $r \\leq J$ and $=\\hat{c}_{r-J} /\\left(\\lambda s^{2}\\right)$ if $r>J$. Then, differentiating (57) with respect to $s$, we obtain:\n\n$$\nH \\frac{d y}{d s}=-A \\frac{d \\lambda}{d s}-B .\n$$\n\nAt the same time, differentiating the constraint $s g(y(s))=q$ with respect to $s$, we obtain:\n\n$$\ng(y)+s \\nabla g(y)^{\\top} \\frac{d y}{d s}=0 .\n$$\n\nSolving for $d y / d s$ in (58) and substituting it into (59), we obtain, omitting the dependence on $y$ :\n\n$$\n\\frac{d \\lambda}{d s} s \\nabla g^{\\top}\\left(-H^{-1} A\\right)=-g-s \\nabla g^{\\top}\\left(-H^{-1} B\\right) .\n$$\n\nNow, observe that for all $y \\in \\mathbb{R}_{++}^{J+K}$, by assumption on $g, H(y)$ is symmetric negative definite; moreover, by the positivity of cross-derivative, all off-diagonal entries of $H(y)$ are strictly positive. Then, $-H(y)$ is a Stieltjes matrix and, consequently, all elements of $H(y)^{-1}$ are negative. It immediately follows from (60) that whenever $q>0$ and $s>0, d \\lambda / d s<0$. □\n\nAs such, the seller's problem for any given $s$ is analogous to Mussa and Rosen (1978). Since $w$ and $s$ are independently distributed, we can drop the dependence on $s$ and define the virtual value as\n\n$$\n\\varphi(w) \\triangleq w-\\frac{1-F_{w}(w)}{f_{w}(w)} .\n$$\n\nSince $\\varphi(w)$ is increasing, all $w$ with $\\varphi(w) \\leq 0$ are excluded and all other $w$ receive the quality level $q(w, s)$ that solves:\n\n$$\n\\varphi(w)=C_{q}(q(w, s), s) .\n$$\n\nThe corresponding optimal transfers are\n\n$$\nt(w, s)=w q(w, s)-\\int_{0}^{w} q(r, s) d r\n$$\n\nLemma 5 shows that it is cheaper to generate an extra unit of (total) quality when you have more tasks. This property is intuitive given that the returns on each task are diminishing. Thus, the optimal quality, and hence the buyer's rent, increase in scale for any given value $w$.\n\nAssumption 2 (Bounded Rent Increase). For all $w, s$, the function $q(w, s)$ defined in (62) satisfies $\\int_{0}^{w} s q_{s}(r, s) d r \\leq w q(w, s)$.\n\nAssumption 2 requires that the buyer's rent does not grow too quickly and, specifically, that the marginal increase of buyer rent from having an additional task is smaller than the average equilibrium value generated by LLM across existing tasks.\n\nProposition 10 (Optimal Menu of Token Allocations). Under Assumption 2, an optimal menu is\n\n$$\n\\left(\\left(x_{i}(w, s)\\right)_{i \\in[0,1]}, z(w, s), t(w, s)\\right)_{(w, s)},\n$$\n\nwhere for each $(w, s),\\left(\\left(x_{i}(w, s)\\right)_{i \\in[0,1]}, z(w, s)\\right)$ are cost-minimizing tokens from (55) that deliver quality $q(w, s)$ as defined in (62), and $t(w, s)$ is as defined in (63).\n\nProof. If each type reports truthfully, then the menu attains the profits of the observablescale benchmark and is thus optimal.\n\nIf type $(w, s)$ deviates to $(w, \\tilde{s})$ with $\\tilde{s} \\leq s$, then, under the proposed menu, he obtains exactly the same payoff as type $(w, \\tilde{s})$, because he processes the same number of tasks with the same willingness to pay for quality. By Lemma $5, C_{q}(q, s)$ is decreasing in $s$ for all $q$, and thus $q(w, s)$ is increasing in $s$ for all $w$. Therefore, the rents accrued by type $(w, s)$ under truth-telling,\n\n$$\nU(w, s)=\\int_{0}^{w} q(r, s) d r\n$$\n\nare increasing in $s$ for all $w$. Therefore, $(w, s)$ does not want to deviate to $(w, \\tilde{s})$ with $\\tilde{s} \\leq s$. Furthermore, by incentive compatibility within a given $\\tilde{s},(w, \\tilde{s})$ does not want to deviate to $(\\tilde{w}, \\tilde{s})$. Therefore, $(w, s)$ does not want to deviate to any $(\\tilde{w}, \\tilde{s})$ with $\\tilde{s} \\leq s$.\n\nIf type $(w, s)$ deviates to $(\\tilde{w}, \\tilde{s})$ with $\\tilde{s}>s$, then he obtains gross payoff $w q(\\tilde{w}, \\tilde{s}) s / \\tilde{s}$ and pays the transfer $t(\\tilde{w}, \\tilde{s})$. Therefore, the optimal double deviation strategy for a misreporting type solves\n\n$$\n\\max _{\\tilde{w} \\geq 0}\\left[w q(\\tilde{w}, \\tilde{s}) \\frac{s}{\\tilde{s}}-\\tilde{w} q(\\tilde{w}, \\tilde{s})+\\int_{0}^{\\tilde{w}} q(r, \\tilde{s}) d r\\right] .\n$$\n\nBecause the mechanism incentivizes truthful reporting by any $\\tilde{s}$-truthtelling type, including the type ( $w s / \\tilde{s}, \\tilde{s}$ ), it follows that\n\n$$\n\\tilde{w}^{*}=\\frac{w s}{\\tilde{s}}, \\quad U(w ; s, \\tilde{s})=\\int_{0}^{\\frac{w s}{\\tilde{s}}} q(r, \\tilde{s}) d r\n$$\n\nThe condition that discourages local deviations to $\\tilde{s}>s$ is:\n\n$$\ns \\int_{0}^{w} q_{s}(r, s) d r \\leq w q(w, s)\n$$\n\nfor all $(w, s)$, which is precisely Assumption 2. Furthermore, observe that for $\\tilde{s}>s$ :\n\n$$\n\\begin{aligned}\nU_{\\tilde{s}}(w ; s, \\tilde{s}) & =\\int_{0}^{\\frac{w s}{\\tilde{s}}} q_{s}(r, \\tilde{s}) d r-\\frac{w s}{\\tilde{s}^{2}} q\\left(\\frac{w s}{\\tilde{s}}, \\tilde{s}\\right) \\\\\n& =\\frac{1}{\\tilde{s}}\\left(\\tilde{s} \\int_{0}^{\\frac{w s}{\\tilde{s}}} q_{s}(r, \\tilde{s}) d r-\\frac{w s}{\\tilde{s}} q\\left(\\frac{w s}{\\tilde{s}}, \\tilde{s}\\right)\\right) \\leq 0\n\\end{aligned}\n$$\n\nwhere the inequality holds by Assumption 2 with $(w, s)$ replaced by $(s w / \\tilde{s}, \\tilde{s})$. Therefore, the upward global deviations in $\\tilde{s}$ are suboptimal and the result follows. $\\square$","text_sha256":"f544c919c9630a8f9268538385fda241ab5788e0fda6f48f29ed66c3a3d05461"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0036","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nArmstrong, M. (1996): \"Multiproduct Nonlinear Pricing,\" Econometrica, 64, 51-76.\nBabaioff, M., R. Kleinberg, and R. Paes Leme (2012): \"Optimal Mechanisms for Selling Information,\" in Proceedings of the 13th ACM Conference on Electronic Commerce, 92-109.\n\nBergemann, D., M. Bojko, P. Dütting, R. Paes Leme, H. Xu, and S. Zuo (2024): \"Data-Driven Mechanism Design: Jointly Eliciting Preferences and Information,\" arXiv preprint arXiv:2412.16132.\n\nBergemann, D., A. Bonatti, and A. Smolin (2018): \"The Design and Price of Information,\" American Economic Review, 108, 1-48.\n\nCalzolari, G. and V. Denicolò (2013): \"Competition with Exclusive Contracts and Market-Share Discounts,\" American Economic Review, 103, 2384-2411.\n\n- (2015): \"Exclusive Contracts and Market Dominance,\" American Economic Review, 105, 3321-3351.\n\nCastro-Pires, H., H. Chade, and J. Swinkels (2024): \"Disentangling Moral Hazard and Adverse Selection,\" American Economic Review, 114, 1-37.\n\nDaskalakis, C., A. Deckelbaum, and C. Tzamos (2017): \"Strong Duality for a Multiple-Good Monopolist,\" Econometrica, 85, 735-767.\n\nDemirer, M., A. Fradkin, N. Tadelis, and S. Peng (2025): \"The Emerging Market for Intelligence: Pricing, Supply, and Demand for LLMs,\" Tech. rep., National Bureau of Economic Research.\n\nDevanur, N. R., K. Goldner, R. R. Saxena, A. Schvartzman, and S. M. Weinberg (2020): \"Optimal Mechanism Design for Single-Minded Agents,\" in Proceedings of the 21st ACM Conference on Economics and Computation, 193-256.\n\nDoligalski, P., P. Dworczak, J. Krysta, and F. Tokarski (2025): \"Incentive separability,\" Journal of Political Economy Microeconomics, 3, 539-567.\n\nDuetting, P., V. Mirrokni, R. Paes Leme, H. Xu, and S. Zuo (2024): \"Mechanism design for large language models,\" in Proceedings of the ACM on Web Conference 2024, 144-155.\n\nFiat, A., K. Goldner, A. R. Karlin, and E. Koutsoupias (2016): \"The Fedex Problem,\" in Proceedings of the 2016 ACM Conference on Economics and Computation, 21-22.\n\nFish, S., Y. A. Gonczarowski, and R. I. Shorrer (2024): \"Algorithmic Collusion by Large Language Models,\" arXiv preprint arXiv:2404.00806.\n\nHaghpanah, N. and R. Siegel (2024): \"Screening Two Types,\" Tech. rep., Penn State.\nJullien, B. (2000): \"Participation Constraints in Adverse Selection Models,\" Journal of Economic Theory, 93, 1-47.\n\nKaplan, J., S. McCandlish, T. Henighan, T. B. Brown, B. Chess, R. Child, S. Gray, A. Radford, J. Wu, and D. Amodei (2020): \"Scaling Laws for Neural Language Models,\" arXiv preprint arXiv:2001.08361.\n\nLaffont, J.-J. and J. Tirole (1990): \"The regulation of multiproduct firms: Part I: Theory,\" Journal of Public Economics, 43, 1-36.\n\nMahmood, R. (2024): \"Pricing and Competition for Generative AI,\" arXiv preprint arXiv:2411.02661.\n\nMussa, M. and S. Rosen (1978): \"Monopoly and Product Quality,\" Journal of Economic Theory, 18, 301-317.\n\nRochet, J.-C. and L. A. Stole (2003): \"The Economics of Multidimensional Screening,\" Econometric Society Monographs, 35, 150-197.\n\nTirole, J. (1988): The Theory of Industrial Organization, Cambridge: MIT Press.\nWu, Y., Z. Sun, S. Li, S. Welleck, and Y. Yang (2025): \"Inference Scaling Laws: An Empirical analysis of Compute-Optimal Inference for LLM Problem-Solving,\" in The Thirteenth International Conference on Learning Representations.\n\nYang, K. H. (2022): \"Selling Consumer Data for Profit: Optimal Market-Segmentation Design and Its Consequences,\" American Economic Review, 112, 1364-1393.\n\n[^0]:    *Bergemann: Department of Economics, Yale University, dirk.bergemann@yale.edu. Bonatti: MIT Sloan, bonatti@mit.edu. Smolin: Toulouse School of Economics, alexey.v.smolin@gmail.com. We thank Mark Armstrong, Mert Demirer, Scott Kominers, Antonio Russo, Ron Siegel, and Frank Yang for valuable comments and discussions, as well as audiences at the Triangle Conference, USC, Caltech, Virginia, the Chicago workshop, the Sciences Po workshop, the NBER Market Design workshop, VSET, Brown, LSE, the TSE Digital Economics Conference, the IESE Economics of AI Conference, and Cambridge. An extended abstract of an earlier version of this paper appeared as \"The Economics of Large Language Models: Token Allocation, Fine-Tuning, and Optimal Pricing\" in the proceedings of EC'25. Dirk Bergemann gratefully acknowledges financial support from NSF SES 2049754 and ONR MURI. Alessandro Bonatti gratefully acknowledges financial support from NSF SES 2519401. Alex Smolin gratefully acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (grant ANR-17-EURE-0010) and through the AI Interdisciplinary Institute ANITI (grant ANR-23-IACL-0002).\n\n[^1]:    ${ }^{1}$ For the practical challenges associated with LLM pricing, see https://www.wsj.com/articles/no-one-knows-how-to-price-ai-tools-f346ea8a.\n\n[^2]:    ${ }^{2}$ That is, for all $j$ and $x_{i,-j} \\in \\mathbb{R}_{+}^{J-1}$ such that $\\Psi\\left(0, x_{i,-j}\\right)>0, \\lim _{x_{i j} \\downarrow 0} \\Psi_{j}\\left(x_{i j}, x_{i,-j}\\right)=+\\infty$.\n    ${ }^{3}$ That is, there exists $z_{0} \\geq 0, z_{0} \\neq 0$, such that $\\lim _{r \\downarrow 0} \\Phi\\left(r z_{0}\\right) / r^{1-\\sigma}=+\\infty$. Sufficient conditions for this property are (i) $\\Phi(0)>0$ or (ii) $\\Phi$ is homogeneous of degree $\\hat{\\sigma} \\in(0,1)$ with $\\sigma+\\hat{\\sigma}<1$.\n\n[^3]:    ${ }^{4}$ For tractability, we will often study the case $g(x, 0)=0$. This can be viewed as a normalization that doesn't affect the economics of the problem.\n\n[^4]:    ${ }^{5}$ The parameter counts of leading models are estimated to be in the several-billion range; see https: //codingscape.com/blog/most-powerful-llms-large-language-models.\n    ${ }^{6}$ Recent progress has leveraged this channel; see, for instance, https://arcprize.org/blog/oai-o3-pub-breakthrough.\n\n[^5]:    ${ }^{7}$ Note that fine-tuning typically does not change the model's architecture or size and therefore should not directly affect marginal inference costs.\n\n[^6]:    ${ }^{8}$ In Section 5, we show that if $\\Phi(Z)$ is also a homogeneous function, then $C(Q)$ is simply a power function.\n\n[^7]:    ${ }^{9}$ Equivalently, we could assume that the buyer can combine tokens from two models on any task, with a performance given by the sum of individual gain functions.\n\n[^8]:    ${ }^{10}$ See Calzolari and Denicolò (2013) for a related analysis of nonlinear competition between a dominant firm and a competitive fringe under private information about demand.","text_sha256":"a87faabcf5086a74fa54b3e4708ee585016ce6f691a68e365d1085e40f378013"}
{"schema_version":"1.0","chunk_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10:0037","work_id":"alex-smolin:menu-pricing-of-large-language-models","paper_id":"alex-smolin:menu-pricing-of-large-language-models:2026-03-10","title":"Menu Pricing of Large Language Models","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-03-10","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md","source_record":"https://arxiv.org/abs/2502.07736","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"[^9]:    ${ }^{11}$ Indeed, $\\underline{\\hat{\\theta}}=\\left[\\left(\\frac{1-\\sigma}{1-\\sigma-\\hat{\\sigma}_{L}}\\right)^{1-\\sigma-\\hat{\\sigma}_{L}}\\left(\\frac{\\sigma}{\\sigma+\\hat{\\sigma}_{L}}\\right)^{\\sigma+\\hat{\\sigma}_{L}}\\right]^{(1-\\sigma) / \\hat{\\sigma}_{L}}$, and thus, by the strict concavity of the logarithm and Jensen's inequality, $\\ln \\left(\\frac{\\hat{\\theta}}{\\underline{\\theta}}\\right)=\\frac{1-\\sigma}{\\hat{\\sigma}_{L}}\\left[\\left(1-\\sigma-\\hat{\\sigma}_{L}\\right) \\ln \\left(\\frac{1-\\sigma}{1-\\sigma-\\hat{\\sigma}_{L}}\\right)+\\left(\\sigma+\\hat{\\sigma}_{L}\\right) \\ln \\left(\\frac{\\sigma}{\\sigma+\\hat{\\sigma}_{L}}\\right)\\right]<0$.\n\n[^10]:    ${ }^{12}$ Calzolari and Denicolò (2015) show that a dominant firm with a competitive advantage can profitably impose exclusive dealing on privately informed buyers; the mechanism here is analogous, though exclusivity arises from the optimal nonlinear tariff rather than from contractual restrictions.\n\n[^11]:    ${ }^{13}$ For instance, a message to GPT-4o-mini costs 9 points, while a message to Claude Opus 4 costs 4,105 points, a ratio of roughly 450×.\n\n[^12]:    ${ }^{14}$ As in the theory, the overage charge interacts with the multiplier system: a query to a 1 × model costs \\$0.04 in the overage region, while a query to a $3 \\times$ model (Claude Opus 4.5) costs \\$0.12.\n\n[^13]:    ${ }^{15}$ By homogeneity, for all $x \\in \\mathbb{R}_{+}^{J}$ and $r>0, \\nabla \\Psi(r x)=r^{\\sigma-1} \\nabla \\Psi(x)$, so any token ray corresponds to a gradient ray. By strict concavity, if $x_{1} \\neq x_{2}$, then $\\nabla \\Psi\\left(x_{1}\\right) \\neq \\nabla \\Psi\\left(x_{2}\\right)$, so distinct token rays map to distinct gradient rays.\n\n[^14]:    ${ }^{16}$ In the general case of heterogeneous $\\sigma_{l}$, the optimal task split and the resulting buyer's payoff do not admit tractable closed-form solutions. In particular, the fine details of the profile $w$ could matter.\n\n[^15]:    ${ }^{17}$ Non-monotone partitions may also be optimal, but $\\theta_{l}\\left(I_{l}^{*}\\right)$ are uniquely determined.\n\n[^16]:    ${ }^{18}$ The Inada condition is imposed only for simplicity of arguments in Lemma 5.","text_sha256":"dc7812fef8b565ca82166cb8178dd3cb00bb4286f5019d549e1c67f475fa5f84"}
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{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0001","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Laura Doval; Alex Smolin.\n> Canonical citation: Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"6582e63744cb238a6b1a64b4ea7db15d546d4fe66ec86a5212879ab7ea8eb9cc"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0002","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Calibrated Mechanism Design","text":"# Calibrated Mechanism Design\n\n**Authors:** Laura Doval; Alex Smolin\n\n**Manuscript date:** 2026-02-18\n\n#### Abstract\n\nWe study mechanism design when a designer repeatedly uses a fixed mechanism to interact with strategic agents who learn from observing their allocations. We introduce a static framework, calibrated mechanism design, requiring mechanisms to remain incentive compatible given the information they reveal about an underlying state through repeated use. In single-agent settings, we prove implementable outcomes correspond to two-stage mechanisms: the designer discloses information about the state, then commits to a state-independent allocation rule. This yields a tractable procedure to characterize calibrated mechanisms, combining information design and mechanism design. In private values environments, full transparency is optimal and correlationbased surplus extraction fails. We provide a microfoundation by showing calibrated mechanisms characterize exactly what is implementable when an infinitely patient agent repeatedly interacts with the same mechanism. Dynamic mechanisms that condition on histories expand implementable outcomes only by weakening incentive constraints, but not by enriching the designer's ability to obfuscate learning.\n\n[^0]","text_sha256":"a5985af64c5e6dfc96c8e30e6aa763bc9a5aafc6ccbc222628d7db73bb7e60d2"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0003","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nMany economic institutions rely on mechanisms that remain fixed while agents interact with them repeatedly. Online platforms commit to stable auction formats for advertising slots, lenders use persistent scoring algorithms for loan decisions, and regulators establish durable rules for market participants. When the mechanism's operation depends on information known only to the designer-such as the platform's data about match values, the lender's assessment of credit market conditions, or the regulator's understanding of market fundamentals-participants may infer this information by observing their outcomes across repeated interactions. This learning creates a fundamental constraint: the information a mechanism reveals through repeated use limits what outcomes it can implement in the long run. Participants can use the information gleaned from past interactions when deciding whether and how to participate, tightening the designer's incentive constraints. A lender whose approval decisions depend on unobserved credit market conditions will gradually reveal these conditions to borrowers through his lending decisions, constraining the lender's ability to provide credit efficiently. We study how this endogenous information leakage shapes the set of implementable outcomes in mechanism design.\n\nA simple example illustrates how learning prevents the designer from exploiting his information. Consider a seller who repeatedly offers a good whose demand depends on an unobserved state, which can be either low $(L)$ or high $(H)$. Each state is equally likely. The seller faces a buyer whose value for the good can take one of two values, 1/2 or 1. The probability that the buyer's value is 1 is higher when the demand state is high. Table 1 summarizes the value distribution conditional on the demand state:\n\n|  | $v=1 / 2$ | $v=1$ |\n| :--- | :--- | :--- |\n| $L$ | 2/3 | 1/3 |\n| $H$ | 1/3 | 2/3 |\n\nTable 1: Value distribution conditional on demand state.\n\nSuppose the seller can design the terms of trade, that is, the probability with which he allocates the good to the buyer $(q \\in[0,1])$ and the payment the buyer makes to the seller $(t \\in \\mathbb{R})$. The buyer's payoff is $v q-t$, and the seller's is $t$. The buyer can always choose to not trade with the seller and ensure a payoff of 0.\n\nSuppose first the buyer and the seller interact only once. Table 2 depicts an optimal mechanism for the seller in this case:\n\n|  | $v=1 / 2$ | $v=1$ |\n| :--- | :--- | :--- |\n| $L$ | $(1,0)$ | $(1,0)$ |\n| $H$ | (1,3/2) | (1,3/2) |\n\nTable 2: Trade probabilities and payments as a function of buyer's value and demand state.\n\nIn this mechanism, the buyer gets the good for free when the demand state is $L$ and pays a price of 3/2 when it is $H$. If this mechanism were offered once without the buyer observing the demand state, the buyer obtains a payoff of 0 from participating and truthfully reporting her type. Unsurprisingly,\nthe seller extracts the buyer's surplus: the seller knows the demand state, which is correlated with the buyer's type, and exploits this information in the design of his mechanism (cf. Crémer and McLean, 1988).\n\nSuppose now the buyer interacts repeatedly with the mechanism, but the state remains fixed. If the buyer observes nothing from her interaction with the mechanism, the buyer is willing to participate and truthfully report her value into the mechanism, no matter how many times it is offered: In each period, she anticipates getting a (continuation) payoff of 0 from engaging with the mechanism. Suppose, instead, the buyer observes her allocation in the mechanism. If the demand state is $L$, the buyer gets the good for free at the end of the first period, and from now on knows this is what she will get in the mechanism. If the demand state is $H$, the buyer gets the good and pays a price of 3/2 as she agreed to when she decided to participate in period 1, but anticipating a price of 3/2 from then onwards, never again participates in the mechanism. Thus, whereas the seller can implement the outcomes in Table 2 when the buyer does not observe her allocations, this is no longer the case when she can.\n\nThis paper develops a framework for mechanism design in which agents' ability to learn about the designer's information from repeatedly playing a mechanism constrains implementable outcomes. In our framework, allocations depend on agents' reports and on a state known only to the designer. Through repeated participation, agents observe their allocations and gradually learn about this state. A mechanism therefore serves a dual role: it determines allocations based on reports, and it acts as an information structure that reveals the underlying state. The more the mechanism conditions on the state, the more information it leaks, and the tighter the constraints on implementable outcomes.\n\nWe approach our analysis in two steps. First, we introduce a static solution concept for mechanism design that directly models the feedback between the mechanism, the information it reveals, and participants' behavior. This solution concept allows us to tractably capture the limits on the set of implementable outcomes implied by agents' learning, while abstracting from the dynamics of experimentation. Second, we provide a dynamic microfoundation showing this static solution concept precisely captures the implementable outcomes when an infinitely patient agent repeatedly interacts with the same mechanism.","text_sha256":"c55ee4f92d7e0db9c58aee107d1ed7844df4914853d65f916bf6ca4ac67c0a08"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0004","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"In Section 2, we introduce a static solution concept-calibrated mechanism design-requiring that mechanisms remain incentive compatible and individually rational given the information they reveal about the state through their allocations. We formalize this requirement through the notion of a calibrated mechanism. We couple each mechanism with an information structure that describes what participants learn about the state from the mechanism. The information structure reveals to each agent an interim allocation rule-the mapping from her type reports to lotteries over her allocations-capturing what she would learn from repeatedly observing her outcomes in the mechanism. We require the information structure to be calibrated in the sense of Foster and Vohra (1997): the interim allocation rule each agent observes must accurately describe the allocation probabilities she faces. Throughout the paper, we study calibrated mechanism design: the designer chooses a mechanism that remains incentive compatible and individually rational when participants have access to the mechanism's calibrated information structure before playing. Calibration imposes a constraint\non the designer relative to standard mechanism design: the more the allocation rule depends on the state, the more informative the calibrated information structure becomes, and hence the more incentive and participation constraints the designer must satisfy.\n\nIn private values environments, the constraint that the mechanism must remain incentive compatible and individually rational given the information it reveals about the state pushes the designer to full transparency. We show in Theorem 1 that, under the calibration constraint, the designer can do no better than inducing in each state the optimal direct mechanism when there is common knowledge of that state. In particular, in settings with transferable utility in which the designer has statistical information about the agents' types, Theorem 1 implies the designer cannot extract full surplus.\n\nIn Section 3, we characterize optimal calibrated mechanisms through a tractable class we dub twostage mechanisms. In a two-stage mechanism, the designer first discloses information about the state to the agent-inducing a belief about the state-then commits to an allocation rule that depends only on the agent's report, not the state itself. Theorem 2 shows that in single-agent settings, calibrated mechanisms and two-stage mechanisms implement exactly the same outcome distributions. This equivalence yields a practical algorithm for finding optimal calibrated mechanisms, combining tools from information design and mechanism design: for each possible belief the designer might induce, solve a standard mechanism design problem given that belief; then choose the optimal information disclosure by concavifying the resulting value function.\n\nIn the case of multiple agents, Proposition 1 shows calibrated mechanisms admit a similar representation via generalized two-stage mechanisms: Like two-stage mechanisms, the designer individually discloses to each agent a belief about the state and offers an incentive compatible and individually rational interim allocation rule that no longer conditions on the state. Whereas the designer observes the disclosed belief profile, each agent only observes the belief disclosed to her. ${ }^{1}$ Moreover, each agent learns only her own interim allocation rule-how her reports map to her allocations-rather than the complete mapping from type profiles to allocations. This partial observability requires additional consistency conditions to ensure agents' interim allocation rules are mutually compatible. In contrast to the single-agent case, not every generalized two-stage mechanism induces a calibrated mechanism, as generalized two-stage mechanisms may reveal strictly less information than calibrated mechanisms.\n\nIn Section 4, we study optimal calibrated mechanism design in the canonical setting of quasilinear utilities, single-dimensional types and allocations. In Section 4.1, we study the single-agent case. We show that if the order of types is state independent, then optimal two-stage mechanisms fully reveal the state, whereas this conclusion can be reversed when the order of types is state-dependent. In Section 4.2, we compare optimal calibrated mechanism design against the Myersonian benchmark. We provide sufficient conditions under which the designer realizes the payoff of the Myersonian benchmark under the calibration constraint; under these conditions, the optimal Myersonian mechanism satisfies the agent's incentive constraints state-by-state. Building on that result, we analyze multi-agent applications in Section 4.3.\n\n[^1]Section 5 provides a microfoundation for calibrated mechanism design. We analyze an infinitehorizon game where an infinitely patient agent repeatedly plays the same mechanism. ${ }^{2}$ The state remains fixed, but the agent's type is redrawn each period independently of the state. ${ }^{3}$ Our notion of implementation is based on the long-run expected frequency of allocation-type-state tuples when the agent best responds to the mechanism. Theorem 3 shows that the implementable outcome distributions are precisely those induced by incentive compatible two-stage mechanisms. This result validates our static framework: calibrated mechanism design captures exactly what is implementable through repeated play.","text_sha256":"37e0807af1ad66fe2646a99c903cdf9ca724ff69c8306ee1a469894597a458d1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0005","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"We then ask whether giving the designer additional flexibility helps. In a dynamic mechanism, the designer can condition each period's allocation on the complete history of past reports and allocations, rather than using the same mechanism repeatedly. Theorem 4 shows that dynamic mechanisms expand implementable outcomes in a specific way: they correspond to two-stage mechanisms with weaker incentive compatibility and individual rationality conditions. The designer can now exploit the ability to monitor the frequency of type reports over time, which allows him to punish detectable deviations-reporting strategies whose frequency distribution differs from the true type distribution. Instead, the mechanism must be robust to undetectable ones. Importantly, in environments with transferable utility, this distinction vanishes: As shown in Rahman (2024), eliminating profitable undetectable deviations is equivalent to incentive compatibility, so dynamic mechanisms implement exactly the same distributions over physical allocations, types, and states as our static calibrated mechanisms.\n\nRelated Literature The paper lies at the intersection of four literatures: rational expectations equilibria, (public) information disclosure in mechanism design, the computer science literature on learning in repeated auctions, and dynamic implementation.\n\nThe definition of a calibrated mechanism is in the spirit of rational expectations equilibria (Radner, 1979; Green, 1977; Kreps, 1977). Indeed, requiring a mechanism to remain incentive compatible given the information it reveals about the state mirrors the rational-expectations requirement that prices clear markets given the information they convey. Unlike rational expectations equilibrium, where the only role of prices is to clear the market, calibrated mechanisms are chosen by a designer who understands the incentive implications of the mechanism's information leakage and trades this off against the value of conditioning the mechanism on the state. Similar to our analysis in Section 5, some papers in the literature have studied the question of whether rational expectations equilibria emerge from learning dynamics (see, for instance, Milgrom, 1981; Blume et al., 1982).\n\nFollowing Milgrom and Weber (1982), a literature has studied whether a designer should publicly disclose information he knows before a mechanism is played. Ottaviani and Prat (2001) show revealing a signal affiliated with the buyer's value is optimal in a single-agent screening problem. When considering the case of an informed principal, they consider what we call two-stage mechanisms to bound the monopolist's profits. Szabadi (2018) and Yamashita (2018) study the optimal release of\n\n[^2]public information followed by an optimal mechanism conditional on that disclosure, while Fu et al. (2012) study this question in the context of a second price auction. In those papers, the restriction to public disclosure and the independence of the mechanism on information other than the disclosed one is a constraint on the class of mechanisms the designer can use. Instead, we show this class of mechanisms is without loss when the designer faces our calibration constraint in the single-agent case, but it may not be in the multi-agent case. Note, however, that when full or no disclosure are optimal in the Myersonian benchmark the distinction between private and public disclosure is immaterial. For that reason, the results on the achievability of the Myersonian benchmark are similar across their and our work. Daskalakis et al. (2016) lift the restriction to public disclosure and study the Myersonian benchmark in an auction setting, showing that the complexity of that problem is the same as that of a multi-product monopolist (cf. Guesnerie and Laffont, 1984). ${ }^{4}$\n\nMotivated by the prevalence of fixed auction formats with which bidders interact repeatedly, a literature in computer science studies the properties of bidder learning algorithms and the implications for the auctioneer (see, for instance, Golrezaei et al., 2019; Nedelec et al., 2019; Kanoria and Nazerzadeh, 2020, and Nedelec et al., 2022 for a survey treatment). A common finding is that learning bidders can take advantage of \"naive\" auction formats which are no longer incentive compatible when bidders learn. Inspired by this literature, we develop a framework which allows us to systematically study the question of optimal mechanism design in the presence of learning agents.\n\nOur dynamic implementation results relate to the literature that studies whether a mechanism can be implemented either by linking decisions (Jackson and Sonnenschein, 2007; Ball and Kattwinkel, 2023) or in the patient limit of a repeated interaction (Renou and Tomala, 2015; Margaria and Smolin, 2018; Meng, 2021). Both strands identify cyclical monotonicity as the condition for implementation (cf. Rochet, 1987). Rahman (2024) shows that cyclical monotonicity is equivalent to the absence of profitable undetectable deviations.\n\nBy focusing on what agents learn from the designer's information, our paper is distinct from the literature on mechanism design with interdependent payoffs which focuses on agents' learning about others' types through their actions in the mechanism (Green and Laffont, 1987; Niemeyer, 2022; Häfner et al., 2025). Moreover, by focusing in the case of a designer with commitment, we are distinct from the literature on the informed principal (Myerson, 1983; Maskin and Tirole, 1990).\n\nLastly, our paper contributes to two literatures. First, by studying the informational role of the mechanism, we contribute to the literature on feedback in auctions, which analyzes how different feedback rules affect bidders' information about other agents, and ultimately behavior in first price auctions (see, for instance, Esponda, 2008; Bergemann and Hörner, 2018; Cesa-Bianchi et al., 2024). Second, by showing the designer's problem involves solving information and mechanism design problems, our paper joins a recent literature that highlights the dual role of the mechanism as an information structure and an allocation rule (Calzolari and Pavan, 2006; Dworczak, 2020; Doval and Skreta, 2022).\n\n[^3]","text_sha256":"7e3569860fc83693ef523d3a01c9f40e76c5c0539ecc28e638cf72d1dc95d283"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0006","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Calibrated Mechanism Design","text":"## 2 Calibrated Mechanism Design\n\nIn this section, we introduce the static setting and solution concept that captures the impact of agents' learning from the mechanism on the set of implementable outcomes. We defer to Section 5 the analysis of the dynamic game whose outcomes our static solution concept captures.\n\nPrimitives A designer (he) interacts with $N$ privately informed agents (she) to determine an allocation. Let $\\Theta_{i}$ denote the set of types of agent $i$, and $\\Theta \\equiv \\times_{i=1}^{N} \\Theta_{i}$. Each agent knows her type, but not those of other agents. The allocation space is given by $A \\equiv \\times_{i=1}^{N} A_{i} .{ }^{5}$ Finally, let $\\Omega$ denote a set of states, which are known to the designer, but not to the agents. The sets $\\Theta_{i}, A_{i}$, and $\\Omega$ are assumed to be finite throughout. ${ }^{6}$ Agent $i$ 's payoffs are given by $u_{i}: A_{i} \\times \\Theta_{i} \\times \\Omega \\rightarrow \\mathbb{R}$. That is, agent $i$ cares about her dimension of the allocation, her type, and the state, and not about other agents' allocations or types.\n\nDenote by $\\mu_{0}$ the distribution over $\\Omega$. For each $\\omega \\in \\Omega$, let $f(\\cdot \\mid \\omega) \\in \\Delta(\\Theta)$ denote the type distribution. We assume throughout the types are independently distributed conditional on the state, that is,\n\n$$\nf(\\theta \\mid \\omega)=\\prod_{i=1}^{N} f_{i}\\left(\\theta_{i} \\mid \\omega\\right),\n$$\n\nfor all $\\theta \\in \\Theta$ and $\\omega \\in \\Omega$. Together with the assumption on agents' payoffs, the assumption on $f(\\cdot \\mid \\omega)$ allows us to isolate the effect of learning about the state from that of learning about others' types (perhaps because others' types provide additional information about the state).\n\nMechanisms We model mechanisms as mappings\n\n$$\n\\phi: \\Theta \\times \\Omega \\times[0,1] \\rightarrow \\Delta(A),\n$$\n\nwhere $\\varepsilon \\in[0,1]$ is a uniformly distributed random variable, which we refer to as the randomization device.\n\nSeveral comments are in order. First, to understand how a mechanism works, the timing of when the different random variables is drawn is important. In particular, we assume that both the state $\\omega$ and the realization of the randomization device $\\varepsilon$ are independently drawn at the beginning, but not observed by the agents. This determines the direct mechanism $\\phi(\\cdot, \\omega, \\varepsilon): \\Theta \\rightarrow \\Delta(A)$ to which the agents send type reports, which in turn determines the lottery from which the allocation is drawn. Thus, the allocation is random in our setting for two reasons: on the one hand, the agents do not know the realization of $(\\omega, \\varepsilon)$, and hence the direct mechanism $\\phi(\\cdot, \\omega, \\varepsilon)$ they face. Second, conditional on $(\\omega, \\varepsilon)$, the allocation may be drawn at random. Mathematically, we could have subsumed all sources of randomness in the allocation into the randomization device. However, as we explain next, the definition in Equation 2 allows us to distinguish the source of randomness in the allocation that is informative about the state from that which is not.\n\n[^4]Second, it is useful to consider the reason for the randomization device in the definition of a mechanism. For simplicity, consider the case of the designer facing a single agent. If the agent had repeated access to the mechanism, the agent would stand to learn the mapping $\\phi(\\cdot, \\omega, \\varepsilon): \\Theta \\rightarrow \\Delta(A)$ by experimenting with different reports into the mechanism and observing the resulting allocations. ${ }^{7}$ Without the randomization device, the agent would stand to learn a partition of the set of states, where states in the same cell of the partition induce the same direct mechanism $\\phi(\\cdot, \\omega, \\varepsilon)$. By allowing the designer to rely on the randomization device, we allow him to obfuscate the agent's learning beyond a simple partitional structure. Contrast this with the Myersonian benchmark in which without loss of generality the designer would offer mechanisms that do not rely on such devices, that is, $\\phi_{\\mathrm{My}}: \\Theta \\times \\Omega \\rightarrow \\Delta(A)$. Indeed, the Myersonian designer is not concerned with the agents' learning: without loss of generality, he does not disclose anything about the state to the agents, so that the question of how to optimally release information about the state is moot.\n\nLastly, note that we assume the mechanism asks the agents for type reports. In Appendix D, we show that the revelation principle holds in the setting of this section: it is without loss of generality to focus on direct and incentive compatible mechanisms that induce full participation.\n\nCalibrated information structures We now describe how a mechanism induces an information structure, which we define using the language in Green and Stokey (2022) and Gentzkow and Kamenica (2017). An information structure is a mapping ${ }^{8}$\n\n$$\n\\pi: \\Omega \\times[0,1] \\rightarrow S_{1}^{*} \\times \\cdots \\times S_{N}^{*}\n$$\n\nwhere $\\varepsilon \\in[0,1]$ is a uniformly distributed random variable-in fact, it is the same as in the definition of a mechanism-and\n\n$$\nS_{i}^{*}=\\Delta\\left(A_{i}\\right)^{\\Theta_{i}}\n$$\n\nis the set of agent $i$ 's interim allocation rules. ${ }^{9}$ We choose this language for the information structure to capture the idea that if agent $i$ plays the mechanism repeatedly, she stands to learn how her reports influence her allocation probabilities, i.e., her interim allocation rule. The interim allocation rule, in turn, depends on the mechanism and the strategies of others. Below, we require the interim allocation rule is well-calibrated with the mechanism and others' strategies:\n\nDefinition 1 (Calibrated information structures). We say that the information structure is calibrated to mechanism $\\phi$ if for all $(\\omega, \\varepsilon) \\in \\Omega \\times[0,1]$ such that $\\pi(\\omega, \\varepsilon)=\\left(s_{1}^{*}, \\ldots, s_{N}^{*}\\right)$ we have that for all $i \\in\\{1, \\ldots, N\\}$,","text_sha256":"2eb0b2572cc315afb5730fa23f18081e09f7c5397c323bd433f6ad8f21049092"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0007","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Calibrated Mechanism Design","text":"[^5]all $\\theta_{i} \\in \\Theta_{i}$, and all $a_{i} \\in A_{i}$\n$$\ns_{i}^{*}\\left(a_{i} \\mid \\theta_{i}\\right)=\\mathbb{E}_{\\tilde{\\theta}_{-i} \\sim f_{-i}(\\cdot \\mid \\omega)}\\left[\\sum_{a_{-i} \\in A_{-i}} \\phi\\left(\\theta_{i}, \\tilde{\\theta}_{-i}, \\omega, \\varepsilon\\right)\\left(a_{i}, a_{-i}\\right)\\right] .\n$$\nWe denote by $\\pi_{\\phi}$ the information structure calibrated to mechanism $\\phi$.\nIn words, the information structure is calibrated if whenever agent $i$ observes that her interim allocation rule in the mechanism is $s_{i}^{*}$, then $s_{i}^{*}$ describes the true probabilities with which agent $i$ gets different allocations $a_{i}$ as a function of her different type reports $\\theta_{i}^{\\prime}$ in the mechanism. As the right hand side of Equation 3 shows, these probabilities depend on: (i) the mechanism $\\phi(\\cdot, \\omega, \\varepsilon)$, and (ii) others' type reports. Implicit in the definition is that other agents are submitting their reports truthfully. While this is a simplification, ${ }^{10}$ it turns out to not be an issue because we study incentive compatible and individually rational mechanisms in the sense we define next.\n\nInformation leakage from a mechanism To close our model, we consider how the mechanism and its induced information structure affect agents' incentives. The mechanism $\\phi$ and the calibrated information structure $\\pi_{\\phi}$ induce the following game of incomplete information among the agents, where we use Bayes Nash equilibrium as the solution concept. In this game, nature draws (i) the state $\\omega$ from distribution $\\mu_{0}$, (ii) $\\varepsilon \\in[0,1]$ according to the uniform distribution, and (iii) the type profile $\\theta$ from $f(\\cdot \\mid \\omega)$. Then, each agent $i$ observes her type $\\theta_{i}$ and her signal $s_{i}^{*}=\\pi_{\\phi, i}(\\omega, \\varepsilon)$. Finally, agents simultaneously decide whether to participate in the mechanism, and conditional on participating what type report to send. Conditional on an agent choosing not to participate, each agent $i$ gets outside option $a_{i \\varnothing} .{ }^{11}$\n\nFormally, given the mechanism $\\phi$ and its calibrated information structure $\\pi_{\\phi}$, we say that the mechanism is incentive compatible if for all agents $i$, types $\\theta_{i} \\in \\Theta_{i}$, signals $s_{i}^{*} \\in S_{i}^{*}$ on the support of $\\pi_{\\phi, i}$, the following holds:\n\n$$\n\\theta_{i} \\in \\arg \\max _{\\theta_{i}^{\\prime} \\in \\Theta_{i}} \\mathbb{E}_{\\left(\\omega, \\varepsilon, \\theta_{-i}\\right)}\\left[u_{i}\\left(\\phi\\left(\\theta_{i}^{\\prime}, \\theta_{-i}, \\omega, \\varepsilon\\right), \\theta_{i}, \\omega\\right) \\mid\\left(\\theta_{i}, s_{i}^{*}\\right)\\right]\n$$\n\nwhere we abuse notation and implicitly (linearly) extend the agent's payoff function to account for lotteries over allocations (conditional on $\\left(\\theta_{-i}, \\omega, \\varepsilon\\right)$ ). Furthermore, we say that the mechanism is individually rational if for all agents $i$, types $\\theta_{i} \\in \\Theta_{i}$, and signals $s_{i}^{*} \\in S_{i}^{*}$ on the support of $\\pi_{\\phi, i}$, the following holds:\n\n$$\n\\mathbb{E}_{\\left(\\omega, \\varepsilon, \\theta_{-i}\\right)}\\left[u_{i}\\left(\\phi\\left(\\theta_{i}, \\theta_{-i}, \\omega, \\varepsilon\\right), \\theta_{i}, \\omega\\right)-u_{i}\\left(a_{i \\phi}, \\theta_{i}, \\omega\\right) \\mid\\left(\\theta_{i}, s_{i}^{*}\\right)\\right] \\geq 0 .\n$$\n\nImportantly, the agents' incentive and participation constraints must hold for each of their types and each of their private signals, reflecting the agents have access to the information leaked by the mechanism before they play in it. Note, however, the mechanism need not elicit the agents' observed\n\n[^6]signals, as the mechanism \"knows\" each agent's signal realization.\n\nCalibrated Mechanism Design In the rest of the paper, we study the problem of calibrated mechanism design in which the designer selects a mechanism $\\phi$ that satisfies Equations $\\operatorname{IC}\\left(\\theta_{i}, s_{i}^{*}\\right)$ and $\\operatorname{IR}\\left(\\theta_{i}, s_{i}^{*}\\right)$ for all $\\left(i, \\theta_{i}, s_{i}^{*}\\right)$, when the signals are drawn according to the calibrated information structure $\\pi_{\\phi}$.\n\nDefinition 2 (Calibrated Mechanism Design). Let $w: A \\times \\Theta \\times \\Omega \\rightarrow \\mathbb{R}$ denote the designer's payoff and let $\\mathcal{M}_{\\mathrm{ca}}$ denote the set of mechanisms that are incentive compatible and individually rational when agents have access to the calibrated information structure. The calibrated mechanism design problem is as follows:\n\n$$\n\\max _{\\phi \\in \\mathcal{M}_{\\mathrm{cal}}} \\mathbb{E}_{(\\omega, \\varepsilon, \\theta)}[w(\\phi(\\theta, \\omega, \\varepsilon), \\theta, \\omega)] .\n$$\n\nWe refer to elements of $\\mathcal{M}_{\\text {cal }}$ as calibrated mechanisms and the solution to $O P T_{\\mathrm{cal}}$ as the optimal calibrated mechanism.\n\nThree comments are in order:\nFirst, calibration imposes a constraint on the designer vis-à-vis standard mechanism design. After all, the incentive and participation constraints faced by the designer are endogenous to the mechanism. The more the designer's mechanism depends on the state, the more informative the calibrated information structure is, and the more incentive constraints the designer faces. Only when each agent's interim allocation rule is constant in $\\omega$ does the mechanism not leak information and the incentive and participation constraints reduce to the standard ones.\n\nSecond, in the single-agent setting, the calibration constraint admits two complementary interpretations. Throughout the paper, we emphasize the learning-by-experimentation interpretation: the calibrated information structure represents what the agent can ultimately infer by repeatedly interacting with the mechanism. Accordingly, the designer should ensure incentive compatibility with respect to the full information the agent eventually obtains. At the same time, calibration can also be interpreted as a transparency requirement. Indeed, upon observing signal $s^{*}: \\Theta \\rightarrow \\Delta(A)$, the agent knows the consequences of her choices in the mechanism, even if she does not know the state. ${ }^{12}$","text_sha256":"9c592f2c38b806e01da24162c9171dcb6e41719c7bbd368152de661ed0d2fdf3"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0008","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Calibrated Mechanism Design","text":"With multiple agents, these interpretations differ. The natural extension of the transparency requirement is that agents learn the mapping from profiles of type reports to lotteries over profiles of allocations before playing the mechanism. By contrast, the calibrated information structure reveals to each agent her interim allocation rule, that is, the mappings from her own reports to lotteries over her own allocations. As we discuss in the next section, the gap between these two interpretations is the gap between the designer publicly or privately disclosing information about the state to the agents.\n\nLastly, the definition of calibration assumes agents only learn about the state through their allocations\n\n[^7]in the mechanism, and not their payoffs. ${ }^{13,14}$ This assumption allows us to focus on the information that the mechanism leaks regardless of payoff assumptions. This allows us to avoid situations in which the mechanism does not condition the allocation on the state, but the agents learn because they have different payoffs from the same allocation in different states; or the mechanism conditions on the state, but this information is not payoff relevant to (some types of) the agent. Our microfoundation in Section 5.1 in fact deals with this last wrinkle: We show that even if the agent extracts less information than that in the calibrated information structure, she learns enough that her payoff is as if she had access to the calibrated information structure.\n\nWe conclude this section by illustrating how our static solution concept captures the dynamics we alluded to in the introductory example:\n\nExample 1 (Selling a good under demand uncertainty). Consider again the example in the introduction, in which a buyer with binary values $v \\in\\{1 / 2,1\\}$ faces a seller who knows whether demand is high $(\\omega=H)$ or low $(\\omega=L)$. The left panel of Table 3 describes the probabilities of trade and payments of the optimal (Myersonian) mechanism. In the introduction, we discussed this mechanism fails to extract full surplus in the long run as the buyer would quit the mechanism after seeing her allocation is (1,3/2). We now describe this in the language of calibration.\n\nThe right panel of Table 3 describes the information structure induced by the surplus extraction mechanism. Because in this mechanism the buyer's allocation does not depend on her values, we describe signals as allocations. The calibrated information structure is fully informative: when the state is $L$, the buyer sees signal $(1,0)$ with probability 1 , and when the state is $H$, she sees signal $(1,3 / 2)$ with probability 1.\n\n|  | $\\nu=1 / 2$ | $v=1$ |  | $(1,0)$ | (1,3/2) |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| $\\omega=L$ | $(1,0)$ | $(1,0)$ | $\\omega=L$ | 1 | 0 |\n| $\\omega=H$ | (1,3/2) | (1,3/2) | $\\omega=H$ | 0 | 1 |\n\nTable 3: Trade probabilities and payments in optimal Myersonian mechanism (left); calibrated information structure (right). We describe signals as allocations, because the mechanism does not screen the buyer's values.\n\nWhen the buyer has access to the calibrated information structure before playing the mechanism, the surplus extraction mechanism does not satisfy the buyer's participation constraints, which must hold for each buyer value and each signal she observes. In particular, when the buyer sees signal (1,3/2), she knows her payoff in the mechanism is negative and quits. Thus, the calibration constraint prevents the seller from extracting the buyer's surplus. In this case, the restriction induced by calibration endogenously provides the buyer with withdrawal rights, which, as Haberman and Jagadeesan (2025) show, prevent sellers from employing Crémer-McLean-style schemes. ${ }^{15}$\n\n[^8]Consider now the mechanism in the left panel of Table 4, which corresponds to posting a price of 1/2 when the state is $L$ and a price of 1 when the state is $H$. The right panel of Table 4 depicts the calibrated information structure. Note that when the state is $H$, the information structure sends with probability 1 the interim allocation rule \\{(1/2, (0, 0)), (1, (1, 1))\\}, representing that if the buyer reports her value is 1/2 she gets nothing and pays nothing, whereas if her report is 1 , she obtains the good at a price of 1 .\n\n|  | $\\nu=1 / 2$ | $v=1$ |  | \\{(1,1/2)\\} | \\{(1/2,(0,0)),(1,(1,1))\\} |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| $\\omega=L$ | (1,1/2) | (1,1/2) | $\\omega=L$ | 1 | 0 |\n| $\\omega=H$ | $(0,0)$ | $(1,1)$ | $\\omega=H$ | 0 | 1 |\n\nTable 4: Trade probabilities and payments in optimal calibrated mechanism (left); calibrated information structure (right)\n\nNote that the mechanism is incentive compatible and individually rational when the buyer has access to the calibrated information structure. As the results that follow allow us to establish, this is indeed the optimal calibrated mechanism.\n\nPrivate value environments A natural case to consider is that when agents' payoffs are state independent, that is, for each agent $i$, the agent's utility function can be written as $u_{i}\\left(a_{i}, \\theta_{i}\\right)$. Under private values, the state describes either statistical information about the agents' types as in Example 1, or a payoff-relevant variable for the designer.\n\nTheorem 1 collects our main characterization result for this case. To state it, let $\\phi_{\\text {full }}$ denote the following mechanism: For each $(\\omega, \\varepsilon) \\in \\Omega \\times[0,1], \\phi_{\\text {full }}(\\cdot, \\omega, \\varepsilon): \\Theta \\rightarrow \\Delta(A)$ is the designer optimal incentive compatible and individually rational direct mechanism when it is common knowledge that the state is $\\omega$.\n\nTheorem 1 (Private values). Under private values, the designer's payoff under the optimal calibrated mechanism is the same payoff he would obtain by choosing $\\phi_{\\text {full }}$.","text_sha256":"6228a326b1af31d86bb6252850f8c0097fd5c21700681c08932159a700029017"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0009","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Calibrated Mechanism Design","text":"That is, in private values environments, the calibration constraint pushes the designer toward full transparency. In particular, in settings with transferable utility in which the designer possesses statistical information about the agents' types, Theorem 1 implies the designer cannot engage in Crémer-McLean style schemes under calibration, and hence extract full surplus. Whereas the implication of calibrated mechanism design in private values environments is powerful, the result is fairly intuitive: The designer benefits from making the mechanism opaque by pooling states inasmuch as it weakens the incentive or participation constraints of the agents. Under private values, however, agents' incentive constraints depend on the state only through the mechanism, and calibration imposes constraints on the mechanism state-by-state. ${ }^{16}$\n\n[^9]","text_sha256":"43dd512441808fa09f5ff337fab98d8cd6223c92faa9e465c9cb2018ab51e28f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0010","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Two-stage mechanisms","text":"## 3 Two-stage mechanisms\n\nIn this section, we introduce an alternative representation of calibrated mechanisms that we use throughout our illustrations. We introduce it first for the case of a single agent and then for multiple agents.\n\nSingle-agent case and two-stage mechanisms We find it instructive to first consider the case $N=1$, and for simplicity drop the subscripts 1 from the notation. Consider a mechanism $\\phi$ and its calibrated information structure $\\pi_{\\phi}$. When the agent of type $\\theta$ observes signal $s^{*}$, two things happen: On the one hand, the agent updates her prior, $\\mu_{0}(\\omega \\mid \\theta),^{17}$ to some belief $\\mu\\left(\\theta, s^{*}\\right) \\in \\Delta(\\Omega)$. On the other hand, the agent learns that she faces allocation rule $s^{*}$ in the mechanism. Thus, her payoff in the mechanism when her type is $\\theta$, observes signal $s^{*}$, and reports $\\theta^{\\prime}$ can be written as follows:\n\n$$\n\\mathbb{E}_{(\\omega, \\varepsilon)}\\left[u\\left(\\phi\\left(\\theta^{\\prime}, \\omega, \\varepsilon\\right), \\theta, \\omega\\right) \\mid\\left(\\theta, s^{*}\\right)\\right]=\\sum_{a \\in A} s^{*}\\left(a \\mid \\theta^{\\prime}\\right)\\left(\\sum_{\\omega \\in \\Omega} \\mu\\left(\\omega \\mid \\theta, s^{*}\\right) u(a, \\theta, \\omega)\\right) .\n$$\n\nIn other words, the information structure $\\pi_{\\phi}$ provides the agent with all the necessary information to evaluate her payoffs in the mechanism: her belief about the state and her allocation rule. This allocation rule $s^{*}: \\Theta \\rightarrow \\Delta(A)$ satisfies two properties. First, because under calibration $s^{*}$ is the true interim allocation rule faced by the agent, she learns no further information about the state beyond that contained in $\\mu\\left(\\theta, s^{*}\\right)$. Second, Equations $\\operatorname{IC}\\left(\\theta_{i}, s_{i}^{*}\\right)$ and $\\operatorname{IR}\\left(\\theta_{i}, s_{i}^{*}\\right)$ imply the allocation rule is incentive compatible and individually rational when the agent holds belief $\\mu\\left(\\theta, s^{*}\\right)$.\n\nThe above discussion suggests an alternative representation of a calibrated mechanism, which we dub a two-stage mechanism and define as follows:\n\nDefinition 3 (Two-stage mechanisms). A two-stage mechanism is a mapping $\\psi: \\Theta \\times \\Omega \\rightarrow \\Delta(A \\times \\Delta(\\Omega))$ such that a Bayes plausible Blackwell experiment $\\beta: \\Omega \\rightarrow \\Delta(\\Delta(\\Omega))$ and an allocation rule $\\alpha: \\Theta \\times \\Delta(\\Omega) \\rightarrow$ $\\Delta(A)$ exist such that for all $(\\theta, \\omega) \\in \\Theta \\times \\Omega$ and all measurable subsets $\\tilde{\\Delta} \\subset \\Delta(\\Omega),{ }^{18}$\n\n$$\n\\psi(\\{a\\} \\times \\tilde{\\Delta} \\mid \\theta, \\omega)=\\int_{\\tilde{\\Delta}} \\alpha(a \\mid \\theta, \\mu) \\beta(d \\mu \\mid \\omega)\n$$\n\nWe say the two-stage mechanism is incentive compatible and individually rational if on the support of $\\mu_{0} \\otimes \\beta$, the allocation rule $\\alpha(\\cdot \\mid \\cdot, \\mu): \\Theta \\rightarrow \\Delta(A)$ is incentive compatible and individually rational conditional on the agent observing $\\mu$.\n\nIn a two-stage mechanism, the designer first discloses information about $\\omega$ in the form of a belief $\\mu$ about $\\Omega$, and conditional on that belief-but not the state-offers a direct mechanism $\\alpha(\\cdot \\mid \\cdot, \\mu)$ : $\\Theta \\rightarrow \\Delta(A)$. Two aspects of two-stage mechanisms are worth highlighting: First, the disclosure is type-independent. The designer discloses information to the agent without first communicating\n\n[^10]with the agent. Second, because the direct mechanism $\\alpha(\\cdot \\mid \\cdot, \\mu)$ does not depend on $\\omega$, observing the allocation reveals no further information about the state.\n\nLastly, when we say the experiment $\\beta$ is Bayes plausible, we mean that the distribution of posteriors induced by $\\beta$ has mean $\\mu_{0}$, and hence we can interpret $\\mu$ as the designer's belief about the state conditional on observing $\\mu$. ${ }^{19}$ Whereas the designer and the agent do not necessarily have the same beliefs about the state, the agent's beliefs about the state conditional on observing $\\mu$ obtain from a known transformation from those of the designer (Alonso and Câmara, 2016; Laclau and Renou, 2017). Thus, ensuring Bayes plausibility with respect to $\\mu_{0}$ suffices.\n\nTheorem 2 shows that (incentive compatible and individually rational) calibrated mechanisms and two-stage mechanisms implement the same distributions over outcomes $\\vartheta \\in \\Delta(A \\times \\Theta \\times \\Omega)$ :\n\nTheorem 2 (Two-stage and calibrated mechanisms). Suppose $N=1$. An outcome distribution $\\vartheta \\in$ $\\Delta(A \\times \\Theta \\times \\Omega)$ is implementable by an incentive compatible and individually rational calibrated mechanism if and only if it is implementable by an incentive compatible and individually rational two-stage mechanism. That is, if and only if\n\n$$\n\\vartheta(a, \\theta, \\omega)=\\mu_{0}(\\omega) f(\\theta \\mid \\omega) \\int_{\\Delta(\\Omega)} \\alpha(a \\mid \\theta, \\mu) \\beta(d \\mu \\mid \\omega)\n$$\n\nfor some Bayes plausible $\\beta: \\Omega \\rightarrow \\Delta(\\Delta(\\Omega))$ and incentive compatible and individually rational $\\alpha$ : $\\Theta \\times \\Delta(\\Omega) \\rightarrow \\Delta(A)$.","text_sha256":"24635a7596f72d2566e67811310f507e08152c00fa0f841eb73f8c2168a9b5f8"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0011","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Two-stage mechanisms","text":"The proof of this and all results in this section can be found in Appendix B.\nIn the single-agent case, Theorem 2 shows that the calibrated mechanism design problem is equivalent to a standard mechanism design problem in which we restrict the designer to using a specific class of mechanisms; namely, incentive compatible and individually rational two-stage mechanisms. As we explained above, a mechanism $\\phi$ and its calibrated information structure $\\pi_{\\phi}$ can be seen as actually inducing a joint distribution over $A \\times \\Theta \\times \\Omega \\times \\Delta(\\Omega)$. Theorem 2 implies this joint distribution admits two conditional independence properties. First, the allocation is conditionally independent of the state, conditional on the agent's type and the induced belief. ${ }^{20}$ This follows from the signals $s^{*}$ carrying no further information about the state than that what is contained in the agent's belief. Second, the designer disclosed belief is conditionally independent of the agent's type conditional on the state. In the static setting of Section 2, this is because the calibrated information structure discloses information to the agent uniformly across her types. In the dynamic setting of Section 5.1, this type-independent disclosure arises endogenously because the agent's experimentation opportunities are independent of her type.\n\nTwo-stage mechanisms solve calibrated mechanism design Theorem 2 is of practical import as it provides a recipe of sorts for characterizing the designer's optimal calibrated mechanism (see the applications in Section 4). For each $\\mu \\in \\Delta(\\Omega)$, the designer chooses a mechanism $\\alpha(\\cdot \\mid \\cdot, \\mu): \\Theta \\rightarrow \\Delta(A)$ that maximizes his expected payoff when the designer believes $\\mu$ is the distribution of states, and\n\n[^11]subject to the agent's incentive compatibility and individually rational constraints conditional on the designer's belief being $\\mu$. Proceeding in this way, we obtain the designer's value function $W: \\Delta(\\Omega) \\rightarrow \\mathbb{R}$. The optimal Blackwell experiment obtains from the concavification of $W$. We illustrate this procedure with two examples:\n\nExample 1 (continued). Consider again the seller-buyer example, in which the buyer is privately informed about her value for the good and the seller knows the demand state. By Theorem 2, we can find the seller's optimal calibrated mechanism as follows. First, equate $\\mu$ with the probability that the state is $H$. For each $\\mu \\in[0,1]$, consider the following problem:\n\n$$\n\\begin{aligned}\nW(\\mu) \\equiv \\max _{(q, t): V \\rightarrow[0,1] \\times \\mathbb{R}} \\mu\\left(\\frac{2}{3} t(1)+\\frac{1}{3} t(1 / 2)\\right)+(1-\\mu)\\left(\\frac{1}{3} t(1)+\\frac{2}{3} t(1 / 2)\\right) \\\\\n\\text { s.t. }\\left\\{\\begin{array}{ll}\n(\\forall v \\in\\{1 / 2,1\\}) & v q(v)-t(v) \\geq 0 \\\\\n\\left(\\forall v, v^{\\prime} \\in\\{1 / 2,1\\}, v \\neq v^{\\prime}\\right) & v q(v)-t(v) \\geq v q\\left(v^{\\prime}\\right)-t\\left(v^{\\prime}\\right)\n\\end{array} .\\right.\n\\end{aligned}\n$$\n\nThat is, the seller chooses an incentive compatible and individually rational selling mechanism that maximizes his expected revenue when his belief is $\\mu$. Because $\\omega$ is not payoff relevant to the buyer-it is just statistical information about the buyer's valuation-and the mechanism does not depend on state, the buyer's belief about $\\omega$ does not enter her incentive constraints.\n\nThe solution to the seller's problem in Equation 6 is simple: the seller posts a price of $1 / 2$ when $\\mu \\leq 1 / 2$ and a price of 1 when $\\mu>1 / 2$. Hence, the seller's value function is given by\n\n$$\nW(\\mu)=\\max \\left\\{\\frac{1}{2}, \\mu \\frac{2}{3}+(1-\\mu) \\frac{1}{3}\\right\\},\n$$\n\nand is illustrated by the solid line in blue on Figure 1. In words, the seller either sells the good at a price of 1/2 and the buyer buys with probability 1, or he sells the good at a price of 1 and the buyer buys whenever her value is 1 , which happens with the probability in the second argument of the max.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: Seller's payoff in Example 1.\n\nThe optimal calibrated mechanism can be read from the concavification of $W$, which is the dashed, red line in Figure 1: The seller first reveals the state to the agent, and offers a price of $1 / 2$ when $\\omega=L$ and a price of 1 when $\\omega=H$.\n\nExample 1 illustrates a more general principle that provides additional intuition for Theorem 1. In the private values case and when $N=1$, the designer's value function $W: \\Delta(\\Omega) \\rightarrow \\mathbb{R}$ is convex. As Equation 6 illustrates, the designer maximizes a linear function in beliefs subject to constraints that do not depend on the induced belief. Convexity of $W$ implies full disclosure is (weakly) optimal, and Theorem 1 follows.\n\nExample 2 (Horizontal differentiation). Consider a seller who owns a good of unknown type, $\\omega \\in\\{L, R\\}$, and a buyer whose private information is indexed by $\\Theta=\\left\\{\\theta_{1}, \\theta_{2}, \\theta_{3}\\right\\}$. Assume the good's type (the state) and the buyer's types are independent, and equally likely. Table 5 describes the buyer's value for the seller's good as a function of hers and the good's type, $v(\\theta, \\omega)$. When the good is $\\omega=L$, the buyer of type $\\theta_{3}$ has the highest value for the good, whereas when the good is $\\omega=R$, the buyer of type $\\theta_{3}$ has the lowest value for the good.\n\n|  | $\\theta_{1}$ | $\\theta_{2}$ | $\\theta_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega=L$ | 1 | 2 | 3 |\n| $\\omega=R$ | 2 | 2 | 1 |\n\nTable 5: Buyer's values.\n\nSuppose the buyer's utility is quasilinear, that is, $u(q, t, \\theta, \\omega)=q v(\\theta, \\omega)-t$, and the seller wishes to maximize his revenue. Furthermore, assume the buyer's outside option is no trade.\n\nConsider first the optimal mechanism the designer would offer absent the calibration constraint, depicted in the top panel of Table 6. This mechanism asks types $\\theta_{2}$ and $\\theta_{3}$ for a payment of 2 and allocates the good with probability 1 , regardless of its kind. Instead, it asks the buyer of $\\theta_{1}$ to pay 1 in exchange for getting the good only when it is of her favorite kind $(\\omega=R)$.\n\n|  | $\\theta_{1}$ | $\\theta_{2}$ | $\\theta_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega=L$ | $(0,1)$ | $(1,2)$ | $(1,2)$ |\n| $\\omega=R$ | $(1,1)$ | $(1,2)$ | $(1,2)$ |","text_sha256":"c59ac7add2a38fa256a7aa1b8e63d2674bffc1d559bd1de4748f6378d66d179c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0012","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Two-stage mechanisms","text":"|  | $\\left\\{\\left(\\theta_{1},(0,1)\\right),\\left(\\theta_{2},(1,2)\\right),\\left(\\theta_{3},(1,2)\\right)\\right\\}$ | $\\left\\{\\left(\\theta_{1},(1,1)\\right),\\left(\\theta_{2},(1,2)\\right),\\left(\\theta_{3},(1,2)\\right)\\right\\}$ |\n| :--- | :--- | :--- |\n| $\\omega=L$ | 1 | 0 |\n| $\\omega=R$ | 0 | 1 |\n\nTable 6: Trade probabilities and transfers in the optimal mechanism (top); calibrated information structure (bottom).\n\nThe bottom panel of Table 6 depicts the information structure calibrated to the optimal mechanism. It sends two signals: when the good is $L$, the buyer can choose to either not get the good and pay 1 , or get the good and pay 2. Instead, when the good is R, the buyer is choosing between paying 1 or 2 to obtain the good with probability 1.\n\nUnder the calibrated information structure, the optimal mechanism is neither incentive compatible nor individually rational. When the good is $R$, the buyer would prefer to choose $(1,1)$ regardless of her type. Instead, when the good is $L$, the buyer of $\\theta_{1}$ would quit the mechanism instead of paying 1 and getting nothing.\n\nTo characterize the optimal calibrated mechanism, we rely again on two-stage mechanisms. Equate $\\mu$ with the probability that the good is $R$. Note that because states and types are independent, if the seller assigns probability $\\mu$ to the state being $R$, so does the buyer (and vice versa). For each $\\mu \\in[0,1]$, the seller solves the following problem\n\n$$\n\\begin{aligned}\nW(\\mu) & \\equiv \\max _{(q, t): \\Theta \\rightarrow[0,1] \\times \\mathbb{R}} \\sum_{\\theta \\in \\Theta} \\frac{1}{3} t(\\theta) \\\\\n& \\text { s.t. }\\left\\{\\begin{array}{ll}\n\\left(\\forall \\theta \\in\\left\\{\\theta_{1}, \\theta_{2}, \\theta_{3}\\right\\}\\right) & q(\\theta) \\mathbb{E}_{\\mu} v(\\theta, \\cdot)-t(\\theta) \\geq 0 \\\\\n\\left(\\forall \\theta, \\theta^{\\prime} \\in\\left\\{\\theta_{1}, \\theta_{2}, \\theta_{3}\\right\\}, \\theta^{\\prime} \\neq \\theta\\right) & q(\\theta) \\mathbb{E}_{\\mu} v(\\theta, \\cdot)-t(\\theta) \\geq q\\left(\\theta^{\\prime}\\right) \\mathbb{E}_{\\mu} v(\\theta, \\cdot)-t\\left(\\theta^{\\prime}\\right)\n\\end{array} .\\right.\n\\end{aligned}\n$$\n\nIn this case, the seller's objective function does not depend on the induced belief $\\mu$ as types and states are independent. Instead, the buyer's incentive and individual rationality constraints do depend on $\\mu$ as the state is payoff relevant. The solution to the problem in Equation 7 is a posted price, whose value depends on $\\mu$. For instance, when $\\mu \\in\\{0,1\\}$, the optimal price is 2 and the seller's revenue is 4/3. Instead, when $\\mu=2 / 3$, the optimal price is 5/3 and profits are maximal and equal to 5/3. Indeed, when $\\mu=2 / 3$, the heterogeneity across buyer types is minimized (and hence, their rents), and by setting $p=5 / 3$ all buyer types buy. The blue line in Figure 2 depicts the seller's expected profit as a function of his belief $\\mu$.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 2: Seller's profit in the two-stage mechanism\n\nThe optimal calibrated mechanism can be read from the concavification of $W$ at $\\mu_{0}=1 / 2$, depicted by the dashed red line in Figure 2. The seller provides the buyer with partial information about the good: He either reveals the good is $L$ and sells the good at a price of 2 , or he obfuscates the good-inducing a belief of 2/3-and sets a price of5/3.\n\nAnother consequence of Theorem 2 is that without loss of generality, we can focus on calibrated mechanisms with finite calibrated information structures:\n\nCorollary 1 (Support of calibrated information structures). It is without loss of generality to restrict attention to two-stage mechanisms that induce at most $|\\Omega|$ beliefs.\n\nIn other words, it is without loss of generality to focus on calibrated mechanisms that induce at most $|\\Omega|$ allocation rules.\n\nMultiple agents and generalized two-stage mechanisms In the case of multiple agents, we can also interpret a calibrated mechanism as conveying to each agent $i$ both the information she should have about the state upon seeing signal $s_{i}^{*}, \\mu_{i}\\left(\\theta_{i}, s_{i}^{*}\\right)$, and her interim allocation rule, $s_{i}^{*}: \\Theta_{i} \\rightarrow \\Delta\\left(A_{i}\\right)$. However, two differences arise relative to the single-agent case: First, each agent $i$ receives her information privately from that of other agents. Second, even if the agents put together the information they receive, this is not enough to learn the ex-post allocation rule, that is, the map from type profiles to allocations. After all, each agent $i$ observes her interim allocation rule alone. These differences are natural when we think of calibrated mechanisms as capturing the information agents stand to learn from experimenting with the mechanism: There is no reason all agents will learn the same information, and from observing her own allocations, and not those of others, an agent can only learn about her interim allocation rule, not the ex-post one.\n\nThese observations together imply that to describe the analogue of a two-stage mechanism in multiagent settings we need to (i) allow for agent-by-agent information disclosure, and (ii) keep track that the interim allocation rules are consistent with the same ex-post allocation rule. These considerations motivate the following generalization of a two-stage mechanism:\n\nDefinition 4 (Generalized two-stage mechanism). A generalized two-stage mechanism is a mapping $\\psi: \\Theta \\times \\Omega \\rightarrow \\Delta\\left(\\Delta(\\Omega)^{N} \\times A\\right)$ for which a tuple of mappings\n\n$$\n\\beta: \\Omega \\rightarrow \\Delta\\left(\\Delta(\\Omega)^{N}\\right), \\quad \\alpha_{i}: \\Theta_{i} \\times \\Delta(\\Omega) \\rightarrow \\Delta\\left(A_{i}\\right), \\quad \\alpha: \\Theta \\times \\Omega \\times \\Delta(\\Omega)^{N} \\rightarrow \\Delta(A),\n$$\n\nexist such that:","text_sha256":"010a121762273a49e45f2b050963cca81fa2c8c9439103f2330290e0ad74795a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0013","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Two-stage mechanisms","text":"1. For all $(\\theta, \\omega) \\in \\Theta \\times \\Omega$, and all measurable subsets $\\left(\\tilde{\\Delta}_{i}\\right)_{i=1}^{N} \\subset \\Delta(\\Omega)^{N}$, we have\n$$\n\\psi\\left(\\times_{i=1}^{N} \\tilde{\\Delta}_{i} \\times\\{a\\} \\mid \\theta, \\omega\\right)=\\int_{\\times_{i=1}^{N} \\tilde{\\Delta}_{i}} \\alpha\\left(a \\mid \\theta, \\omega, \\mu_{1}, \\ldots, \\mu_{N}\\right) \\beta\\left(d\\left(\\mu_{1}, \\ldots, \\mu_{N}\\right) \\mid \\omega\\right)\n$$\n2. The Blackwell experiment $\\beta$ is Bayes plausible,\n3. For all $i \\in\\{1, \\ldots, N\\}$, the interim allocation rule $\\alpha_{i}$ satisfies that for all measurable subsets $\\tilde{\\Delta}$ of $\\Delta(\\Omega)$ and all $\\left(a_{i}, \\theta_{i}, \\omega\\right) \\in A_{i} \\times \\Theta_{i} \\times \\Omega$\n$$\n\\int_{\\tilde{\\Delta} \\times \\Delta(\\Omega)^{N-1}}\\left\\{\\alpha_{i}\\left(a_{i} \\mid \\theta_{i}, \\mu_{i}\\right)-\\mathbb{E}_{f_{-i}(\\cdot \\mid \\omega)}\\left[\\sum_{a_{-i} \\in A_{-i}} \\alpha\\left(a_{i}, a_{-i} \\mid \\theta_{i}, \\theta_{-i}, \\omega, \\mu_{i}, \\mu_{-i}\\right)\\right]\\right\\} \\beta\\left(d\\left(\\mu_{i}, \\mu_{-i}\\right) \\mid \\omega\\right)=0 .\n$$\nWe say the generalized two-stage mechanism is incentive compatible and individually rational if for all $i \\in\\{1, \\ldots, N\\}$, on the support of $\\mu_{0} \\otimes \\beta, \\alpha_{i}\\left(\\cdot \\mid \\cdot, \\mu_{i}\\right)$ is incentive compatible and individually rational for agent $i$ when she learns $\\mu_{i}$.\n\nAs anticipated, generalized two-stage mechanisms differ from two-stage mechanisms in three ways when $N>1$. First, because disclosures are private, the experiment $\\beta$ now outputs a profile of beliefs, one for each agent. As shown in Arieli et al. (2024), $\\beta$ is Bayes plausible if and only if for each agent $i$, the marginal Blackwell experiment $\\beta_{i}$ is Bayes plausible. Second, while the individual interim allocation rule $\\alpha_{i}$ only depends on the disclosed belief to agent $i, \\mu_{i}$, and not the state, the ex-post\nallocation rule $\\alpha$ may depend on the state, even conditional on the belief profile $\\left(\\mu_{1}, \\ldots, \\mu_{N}\\right)$. The reason is that this belief profile is no longer a sufficient statistic for the ex-post allocation rule as each agent $i$ only observes their interim allocation. Third and relatedly, we need to keep track of both the interim allocation rules $\\left(\\alpha_{i}\\right)_{i=1}^{N}$ and the ex-post allocation rule $\\alpha$ to check that the interim allocation rules are consistent with the same mechanism. An interesting question for future work would be to characterize which interim allocation rules $\\left(\\alpha_{i}\\right)_{i=1}^{N}$ are consistent with some ex-post allocation rule $\\alpha$, so that one could focus on the interim allocation rules alone.\n\nAs we show in Proposition 1, a calibrated mechanism induces a generalized two-stage mechanism:\nProposition 1. If outcome distribution $\\vartheta \\in \\Delta(A \\times \\Theta \\times \\Omega)$ is implementable by an incentive compatible and individually rational calibrated mechanism, then it is implementable by an incentive compatible and individually rational generalized two-stage mechanism.\n\nIn contrast to the single-agent case, not every outcome distribution implemented by a generalized two-stage mechanism can be implemented by a calibrated mechanism. On the one hand, no agent's beliefs are a sufficient statistic for the information the mechanism leaks about the state, so that the allocation rule $\\bar{\\alpha}$ may still leak information about the state or others' beliefs, which in turn leak information about the state. On the other hand, because in a calibrated mechanism each agent learns her interim allocation rule conditional on $(\\omega, \\varepsilon)$, the incentive and participation constraints associated to a generalized two-stage mechanism are weaker than those implied by a calibrated mechanism whenever multiple interim allocation rules underlie the same belief: Even if the average interim allocation rule $\\alpha_{i}$ is incentive compatible and individually rational, each of the interim allocation rules underlying that average need not be.","text_sha256":"9b74ee311bb174c9ea8dc8174f18bb71ff4fd5ef91bac9e41ff1ec921db7dcfc"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0014","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Applications","text":"## 4 Applications\n\nIn this section, we study optimal calibrated mechanism design in canonical mechanism design settings with quasilinear utilities. We first consider the case of a single agent, with single-dimensional types and allocations, and supermodular payoffs. In Section 4.1, we show that if the order of types is state independent, then optimal two-stage mechanisms fully reveal the state, whereas this conclusion can be reversed when the order of types is state-dependent. In Section 4.2, we compare optimal calibrated mechanism design against the Myersonian benchmark. Lastly, we analyze a multi-agent application in Section 4.3.","text_sha256":"c047d286cdf362beee6b5a36e123bf0018cb88fe7e0c63e9ce6cfd1b3f6e8b50"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0015","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.1 Calibrated Screening","text":"### 4.1 Calibrated Screening\n\nWe consider the following version of the model in Section 2. Suppose $N=1$ and let $\\Theta=[\\underline{\\theta}, \\bar{\\theta}]$ denote the set of types. Assume $\\theta$ is distributed according to a full support distribution $F$ with density $f$. Hence, throughout, we consider the case in which the agent's type is independent of $\\omega$. Denote the set of allocations by $A=[0, \\bar{q}] \\times \\mathbb{R}$, where $q \\in[0, \\bar{q}]$ is the (physical) allocation and $t \\in \\mathbb{R}$ is a payment from the agent to the designer. ${ }^{21}$\n\n[^12]The agent's and the designer's payoffs are given by $u(q, \\theta, \\omega)-t$ and $w(q, \\theta, \\omega)+t$, respectively. Assume that if the agent does not participate, then the outside option is $a_{\\varnothing}=(0,0)$, and that this yields a payoff of 0 to both the designer and the agent. Throughout, we assume that for each $\\omega \\in \\Omega$, the family of functions $\\{\\theta \\mapsto u(q, \\theta, \\omega): q \\in[0, \\bar{q}]\\}$ is equi-Lipschitz on $\\Theta$ : a positive constant $L_{\\omega}$ exists such that for all $\\theta, \\theta^{\\prime} \\in \\Theta$ and $q \\in[0, \\bar{q}],\\left|u(q, \\theta, \\omega)-u\\left(q, \\theta^{\\prime}, \\omega\\right)\\right| \\leq L_{\\omega}\\left|\\theta-\\theta^{\\prime}\\right| .{ }^{22}$ Furthermore, the analysis that follows restricts attention to mechanisms that do not randomize on the allocation (beyond the inherent randomness of $\\Omega \\times[0,1]$ ). Remark 1 at the end of this section discusses settings in which this is not a restriction and how to generalize the observations herein when random allocations are allowed.\n\nOur goal is to characterize the designer optimal calibrated mechanism and how its properties depend on how the state affects the order of types.\n\nState-independent type ranking We consider first the case in which the order of types is independent of the state. Formally, assume that for all $\\omega \\in \\Omega$, the function $u(\\cdot, \\omega)$ is supermodular in $(q, \\theta)$. That is, in all states, the agent with higher value of $\\theta$ values $q$ more. These assumptions are satisfied, for instance, for $u(q, \\theta, \\omega)=\\theta \\omega q$ or $u(q, \\theta, \\omega)=(\\theta+\\omega) q$.\n\nBy Theorem 2, we can characterize the optimal calibrated mechanism via two-stage mechanisms. To do so, we solve the problem \"backward\": For each $\\mu \\in \\Delta(\\Omega)$ the designer may induce about the state, the designer chooses an optimal direct mechanism $\\left(q_{\\mu}, t_{\\mu}\\right): \\Theta \\rightarrow A$. This determines the designer's value function $W: \\Delta(\\Omega) \\rightarrow \\mathbb{R}$. We obtain the designer's optimal Blackwell experiment by studying the properties of $W$.\n\nGiven belief $\\mu$, define the agent's and the designer's (expected) payoff at $(q, t, \\theta)$ as follows:\n\n$$\nu(q, \\theta \\mid \\mu) \\equiv \\sum_{\\omega \\in \\Omega} \\mu(\\omega) u(q, \\theta, \\omega), w(q, \\theta \\mid \\mu) \\equiv \\sum_{\\omega \\in \\Omega} \\mu(\\omega) w(q, \\theta, \\omega)\n$$\n\nThus, conditional on inducing belief $\\mu$, the designer's problem can be written as follows:\n\n$$\n\\begin{aligned}\nW(\\mu) \\equiv \\max _{(q, t): \\Theta \\rightarrow A} \\int_{\\Theta}[w(q(\\theta), \\theta, \\mu)+t(\\theta)] F(d \\theta) \\\\\n\\text { s.t. }\\left\\{\\begin{array}{ll}\n(\\forall \\theta \\in \\Theta) & u(q(\\theta), \\theta \\mid \\mu)-t(\\theta) \\geq 0 \\\\\n\\left(\\forall \\theta, \\theta^{\\prime} \\in \\Theta\\right) & u(q(\\theta), \\theta \\mid \\mu)-t(\\theta) \\geq u\\left(q\\left(\\theta^{\\prime}\\right), \\theta \\mid \\mu\\right)-t\\left(\\theta^{\\prime}\\right)\n\\end{array} .\\right.\n\\end{aligned}\n$$\n\nOur assumptions imply that $u(\\cdot \\mid \\mu)$ is supermodular in $(q, \\theta)$. It follows that the designer can only choose among those $q: \\Theta \\rightarrow[0, \\bar{q}]$ that are (weakly) increasing in $\\theta$. Let $Q_{\\uparrow}$ denote the set of all such $q(\\cdot)$. Furthermore, at the optimum, the participation constraint of $\\theta=\\underline{\\theta}$ binds.\n\nDefine the virtual surplus at $(q, \\theta, \\omega)$ as follows:\n\n$$\nJ((q, \\theta, \\omega) ; F)=w(q, \\theta, \\omega)+u(q, \\theta, \\omega)-u_{2}(q, \\theta, \\omega) \\frac{1-F(\\theta)}{f(\\theta)},\n$$\n\nwhere $u_{2}$ is the derivative of $u$ against its second coordinate; the equi-Lipschitz assumption implies it exists almost everywhere. Then, conditional on inducing belief $\\mu$, the designer's payoff can be written\n\n[^13]as follows:\n$$\nW(\\mu)=\\max _{q \\in Q_{\\uparrow}} \\int_{\\Theta} \\mathbb{E}_{\\mu}[J((q(\\theta), \\theta, \\omega) ; F)] F(d \\theta) .\n$$\nNote the objective is linear in $\\mu$ and the constraint set is independent of $\\mu$. We conclude that $W$ is convex, as it is the maximum of linear functionals in $\\mu$. It follows that full disclosure is an optimal experiment for the designer. Equivalently, an optimal calibrated mechanism exists in which the designer chooses the mechanism $\\phi_{\\text {full }}^{D}$, where for all $\\omega \\in \\Omega$, $\\phi_{\\text {full }}^{D}(\\cdot, \\omega, \\cdot): \\Theta \\times[0,1] \\rightarrow A$ is the optimal deterministic mechanism when it is common knowledge that the state is $\\omega$.\n\nProposition 2 summarizes the above discussion:\nProposition 2 (State-Independent Type Ranking). In a single-dimensional screening problem with state-independent type ranking, the designer can do no better than choosing $\\phi_{\\text {ful }}^{D}$ among deterministic mechanisms.\n\nBy Proposition 2, in screening problems with state-independent ranking of types across states, the calibration constraint makes any pooling of mechanisms across states unprofitable. ${ }^{23}$ Remarkably, this result holds for any designer objective, such as profit, revenue, or efficiency. It also requires no regularity assumptions on the type distribution, as we do not obtain the result by looking at the relaxed problem. Instead, our argument relies on the restriction to deterministic mechanisms (conditional on the induced belief), which in turn delivers that the set of implementable allocations does not depend on the induced belief. Remark 1 discusses conditions under which (i) the restriction to deterministic mechanisms is without loss of optimality, and (ii) the set of implementable allocations does not depend on the induced belief, even when randomized mechanisms are allowed. Readers interested in the case of state-dependent ranking can skip this remark with little loss of continuity.","text_sha256":"b63443dd059f45f366d41ac3157e361a9e96258d14508097d8c344f49abb61ab"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0016","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.1 Calibrated Screening","text":"Remark 1 (Proposition 2 without deterministic mechanisms). Under our assumptions, deterministic mechanisms are without loss of optimality if the agent's payoff is linear in $q$ and the designer's payoff is concave in $q$. (See Section 4.2 for yet another condition.) However, the driving force behind Proposition 2 is that the designer's constraint set does not depend on the induced belief. The state-bystate supermodularity assumption and the restriction to deterministic mechanisms is one way to ensure this is the case. We now discuss two other cases in which the designer's constraint set does not depend on the induced belief and thus Proposition 2 holds for the optimal (not necessarily deterministic) calibrated mechanism.\n\nFirst, suppose the agent's payoff is linear in $q$, so that $u(q, \\theta, \\omega)=q v(\\theta, \\omega)$, where $v(\\cdot, \\omega)$ is increasing for all $\\omega$. Then, the set of implementable lotteries over $q$ when the belief is $\\mu$ is given by:\n\n$$\nQ_{\\uparrow, \\text { random }}=\\left\\{\\xi: \\Theta \\rightarrow \\Delta([0, \\bar{q}]): \\mathbb{E}_{\\xi(\\theta)}[q] \\text { is increasing in } \\theta\\right\\} .\n$$\n\nIn this case, we obtain that the designer cannot do any better than choosing $\\phi_{\\text {ful }}$, which is the\n\n[^14]mechanism that implements in each state $\\omega$ the optimal mechanism under common knowledge that the state is $\\omega$. This relates to the results in Szabadi (2018) and Yamashita (2018), who study the optimal mechanism design preceded by public information disclosure. Both papers consider settings in which the agent's payoff is linear in $q$ and obtain that full disclosure is optimal when the ranking of types is independent of the state. ${ }^{24}$\n\nSecond, suppose the agent's payoff has the form\n\n$$\nu(q, \\theta, \\omega)=b(\\theta) c(\\omega) v(q)+k_{1}(q, \\omega)+k_{2}(\\theta, \\omega),\n$$\n\nwhere $b$ is increasing in $\\theta$ and $c(\\cdot)$ does not change sign on $\\Omega$. Under this assumption, $u(q, \\theta \\mid \\mu)$ satisfies monotonic expectational differences for all $\\mu \\in \\Delta(\\Omega)$ (see, e.g., Kartik et al., 2024). Consequently, one can define a linear order $\\succeq$ over $\\Delta([0, \\bar{q}])$ as follows: $\\xi \\succeq \\xi^{\\prime}$ if $u(\\xi, \\theta \\mid \\mu)-u\\left(\\xi^{\\prime}, \\theta \\mid \\mu\\right)$ is increasing in $\\theta$, where $u(\\xi, \\theta \\mid \\mu)$ is the linear extension of $u(\\cdot, \\theta \\mid \\mu)$ to $\\Delta([0, \\bar{q}])$. This linear order implies the ranking of types is state independent. Indeed, the analog of $Q_{\\uparrow, \\text { random }}$ is the set of all $\\xi: \\Theta \\rightarrow \\Delta([0, \\bar{q}])$ such that $\\theta \\geq \\theta^{\\prime}$ implies $\\xi(\\theta) \\succeq \\xi\\left(\\theta^{\\prime}\\right)$, which is again independent of the induced belief.\n\nState-dependent type ranking Example 2 illustrates that when the ranking of types is not uniform across states, full transparency may not be optimal. ${ }^{25}$ We now provide a more systematic analysis of this phenomenon, using the previous results. To provide the starkest contrast with Proposition 2, we consider a setting that shares a key feature of Example 2: we can partition $\\Delta(\\Omega)$ into two regions such that within each region the ranking of types-as determined by $u(q, \\theta \\mid \\mu)$-is the same, but it differs across regions.\n\nConcretely, suppose that $\\Omega=\\left\\{\\omega_{1}, \\omega_{2}\\right\\}$. Furthermore, assume\n\n$$\nu(q, \\theta, \\omega)=\\left\\{\\begin{array}{ll}\nq \\theta & \\text { if } \\omega=\\omega_{2} \\\\\nq(c-b \\theta) & \\text { otherwise, }\n\\end{array},\\right.\n$$\n\nwhere $c \\in \\mathbb{R}$, and $b>0$. Identify beliefs with the probability that the state is $\\omega_{2}$ and define\n\n$$\n\\hat{\\mu}=\\frac{b}{1+b} .\n$$\n\nFor $\\mu<\\hat{\\mu}$, we have that $u(q, \\theta \\mid \\mu)$ is decreasing in $\\theta$, whereas if $\\mu>\\hat{\\mu}$, then $u(q, \\theta \\mid \\mu)$ is increasing in $\\theta$.\nConsider now the designer's optimal payoff $W: \\Delta(\\Omega) \\rightarrow \\mathbb{R}$ as a function of the different beliefs he may induce. When $\\mu \\geq \\hat{\\mu}$, the designer's payoff can be obtained by solving the program in Equation 9 as before. Instead, when $\\mu<\\hat{\\mu}$, the designer's payoff can be obtained by solving a problem analogous to Equation 9, but where the space of implementable allocations is the set of decreasing $q, Q_{\\downarrow}$, and the participation constraint of $\\bar{\\theta}$ binds. It follows that $W$ is convex on $[0, \\hat{\\mu})$ and $(\\hat{\\mu}, 1]$. Thus, the support of designer's optimal experiment is included in \\{0, $\\hat{\\mu}, 1\\}$.\n\n[^15]Proposition 3 (State-dependent type ranking). Suppose the agent's payoff satisfies the assumptions above. If $(1-\\hat{\\mu}) W(0)+\\hat{\\mu} W(1) \\geq W(\\hat{\\mu})$, full transparency is optimal. Otherwise, full transparency is not optimal: if $\\mu_{0}<\\hat{\\mu}$, it is optimal to split $\\mu_{0}$ to 0 and $\\hat{\\mu}$; if $\\mu_{0}>\\hat{\\mu}$, it is optimal to split $\\mu_{0}$ to $\\hat{\\mu}$ and 1 . In particular, if $W(\\hat{\\mu})>\\max \\{W(0), W(1)\\}$, then full transparency is not optimal.\n\nBy Proposition 3, whether full transparency is optimal depends on the designer and agent's payoffs and the type distribution, but only through their impact on the value the function $W$ takes at points $\\{0, \\hat{\\mu}, 1\\}$. At $\\hat{\\mu}$, the agent earns no rents-as $u(\\cdot \\mid \\hat{\\mu})$ is constant across types-which pushes against full transparency. At the same time, efficiency may dictate the designer to condition the allocation rule on the state, which favors information disclosure. The piecewise convexity of $W$ implies that if $W(\\hat{\\mu})$ dominates $W$ at the extreme beliefs, the rent extraction motive dominates and the designer does not engage in full disclosure.","text_sha256":"295a0bb6abec11efde819dccc9f15b46b6f5c2c7db1c1b725964990ca5b9bc56"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0017","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.2 Comparison with Myersonian Mechanism Design","text":"### 4.2 Comparison with Myersonian Mechanism Design\n\nWe now compare optimal calibrated mechanism design and the Myersonian benchmark. In the Myersonian benchmark, the designer is not concerned with the information the mechanism reveals about the state, and hence provides a natural upper bound on the designer's payoffs in calibrated mechanism design. The gap between the designer's optimal payoff across both benchmarks quantifies the loss from the calibration constraint. If no gap exists, the calibration constraint is non-binding and an optimal calibrated mechanism can be found solving the Myersonian benchmark. Instead, if a gap exists, the optimal mechanism in the Myersonian benchmark reveals information about the state in a way that it fails to be incentive compatible or individually rational under calibration.\n\nMyersonian benchmark In the Myersonian benchmark, the designer chooses a direct mechanism $(\\xi, t): \\Theta \\times \\Omega \\rightarrow \\Delta([0, \\bar{q}]) \\times \\mathbb{R}$ subject to incentive and participation constraints that must hold on average across states under the prior $\\mu_{0} .{ }^{26}$ Formally,\n\n$$\n\\begin{aligned}\n& W_{\\mathrm{My}} \\equiv \\max _{(q, t): \\Theta \\times \\Omega \\rightarrow[0, \\bar{q}] \\times \\mathbb{R}} \\int_{\\Theta} \\mathbb{E}_{\\mu_{0}}[w(\\xi(\\theta, \\omega), \\theta, \\omega)+t(\\theta, \\omega)] F(d \\theta) \\\\\n& \\text { s.t. }\\left\\{\\begin{array}{l}\n(\\forall \\theta \\in \\Theta) \\mathbb{E}_{\\mu_{0}}[u(\\xi(\\theta, \\omega), \\theta, \\omega)-t(\\theta, \\omega)] \\geq 0 \\\\\n\\left(\\forall \\theta, \\theta^{\\prime} \\in \\Theta\\right) \\mathbb{E}_{\\mu_{0}}[u(\\xi(\\theta, \\omega), \\theta, \\omega)-t(\\theta, \\omega)] \\geq \\mathbb{E}_{\\mu_{0}}\\left[u\\left(\\xi\\left(\\theta^{\\prime}, \\omega\\right), \\theta, \\omega\\right)-t\\left(\\theta^{\\prime}, \\omega\\right)\\right]\n\\end{array},\\right.\n\\end{aligned}\n$$\n\nwhere $w(\\xi, \\theta, \\omega)$ and $u(\\xi, \\theta, \\omega)$ are the linear extensions of $w(\\cdot, \\theta, \\omega)$ and $u(\\cdot, \\theta, \\omega)$, respectively.\nProgram $\\mathrm{OPT}_{\\mathrm{My}}$ is a mechanism design problem with a multidimensional allocation, corresponding to assigning (a distribution over) $q$ in each state. As a result, the distinction between the Myersonian benchmark and optimal calibrated design shows in the monotonicity requirements the allocation $\\xi(\\theta, \\omega)$ must satisfy for a transfer $t: \\Theta \\times \\Omega \\rightarrow \\mathbb{R}$ to exist that implements $\\xi(\\theta, \\omega)$. Indeed, implementability of $\\xi: \\Theta \\times \\Omega \\rightarrow \\Delta([0, \\bar{q}])$ is equivalent to integral monotonicity (Rochet, 1987; Pavan et al., 2014): ${ }^{27}$\n\n$$\n\\left(\\forall \\theta, \\theta^{\\prime} \\in \\Theta\\right) \\int_{\\theta^{\\prime}}^{\\theta} \\int_{\\Omega}\\left[u_{2}(\\xi(s, \\omega), s, \\omega)-u_{2}\\left(\\xi\\left(\\theta^{\\prime}, \\omega\\right), s, \\omega\\right)\\right] d \\mu_{0} d s \\geq 0\n$$\n\n[^16]where recall $u_{2}$ is the derivative of $u$ in its second coordinate.\n\nComparison with calibrated mechanism design To facilitate the comparison with Proposition 2, we focus on deterministic mechanisms $(q, t): \\Theta \\times \\Omega \\rightarrow[0, \\bar{q}] \\times \\mathbb{R}$. Remarkably, even if the agent's payoff net of transfers, $u(q, \\theta, \\omega)$, is supermodular in $(q, \\theta)$ for all $\\omega \\in \\Omega$, the characterization of the set of implementable $q(\\cdot)$ cannot be simplified beyond integral monotonicity without further assumptions. Because integral monotonicity is a global, implicitly defined constraint, verifying implementability and computing the optimal mechanism is more computationally involved in the Myersonian benchmark than in calibrated mechanism design. Indeed, Proposition 2 implies the optimal deterministic calibrated mechanism coincides with the state-by-state optimal deterministic mechanism under this assumptions. In other words, the optimal deterministic calibrated mechanism can be obtained by selecting allocations $q(\\cdot)$ that satisfy\n\n$$\nQ_{\\mathrm{cal}}=\\{q: \\Theta \\times \\Omega \\rightarrow[0, \\bar{q}]:(\\forall \\omega \\in \\Omega) q(\\cdot, \\omega) \\text { is increasing }\\} .\n$$\n\nThat is, the allocation in the optimal calibrated mechanism must satisfy monotonicity state-by-state. Instead, the optimal deterministic Myersonian mechanism can be obtained by selecting allocations $q(\\cdot)$ that satisfy Equation IM, which we denote by $Q_{\\mathrm{My}}$.\n\nWhen $u(q, \\theta, \\omega)$ is supermodular in $(q, \\theta)$ for all $\\omega \\in \\Omega$, the above discussion implies the designer's optimal payoff in the Myersonian and calibration settings can be written as follows:\n\n$$\n\\begin{aligned}\nW_{\\mathrm{My}}^{D} & =\\max _{q \\in Q_{\\mathrm{My}}} \\int_{\\Theta} \\mathbb{E}_{\\mu_{0}}[J(q(\\theta, \\omega), \\theta, \\omega ; F)] F(d \\theta), \\\\\nW_{\\mathrm{cal}}^{D} & =\\max _{q \\in Q_{\\mathrm{cal}}} \\int_{\\Theta} \\mathbb{E}_{\\mu_{0}}[J(q(\\theta, \\omega), \\theta, \\omega ; F)] F(d \\theta),\n\\end{aligned}\n$$\n\nwhere the superscript $D$ in the objective is a reminder that we restrict attention to mechanisms that are deterministic conditional on the state, or the induced belief.\n\nBy reducing the comparison across settings to monotonicity requirements on the space of allocations, the above expressions provide us with an immediate way of comparing the designer's payoffs across settings. In particular, when the optimal Myersonian mechanism satisfies the state-by-state monotonicity constraints, we have that the calibration constraint entails no loss to the designer. We record this observation for future use:\n\nObservation 1. Suppose $u(q, \\theta, \\omega)$ is supermodular in $(q, \\theta)$ for all $\\omega \\in \\Omega$. Then, if the allocation rule in the Myersonian benchmark satisfies monotonicity state-by-state, $W_{\\text {cal }}^{D}=W_{M y}^{D}$.\n\nTwo natural questions are under what conditions the solution to OPT $_{\\text {My }}$ is deterministic and satisfies state-by-state monotonicity. We answer them simultaneously by studying the relaxed program. Inspection of Equation 10 reveals that if the virtual surplus is supermodular in $(q, \\theta)$ for every $\\omega$, then the solution $q_{\\text {rel }}$ to the relaxed problem\n\n$$\nW_{\\mathrm{rel}}=\\max _{q: \\Theta \\times \\Omega \\rightarrow[0, \\bar{q}]} \\int_{\\Theta} \\mathbb{E}_{\\mu_{0}}[J(q(\\theta, \\omega), \\theta, \\omega ; F)] F(d \\theta),\n$$","text_sha256":"d9ec572184ea97e55ac72b86b5803a17d9c954801074c98e8d557e117d094825"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0018","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.2 Comparison with Myersonian Mechanism Design","text":"satisfies monotonicity state-by-state by Topkis' theorem. Moreover, a stochastic mechanism is\nequivalent to a deterministic mechanism which depends on the random reports of a fictitious agent (Pavan et al., 2014). The virtual surplus in this fictitious setting coincides with that in the integrand on the right-hand side of Equation 11-the type reports of the fictitious agent are payoff irrelevant-and is maximized by $q_{\\text {rel }}$.\n\nProposition 4 (Sufficient condition for no gap). Suppose the virtual surplus $J((q, \\theta, \\omega) ; F)$ is supermodular in $(q, \\theta)$ for all $\\omega$. Then, the designer's payoffs under the optimal Myersonian and calibrated mechanisms coincide.\n\nBy contrast to Proposition 2, Proposition 4 relies on assumptions on the type distribution and the designer's payoff. As Example 2 illustrates, the supermodularity of the virtual surplus can fail when the type distribution is not regular, creating a gap between the designer's payoff at the optimal Myersonian and calibrated mechanisms. Example 3 illustrates such a gap can also arise when the designer's payoff is not supermodular:\n\nExample 3 (Payoff gap when $w$ is not supermodular). Suppose states are binary, $\\Omega=\\left\\{\\omega_{L}, \\omega_{H}\\right\\}=\\{1,3\\}$, and equally likely. Suppose types are uniformly distributed, $\\theta \\sim U[0,1]$. Finally, let $q \\in[0,1]$ denote the probability the seller's good is allocated. Payoffs are given by:\n\n$$\n\\begin{aligned}\nu(q, \\theta, \\omega) & =q \\theta \\omega \\\\\nw(q, \\theta, \\omega) & =2(1-2 \\theta) q .\n\\end{aligned}\n$$\n\nNote that $w$ is increasing in $q$ when $\\theta<1 / 2$ and decreasing in $q$ when $\\theta>1 / 2 .{ }^{28}$ In this case, the virtual surplus evaluated at different states is:\n\n$$\nJ((q, \\theta, \\omega) ; F)=\\left\\{\\begin{array}{ll}\n(1-2 \\theta) q & \\text { if } \\omega=\\omega_{L} \\\\\n(2 \\theta-1) q & \\text { otherwise }\n\\end{array} .\\right.\n$$\n\nIn the Myersonian benchmark, implementable allocations are elements of $Q_{M y}$, which in this case is equivalent to requiring that $\\mathbb{E}_{\\mu_{0}}[q(\\cdot, \\omega) \\omega]$ is increasing. The optimal Myersonian allocation obtains from pointwise maximizing the virtual surplus, and is given by:\n\n$$\nq_{M y}(\\theta, \\omega)=\\left\\{\\begin{array}{ll}\n1 & \\text { if } \\omega=\\omega_{L} \\text { and } \\theta<1 / 2 \\\\\n1 & \\text { if } \\omega=\\omega_{H} \\text { and } \\theta>1 / 2 \\\\\n0 & \\text { otherwise }\n\\end{array} .\\right.\n$$\n\nThe designer's payoff under the Myersonian mechanism is 1/4.\nBy Proposition 2, $q_{M y}$ cannot be implemented by a calibrated mechanism as it is not increasing state-bystate. Intuitively, when $\\omega=\\omega_{L}$, types above 1/2 would learn from the calibrated information structure that they do not obtain the good, whereas types below 1/2 do, and would misreport their types.\n\nInstead, in the optimal calibrated mechanism, the designer sets $q_{\\text {cal }}\\left(\\theta, \\omega_{H}\\right)=\\mathbb{1}[\\theta \\geq 1 / 2]$ and sets\n\n[^17]$q\\left(\\theta, \\omega_{L}\\right)$ to be constant in $\\theta$. The designer's payoff under calibration is $W_{\\text {cal }}=1 / 8<W_{M y}$.","text_sha256":"1637c3e99880f76b74162a328fbeb7cacbd7a98d9e48c4861a62bdbdc91f5f36"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0019","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.3 Optimal Calibrated Auction","text":"### 4.3 Optimal Calibrated Auction\n\nIn this section, we consider a multiple agent application and study the design of the optimal calibrated auction. Proposition 1 implies the optimal calibrated auction induces a generalized two-stage mechanism, and hence the optimal generalized two-stage mechanism provides an upper bound on the designer's optimal payoff under calibration. However, computing the optimal generalized two-stage mechanism is complicated because (i) no tractable characterization of joint distributions over posterior beliefs is available, and (ii) the allocation rule may condition on the state and not only the agents' beliefs. For that reason, our analysis below relies on Observation 1: We show the optimal Myersonian auction can be implemented by fully revealing the state, and hence, remains incentive compatible and individually rational when the agents have access to the calibrated information structure. Below, we first specialize our multi-agent model and notation to the auction application and then link our assumptions to online advertising.\n\nSuppose there is a single good for sale and the state is multidimensional, $\\omega=\\left(\\omega_{i}, \\omega_{0 i}\\right)_{i \\in[N]} \\in \\mathbb{R}_{+}^{2 N}$, and distributed according to prior distribution $\\mu_{0}$. Suppose that for all $i \\in[N], \\Theta_{i}=[0,1]$, with $\\theta_{i} \\sim F_{i}$ with full-support density $f_{i}$. That is, we are assuming agents' types are independent of the state, and hence, independent across each other. Denote by $q_{i} \\in[0,1]$ the probability agent $i$ is allocated the good, and note that feasibility implies that $0 \\leq \\sum_{i=1}^{N} q_{i} \\leq 1$.\n\nWe assume the agents' and the designer's utilities are quasilinear in transfers. Agent $i$ 's payoff net of transfers is $u_{i}\\left(q_{i}, \\theta_{i}, \\omega\\right)=q_{i}\\left(\\omega_{i} \\theta_{i}+\\omega_{0 i}\\right)$. Thus, state components $\\omega_{i}$ capture the value responsiveness to agent's private information, whereas state components $\\omega_{0 i}$ capture the overall shift. The state components can be correlated (and asymmetric) across agents, allowing for interdependent values. The designer's payoff net of transfers is $w(q, \\theta, \\omega)=\\sum_{i} q_{i} w_{i}(\\theta, \\omega)$ for some functions $\\left(w_{i}\\right)_{i \\in[N]}$. Below, we study the designer-optimal calibrated mechanism.\n\nTo fix ideas, consider the following mapping to an online advertising environment. The designer is an advertising platform, and the good is an advertising slot on a given webpage targeted to a selected category of users in a given week. Agents are firms that wish to display their ads, and their private types represent the expected revenue from a click on their ad. State components $\\omega_{i}$ could capture individual click-through rates or match values, while state components $\\omega_{0 i}$ could capture individual display values, that is, the expected revenue from an ad being displayed irrespective of whether it is clicked (for instance, due to brand-building effects). The state is observed through proprietary data available to the platform and can be used in the design of the auction. The platform values the resulting revenue but may also have additional efficiency considerations, summarized by $w_{i}$.\n\nAs anticipated, we characterize the optimal calibrated mechanism by showing that it coincides with the Myersonian optimal one. To this end, consider the Myersonian problem, in which the designer chooses $(q(\\theta, \\omega), t(\\theta, \\omega)) \\in[0,1]^{N} \\times \\mathbb{R}^{N}$. Because agent $i$ 's payoff is linear in $\\theta_{i}$, arguments analogous to those in Section 4.2 imply a feasible $q(\\theta, \\omega)$ is implementable if and only if for all $i, \\mathbb{E}_{F_{-i}, \\mu_{0}}\\left[q_{i}\\left(\\theta_{i}, \\theta_{-i}, \\omega\\right) \\omega_{i}\\right]$ is increasing in $\\theta_{i}$. In a slight abuse of notation, denote by $Q_{\\mathrm{My}}$ the set of all such functions and define\nthe virtual surplus as\n\n$$\nJ((q, \\theta, \\omega) ; F)=\\sum_{i=1}^{N} q_{i}(\\theta, \\omega)\\left(w_{i}(\\theta, \\omega)+\\left(\\theta_{i}-\\frac{1-F_{i}\\left(\\theta_{i}\\right)}{f_{i}\\left(\\theta_{i}\\right)}\\right) \\omega_{i}+\\omega_{0 i}\\right) .\n$$\n\nStandard arguments imply the individual rationality constraint of $\\theta_{i}=0$ binds for all $i$, and an optimal mechanism solves\n\n$$\nW_{\\mathrm{My}}=\\max _{q \\in Q_{\\mathrm{My}}} \\int_{[0,1]^{N}} \\mathbb{E}_{\\mu_{0}}[J(q(\\theta, \\omega), \\theta, \\omega ; F)] f(\\theta) d \\theta .\n$$\n\nProposition 5 (No gap in regular auctions). Suppose that (i) for all $i \\in\\{1, \\ldots, N\\}, F_{i}$ is Myerson regular, and for all $i, j, \\theta$, and $\\omega, w_{i \\theta_{i}}(\\theta, \\omega) \\geq 0, w_{i \\theta_{i}}(\\theta, \\omega) \\geq w_{j \\theta_{i}}(\\theta, \\omega) .{ }^{29}$ Then, $W_{\\text {cal }}=W_{M y}$.\n\nThe proof of Proposition 5 in Appendix D. 2 shows that under our assumptions the optimal Myersonian mechanism can be obtained by solving the relaxed program. Importantly, the assumption that $w_{i \\theta_{i}}(\\theta, \\omega) \\geq w_{j \\theta_{i}}(\\theta, \\omega)$ ensures that an increase in agent $i$ 's type increases the designer's payoff of giving the object to agent $i$ by more than the value of giving it to other agents. This, in turn, ensures agent $i$ 's allocation probability is increasing in her type.\n\nViewed through the lens of the online advertising example, Proposition 5 implies that in regular environments, while the advertising platform benefits from having the data on click-through rates and display values, it does not benefit from the informational advantage over bidders that such data entails. Its objective is maximized by making the click-through rates and display values readily available to bidders and running optimal auctions in all instances.","text_sha256":"41da51329522a97bc41460d3c5975272632c5f8c5b08132b4dfe023d8526a15d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0020","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Microfoundation","text":"## 5 Microfoundation\n\nIn this section, we provide a microfoundation for calibrated mechanism design by analyzing the outcome distributions that can arise when an agent repeatedly engages with the same mechanism (Section 5.1) and contrast this to what can be implemented when the designer can offer the agent a fully dynamic mechanism (Section 5.2). To keep the presentation simple, we present the results with minimal notation, and refer the reader to Appendix C for details.\n\nThroughout, we consider the case of a single agent, whose type (i) is redrawn each period from the same distribution and (ii) is independent of the state. The reason for (i) is as follows. When the designer offers the agent a fully dynamic mechanism, the revelation principle implies that it is without loss of generality for the designer to ask the agent for type reports. Moreover, logic similar to that in Myerson (1986) implies that the designer only elicits one type report when the agent's type is persistent, and hence, the agent has no possibility of experimenting with the mechanism. Hence, to put repeated and dynamic mechanisms on a more similar footing, assuming the agent's type is redrawn each period is necessary. However, when the agent's type is repeatedly drawn from a distribution that depends on the state, the agent learns about the state both through her own type and her allocations in the mechanism. ${ }^{30}$ Thus, we assume (ii) so that the agent learns about the state only through her\n\n[^18]interaction with the mechanism. Lastly, we consider the single-agent case as extending the results in this section to multiple agents requires addressing subtle issues in strategic experimentation, which we plan to pursue in future work.","text_sha256":"a64b55710b82ac2fd932b2563c26489b55dd9aec8fd77a8e38d846d6596d0684"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0021","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.1 Repeated Interactions with a Mechanism","text":"### 5.1 Repeated Interactions with a Mechanism\n\nWe consider first the case in which the agent interacts repeatedly with the same mechanism $\\phi$ in each period of an infinite horizon interaction. In line with Section 2, a repeated mechanism is a mapping\n\n$$\n\\phi: M \\times \\Omega \\times \\mathcal{E} \\rightarrow \\Delta(A),\n$$\n\nwhere $M$ is a finite set of messages and $\\mathcal{E}$ is a finite set endowed with some measure, denoted $\\eta$. The results in Section 3 imply that assuming $\\mathcal{E}$ is finite is without loss of generality and it simplifies the proofs. In contrast to Section 2, we allow the mechanism to have an arbitrary message space. The reason is that we cannot invoke the revelation principle when the designer offers the same mechanism repeatedly: unless the agent's best response is the same across periods, the composition of the mechanism with the agent's reporting strategy yields a time-dependent, direct mechanism. To avoid keeping track of participation and reporting strategies separately in what follows, we assume a message $m_{\\varnothing} \\in M$ exists such that for all $(\\omega, \\varepsilon) \\in \\Omega \\times \\mathcal{E}, \\phi\\left(m_{\\varnothing}, \\omega, \\varepsilon\\right)=\\delta_{a_{\\varnothing}}$.\n\nTiming Given $\\phi$, the agent faces the following extensive form. Nature draws $(\\omega, \\varepsilon)$ once at the beginning, unobserved to the agent. In each period, nature first draws the agent's type, which the agent observes. The agent then sends a message $m$ into the mechanism. The mechanism then draws the allocation from $\\phi(\\cdot \\mid m, \\omega, \\varepsilon)$, which the agent observes.\n\nGiven the mechanism $\\phi$ and the extensive form game it induces, the agent's strategy specifies for each period $t$ and each period- $t$ type $\\theta \\in \\Theta$, a distribution over $M$, as a function of the agent's past observations, which include her past types, messages, and allocations. Importantly, we assume the agent does not observe her payoffs to focus on the agent learning through the mechanism.\n\nWe assume the agent is infinitely patient, that is, she has limit-of-means preferences. Her average payoff through period $T$ when the realization is $(\\omega, \\varepsilon)$ and the type-message-allocation sequence is $\\left(\\theta_{t}, m_{t}, a_{t}\\right)_{t=1}^{T}$ is given by:\n\n$$\nU_{T}\\left(\\left(\\theta_{t}, m_{t}, a_{t}\\right)_{t=1}^{T}, \\omega, \\varepsilon\\right)=\\frac{1}{T} \\sum_{t=1}^{T} u\\left(a_{t}, \\theta_{t}, \\omega\\right) .\n$$\n\nA strategy $\\sigma$ is a best response for the agent if for all alternative strategies $\\sigma^{\\prime}$, we have that\n\n$$\n\\lim \\inf _{T \\rightarrow \\infty} \\mathbb{E}_{\\sigma}\\left[U_{T}\\right] \\geq \\lim \\sup _{T \\rightarrow \\infty} \\mathbb{E}_{\\sigma^{\\prime}}\\left[U_{T}\\right],\n$$\n\nwhere $\\mathbb{E}_{\\sigma}$ is the expectation relative to the measure induced over the terminal histories by the prior on $\\Omega$, the distribution on $\\mathcal{E}$, the agent's type distribution $f$, the mechanism $\\phi$, and the agent's reporting strategy $\\sigma .{ }^{31}$\n\n[^19]Implementation Our notion of implementation is based on the induced occupation measure on $A \\times \\Theta \\times \\Omega$, that is, the (limit) expected frequency of tuples ( $a, \\theta, \\omega$ ) when the agent best responds to the mechanism. For this reason, we restrict attention to mechanisms $\\phi$ for which (i) a best-response $\\sigma$ exists, and (ii) its induced occupation measure $v_{\\sigma}$ over $A \\times \\Theta \\times M \\times \\Omega \\times \\mathcal{E}$ exists, defined as follows ${ }^{32}$\n\n$$\nv_{\\sigma}(a, \\theta, m, \\omega, \\varepsilon)=\\lim _{T \\rightarrow \\infty} \\frac{1}{T} \\mathbb{E}_{\\sigma}\\left[\\sum_{t=1}^{T} \\mathbb{1}\\left[\\left(a_{t}, \\theta_{t}, m_{t}, \\omega^{\\prime}, \\varepsilon^{\\prime}\\right)=(a, \\theta, m, \\omega, \\varepsilon)\\right]\\right]=\\lim _{T \\rightarrow \\infty} v_{\\sigma}^{T}(a, \\theta, m, \\omega, \\varepsilon),\n$$\n\nwhere the last identity defines $v_{\\sigma}$ as the limit of the up to period $T$ occupation measures $v_{\\sigma}^{T}$, which are always well-defined.\n\nUnder our definition of best response, which is the same as in Hart (1985), existence of a best response implies the agent's payoff at the best-response strategy is well-defined. ${ }^{33}$ Even if the occupation measure in Equation 14 is enough to calculate the agent's payoffs, that the agent's payoffs are welldefined does not mean the occupation measure is well-defined. Because outcome distributions-and not payoffs-are usually the focus of mechanism design, we require that both the mechanism has a best response and it induces a well-defined occupation measure.\n\nDefinition 5 (Implementation). Outcome distribution $\\vartheta \\in \\Delta(A \\times \\Theta \\times \\Omega)$ can be implemented by a repeated mechanism if a mechanism $\\psi$ and a best-response strategy $\\sigma$ exist such that\n\n$$\n\\vartheta(a, \\theta, \\omega)=\\sum_{\\varepsilon \\in \\mathcal{E}, m \\in M} v_{\\sigma}(a, \\theta, m, \\omega, \\varepsilon) .\n$$\n\nWe are now ready to state the main result of this section. Theorem 3 shows that the outcome distributions implemented by repeated mechanisms can be implemented by two-stage mechanisms, and hence by calibrated mechanisms:\n\nTheorem 3 (Microfoundation of Calibrated Mechanism Design). Outcome distribution $\\vartheta \\in \\Delta(A \\times \\Theta \\times \\Omega)$ is implementable by a repeated mechanism if and only if $\\vartheta$ can be implemented by an incentive compatible and individually rational two-stage mechanism, that is for all $(a, \\theta, \\omega) \\in A \\times \\Theta \\times \\Omega$,\n\n$$\n\\vartheta(a, \\theta, \\omega)=\\mu_{0}(\\omega) f(\\theta) \\int_{\\Delta(\\Omega)} \\alpha(a \\mid \\theta, \\mu) \\beta(d \\mu \\mid \\omega)\n$$\n\nwhere $\\beta: \\Omega \\rightarrow \\Delta(\\Delta(\\Omega))$ is Bayes plausible and $\\alpha(\\cdot \\mid \\cdot, \\mu): \\Theta \\rightarrow \\Delta(A)$ is incentive compatible and individually rational on the support of $\\mu_{0} \\otimes \\beta$.\n\nThe proof of this and all results in this section can be found in Appendix C.\nTheorem 3 provides a microfoundation for calibrated mechanism design. Whenever the designer is concerned with agents learning from the outcome of the mechanism and cares only about the long-run outcome distribution, it is as if he is designing a two-stage mechanism.\n\nWe now provide a proof sketch for Theorem 3, which is also useful to understand the proof of the result","text_sha256":"7d2b88392f93c0da4d1a3c96f98aef6ce24863417b727755540b12a79fbf3ee5"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0022","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.1 Repeated Interactions with a Mechanism","text":"[^20]in the next section. For simplicity, let $\\tilde{\\Omega}=\\Omega \\times \\mathcal{E}$ with elements $\\tilde{\\omega}$. Suppose repeated mechanism $\\phi$ implements $\\vartheta$, and let $v_{\\sigma} \\in \\Delta(A \\times \\Theta \\times M \\times \\tilde{\\Omega})$ denote the induced occupation measure. As the analysis so far illustrates, tracking the joint distribution over allocations, types, states, and beliefs is important to show that $\\vartheta$ can be implemented via a two-stage mechanism. To this end, we extend the up to period $T$ occupation measures, $v_{\\sigma}^{T} \\in \\Delta(A \\times \\Theta \\times M \\times \\tilde{\\Omega})$, to account for the frequency of beliefs through period $T$. In fact, we define two sequences of extended occupation measures over $A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega})$ : the first, $\\bar{v}_{\\sigma}^{T, 1}$, calculates the frequency of a tuple $(a, \\theta, m, \\tilde{\\omega}, \\mu)$ by counting the beliefs at the beginning of period $t$ and the second, $\\bar{v}_{\\sigma}^{T, 2}$, by counting the beliefs at the end of period $t$. Whereas the martingale property of beliefs implies these two sequences have the same (subsequential) limits, they have different conditional independence properties, which we use to derive the representation of $\\vartheta$ via a two-stage mechanism. Suppose for simplicity that $\\bar{v}_{\\sigma}^{T, 1}$ (and hence, $\\bar{v}_{\\sigma}^{T, 2}$ ), have limit $\\bar{v}_{\\sigma}$, though this assumption is not needed for the proof. ${ }^{34}$ A consequence of the martingale property of beliefs is that only the long-run beliefs of the agent are in the support of $\\bar{v}_{\\sigma}$.\n\nThe proof consists of three steps. In the first step, we show that $\\bar{v}_{\\sigma}$ admits the following decomposition:\n\n$$\n\\bar{v}_{\\sigma}(\\{(a, \\theta, m, \\tilde{\\omega})\\} \\times \\tilde{\\Delta})=\\int_{\\tilde{\\Delta}} \\mu(\\tilde{\\omega}) f(\\theta) \\rho(m \\mid \\theta, \\mu) \\alpha^{\\prime}(a \\mid m, \\mu) \\tau(d \\mu)\n$$\n\nwhere (i) $\\tau \\in \\Delta(\\Delta(\\tilde{\\Omega}))$ has mean $\\mu_{0} \\otimes \\eta$, where recall $\\eta$ is the measure on $\\mathcal{E}$, and (ii) $\\rho: \\Theta \\times \\Delta(\\tilde{\\Omega}) \\rightarrow$ $\\Delta(M)$ is a \"Markovian reporting strategy\", and (iii) $\\alpha^{\\prime}$ is almost the allocation rule in the two-stage mechanism, and hence the prime notation. Moreover, on the support of $\\mu, \\alpha^{\\prime}(\\cdot \\mid \\cdot, \\mu)$ coincides with $\\phi(\\cdot \\mid \\cdot, \\tilde{\\omega})$, implying that $\\phi(\\cdot \\mid \\cdot, \\tilde{\\omega})$ is constant in $\\tilde{\\omega}$ on the support of $\\mu$. This is the step which exploits the different conditional independence properties of $\\bar{v}_{\\sigma}^{T, 1}$ and $\\bar{v}_{\\sigma}^{T, 2}$. We use $\\bar{v}_{\\sigma}^{T, 1}$ to show the conditional independence of types and beliefs-all agent types in period $t$ have the same belief at the beginning of period $t$-and $\\bar{v}_{\\sigma}^{T, 2}$ to show the conditional independence of the allocation and the state-the belief at the end of period $t$ contains all the information about the state contained in the allocation.\n\nIn the second step, we show that $\\rho: \\Theta \\times \\Delta(\\tilde{\\Omega}) \\rightarrow \\Delta(M)$ is indeed a best response for the agent when her type is $\\theta$ and her belief is $\\mu$. In other words, the support of $\\rho(\\cdot \\mid \\theta, \\mu)$ is contained in\n\n$$\n\\arg \\max _{m \\in M} \\sum_{\\tilde{\\omega} \\in \\tilde{\\Omega}} \\mu(\\tilde{\\omega}) \\sum_{a \\in A} \\phi(a \\mid m, \\tilde{\\omega}) u(a, \\theta, \\omega),\n$$\n\nfor beliefs on the support of $\\tau$. Hence, we can use $\\rho$ and $\\phi$ to define a direct mechanism $\\alpha(\\cdot \\mid \\cdot, \\mu): \\Theta \\rightarrow$ $\\Delta(A)$ that satisfies the agent's participation and incentive constraints when her belief is $\\mu$. Together, Steps 1 and 2 allow us to obtain the representation of $\\vartheta$ as in Equation $15 .^{35}$\n\nWhereas the above steps are enough to show that $\\vartheta$ is implementable by some individually rational and incentive compatible two-stage mechanism, they do not necessarily imply that the distribution over posteriors $\\tau$ is the one induced by the information structure calibrated to $\\phi, \\pi_{\\phi}$. The last step of the proof shows that even if this is not the case, the agent adequately learns the information contained in $\\pi_{\\phi}$ in the sense of Aghion et al. (1991). Indeed, Lemma C. 4 shows a strategy exists that approximately\n\n[^21]delivers the payoff from learning $\\pi_{\\phi}$, so that the agent's payoff under $\\sigma$ is at least the payoff she would obtain if she had learned $\\pi_{\\phi}$. Because the payoff from learning $\\pi_{\\phi}$ is the maximal payoff the agent can possibly attain, we conclude that the payoff under $\\sigma$ is the payoff the agent would attain when facing the calibrated information structure $\\pi_{\\phi}$ (and best responding to it).","text_sha256":"f3dffd728c6b75af48ed60f9deb27822faaa8743399b80cb4fed660b1c4f259f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0023","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.2 Dynamic Mechanisms","text":"### 5.2 Dynamic Mechanisms\n\nIn this section, we consider the case in which the designer can offer the agent a dynamic mechanism, that is, one that conditions the allocation in each period on the history of past allocations and reports. The analysis herein allows us to describe the limits implied by calibration on the set of implementable outcomes.\n\nDynamic mechanisms A dynamic mechanism $\\varphi=\\left(\\varphi_{t}\\right)_{t \\in \\mathbb{N}}$ is a sequence of mappings that condition on the state, the history of participation decisions, type reports and allocations, and today's report, and output an allocation. Formally, expand the set of type reports and allocations by a non-participation message and the outside option, which we denote by $\\Theta A_{\\varnothing}=\\Theta \\times A \\cup\\left\\{\\left(\\varnothing, a_{\\varnothing}\\right)\\right\\} .{ }^{36}$ For each $t \\in \\mathbb{N}$, define the mechanism in period $t, \\varphi_{t}: \\Omega \\times\\left(\\Theta A_{\\varnothing}\\right)^{t-1} \\times \\Theta \\rightarrow \\Delta(A)$. Because the designer can flexibly design the mechanism in each period we no longer rely on the randomization device.\n\nA dynamic mechanism induces an extensive-form game for the agent, in which in each period, the agent decides whether to participate, and conditional on participation what type to report. Whenever the agent chooses not to participate, she obtains her outside option $a_{\\varnothing}$. We denote by $p$ the agent's participation strategy and by $\\sigma$ the agent's reporting strategy.\n\nImplementation Our notion of implementation continues to be based on the occupation measure over the set of allocations, types, participation decisions, type reports, and states, induced by the distributions $\\mu_{0}$ and $f$, the mechanism $\\varphi$, and the agent's participation and reporting strategy. However, as we show in Appendix D.3.1, it is without loss to focus on mechanisms such that (i) participation with probability 1 and truthtelling is a best response for the agent, and (ii) the mechanism implements the outside option with probability 1 in all future periods following a non-participation decision by the agent. ${ }^{37}$ Thus, we focus on dynamic mechanisms $\\varphi$ such that (i) a best response exists, and (ii) the occupation measure over $A \\times \\Theta \\times \\Omega$ is well-defined.\n\nIncentives in dynamic mechanisms Dynamic mechanisms allow the designer to condition the agent's allocation on the history of past participation decisions and reports (and allocations), and hence allow the designer to implement outcomes that satisfy weaker notions of truthtelling and participation, which we explain next.\n\nBecause the designer can condition the mechanism on the history of past reports, he can compare the frequency of type reports against the type distribution. So long as the agent is telling the truth, the\n\n[^22]frequency of reports will match the type distribution $f$ over large blocks of time. In fact, any reporting strategy whose expected frequency of reports matches the type distribution will be indistinguishable from truthtelling.\n\nDefinition 6 (Undetectable deviations). An undetectable deviation is a reporting strategy $\\sigma: \\Theta \\rightarrow \\Delta(\\Theta)$ such that for all $\\theta^{\\prime} \\in \\Theta$\n\n$$\n\\sum_{\\theta \\in \\Theta} f(\\theta) \\sigma\\left(\\theta^{\\prime} \\mid \\theta\\right)=f\\left(\\theta^{\\prime}\\right) .\n$$\n\nBy tracking the empirical distribution of type reports, the designer can dissuade the agent from employing detectable deviations. Thus, in a dynamic mechanism, the designer should be concerned with only discouraging undetectable deviations. This leads to a weaker notion of incentive compatibility for allocation rules:\n\nDefinition 7 (Unprofitable undetectable deviations). The allocation rule $\\alpha: \\Theta \\times \\Delta(\\Omega) \\rightarrow \\Delta(A)$ lacks profitable undetectable deviations at belief $\\mu \\in \\Delta(\\Omega)$ if for all undetectable deviations $\\sigma$,\n\n$$\n\\sum_{\\theta \\in \\Theta} f(\\theta) \\sum_{a \\in A} \\alpha(a \\mid \\theta, \\mu) \\sum_{\\omega \\in \\Omega} \\mu(\\omega) u(a, \\theta, \\omega) \\geq \\sum_{\\theta \\in \\Theta} f(\\theta) \\sum_{\\theta^{\\prime} \\in \\Theta} \\sigma\\left(\\theta^{\\prime} \\mid \\theta\\right) \\sum_{a \\in A} \\alpha\\left(a \\mid \\theta^{\\prime}, \\mu\\right) \\sum_{\\omega \\in \\Omega} \\mu(\\omega) u(a, \\theta, \\omega) .\n$$\n\nA two-stage mechanism $\\psi$ with allocation rule $\\alpha$ lacks profitable undetectable deviations if $\\alpha(\\cdot \\mid \\cdot, \\mu)$ lacks profitable undetectable deviations for all beliefs in the support of the mechanism.\n\nTo illustrate the difference between the lack of profitable undetectable deviations and incentive compatibility, consider the following example from Ball and Kattwinkel (2023). Suppose the agent types are binary, $\\left\\{\\theta_{1}, \\theta_{2}\\right\\}$, and equally likely. The set of allocations, $q \\in\\{0,1\\}$, describes whether the agent receives a good. Finally, suppose the agent's payoff is $u(q, \\theta)=q \\theta$ and $\\theta_{1}<\\theta_{2}$. Consider the mechanism that allocates the good to $\\theta_{2}$ : While it is not incentive compatible, it lacks profitable undetectable deviations. The constraint that the deviation must be undetectable implies the gains from $\\theta_{1}$ obtaining the good come at the expense of $\\theta_{2}$ getting the good.\n\nConsider now the agent's participation incentives in the dynamic mechanism: once the agent rejects the mechanism once, the agent obtains her outside option in all continuation histories independent of her participation decision and her types. In other words, whereas the agent can always ensure her outside option by rejecting the mechanism in a given period, she is effectively quitting the mechanism forever for all her types. The following definition introduces the notion of individual rationality satisfied by the mechanism in the long run.\n\nDefinition 8 (Ex ante individual rationality). The allocation rule $\\alpha: \\Theta \\times \\Delta(\\Omega) \\rightarrow \\Delta(A)$ is ex ante individually rational at belief $\\mu \\in \\Delta(\\Omega)$ if\n\n$$\n\\sum_{\\theta \\in \\Theta} f(\\theta) \\sum_{a \\in A} \\alpha(a \\mid \\theta, \\mu) \\sum_{\\omega \\in \\Omega} \\mu(\\omega) u(a, \\theta, \\omega) \\geq \\sum_{\\theta \\in \\Theta} f(\\theta) \\sum_{\\omega \\in \\Omega} \\mu(\\omega) u\\left(a_{\\varnothing}, \\theta, \\omega\\right) .\n$$","text_sha256":"55e10e75235c4fccadac5de43dec3a059027d6f4f343ec44e2299150a0b0c826"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0024","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.2 Dynamic Mechanisms","text":"A two-stage mechanism $\\psi$ with allocation rule $\\alpha$ is ex ante individually rational if $\\alpha(\\cdot \\mid \\cdot, \\mu)$ is ex ante individually rational for all beliefs in the support of the mechanism.\n\nWe are now ready to state the main result of this section:\nTheorem 4 (Implementable Outcomes via Dynamic Mechanisms). A dynamic mechanism exists that implements outcome $\\vartheta \\in \\Delta(A \\times \\Theta \\times \\Omega)$ if and only if $\\vartheta$ can be implemented by an ex ante individually rational two-stage mechanism which lacks profitable undetectable deviations. That is, if and only if for all $(a, \\theta, \\omega) \\in A \\times \\Theta \\times \\Omega$\n\n$$\n\\vartheta(a, \\theta, \\omega)=\\mu_{0}(\\omega) f(\\theta) \\int_{\\Delta(\\Omega)} \\alpha(a \\mid \\theta, \\mu) \\beta(d \\mu \\mid \\omega)\n$$\n\nwhere $\\beta: \\Omega \\rightarrow \\Delta(\\Delta(\\Omega))$ is Bayes plausible and $\\alpha(\\cdot \\mid \\cdot, \\mu): \\Theta \\rightarrow \\Delta(A)$ lacks profitable undetectable deviations and is ex ante individually rational on the support of $\\mu_{0} \\otimes \\beta$.\n\nTheorem 4 characterizes the outcome distributions implementable by dynamic mechanisms as those implemented by two-stage mechanisms that satisfy the incentive constraints: unprofitability of undetectable deviations and ex ante individual rationality. Notably, both notions of incentive constraints apply in the aggregate over the type distribution, which reflects the transient nature of the agent's private information.\n\nComparing Theorem 3 and Theorem 4, we see that dynamic mechanisms allow the designer to weaken the incentive constraints of the agent, but do not allow him to engage in richer-i.e., typedependent-disclosures. Despite dynamic mechanisms implying weaker incentive constraints, we can build on the results of Rochet (1987) and Rahman (2024) to show that in settings with transferable utility, where $a=(q, t)$, dynamic and repeated mechanisms implement the same set of physical allocations $q: \\Theta \\times \\Omega \\rightarrow \\mathbb{R} .^{38}$ Indeed, Rahman (2024) shows the lack of profitable undetectable deviations is equivalent to cyclical monotonicity in Rochet (1987). Thus, in settings with transferable utility, Theorems 3 and 4 imply that dynamic mechanisms do not allow the designer to expand on the set of implementable distributions over $(q, \\theta, \\omega)$.\n\nThe proof of the only if direction is similar to that of Theorem 3, in that we similarly extend the occupation measure to account for the agent's beliefs and show it satisfies the conditional independence properties implied by a two-stage mechanism. In a dynamic mechanism, however, the agent can ensure the payoff of some, but not all deviations. The latter property is what delivers that the two-stage mechanism must lack profitable undetectable deviations.\n\nThe proof of the if direction, instead, harnesses a construction in Margaria and Smolin (2018). The proof proceeds in two steps. In the first step, we analyze a fictitious model without state uncertainty in which a designer faces a privately informed agent, so that implementable outcomes are elements of $\\Delta(A \\times \\Theta)$. We show that if $\\vartheta^{\\prime}(a, \\theta)=f(\\theta) \\alpha^{\\prime}(a \\mid \\theta) \\in \\Delta(A \\times \\Theta)$ is such that $\\alpha^{\\prime}$ lacks profitable undetectable deviations and is ex ante individually rational, then a dynamic mechanism exists that implements $\\vartheta^{\\prime} .^{39}$ This is the step that relies on Margaria and Smolin (2018). We construct a dynamic mechanism, which can be split into blocks of random length. Each block consists of two phases: a reporting\n\n[^23]phase and an adjustment phase. In the reporting phase, the mechanism uses the agent's reports to determine the allocation. Instead, in the adjustment phase, the mechanism simulates type reports so that the frequency of type reports matches the type distribution (in expectation) over the length of the block, whenever this is not the case at the end of the reporting phase. These two steps ensure that the expected frequency of type reports and allocations matches $\\vartheta^{\\prime}$. We then leverage that $\\alpha^{\\prime}(\\cdot \\mid \\theta)$ lacks profitable undetectable deviations to show the agent cannot do better than by telling the truth. Hence, the induced frequency of types and allocations also matches $\\vartheta^{\\prime}$. Moreover, the construction ensures that after any history, truthtelling delivers a continuation payoff equal to the ex ante payoff. Because $\\alpha^{\\prime}$ is ex ante individually rational, we conclude the participation constraints are satisfied.\n\nThe second step uses the above result and the representation of the outcome distribution via a twostage mechanism to construct a dynamic mechanism that implements any outcome distribution that satisfies the properties in Theorem 4. Indeed, one can construct a dynamic mechanism which uses a finite number of steps to disclose information to the agent via the realized allocations, ${ }^{40}$ and then continues as in the above construction to implement the allocation rule $\\alpha(\\cdot \\mid \\cdot, \\mu)$.","text_sha256":"dd0503cdf34c4a572042254477bd3a14ea546a527ddc686c275a081b9e1d877b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0025","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Conclusions","text":"## 6 Conclusions\n\nMany economic institutions-online platforms, lenders, regulators-rely on mechanisms that remain fixed while agents interact with them repeatedly. When the mechanism's operation depends on a state known only to the designer, agents can learn this state from their outcomes, constraining what the mechanism can implement. We introduce calibrated mechanism design, a static solution concept that requires mechanisms to remain incentive compatible given the information they endogenously reveal about the designer's private state through repeated use. In private value environments, the calibration constraint pushes the designer toward full transparency, precluding Crémer-McLean-style schemes under transferable utility. In single agent-settings, calibrated mechanisms are equivalent to two-stage mechanisms. This equivalence yields a practical algorithm for finding optimal calibrated mechanisms, combining tools from information design and mechanism design. We provide a microfoundation by showing calibrated mechanisms characterize exactly what is implementable when an infinitely patient agent repeatedly interacts with the same mechanism, and study the implications on implementable outcomes of allowing the designer to offer fully dynamic mechanisms.\n\nThe most important direction for future work is deepening the analysis of multi-agent settings. On the one hand, understanding when generalized two-stage mechanisms coincide with calibrated mechanisms would enable the study of multi-agent applications, while abstracting from the dynamics of experimentation. On the other hand, extending our microfoundation to the multi-agent case would further ground calibrated mechanism design. More broadly, our framework suggests that any institution whose repeated operation leaks information about its designer's knowledge faces a fundamental tradeoff between conditioning the mechanism on this information and the information this leaks to participants, and calibrated mechanism design offers a disciplined way to analyze it.\n\n[^24]","text_sha256":"f39151a1cd57ae2be0aa9ed174babf0e8c2ee0120ae4cd78e83fb0ccd2f8736f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0026","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAghion, P., P. 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(1981): \"Rational Expectations, Information Acquisition, and Competitive Bidding,\" Econometrica, 921-943.\n\nMilgrom, P. R. and R. J. Weber (1982): \"A Theory of Auctions and Competitive Bidding,\" Econometrica, 1089-1122.\n\nMyerson, R. B. (1983): \"Mechanism Design by an Informed Principal,\" Econometrica, 1767-1797.\n\n- (1986): \"Multistage Games with Communication,\" Econometrica, 323-358.\n\nNedelec, T., C. Calauzènes, N. El Karoui, and V. Perchet (2022): \"Learning in Repeated Auctions,\" Found. Trends Mach. Learn., 15, 176-334.\n\nNedelec, T., N. E. Karoui, and V. Perchet (2019): \"Learning to Bid in Revenue-Maximizing Auctions,\" in Proceedings of the 36th International Conference on Machine Learning, vol. 97, 4781-4789.","text_sha256":"a302d5c62994b1c79c2444db510037f5049c13a2de24a267935f41c7c9c9148e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0027","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"Niemeyer, A. (2022): \"Posterior Implementability in an N-Person Decision Problem,\" Working Paper.\nOttaviani, M. and A. Prat (2001): \"The Value of Public Information in Monopoly,\" Econometrica, 69, 1673-1683.\n\nPavan, A., I. Segal, and J. Toikka (2014): \"Dynamic Mechanism Design: A Myersonian Approach,\" Econometrica, 82, 601-653.\n\nRadner, R. (1979): \"Rational Expectations Equilibrium: Generic Existence and the Information Revealed by Prices,\" Econometrica, 655-678.\n\nRahman, D. M. (2024): \"Detecting Profitable Deviations,\" Journal of Mathematical Economics, 111, 102946.\n\nRenou, L. and T. Tomala (2015): \"Approximate Implementation in Markovian Environments,\" Journal of Economic Theory, 159, 401-442.\n\nRochet, J.-C. (1987): \"A Necessary and Sufficient Condition for Rationalizability in a Quasi-Linear Context,\" Journal of Mathematical Economics, 16, 191-200.\n\nRubin, H. and O. Wesler (1958): \"A Note on Convexity in Euclidean N-Space,\" in Proceedings of the American Mathematical Society, vol. 9, 522-523.\n\nSmolin, A. (2023): \"Disclosure and Pricing of Attributes,\" RAND Journal of Economics, 54, 570-597.\nSzabadi, B. (2018): \"Essays in Microeconomic Theory: Information Design, Partnerships, and Matching,\" Ph.D. thesis, Northwestern University.\n\nYamashita, T. (2018): \"Optimal Public Information Disclosure by Mechanism Designer,\" Working Paper.","text_sha256":"79c8bfe43f1d95af0e899fba40e5f890608b01ac5ba76336b64845542e719962"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0028","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Mathematical conventions","text":"## Mathematical conventions\n\nThroughout the appendix, we take all sets to be Polish spaces, that is, completely metrizable, separable, topological spaces, and endow them with their Borel $\\sigma$-algebra. We endow product spaces with their product $\\sigma$-algebra. For a Polish space $X$, we let $\\mathcal{B}_{X}$ denote its Borel $\\sigma$-algebra and $\\Delta(X)$ the set of all Borel probability measures on $X$, endowed with the weak* topology. Thus, $\\Delta(X)$ is also a Polish space (Aliprantis and Border, 2006), and it is compact, whenever $X$ is compact (Aliprantis and Border, 2006, Theorem 15.11 and Theorem 15.15).\n\nNotational conventions If $X$ is a Polish space, $\\tilde{X}$ denotes a measurable subset of $X$, i.e., an element of the Borel $\\sigma$-algebra on $X$, and $C_{b}(X)$ denotes the set of continuous and bounded functions on $X$. Given a measure $v \\in \\Delta\\left(\\times_{i=1}^{N} Y_{i}\\right)$, we denote by $v_{Y_{j} Y_{k} \\ldots Y_{l}}$ the marginal of $v$ on $Y_{j} Y_{k} \\ldots Y_{l}$. When one of the $Y_{i}=\\Delta\\left(X_{i}\\right)$, we write $\\Delta$ instead of $Y_{i}$ in the subscript, when it is unlikely to generate confusion.\n\nThroughout the appendix, we define different distributions that arise in our proofs. Because we endow product spaces with their product topology and their product Borel $\\sigma$-algebra, it is enough to define these new measures on the measurable rectangles and we follow this convention throughout.\n\nDisintegration We rely on the notion of disintegration in many of our proofs (Bogachev, 2007, Chapter 10.6). We define disintegration in the context of product sets $X \\times Y$, as this is the one that shows up in the proof, but it is more general than this. Given a measure $v \\in \\Delta(X \\times Y), \\lambda: X \\times \\mathcal{B}_{Y} \\rightarrow[0,1]$ is the disintegration of $v$ along $X$ if the following holds\n\n1. For all $\\tilde{Y} \\in \\mathcal{B}_{Y}, x \\mapsto \\lambda_{x}(\\tilde{Y})$ is measurable,\n2. For $v_{X}$-almost everywhere $x \\in X, \\tilde{Y} \\mapsto \\lambda_{x}(\\tilde{Y})$ is a probability measure, and\n3. For every bounded measurable function $g: X \\times Y \\rightarrow \\mathbb{R}$,\n$$\n\\int_{X \\times Y} g(x, y) v(d(x, y))=\\int_{X} \\int_{Y} g(x, y) \\lambda_{x}(d y) v_{X}(d x)\n$$\n\nKallenberg (2017, Theorem 1.23) ensures that $\\left\\{\\lambda_{x}: x \\in X\\right\\}$ exists and is unique $v_{X}$-almost everywhere.","text_sha256":"c28d1b2b75113d4cf46d55fc71f50b5c2bbd150809b5f0185596037635424028"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0029","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Omitted proofs from Section 2","text":"## A Omitted proofs from Section 2\n\nProof of Theorem 1. Suppose the agents' payoffs are state independent and in a slight abuse of notation let $u_{i}\\left(a_{i}, \\theta_{i}\\right)$ denote agent $i$ 's utility.\n\nThe calibrated mechanism design problem is\n\n$$\n\\max _{\\phi: \\Theta \\times \\Omega \\times[0,1] \\rightarrow \\Delta(A)} \\sum_{\\omega \\in \\Omega} \\mu_{0}(\\omega) \\sum_{\\theta \\in \\Theta} f(\\theta \\mid \\omega) \\int_{0}^{1} w(\\phi(\\theta, \\omega, \\varepsilon), \\theta, \\omega) \\lambda(d \\varepsilon),\n$$\n\nsubject to the following constraints holding for all $(\\omega, \\varepsilon) \\in \\Omega \\times[0,1], i \\in[N], \\theta_{i} \\in \\Theta$ and $\\theta_{i}^{\\prime} \\in \\Theta$ :\n\n$$\n\\begin{aligned}\n& \\sum_{a_{i} \\in A_{i}} \\mathbb{E}_{f_{-i}(\\cdot \\mid \\omega)}\\left[\\sum_{a_{-i} \\in A_{-i}} \\phi\\left(\\theta_{i}, \\theta_{-i}, \\omega, \\varepsilon\\right)\\left(a_{i}, a_{-i}\\right)\\right] u_{i}\\left(a_{i}, \\theta_{i}\\right) \\geq \\sum_{\\left(a_{i} \\in A_{i}\\right)} \\mathbb{E}_{f_{-i}(\\cdot \\mid \\omega)}\\left[\\sum_{a_{-i} \\in A_{-i}} \\phi\\left(\\theta_{i}^{\\prime}, \\theta_{-i}, \\omega, \\varepsilon\\right)\\left(a_{i}, a_{-i}\\right)\\right] u_{i}\\left(a_{i}, \\theta_{i}\\right) \\\\\n& \\sum_{a_{i} \\in A_{i}} \\mathbb{E}_{f_{-i}(\\cdot \\mid \\omega)}\\left[\\sum_{a_{-i} \\in A_{-i}} \\phi\\left(\\theta_{i}, \\theta_{-i}, \\omega, \\varepsilon\\right)\\left(a_{i}, a_{-i}\\right)\\right] u_{i}\\left(a_{i}, \\theta_{i}\\right) \\geq u_{i}\\left(a_{i \\varnothing}, \\theta_{i}\\right) .\n\\end{aligned}\n$$\n\nIn other words, for each agent $i$, her interim allocation rule $\\pi_{\\phi, i}(\\omega, \\varepsilon)$ must be an element of $S_{I C / I R, i}^{*}$, where the latter is the set of interim allocation rules $S_{i}^{*}: \\Theta_{i} \\rightarrow \\Delta\\left(A_{i}\\right)$ that satisfy the following incentive compatibility and individual rationality constraints:\n\n$$\n\\begin{gathered}\n\\left(\\forall \\theta_{i}, \\theta_{i}^{\\prime} \\in \\Theta_{i}\\right) \\sum_{a_{i} \\in A_{i}} s_{i}^{*}\\left(a_{i} \\mid \\theta_{i}\\right) u_{i}\\left(a_{i}, \\theta_{i}\\right) \\geq \\sum_{a_{i} \\in A_{i}} s_{i}^{*}\\left(a_{i} \\mid \\theta_{i}^{\\prime}\\right) u_{i}\\left(a_{i}, \\theta_{i}\\right) \\\\\n\\left(\\forall \\theta_{i} \\in \\Theta_{i}\\right) \\sum_{a_{i} \\in A_{i}} s_{i}^{*}\\left(a_{i} \\mid \\theta_{i}\\right) u_{i}\\left(a_{i}, \\theta_{i}\\right) \\geq u_{i}\\left(a_{i \\varnothing}, \\theta_{i}\\right)\n\\end{gathered}\n$$\n\nBecause the individual rationality and incentive constraints must hold for each pair $(\\omega, \\varepsilon)$, the designer's problem is separable across variables for different $\\omega, \\varepsilon$ : the sets of variables $\\phi(\\cdot, \\omega, \\varepsilon)$ appear in different sets of constraints and the objective function is additively separable across those variables. Consequently, the designer's problem can be solved as a collection of independent problems, one for each $\\omega, \\varepsilon$. $\\square$","text_sha256":"8f916fecc845547cff7dcef010a3d869df4767f0414eb18a628d5e09462f2f10"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0030","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B Omitted proofs from Section 3","text":"## B Omitted proofs from Section 3\n\nIn this section, we present the proofs of Theorem 2 and Proposition 1. We proceed as follows: We first prove Proposition 1, as when $N=1$ its proof implies the \"if\" direction of Theorem 2. We then prove the \"only if\" direction of Theorem 2.\n\nProof of Proposition 1. We focus on the case in which types and states are independently distributed, and explain how to extend the proof when they are not.\n\nLet $\\vartheta \\in \\Delta(A \\times \\Theta \\times \\Omega)$ denote the outcome distribution implemented by an incentive compatible and individually rational calibrated mechanism. We show transition probabilities $\\beta: \\Omega \\rightarrow \\Delta\\left(\\Delta(\\Omega)^{N}\\right)$, $\\bar{\\alpha}: \\Theta \\times \\Omega \\times \\Delta(\\Omega)^{N} \\rightarrow \\Delta(A)$, and $\\alpha_{i}: \\Theta_{i} \\times \\Delta(\\Omega) \\rightarrow \\Delta\\left(A_{i}\\right)$ for $i \\in\\{1, \\ldots, N\\}$ exist such that\n\n$$\n\\vartheta(a, \\theta, \\omega)=\\mu_{0}(\\omega) f(\\theta) \\int_{\\Delta(\\Omega)^{N}} \\bar{\\alpha}\\left(a \\mid \\theta, \\omega, \\mu_{1}, \\ldots, \\mu_{N}\\right) \\beta\\left(d\\left(\\mu_{1}, \\ldots, \\mu_{N}\\right) \\mid \\omega\\right),\n$$\n\nand for all $i \\in\\{1, \\ldots, N\\}$, (i) $\\alpha_{i}$ satisfies item 3 of Definition 4, and (ii) on the support of $\\mu_{0} \\otimes \\beta, \\alpha_{i}\\left(\\cdot \\mid \\cdot, \\mu_{i}\\right)$ is incentive compatible and individually rational when agent $i$ holds belief $\\mu_{i}$.\n\nLet $\\pi_{\\omega, i}:[0,1] \\rightarrow S_{i}^{*}$ denote the mapping $\\varepsilon \\mapsto \\pi_{i}(\\omega, \\cdot)$. For $\\tilde{S}_{i}^{*} \\in \\Delta\\left(A_{i}\\right)^{\\Theta_{i}}$, define\n\n$$\n\\operatorname{Pr}_{i}\\left(\\{\\omega\\} \\times \\tilde{S}_{i}^{*}\\right)=\\mu_{0}(\\omega) \\lambda\\left(\\pi_{\\omega, i}^{-1}\\left(\\tilde{S}_{i}^{*}\\right)\\right)=\\int_{\\tilde{S}_{i}^{*}} \\mu_{i}\\left(\\omega \\mid s_{i}^{*}\\right) \\tau_{\\phi, i}\\left(d s_{i}^{*}\\right)\n$$\n\nwhere the second equality follows from disintegration of $\\operatorname{Pr}_{i} \\in \\Delta\\left(\\Omega \\times S_{i}^{*}\\right)$ along $S_{i}^{*}$, and corresponds to the definition of Bayes rule for agent $i$. Define the measurable mappings, $T_{i}: S_{i}^{*} \\rightarrow \\Delta(\\Omega)$ and\n$T: S^{*} \\rightarrow \\Delta(\\Omega)^{N}$ as follows: $T_{i}\\left(s_{i}^{*}\\right)=\\mu_{i}\\left(\\cdot \\mid s_{i}^{*}\\right)$ and $T\\left(s^{*}\\right)=\\left(T_{1}\\left(s_{1}^{*}\\right), \\ldots, T_{N}\\left(s_{N}^{*}\\right)\\right)$.\nDefine a joint distribution $Q \\in \\Delta\\left(A \\times \\Theta \\times \\Omega \\times \\Delta(\\Omega)^{N}\\right)$ as follows:\n\n$$\nQ\\left(\\{(a, \\theta, \\omega)\\} \\times \\times_{i=1}^{N} \\tilde{\\Delta}_{i}\\right)=\\mu_{0}(\\omega) f(\\theta) \\int_{\\pi_{\\omega}^{-1}\\left(T^{-1}\\left(\\times \\tilde{\\Delta}_{i}\\right)\\right)} \\phi(a \\mid \\theta, \\omega, \\varepsilon) \\lambda(d \\varepsilon)\n$$\n\nwhere $\\pi_{\\omega}^{-1}\\left(T^{-1}\\left(\\times \\tilde{\\Delta}_{i}\\right)\\right)=\\cap_{i=1}^{N}\\left\\{\\varepsilon: T_{i}\\left(\\pi_{\\omega, i}(\\varepsilon)\\right) \\in \\tilde{\\Delta}_{i}\\right\\}$.\nWe note the following properties of $Q$. First, consider its marginal over $\\Theta \\times \\Omega \\times \\Delta(\\Omega)^{N}$,\n\n$$\nQ_{\\Theta \\Omega \\Delta^{N}}\\left(\\{(\\theta, \\omega)\\} \\times \\times_{i=1}^{N} \\tilde{\\Delta}_{i}\\right)=\\mu_{0}(\\omega) f(\\theta) \\lambda\\left(\\cap_{i=1}^{N}\\left\\{\\varepsilon: T_{i}\\left(\\pi_{\\omega, i}(\\varepsilon)\\right) \\in \\tilde{\\Delta}_{i}\\right\\}\\right),\n$$\n\nwhich implies that the disintegration of $Q_{\\Theta \\Omega \\Delta^{N}}$ along $\\Theta \\times \\Omega, \\beta: \\Theta \\times \\Omega \\rightarrow \\Delta\\left(\\Delta(\\Omega)^{N}\\right)$ does not depend on $\\theta$. This automatically implies that $Q$ admits the following disintegration:\n\n$$\nQ\\left(\\{(a, \\theta, \\omega)\\} \\times \\times_{i=1}^{N} \\tilde{\\Delta}_{i}\\right)=\\mu_{0}(\\omega) f(\\theta) \\int_{\\times_{i=1}^{N} \\tilde{\\Delta}_{i}} \\bar{\\alpha}\\left(a \\mid \\theta, \\omega, \\mu_{1}, \\ldots, \\mu_{N}\\right) \\beta\\left(d\\left(\\mu_{1}, \\ldots, \\mu_{N}\\right) \\mid \\omega\\right)\n$$\n\nwhich, in turn, delivers Equation B.1. Moreover, note that the marginal of $\\beta$ on the beliefs of agent $i$, $\\beta_{i}: \\Omega \\rightarrow \\Delta(\\Delta(\\Omega))$, satisfies\n\n$$\n\\beta_{i}\\left(\\tilde{\\Delta}_{i} \\mid \\omega\\right)=\\lambda\\left(\\pi_{\\omega, i}^{-1}\\left(T_{i}^{-1}\\left(\\tilde{\\Delta}_{i}\\right)\\right)\\right) .\n$$\n\nConsider now the marginal on $A_{i} \\times \\Theta_{i} \\times \\Omega \\times \\Delta(\\Omega)$ of $Q, Q_{A_{i} \\Theta_{i} \\Omega \\Delta_{i}}$, which satisfies:\n\n$$\n\\begin{aligned}\n& Q_{A_{i} \\Theta i \\Omega \\Delta_{i}}\\left(\\left\\{\\left(a_{i}, \\theta_{i}, \\omega\\right)\\right\\} \\times \\tilde{\\Delta}_{i}\\right)=\\sum_{\\theta_{-i} \\in \\Theta_{-i}} \\sum_{a_{-i} \\in A_{-i}} Q\\left(\\{(a, \\theta, \\omega)\\} \\times \\tilde{\\Delta}_{i} \\times \\Delta(\\Omega)^{N-1}\\right)= \\\\\n& =\\mu_{0}(\\omega) f_{i}\\left(\\theta_{i}\\right) \\int_{\\pi_{\\omega, i}^{-1}\\left(T_{i}^{-1}\\left(\\tilde{\\Delta}_{i}\\right)\\right)}\\left(\\sum_{\\theta_{-i} \\in \\Theta_{-i}} f_{-i}\\left(\\theta_{-i}\\right) \\sum_{a_{-i} \\in A_{-i}} \\phi\\left(a_{i}, a_{-i} \\mid \\theta_{i}, \\theta_{-i}, \\omega, \\varepsilon\\right)\\right) \\lambda(d \\varepsilon) \\\\\n& =\\mu_{0}(\\omega) f_{i}\\left(\\theta_{i}\\right) \\int_{\\pi_{\\omega, i}^{-1}\\left(T_{i}^{-1}\\left(\\tilde{\\Delta}_{i}\\right)\\right)} \\pi_{\\omega, i}(\\varepsilon)\\left(a_{i} \\mid \\theta_{i}\\right) \\lambda(d \\varepsilon)=\\mu_{0}(\\omega) f_{i}\\left(\\theta_{i}\\right) \\int_{T_{i}^{-1}\\left(\\tilde{\\Delta}_{i}\\right)} s_{i}^{*}\\left(a_{i} \\mid \\theta_{i}\\right)\\left(\\lambda \\circ \\pi_{\\omega, i}^{-1}\\right)\\left(d s_{i}^{*}\\right)\n\\end{aligned}\n$$\n\nLastly, $Q_{A_{i} \\Theta_{i} \\Omega \\Delta_{i}}$ admits the following representation via disintegration:\n\n$$\n\\begin{aligned}\n& Q_{A_{i} \\Theta_{i} \\Omega \\Delta_{i}}\\left(\\left\\{\\left(a_{i}, \\theta_{i}, \\omega\\right)\\right\\} \\times \\tilde{\\Delta}_{i}\\right)=\\mu_{0}(\\omega) f_{i}\\left(\\theta_{i}\\right) \\int_{\\tilde{\\Delta}_{i}} \\alpha_{i}\\left(a_{i} \\mid \\theta_{i}, \\omega, \\mu_{i}\\right) \\beta_{i}\\left(d \\mu_{i} \\mid \\omega\\right) \\\\\n& =\\mu_{0}(\\omega) f_{i}\\left(\\theta_{i}\\right) \\int_{\\tilde{\\Delta}_{i}} \\alpha_{i}\\left(a_{i} \\mid \\theta_{i}, \\omega, \\mu_{i}\\right)\\left(\\lambda \\circ \\pi_{\\omega, i}^{-1} \\circ T_{i}^{-1}\\right)\\left(d \\mu_{i}\\right)\n\\end{aligned}\n$$","text_sha256":"0cab74d5228df05a13bf150c71448ad8df8de4e1322270582d5295671b43dbc1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0031","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B Omitted proofs from Section 3","text":"Together with the uniqueness of disintegration and the sufficiency property of beliefs, Equations B. 3 and B. 4 imply that $\\alpha_{i}$ does not depend on $\\omega$. The incentive compatibility and individual rationality of $\\alpha_{i}$ follows from that of the calibrated mechanism.\n\nFinally, consider the case in which $\\theta$ and $\\omega$ are not independent. Then, the experiment $\\beta$ in Equation B. 2 induces a joint distribution over the beliefs of $N$ fictitious agents whose prior over the state is given by $\\mu_{0}$. Agent $i$ 's updated beliefs when her type is $\\theta_{i}$ obtain from a transformation of $\\mu$ (Alonso and Câmara, 2016; Laclau and Renou, 2017). ${ }^{41}$ Thus, up to changing $f(\\theta)$ by $f(\\theta \\mid \\omega)$, and interpreting\n\n[^25]the draw from the Blackwell experiment as the posterior of an agent with prior belief $\\mu_{0}$, the result follows. $\\square$\n\nProof of Theorem 2.\n\"Only if\" direction Similar to the proof of Proposition 1 , we focus on the case in which $\\theta$ and $\\omega$ are independent. Suppose $\\vartheta \\in \\Delta(A \\times \\Theta \\times \\Omega)$ is implemented by an incentive compatible and individually rational two-stage mechanism. That is,\n\n$$\n\\vartheta(a, \\theta, \\omega)=\\mu_{0}(\\omega) f(\\theta) \\int_{\\Delta(\\Omega)} \\alpha(a \\mid \\theta, \\mu) \\beta(d \\mu \\mid \\omega)\n$$\n\nand $\\alpha$ is incentive compatible and individually rational on the support of $\\mu_{0} \\otimes \\beta$. We construct an incentive compatible and individually rational calibrated mechanism that implements $\\vartheta$.\n\nFirst, if $\\vartheta$ satisfies Equation B.5, Rubin and Wesler (1958) and Carathéodory's theorem (Aliprantis and Border, 2006, Theorem 5.32) imply that a finite support $\\beta^{\\prime}: \\Omega \\rightarrow \\Delta\\left(\\left\\{\\mu_{1}, \\ldots, \\mu_{K}\\right\\}\\right)$ exists such that for all $(a, \\theta, \\omega) \\in A \\times \\Theta \\times \\Omega^{42}$\n\n$$\n\\vartheta(a, \\theta, \\omega)=\\mu_{0}(\\omega) f(\\theta) \\sum_{k=1}^{K} \\alpha\\left(a \\mid \\theta, \\mu_{k}\\right) \\beta^{\\prime}\\left(\\left\\{\\mu_{k}\\right\\} \\mid \\omega\\right) .\n$$\n\nFor each $\\omega \\in \\Omega$, partition $[0,1]=\\cup_{k=1}^{K-2}\\left[b_{k}^{\\omega}, b_{k+1}^{\\omega}\\right) \\cup\\left[b_{K-1}^{\\omega}, 1\\right]$, where $b_{1}=0$, and for all $k \\in\\{1, \\ldots, K-2\\}$, $b_{k+1}^{\\omega}=\\sum_{l=1}^{k} \\beta^{\\prime}\\left(\\left\\{\\mu_{l}\\right\\} \\mid \\omega\\right)$. Define for $\\varepsilon \\in\\left[b_{k}^{\\omega}, b_{k+1}^{\\omega}\\right)$\n\n$$\n\\phi(a \\mid \\theta, \\omega, \\varepsilon)=\\alpha\\left(a \\mid \\theta, \\mu_{k}\\right) .\n$$\n\nThe calibrated information structure is $\\pi_{\\phi}(\\omega, \\varepsilon)=\\phi(\\cdot \\mid \\cdot, \\omega, \\varepsilon)=\\alpha\\left(\\cdot \\mid \\theta, \\mu_{k}\\right)$ for $\\varepsilon \\in\\left[b_{k}^{(w}, b_{k+1}^{(w)}\\right.$ if $m \\leq K-2$ or $\\varepsilon \\in\\left[b_{K-1}^{\\omega}, 1\\right]$.\n\nWe now show that for all $\\theta$ and all $s \\in \\operatorname{supp} \\pi_{\\phi}$, the mechanism $\\phi$ is incentive compatible and individually rational. Note that $\\pi_{\\phi}$ has finite support, and let $s \\in \\operatorname{supp} \\pi_{\\phi}$ and let $\\mu(\\cdot \\mid s)$ denote the updated posterior. Then, $k$ exists such that the following holds:\n\n$$\n\\mu(\\omega \\mid s)=\\frac{\\mu_{0}(\\omega) \\lambda(\\{\\varepsilon: \\pi(\\omega, \\varepsilon)=s\\})}{\\sum_{\\omega^{\\prime} \\in \\Omega} \\mu_{0}\\left(\\omega^{\\prime}\\right) \\lambda\\left(\\left\\{\\varepsilon: \\pi\\left(\\omega^{\\prime}, \\varepsilon\\right)=s\\right\\}\\right)}=\\frac{\\mu_{0}(\\omega)\\left(b_{k+1}^{\\omega}-b_{k}^{\\omega}\\right)}{\\sum_{\\omega^{\\prime} \\in \\Omega} \\mu_{0}\\left(\\omega^{\\prime}\\right)\\left(b_{k+1}^{\\omega^{\\prime}}-b_{k}^{\\omega^{\\prime}}\\right)}=\\mu_{k}(\\omega) .\n$$\n\nMoreover, because $\\phi(\\cdot \\mid \\cdot, \\omega, \\varepsilon)=\\alpha\\left(\\cdot \\mid \\cdot, \\mu_{k}\\right)$, then it satisfies the agent's incentive compatibility and individual rationality constraints when she holds belief $\\mu_{k} .{ }^{43}$ $\\square$\n\n[^26]","text_sha256":"73acf160a774d6a4dc072066da4496c8a50092872befe075b38cc8c4758a284e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0032","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Supplementary Appendix","text":"## Supplementary Appendix","text_sha256":"cba3ae420e71818499268746a37204070253116f466d4f9544a40a2a92546cb1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0033","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C Omitted proofs from Section 5","text":"## C Omitted proofs from Section 5","text_sha256":"5a6decb0d038fa5e8d5314bc2d1c06b9c3e51bb05e59430b9b91dc9c92a4415e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0034","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C. 1 Repeated Mechanisms","text":"## C. 1 Repeated Mechanisms\n\nIn this section, we present the proof of Theorem 3. To do so, we first complete the formal definition of the game induced by repeating mechanism $\\phi: M \\times \\Omega \\times \\mathcal{E} \\rightarrow \\Delta(A)$, by specifying the histories, strategy space, and the distribution over terminal histories induced by the agent's strategy and the mechanism. Having laid this groundwork, we describe the proof strategy, and then provide the formal details of the proof. Throughout this section, we use the shorthand $\\tilde{\\Omega}=\\Omega \\times \\mathcal{E}$, and denote its elements by $\\tilde{\\omega}$.\n\nHistories and strategies Histories through period $t \\in \\mathbb{N}$ are defined as $H^{t} \\equiv(\\Theta \\times M \\times A)^{t-1}$. The set of infinite histories from the agent's point of view is $H^{\\infty}$. The set of terminal histories is $\\mathcal{H}^{\\infty} \\equiv \\tilde{\\Omega} \\times H^{\\infty}$, where recall $\\mathcal{E}$ is finite and endowed with some measure $\\eta$.\n\nThe agent's behavioral strategy is defined as a collection $\\sigma \\equiv\\left(\\sigma_{t}\\right)_{t \\in \\mathbb{N}}$ such that for all $t \\geq 1$\n\n$$\n\\sigma_{t}: H^{t} \\times \\Theta \\rightarrow \\Delta(M) .\n$$\n\nThe tuple of distributions $\\left(\\mu_{0}, \\eta, f\\right)$ together with the mechanism $\\phi$ and the agent's strategy $\\sigma$ determine a joint distribution over $\\mathcal{H}^{\\infty}$ by the Ionescu-Tulcea theorem (Bogachev, 2007, Theorem 10.7.3). We provide more details on this probability distribution below. Denote by $\\mathbb{P}_{\\left(\\mu_{0}, \\eta, f, \\phi, \\sigma\\right)}$ and $\\mathbb{E}_{\\left(\\mu_{0}, \\eta, f, \\phi, \\sigma\\right)}$ the probability distribution over the terminal histories and the expectation with respect to this distribution, respectively. Whenever it is not likely to lead to confusion, we drop the dependence on $\\left(\\mu_{0}, \\eta, f, \\phi, \\sigma\\right)$, and whenever we want to emphasize the dependence on the agent's strategy we note the dependence on $\\sigma$.\n\nThe distribution over terminal histories $\\mathcal{H}^{\\infty}$ For future use, we review the construction of $\\mathbb{P}_{\\sigma}$. For each $t$, the distributions $\\left(\\mu_{0}, \\eta, f\\right)$ together with the mechanism $\\phi$ and the agent's strategy $\\sigma$ determine a distribution over $\\tilde{\\Omega} \\times H^{t}$, which we denote by $\\mathbb{P}_{\\sigma}^{t} \\in \\Delta\\left(\\tilde{\\Omega} \\times H^{t}\\right)$. Note that for any subset $\\tilde{\\mathcal{H}^{t}} \\subset \\tilde{\\Omega} \\times H^{t}$,\n\n$$\n\\mathbb{P}_{\\sigma}^{t}\\left(\\tilde{\\mathcal{H}}^{t}\\right)=\\mathbb{P}_{\\sigma}^{t+1}\\left(\\tilde{\\mathcal{H}}^{t} \\times(\\Theta \\times M \\times A)\\right) .\n$$\n\nMoreover,\n\n$$\n\\mathbb{P}_{\\sigma}^{t+1}\\left(\\tilde{\\omega}, h^{t}, \\theta, m, a\\right)=\\mathbb{P}_{\\sigma}^{t}\\left(\\tilde{\\omega}, h^{t}\\right) f(\\theta) \\sigma_{t}\\left(h^{t}, \\theta\\right)(m) \\phi(a \\mid m, \\tilde{\\omega}) .\n$$\n\nBy the Ionescu-Tulcea theorem, the distribution $\\mathbb{P}_{\\sigma} \\in \\Delta\\left(\\tilde{\\Omega} \\times H^{\\infty}\\right)$ is the unique distribution that satisfies that for all $t \\in \\mathbb{N}, \\tilde{\\mathcal{H}}^{t} \\subset \\tilde{\\Omega} \\times H^{t}$,\n\n$$\n\\mathbb{P}_{\\sigma}\\left(\\tilde{\\mathcal{H}}^{t} \\times \\prod_{s=t+1}^{\\infty}(\\Theta \\times M \\times A)\\right)=\\mathbb{P}_{\\sigma}^{t}\\left(\\tilde{\\mathcal{H}}^{t}\\right) .\n$$\n\n[^27]Belief system The agent's beliefs over $\\tilde{\\Omega}$ at the beginning of each $t$ are determined by the belief system, which in a slight abuse of notation we denote by $\\mu_{t}: H^{t} \\rightarrow \\Delta(\\tilde{\\Omega})$. The belief system satisfies\n\n$$\n\\mathbb{P}_{\\sigma}^{t}\\left(h^{t}\\right) \\mu_{t}\\left(\\tilde{\\omega} \\mid h^{t}\\right)=\\mathbb{P}_{\\sigma}^{t}\\left(\\tilde{\\omega}, h^{t}\\right) .\n$$\n\nThat is, whenever $h^{t}$ is such that $\\mathbb{P}_{\\sigma}\\left(\\left\\{\\tilde{h} \\in H^{\\infty}: \\tilde{h}^{t}=h^{t}\\right\\}\\right)>0$,\n\n$$\n\\mu_{t}\\left(\\tilde{\\omega} \\mid h^{t}\\right)=\\frac{\\mathbb{P}_{\\sigma}^{t}\\left(\\tilde{\\omega}, h^{t}\\right)}{\\mathbb{P}_{\\sigma}^{t}\\left(h^{t}\\right)}=\\mathbb{P}_{\\sigma}^{t}\\left(\\tilde{\\omega} \\mid h^{t}\\right) .\n$$\n\nGiven $\\mathbb{P}_{\\sigma} \\in \\Delta\\left(\\mathcal{H}^{\\infty}\\right)$, define $\\mu_{\\infty}\\left(\\tilde{\\omega} \\mid h^{\\infty}\\right) \\equiv \\mathbb{P}_{\\sigma}\\left(\\tilde{\\omega} \\mid h^{\\infty}\\right)$ to be the belief system conditional on the whole terminal history $h^{\\infty}$.\n\nRemark C. 1 (Belief system and strategies as functions on $\\mathcal{H}^{\\infty}$ ). Whereas the beliefs and strategies are defined on the finite histories, it is sometimes convenient to write them as functions on $\\mathcal{H}^{\\infty}$ that are adapted to $H^{t}$.\n\nA property of the belief system We collect here a property of the belief system which we use in our proofs below.\n\nLemma C. 1 (Martingale property under weak* convergence). $\\mu_{t}\\left(h^{\\infty}\\right) \\xrightarrow{w^{*}} \\mu_{\\infty}\\left(h^{\\infty}\\right) \\mathbb{P}_{\\sigma}$-almost surely. This and the proof of other technical results are in Appendix D.","text_sha256":"b354aefb6e59bdfb3acf335baddeed6ccf95d77cdda5be57539f962d0e8b8212"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0035","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.1.1 Proof of Theorem 3 (necessity)","text":"## C.1.1 Proof of Theorem 3 (necessity)\n\nWe are now ready to present the proof of Theorem 3, starting by the \"only if\" direction. Let $\\vartheta \\in \\Delta(A \\times$ $\\Theta \\times \\Omega$ ) denote the outcome distribution implemented by repeated mechanism $\\phi$ under best response $\\sigma$, and let $v_{\\sigma}$ denote the associated occupation measure, the definition of which we reproduce below for ease of reference:\n\n$$\nv_{\\sigma}(a, \\theta, m, \\tilde{\\omega})=\\lim _{T \\rightarrow \\infty} \\frac{1}{T} \\mathbb{E}_{\\sigma}\\left[\\sum_{t=1}^{T} \\mathbb{1}\\left[\\left(a_{t}, \\theta_{t}, m_{t}, \\tilde{\\omega}^{\\prime}\\right)=(a, \\theta, m, \\tilde{\\omega})\\right]\\right]=\\lim _{T \\rightarrow \\infty} v_{\\sigma}^{T}(a, \\theta, m, \\tilde{\\omega}),\n$$\n\nwhere recall limits are in the weak* sense. We show that $\\vartheta$ can be implemented by an incentive compatible and individually rational two-stage mechanism.\n\nTo this end, we consider two sequences of extended occupation measures on $A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega})$ :\n\n$$\n\\begin{array}{r}\n\\bar{v}_{\\sigma}^{T, 1}(\\{(a, \\theta, m, \\tilde{\\omega})\\} \\times \\tilde{\\Delta})=\\frac{1}{T} \\mathbb{E}_{\\sigma}\\left[\\sum_{t=1}^{T} \\mathbb{1}\\left[\\left(a_{t}, \\theta_{t}, m_{t}, \\tilde{\\omega}^{\\prime}\\right)=(a, \\theta, m, \\tilde{\\omega})\\right] \\mathbb{1}\\left[\\mu_{t} \\in \\tilde{\\Delta}\\right]\\right], \\\\\n\\bar{v}_{\\sigma}^{T, 2}(\\{(a, \\theta, m, \\tilde{\\omega})\\} \\times \\tilde{\\Delta})=\\frac{1}{T} \\mathbb{E}_{\\sigma}\\left[\\sum_{t=1}^{T} \\mathbb{1}\\left[\\left(a_{t}, \\theta_{t}, m_{t}, \\tilde{\\omega}^{\\prime}\\right)=(a, \\theta, m, \\tilde{\\omega})\\right] \\mathbb{1}\\left[\\mu_{t+1} \\in \\tilde{\\Delta}\\right]\\right] .\n\\end{array}\n$$\n\nWe note the following. First, Equation C. 5 counts the beliefs at the beginning of period $t$, while Equation C. 6 counts the beliefs at the end of period $t$ (after the realization of $\\theta, m$, and $a$.) Equation C. 5 is key to obtain the (limit) independence of the belief and type distributions, while Equation C. 6 allows us to obtain the (limit) independence of the allocation and the state, conditional on the induced belief.\n\nSecond, $v_{\\sigma}^{T}$ is the marginal of both $\\bar{v}_{\\sigma}^{T, 1}$ and $\\bar{v}_{\\sigma}^{T, 2}$. Third, by Lemma C.1, $\\mu_{t} \\xrightarrow{w^{*}} \\mu_{\\infty}$, and hence both $\\bar{v}_{\\sigma}^{T, 1}$ and $\\bar{v}_{\\sigma}^{T, 2}$ have the same set of subsequential limits, which we record for future reference below (see Appendix D for the proof):\n\nLemma C.2. The occupation measures $\\bar{v}_{\\sigma}^{T, 1}$ and $\\bar{v}_{\\sigma}^{T, 2}$ have the same set of subsequential limits.\nThe proof of necessity of Theorem 3 proceeds in five steps. First, we show that the marginal of $\\bar{v}_{\\sigma}^{T, 1}$ on $\\Delta(\\tilde{\\Omega})$, which we denote by $\\tau_{\\sigma}^{T}$ weak*-converges to $\\mathbb{P}_{\\sigma} \\circ \\mu_{\\infty}^{-1}$. We denote this limit measure by $\\tau_{\\sigma}$. By Lemma C.2, $\\tau_{\\sigma}$ is also the (limit) marginal of $\\bar{v}_{\\sigma}^{T, 2}$ on $\\Delta(\\tilde{\\Omega})$.\n\nSecond, we show that up to a subsequence $\\bar{v}_{\\sigma}^{T, 1}, \\bar{v}_{\\sigma}^{T, 2} \\xrightarrow{w^{*}} \\bar{v}_{\\sigma}$. Furthermore, transition probabilities $\\tau_{\\sigma} \\in \\Delta(\\Delta(\\tilde{\\Omega})), \\rho: \\Theta \\times \\Delta(\\tilde{\\Omega}) \\rightarrow \\Delta(M), \\alpha^{\\prime}: M \\times \\Delta(\\tilde{\\Omega}) \\rightarrow \\Delta(A)$ exist such that\n\n$$\nv_{\\sigma}(a, \\theta, m, \\tilde{\\omega})=\\int_{\\Delta(\\tilde{\\Omega})} \\mu(\\tilde{\\omega}) f(\\theta) \\rho(m \\mid \\theta, \\mu) \\alpha^{\\prime}(a \\mid m, \\mu) \\tau_{\\sigma}(d \\mu)\n$$\n\nHence, the agent's payoff when faced with mechanism $\\phi$ and playing strategy $\\sigma$ can be written as:\n\n$$\n\\mathbb{E}_{\\bar{v}_{\\sigma}}[u(a, \\theta, \\omega)]=\\int_{\\Delta(\\tilde{\\Omega})} \\sum_{\\theta \\in \\Theta} f(\\theta) \\sum_{m \\in M} \\rho(m \\mid \\theta, \\mu) \\mathbb{E}_{\\tilde{\\omega} \\sim \\mu}\\left[\\sum_{a \\in A} \\alpha^{\\prime}(a \\mid \\mu, m) u(a, \\theta, \\omega)\\right] \\tau_{\\sigma}(d \\mu) .\n$$\n\nThird, we show that for all $\\theta \\in \\Theta$\n\n$$\n\\mathbb{E}_{\\tau_{\\sigma}}\\left\\{\\sum_{m \\in M} \\rho(m \\mid \\theta, \\mu) \\mathbb{E}_{\\tilde{\\omega} \\sim \\mu}\\left[\\sum_{a \\in A} \\alpha^{\\prime}(a \\mid \\mu, m) u(a, \\theta, \\omega)\\right]-\\max _{m \\in M} \\mathbb{E}_{\\tilde{\\omega} \\sim \\mu}\\left[\\sum_{a \\in A} \\alpha^{\\prime}(a \\mid \\mu, m) u(a, \\theta, \\omega)\\right]\\right\\}=0 .\n$$\n\nEquations C. 8 and C. 9 allow us to identify the incentive compatible and individually rational allocation rule of the two-stage mechanism that implements $\\vartheta$.\n\nFourth, whereas the previous steps identify a two-stage mechanism expressed in terms of posterior beliefs over $\\tilde{\\Omega}$, we show how to obtain a two-stage mechanism expressed in terms of posterior beliefs over $\\Omega$. Finally, we show that the agent's payoff in Equation C. 8 coincides with the payoff she would get when best responding to the information structure calibrated to $\\phi$.\n\nStep 1 Having defined the extended occupation measure in Equation C.5, we present here a property we use in our proof. Let $\\tau_{\\sigma}^{T}$ denote the marginal of $\\bar{v}_{\\sigma}^{T, 1}$ on $\\Delta(\\tilde{\\Omega})$. That is, for any measurable subset $\\tilde{\\Delta} \\subset \\Delta(\\tilde{\\Omega})$, define\n\n$$\n\\tau_{\\sigma}^{T}(\\tilde{\\Delta})=\\frac{1}{T} \\mathbb{E}_{\\sigma}\\left[\\sum_{t=1}^{T} \\mathbb{1}\\left[\\mu_{t} \\in \\tilde{\\Delta}\\right]\\right] .\n$$\n\nIn Appendix D, we prove the following:\nLemma C.3. The sequence of measures $\\left(\\tau_{\\sigma}^{T}\\right)_{T \\in \\mathbb{N}}$ defined by Equation C. 10 converges in the weak* sense to the push-forward measure $\\tau_{\\sigma} \\equiv \\mathbb{P}_{\\sigma} \\circ \\mu_{\\infty}^{-1}$, where $\\mu_{\\infty}\\left(h^{\\infty}\\right)=\\mathbb{P}_{\\sigma}\\left(\\cdot \\mid h^{\\infty}\\right)$.\n\nStep 2 To show that Equation C. 7 holds, we show the following properties of $\\bar{v}_{\\sigma}^{T, 1}$ and $\\bar{v}_{\\sigma}^{T, 2}$. On the one hand, $\\bar{v}_{\\sigma}^{T, 1}$ satisfies that for all $g \\in C_{b}(A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega}))$,","text_sha256":"54b5a7bc3e893cd48200bde01be68d65f528dea03c9c62ab7c9611dbd2fbfa62"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0036","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.1.1 Proof of Theorem 3 (necessity)","text":"$$\n\\int_{A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega})} g(a, \\theta, m, \\tilde{\\omega}, \\mu) d \\bar{v}_{\\sigma}^{T, 1}=\\int_{\\Theta \\times M \\times \\Delta(\\tilde{\\Omega})} \\mathbb{E}_{\\mu}\\left[\\mathbb{E}_{\\phi(\\cdot \\mid m, \\tilde{\\omega})}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)]\\right] d \\bar{v}_{\\sigma, \\Theta M \\Delta^{\\prime}}^{T, 1}\n$$\n\nand for all $q \\in C_{b}(\\Theta \\times \\Delta(\\Omega))$,\n\n$$\n\\int_{\\Theta \\times \\Delta(\\tilde{\\Omega})} q(\\theta, \\mu) d \\bar{v}_{\\sigma, \\Theta \\Delta}^{T, 1}=\\int_{\\Delta(\\tilde{\\Omega})} \\int_{\\Theta} f(\\theta) q(\\theta, \\mu) d \\tau_{\\sigma}^{T}\n$$\n\nOn the other hand, $\\bar{v}_{\\sigma}^{T, 2}$ satisfies that for all $g \\in C_{b}(A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega}))$,\n\n$$\n\\int_{A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega})} g(a, \\theta, m, \\tilde{\\omega}, \\mu) d \\bar{v}_{\\sigma}^{T, 2}=\\int_{\\Theta \\times M \\times A \\times \\Delta(\\tilde{\\Omega})} \\mathbb{E}_{\\tilde{\\omega} \\sim \\mu}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)] d \\bar{v}_{\\sigma, \\Theta M A \\Delta}^{T, 2}\n$$\n\nIn the expressions above, the subscripts on $\\bar{v}_{\\sigma}^{T, k}$ next to $\\sigma$ are the spaces over which we take the marginals, and $\\Delta$ is shorthand notation for $\\Delta(\\tilde{\\Omega})$. Because $\\Delta(A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega}))$ is compact (Aliprantis and Border, 2006, Theorem 15.11), $\\bar{v}_{\\sigma}^{T, 1}$ has a convergent subsequence $\\left(\\bar{v}_{\\sigma}^{T_{n}, 1}\\right)_{n \\in \\mathbb{N}}$, which by Lemma C. 2 is also a convergent subsequence of $\\bar{v}_{\\sigma}^{T, 2}$. Let $\\bar{v}_{\\sigma}$ denote the weak* limit along $T_{n}$. The continuity of the projection implies that $v_{\\sigma}$ is the marginal of $\\bar{v}_{\\sigma}$ on $A \\times \\Theta \\times M \\times \\tilde{\\Omega}$, and $\\tau_{\\sigma} \\equiv \\mathbb{P}_{\\sigma} \\circ \\mu_{\\infty}^{-1}$ is the marginal on $\\Delta(\\Omega)$. Equations C. 12 and C. 13 together imply that $\\bar{v}_{\\sigma}$ admits the decomposition in the right hand side of Equation C.7, and the result follows.\n\nTo show Equation C. 11 holds, use that $\\bar{v}_{\\sigma}^{T, 1}$ has finite support to write it as follows:\n\n$$\n\\begin{aligned}\n\\bar{v}_{\\sigma}^{T, 1}(a, \\theta, m, \\tilde{\\omega}, \\mu) & =\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{h^{t} \\in H^{t}} \\mathbb{P}_{\\sigma}^{t}\\left(\\tilde{\\omega}, h^{t}\\right) f(\\theta) \\sigma_{t}\\left(h^{t}, \\theta\\right)(m) \\phi(a \\mid m, \\tilde{\\omega}) \\mathbb{1}\\left[\\mu_{t}=\\mu\\right] \\\\\n& =\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{h^{t} \\in H^{t}} \\mathbb{P}_{\\sigma}^{t}\\left(h^{t}\\right) \\mu(\\tilde{\\omega}) f(\\theta) \\sigma_{t}\\left(h^{t}, \\theta\\right)(m) \\phi(a \\mid m, \\tilde{\\omega}) \\mathbb{1}\\left[\\mu_{t}=\\mu\\right]\n\\end{aligned}\n$$\n\nwhere the second equality uses that $\\mu_{t}\\left(h^{t}\\right)=\\mathbb{P}_{\\sigma}^{t}\\left(\\cdot \\mid h^{t}\\right)$.\nEquation C. 14 implies the following holds for every bounded continuous function $g \\in C_{b}(A \\times \\Theta \\times M \\times$ $\\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega})):$\n\n$$\n\\mathbb{E}_{\\bar{v}_{\\sigma}^{T, 1}}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)]=\\mathbb{E}_{\\bar{v}_{\\sigma, \\Theta M \\Delta}^{T, 1}}\\left[\\mathbb{E}_{\\mu}\\left[\\mathbb{E}_{\\phi(\\cdot \\mid m, \\tilde{\\omega})}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)]\\right]\\right],\n$$\n\nThis completes the proof that Equation C. 11 holds. Letting $V_{g}(\\theta, M, \\mu)=\\mathbb{E}_{\\mu}\\left[\\mathbb{E}_{\\phi(\\cdot \\mid m, \\tilde{\\omega})}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)]\\right]$, we have that\n\n$$\n\\int_{A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega})} g d \\bar{v}_{\\sigma}^{T, 1}=\\int_{\\Theta \\times M \\times \\Delta(\\tilde{\\Omega})} V_{g} d \\bar{v}_{\\sigma, \\Theta M \\Delta}^{T, 1} \\Leftrightarrow \\int_{A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega})}\\left(g-V_{g}\\right) d \\bar{v}_{\\sigma}^{T, 1}=0 .\n$$\n\nBecause $g-V_{g} \\in C_{b}(A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega}))$ and $\\bar{v}_{\\sigma}^{T_{n}, 1} \\xrightarrow{w^{*}} \\bar{v}_{\\sigma}$, we conclude that\n\n$$\n\\mathbb{E}_{\\bar{v}_{\\sigma}}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)]=\\int_{\\Theta \\times M \\times \\Delta(\\tilde{\\Omega})} \\mathbb{E}_{\\mu}\\left[\\mathbb{E}_{\\phi(\\cdot \\mid m, \\tilde{\\omega})}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)]\\right] d \\bar{v}_{\\sigma, \\Theta M \\Delta} .\n$$\n\nTo show that Equation C. 12 holds, note that the marginal of $\\bar{v}_{\\sigma}^{T, 1}$ on $\\Theta \\times \\Delta(\\tilde{\\Omega})$ equals $\\tau_{\\sigma}^{T} \\otimes f$. Indeed, fix\nany continuous function $q \\in C_{b}(\\Theta \\times \\Delta(\\tilde{\\Omega}))$ and note that for all $T$\n\n$$\n\\mathbb{E}_{\\bar{v}_{\\sigma, \\Theta \\Delta}^{T, 1}}[q(\\theta, \\mu)]=\\mathbb{E}_{\\tau_{\\sigma}^{T}}\\left[\\sum_{\\theta \\in \\Theta} f(\\theta) q(\\theta, \\mu)\\right] .\n$$\n\nLetting $V_{q}(\\mu)=\\sum_{\\theta \\in \\Theta} f(\\theta) q(\\theta, \\mu)$, we have that for all $T$\n\n$$\n\\int_{\\Delta(\\tilde{\\Omega}) \\times \\Theta}\\left(q(\\theta, \\mu)-V_{q}(\\mu)\\right) d \\bar{v}_{\\sigma, \\Theta \\Delta}^{T, 1}=0 .\n$$\n\nBecause $q-V_{q} \\in C_{b}(\\Theta \\times \\Delta(\\tilde{\\Omega}))$ and $\\bar{v}_{\\sigma}^{T_{n}, 1} \\xrightarrow{w^{*}} \\bar{v}_{\\sigma}$, we conclude that\n\n$$\n\\int_{\\Theta \\times \\Delta(\\tilde{\\Omega})} q(\\theta, \\mu) d \\bar{v}_{\\sigma, \\Theta \\Delta}=\\int_{\\Delta(\\tilde{\\Omega})} \\int_{\\Theta} f(\\theta) q(\\theta, \\mu) d \\tau_{\\sigma} .\n$$\n\nLastly, to show that Equation C. 13 holds, note that we can write $\\bar{v}_{\\sigma}^{T, 2}$ as follows (once again, we use that for finite $T$, it has finite support):","text_sha256":"713484e7e3293403c96c86875fc6b9c7980f8d9b026c1b371fa489b9f7d2eee0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0037","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.1.1 Proof of Theorem 3 (necessity)","text":"$$\n\\begin{aligned}\n& \\bar{v}_{\\sigma}^{T, 2}(a, \\theta, m, \\tilde{\\omega}, \\mu)=\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{h^{t} \\in H^{t}} \\mathbb{P}_{\\sigma}^{t}\\left(h^{t}\\right) \\mathbb{P}_{\\sigma}^{t}\\left(\\tilde{\\omega} \\mid h^{t}\\right) f(\\theta) \\sigma_{t}\\left(h^{t}, \\theta\\right)(m) \\phi(a \\mid m, \\tilde{\\omega}) \\mathbb{1}\\left[\\mu_{t+1}\\left(h^{t}, \\theta, m, a\\right)=\\mu\\right]= \\\\\n& =\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{h^{t} \\in H^{t}: \\mu_{t+1}\\left(h^{t}, \\theta, m, a\\right)=\\mu} \\mathbb{P}_{\\sigma}^{t+1}\\left(\\tilde{\\omega}, h^{t}, \\theta, m, a\\right)=\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{h^{t} \\in H^{t}: \\mu_{t+1}\\left(h^{t}, \\theta, m, a\\right)=\\mu} \\mu(\\tilde{\\omega}) \\mathbb{P}_{\\sigma}^{t+1}\\left(h^{t}, \\theta, m, a\\right) \\\\\n& =\\mu(\\tilde{\\omega}) \\frac{1}{T} \\sum_{t=1}^{T} \\sum_{h^{t} \\in H^{t}: \\mu_{t+1}\\left(h^{t}, \\theta, m, a\\right)=\\mu} \\mathbb{P}_{\\sigma}^{t+1}\\left(h^{t}, \\theta, m, a\\right)=\\mu(\\tilde{\\omega}) \\bar{v}_{\\sigma, A \\Theta M \\Delta}^{T, 2}(a, \\theta, m, \\mu),\n\\end{aligned}\n$$\n\nwhere the last expression follows from noting that the term multiplying $\\mu(\\omega)$ in the first expression in the third line is $\\sum_{\\tilde{\\omega}} v_{\\sigma}^{T, 2}(a, \\theta, m, \\tilde{\\omega}, \\mu)$.\n\nThen, for every $g \\in C_{b}(A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega}))$, we have that\n\n$$\n\\mathbb{E}_{\\bar{v}_{\\sigma}^{T, 2}}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)]=\\mathbb{E}_{\\bar{v}_{\\sigma, A \\Theta M \\Delta(\\tilde{\\Omega})}^{T, 2}}\\left[\\mathbb{E}_{\\tilde{\\omega} \\sim \\mu}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)]\\right]\n$$\n\nwhich completes the proof that Equation C. 13 holds. Letting $V_{g}(a, \\theta, m, \\mu)=\\mathbb{E}_{\\tilde{\\omega} \\sim \\mu}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)]$, we have that\n\n$$\n\\int\\left(g-V_{g}\\right) d \\bar{v}_{\\sigma}^{T, 2}=0\n$$\n\nBecause $g-V_{g} \\in C_{b}(A \\times \\Theta \\times M \\times \\tilde{\\Omega} \\times \\Delta(\\tilde{\\Omega}))$ and $\\bar{v}_{\\sigma}^{T_{n}, 2} \\xrightarrow{w^{*}} \\bar{v}_{\\sigma}$, we conclude that\n\n$$\n\\mathbb{E}_{\\bar{v}_{\\sigma}}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)]=\\int_{A \\times \\Theta \\times M \\times \\Delta(\\tilde{\\Omega})} \\mathbb{E}_{\\tilde{\\omega} \\sim \\mu}[g(a, \\theta, m, \\tilde{\\omega}, \\mu)] \\bar{v}_{\\sigma, A \\Theta M \\Delta}(d(a, \\theta, m, \\mu)) .\n$$\n\nEquations C. 16 and C. 17 imply $\\bar{v}_{\\sigma}$ admits the following disintegration:\n\n$$\n\\bar{v}_{\\sigma}(\\{(a, \\theta, m, \\tilde{\\omega})\\} \\times \\tilde{\\Delta})=\\int_{\\tilde{\\Delta}} \\mu(\\tilde{\\omega}) f(\\theta) \\alpha^{\\prime}(a \\mid \\theta, m, \\mu) \\rho(m \\mid \\theta, \\mu) \\tau_{\\sigma}(d \\mu)\n$$\n\nwhere we disintegrated $\\bar{v}_{\\sigma, A \\Theta M \\Delta}$ first along $\\Theta \\times \\Delta(\\tilde{\\Omega})$-and used Equation C. 16 to obtain the independence\nof $\\Theta$ and $\\Delta(\\tilde{\\Omega})$-and then further disintegrated the distribution of $A \\times M$ conditional on $\\Theta \\times \\Delta(\\tilde{\\Omega})$. Now, Equation C. 15 implies that the following also holds\n\n$$\n\\bar{v}_{\\sigma}(\\{(a, \\theta, m, \\tilde{\\omega})\\} \\times \\tilde{\\Delta})=\\int_{\\tilde{\\Delta}} \\mu(\\tilde{\\omega}) f(\\theta) \\phi(a \\mid m, \\tilde{\\omega}) \\rho(m \\mid \\theta, \\mu) \\tau_{\\sigma}(d \\mu)\n$$\n\nwhere once again we use the uniqueness of disintegration. Because Equations C. 18 and C. 19 hold for any tuple $(a, \\theta, m, \\tilde{\\omega})$ and measurable subset $\\tilde{\\Delta}$ of $\\Delta(\\tilde{\\Omega})$, we conclude that (i) $\\alpha^{\\prime}(a \\mid \\theta, m, \\mu)$ does not depend on $\\theta \\tau_{\\sigma}$-almost everywhere, and (ii) $\\phi(\\cdot \\mid m, \\tilde{\\omega})$ is constant on $\\tilde{\\omega}$ in the support of $\\mu \\tau_{\\sigma}$-almost everywhere. This concludes the proof of Step 2.\n\nStep 3 We now argue that the agent achieves\n\n$$\nu^{*}(\\mu) \\equiv \\sum_{\\theta \\in \\Theta} f(\\theta) \\max _{m \\in M} \\sum_{\\tilde{\\omega} \\in \\tilde{\\Omega}} \\mu(\\tilde{\\omega}) \\sum_{a \\in A} \\phi(a \\mid m, \\tilde{\\omega}) u(a, \\theta, \\omega)=\\sum_{\\theta \\in \\Theta} f(\\theta) \\max _{m \\in M} \\sum_{\\tilde{\\omega} \\in \\tilde{\\Omega}} \\mu(\\tilde{\\omega}) \\sum_{a \\in A} \\alpha^{\\prime}(a \\mid m, \\mu) u(a, \\theta, \\omega),\n$$\n\non the support of $\\tau_{\\sigma}$, where the second equality follows from Step 2. Toward a contradiction, suppose this is not the case; that is,\n\n$$\n\\mathbb{E}_{\\tau_{\\sigma}}\\left[\\sum_{\\theta \\in \\Theta} f(\\theta) \\sum_{m \\in M} \\rho(m \\mid \\theta, \\mu) \\sum_{\\tilde{\\omega} \\in \\tilde{\\Omega}} \\mu(\\tilde{\\omega}) \\sum_{a} \\phi(a \\mid m, \\tilde{\\omega}) u(a, \\theta, \\omega)\\right]<\\mathbb{E}_{\\tau_{\\sigma}}\\left[\\sum_{\\theta \\in \\Theta} u^{*}(\\mu)\\right]=U^{*} .\n$$\n\nWe show that the agent can achieve a payoff arbitrarily close to $U^{*}$ by playing according to $\\sigma$ until some finite $T$ and then best-responding to her beliefs at time $T$ in every period thereafter; a contradiction.\n\nConsider a strategy $\\sigma^{\\prime}$ which until some period $T$ plays according to $\\sigma$ and after period $T$ best responds to $\\mu_{T}\\left(h^{T}\\right) \\in \\Delta(\\tilde{\\Omega})$. Because payoffs accumulated on a finite number of periods are irrelevant to longrun payoffs, this strategy results in a payoff:\n\n$$\n\\begin{aligned}\n& \\sum_{h^{T} \\in H^{T}} \\mathbb{P}_{\\sigma}^{T}\\left(h^{T}\\right) \\sum_{\\theta \\in \\Theta} f(\\theta) \\max _{m \\in M}\\left[\\sum_{\\tilde{\\omega} \\in \\tilde{\\Omega}} \\mu_{T}(\\tilde{\\omega}) \\sum_{a \\in A} \\phi(a \\mid m, \\tilde{\\omega}) u(a, \\theta, \\omega)\\right]=\\sum_{h^{T} \\in H^{T}} \\mathbb{P}_{\\sigma}^{T}\\left(h^{T}\\right) u^{*}\\left(\\mu_{T}\\left(h^{T}\\right)\\right) \\\\\n& =\\mathbb{E}_{\\mathbb{P}_{\\sigma}^{T} \\circ \\mu_{T}^{-1}}\\left[u^{*}(\\mu)\\right]=\\mathbb{E}_{\\mathbb{P}_{\\sigma} \\circ \\mu_{T}^{-1}}\\left[u^{*}(\\mu)\\right],\n\\end{aligned}\n$$\n\nwhere the last equality follows as $\\mu_{T}$ is adapted to the histories through $T$. Similar arguments to Lemma C. 3 imply that $\\mathbb{P}_{\\sigma} \\circ \\mu_{T}^{-1} \\xrightarrow{w^{*}} \\mathbb{P}_{\\sigma} \\circ \\mu_{\\infty}^{-1} \\equiv \\tau_{\\sigma}$. Noting that $u^{*}: \\Delta(\\tilde{\\Omega}) \\rightarrow \\mathbb{R}$ is continuous and bounded (as it is the maximum of linear functions in beliefs), we obtain that as $T \\rightarrow \\infty$,\n\n$$\n\\mathbb{E}_{\\mathbb{P}_{\\sigma} \\circ \\mu_{T}^{-1}}\\left[u^{*}(\\mu)\\right] \\rightarrow \\mathbb{E}_{\\tau_{\\sigma}}\\left[u^{*}(\\mu)\\right] .\n$$","text_sha256":"31225cbba99d5f7c6a09ee1c21f325ee2fca6b0b29c4d018227ba7172dbbf3da"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0038","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.1.1 Proof of Theorem 3 (necessity)","text":"It follows that for every $\\delta>0$, we can find $T$ large enough so that $\\left|\\mathbb{E}_{\\mathbb{P}_{\\sigma^{\\circ}} \\mu_{T}^{-1}}\\left[u^{*}(\\mu)\\right]-U^{*}\\right|<\\delta$, contradicting the optimality of $\\sigma$.\n\nWe conclude that $\\alpha: \\Theta \\times \\Delta(\\tilde{\\Omega}) \\rightarrow \\Delta(A)$ defined as follows:\n\n$$\n\\alpha(a \\mid \\theta, \\mu)=\\sum_{m \\in M} \\rho(m \\mid \\theta, \\mu) \\alpha^{\\prime}(a \\mid m, \\mu),\n$$\n\nis incentive compatible and individually rational $\\tau_{\\sigma}$-almost everywhere.\n\nStep 4 We now show how to derive a two-stage mechanism $\\beta^{*}: \\Omega \\rightarrow \\Delta(\\Delta(\\Omega))$ and an allocation rule $\\alpha^{*}: \\Theta \\times \\Delta(\\Omega) \\rightarrow \\Delta(A)$ that implement $\\vartheta$. First, note that the agent's payoff when her type is $\\theta$ and the induced belief is $\\mu \\in \\Delta(\\tilde{\\Omega})$, can be written as\n\n$$\n\\sum_{\\tilde{\\omega} \\in \\tilde{\\Omega}} \\mu(\\tilde{\\omega}) \\sum_{a \\in A} \\alpha(a \\mid \\theta, \\mu) u(a, \\theta, \\omega)=\\sum_{\\omega \\in \\Omega} \\mu_{\\Omega}(\\omega) \\sum_{a \\in A} \\alpha(a \\mid \\theta, \\mu) u(a, \\theta, \\omega),\n$$\n\nwhere $\\mu_{\\Omega}$ is the marginal of $\\mu$ on $\\Omega$ and the equality follows because the realization of $\\varepsilon$ is payoffirrelevant. By Step 3, $\\alpha(\\cdot \\mid \\theta, \\mu)$ is individually rational and incentive compatible when the agent holds belief $\\mu_{\\Omega}$.\n\nFurthermore, for each $(a, \\theta, \\omega) \\in A \\times \\Theta \\times \\Omega$, we have\n\n$$\n\\vartheta(a, \\theta, \\omega)=\\sum_{\\varepsilon \\in \\mathcal{E}} \\int_{\\Delta(\\tilde{\\Omega})} \\mu(\\omega, \\varepsilon) f(\\theta) \\alpha(a \\mid \\theta, \\mu) \\tau_{\\sigma}(d \\mu)=\\int_{\\Delta(\\tilde{\\Omega})} \\mu_{\\Omega}(\\omega) f(\\theta) \\alpha(a \\mid \\theta, \\mu) \\tau_{\\sigma}(d \\mu)\n$$\n\nFor each $\\theta \\in \\Theta$, consider the joint distribution $Q_{\\theta} \\in \\Delta(A \\times \\Delta(\\Omega))$ defined as follows:\n\n$$\nQ_{\\theta}(\\{a\\} \\times \\tilde{\\Delta})=\\int_{\\Delta(\\tilde{\\Omega})} \\mathbb{1}\\left[\\mu_{\\Omega} \\in \\tilde{\\Delta}\\right] \\alpha(a \\mid \\theta, \\mu) \\tau_{\\sigma}(d \\mu)=\\int_{\\tilde{\\Delta}} \\alpha^{*}\\left(a \\mid \\theta, \\mu_{\\Omega}\\right) \\tau^{*}\\left(d \\mu_{\\Omega}\\right)\n$$\n\nwhere the third equality follows from disintegration (note $\\alpha^{*}\\left(\\cdot \\mid \\cdot, \\mu_{\\Omega}\\right)=\\mathbb{E}\\left[\\alpha(\\cdot \\mid \\cdot, \\tilde{\\mu}) \\mid \\tilde{\\mu}_{\\Omega}=\\mu_{\\Omega}\\right]$ ). By the first argument in Step 4, $\\alpha^{*}\\left(\\cdot \\mid \\cdot, \\mu_{\\Omega}\\right)$ is individually rational and incentive compatible when the agent holds $\\mu_{\\Omega}$. We obtain that\n\n$$\n\\vartheta(a, \\theta, \\omega)=\\int_{\\Delta(\\Omega)} \\mu_{\\Omega}(\\omega) f(\\theta) \\alpha^{*}\\left(a \\mid \\theta, \\mu_{\\Omega}\\right) \\tau^{*}\\left(d \\mu_{\\Omega}\\right)\n$$\n\nDefining for all $\\omega \\in \\Omega$ and measurable subsets $\\tilde{\\Delta} \\in \\Delta(\\Omega)$,\n\n$$\n\\beta^{*}(\\tilde{\\Delta} \\mid \\omega)=\\int_{\\tilde{\\Delta}} \\frac{\\mu(\\omega)}{\\mu_{0}(\\omega)} \\tau^{*}(d \\mu)\n$$\n\nSteps 2-4 together imply that $\\vartheta$ can be implemented by the incentive compatible and individually rational two-stage mechanism $\\left(\\beta^{*}, \\alpha^{*}\\right)$.\n\nStep 5: The agent adequately learns Finally, we argue that the agent earns the same payoff as if she had access to the information structure calibrated to $\\phi, \\pi_{\\phi}$.\n\nLemma C.4. Let $\\tau_{\\phi}$ denote the belief distribution induced by the calibrated information structure $\\pi_{\\phi}$. Then, the agent's payoff under $\\sigma$ equals\n\n$$\nU\\left(\\tau_{\\phi}\\right) \\equiv \\mathbb{E}_{\\tau_{\\phi}}\\left[\\sum_{\\theta \\in \\Theta} f(\\theta) \\max _{m \\in M} \\sum_{\\tilde{\\omega} \\in \\tilde{\\Omega}} \\mu(\\tilde{\\omega}) \\sum_{a \\in A} \\phi(a \\mid m, \\tilde{\\omega}) u(a, \\theta, \\omega)\\right] .\n$$\n\nThe proof of this is standard, and hence we defer it to Appendix D.","text_sha256":"5826c7bbeb728c6f3e0aa5c7482a037bb92d50c08769d9220488aefecf177e81"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0039","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.1.2 Proof of Theorem 3 (sufficiency)","text":"## C.1.2 Proof of Theorem 3 (sufficiency)\n\nSuppose $\\vartheta \\in \\Delta(A \\times \\Theta \\times \\Omega)$ is implemented by an incentive compatible and individually rational twostage mechanism. That is,\n\n$$\n\\vartheta(a, \\theta, \\omega)=\\mu_{0}(\\omega) f(\\theta) \\int_{\\Delta(\\Omega)} \\alpha(a \\mid \\theta, \\mu) \\beta(d \\mu \\mid \\omega)\n$$\n\nand $\\alpha$ is incentive compatible and individually rational on the support of $\\mu_{0} \\otimes \\beta$. As in the proof of Theorem 2, a finite support $\\beta^{\\prime}: \\Omega \\rightarrow \\Delta\\left(\\left\\{\\mu_{1}, \\ldots, \\mu_{K}\\right\\}\\right)$ exists such that $\\left(\\beta^{\\prime}, \\alpha\\right)$ implement $\\vartheta$. As in Green and Stokey (2022), the experiment $\\beta^{\\prime}$ can be generated by a finite information structure $\\pi: \\Omega \\times \\mathcal{E} \\rightarrow$ $\\Delta(\\Omega)$, where (i) $\\mathcal{E}$ is finite, (ii) $\\mathcal{E}$ is independent of $\\Omega$, and (iii) $\\mu=\\pi(\\omega, \\varepsilon)$.\n\nConstruct a mechanism $\\phi: \\Theta \\times \\Omega \\times \\mathcal{E} \\rightarrow \\Delta(A)$ such that $\\phi(\\cdot \\mid \\theta, \\omega, \\varepsilon)=\\alpha(\\cdot \\mid \\theta, \\pi(\\omega, \\varepsilon))$. (Note that $\\pi$ is information structure calibrated to $\\phi$, but expressed in beliefs.) Consider now the extensive form game induced by such a mechanism. ${ }^{44}$\n\nIf the agent truthfully reports her type, then the occupation measure induces outcome distribution $\\vartheta$. Hence, under truthtelling, the agent's payoff is:\n\n$$\n\\begin{aligned}\n& U\\left(\\sigma_{\\text {truth }}\\right)=\\sum_{(a, \\theta, \\omega) \\in A \\times \\Theta \\times \\Omega} \\vartheta(a, \\theta, \\omega) u(a, \\theta, \\omega)=\\mathbb{E}_{\\tau_{\\phi}}\\left[\\sum_{\\theta \\in \\Theta} f(\\theta) \\sum_{a \\in A} \\alpha(a \\mid \\theta, \\mu) \\sum_{\\omega \\in \\Omega} \\mu(\\omega) u(a, \\theta, \\omega)\\right] \\\\\n& =\\mathbb{E}_{\\tau_{\\phi}}\\left[\\sum_{\\theta \\in \\Theta} f(\\theta) \\max \\left\\{\\max _{\\theta^{\\prime} \\in \\Theta} \\sum_{a \\in A} \\alpha\\left(a \\mid \\theta^{\\prime}, \\mu\\right) \\sum_{\\omega \\in \\Omega} \\mu(\\omega) u(a, \\theta, \\omega), \\sum_{\\omega \\in \\Omega} \\mu(\\omega) u\\left(a_{\\varnothing}, \\theta, \\omega\\right)\\right\\}\\right]= \\\\\n& =\\mathbb{E}_{\\mathcal{E}}\\left[\\sum_{\\theta \\in \\Theta} f(\\theta) \\max \\left\\{\\max _{\\theta^{\\prime} \\in \\Theta} \\sum_{\\omega \\in \\Omega} \\mu(\\omega) \\sum_{a \\in A} \\phi\\left(a \\mid \\theta^{\\prime}, \\omega, \\varepsilon\\right) u(a, \\theta, \\omega), \\sum_{\\omega \\in \\Omega} \\mu(\\omega) u\\left(a_{\\varnothing}, \\theta, \\omega\\right)\\right\\}\\right],\n\\end{aligned}\n$$\n\nwhere (i) $\\tau_{\\phi}$ is the belief distribution induced by the information structure $\\pi$, and (ii) the first equality is by definition of the occupation measure, the second is the definition that $\\vartheta$ is implemented by the two-stage mechanism, the third follows from incentive compatibility and individual rationality of $\\alpha$, and the fourth is definitional.\n\nMoreover, the payoff in the last line of Equation C. 23 is the payoff the agent obtains by using the \"learning\" strategy in Lemma C.4, which first extracts all the mechanism can teach her about the state and then uses that information to optimize over her participation and reporting strategies. It follows that truthtelling (and participation) are optimal and $\\vartheta$ is implemented by repeated mechanism $\\phi$.","text_sha256":"3909f2dc328902e89e6be59f62bf85ce34a716a3bae448276d350f0a9f3eab1c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0040","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C. 2 Dynamic Mechanisms","text":"## C. 2 Dynamic Mechanisms\n\nIn this section, we present the proof of Theorem 4. To do so, we first complete the formal definition of the game, by specifying the histories, strategy space, and the distribution over terminal histories induced by the agent's strategy and the mechanism. Having laid this groundwork, we describe the proof strategy, and then provide the formal details of the proof.\n\n[^28]Mechanisms, histories, and strategies A dynamic mechanism $\\left(\\varphi_{t}\\right)_{t \\in \\mathbb{N}}$ is a sequence of mappings that condition on the state, the agent's report history, the allocation history, and today's report and output an allocation. By the revelation principle, it is without loss of generality to restrict attention to mechanisms that solicit type reports.\n\nAs in the main text, we expand the set of type reports and allocations by the non-participation decision and the outside option, which we denote by $\\Theta A_{\\varnothing} \\equiv A \\times \\Theta \\cup\\left\\{\\left(\\varnothing, a_{\\varnothing}\\right)\\right\\}$. Then, $\\hat{H}^{t}=\\left(\\Theta A_{\\varnothing}\\right)^{t-1}$ denotes the histories of reports (inclusive of the non-participation decision) and allocations at the beginning of time $t \\in \\mathbb{N}$, and let $\\hat{\\mathcal{H}}^{t}=\\Omega \\times \\hat{H}^{t}$. Similarly, let $\\hat{H}^{\\infty}=\\times_{t \\in \\mathbb{N}}\\left(\\Theta A_{\\varnothing}\\right)$ denote the set of all possible report-allocation outcome paths, and let $\\hat{\\mathcal{H}}^{\\infty}=\\Omega \\times \\hat{H}^{\\infty}$. A dynamic mechanism is then a collection of mappings $\\left(\\varphi_{t}\\right)_{t \\in \\mathbb{N}}$ such that $\\varphi_{t}: \\hat{\\mathcal{H}}^{t} \\times \\Theta \\rightarrow \\Delta(A)$.\n\nTo define the agent's strategy, let $H^{t}=\\Theta^{t-1} \\times \\hat{H}^{t-1}$, where the coordinates denote the sequence of realized types, reports (inclusive of participation decisions), and allocations through period $t-1$. A behavioral strategy is a mapping $\\left(p_{t}, \\sigma_{t}\\right): H^{t} \\times \\Theta \\rightarrow[0,1] \\times \\Delta(\\Theta)$.\n\nThe distribution over terminal histories $\\mathcal{H}^{\\infty}$ To obtain the complete description of the paths on the tree we need to append $\\Omega$ to $H^{t}$; hence the paths through period $t-1$ are $\\Omega \\times H^{t} \\equiv \\mathcal{H}^{t}$. The distributions over states, agent's types, the agent's strategy, and the mechanism induce a distribution over the terminal histories $\\mathcal{H}^{\\infty} \\equiv \\Omega \\times H^{\\infty}$, which we denote by $\\mathbb{P}_{(p, \\sigma)} \\in \\Delta\\left(\\Omega \\times H^{\\infty}\\right)$. We denote by $\\mathbb{E}_{(p, \\sigma)}$ the expectation under this measure. The distribution $\\mathbb{P}_{(p, \\sigma)} \\in \\Delta\\left(\\Omega \\times H^{\\infty}\\right)$ is the unique distribution that satisfies that for all $t \\in \\mathbb{N}, \\tilde{\\mathcal{H}}^{t} \\subset \\Omega \\times \\mathcal{E} \\times H^{t}$,\n\n$$\n\\mathbb{P}_{(p, \\sigma)}\\left(\\tilde{\\mathcal{H}}^{t} \\times \\prod_{s=t+1}^{\\infty}\\left(\\Theta \\times \\Theta A_{\\varnothing}\\right)=\\mathbb{P}_{\\sigma}^{t}\\left(\\tilde{\\mathcal{H}}^{t}\\right),\\right.\n$$\n\nwhere the distributions $\\left(\\mathbb{P}_{(p, \\sigma)}^{t}\\right)_{t \\in \\mathbb{N}}$ satisfy (under participation and truthtelling)\n\n$$\n\\mathbb{P}_{(p, \\sigma)}^{t+1}\\left(\\omega, h^{t}, \\theta, \\theta^{\\prime}, a\\right)=\\mathbb{P}_{(p, \\sigma)}^{t}\\left(\\omega, h^{t}\\right) f(\\theta) \\mathbb{1}\\left[\\theta^{\\prime}=\\theta\\right] \\varphi_{t}\\left(a \\mid \\omega, \\hat{h}^{t}, \\theta^{\\prime}\\right) .\n$$\n\nImplementation We focus on incentive-compatible mechanisms $\\varphi$ for which (i) a best response, ( $p, \\sigma$ ), exists, and (ii) the occupation measure $v_{\\sigma} \\in \\Delta(A \\times \\Theta \\times \\Omega)$ exists, where\n\n$$\nv_{(p, \\sigma)}(a, \\theta, \\omega)=\\lim _{T \\rightarrow \\infty} \\frac{1}{T} \\mathbb{E}_{(p, \\sigma)}\\left[\\sum_{t=1}^{T} \\mathbb{1}\\left[\\left(a_{t}, \\theta_{t}, \\omega^{\\prime}\\right)=(a, \\theta, \\omega)\\right]\\right],\n$$\n\nwhere the limit is in the weak* sense. In contrast to Appendix C.1, we do not keep track of the agent's type reports in the occupation measure, only the agent's types. Under $(p, \\sigma)$ only truthtelling histories have positive probability.","text_sha256":"22158bc9d71659d494be443469c84f0236e01d795f29c12d9c63570e9ed95841"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0041","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.2.1 Proof of Theorem 4 (necessity)","text":"## C.2.1 Proof of Theorem 4 (necessity)\n\nLet $\\vartheta \\in \\Delta(A \\times \\Theta \\times \\Omega)$ denote the outcome distribution implemented by an incentive compatible dynamic mechanism $\\varphi$, and let $v_{(p, \\sigma)}$ denote the occupation measure under the agent's truthtelling strategy. Below, we show that $v_{(p, \\sigma)}$, and hence $\\vartheta$, can be implemented by a two-stage mechanism which lacks profitable undetectable deviations and is ex ante individually rational.\n\nAnalogously to the proof of Theorem 3, we define two sequences of extended occupation measures on $A \\times \\Theta \\times \\Omega \\times \\Delta(\\Omega)$ defined as follows. Letting $\\tilde{\\Delta}$ denote a measurable subset of $\\Delta(\\Omega)$, define\n\n$$\n\\begin{aligned}\n& \\bar{v}_{(p, \\sigma)}^{T, 1}(\\{(a, \\theta, \\omega)\\} \\times \\tilde{\\Delta})=\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{h^{t} \\in H^{t}} \\mathbb{P}_{(p, \\sigma)}^{t}\\left(\\omega, h^{t}\\right) f(\\theta) \\sigma_{t}\\left(h^{t}, \\theta\\right)(\\theta) \\varphi\\left(\\omega, \\hat{h}^{t}, \\theta\\right)(a) \\mathbb{1}\\left[\\mu_{t}\\left(h^{t}\\right) \\in \\tilde{\\Delta}\\right] \\\\\n& \\bar{v}_{(p, \\sigma)}^{T, 2}(\\{(a, \\theta, \\omega)\\} \\times \\tilde{\\Delta})=\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{h^{t} \\in H^{t}} \\mathbb{P}_{(p, \\sigma)}^{t}\\left(\\omega, h^{t}\\right) f(\\theta) \\sigma_{t}\\left(h^{t}, \\theta\\right)(\\theta) \\varphi\\left(\\omega, \\hat{h}^{t}, \\theta\\right)(a) \\mathbb{1}\\left[\\mu_{t+1}\\left(h^{t}, \\theta, \\theta, a\\right) \\in \\tilde{\\Delta}\\right]\n\\end{aligned}\n$$\n\nThe proof proceeds similarly to that in Appendix C.1. First, we show that the occupation measure $v_{(p, \\sigma)} \\in \\Delta(A \\times \\Theta \\times \\Omega)$ admits the following decomposition\n\n$$\nv_{(p, \\sigma)}(a, \\theta, \\omega)=\\int_{\\Delta(\\Omega)} f(\\theta) \\mu(\\omega) \\alpha(a \\mid \\theta, \\mu) \\tau_{(p, \\sigma)}(d \\mu)\n$$\n\nwhere $\\tau_{(p, \\sigma)}$ is the distribution over terminal beliefs (cf. Lemma C.3) and the transition probability $\\alpha: \\Theta \\times \\Delta(\\Omega) \\rightarrow \\Delta(A)$ is our candidate allocation rule. Consequently, the agent's equilibrium payoff can be written as follows:\n\n$$\n\\sum_{(a, \\theta, \\omega) \\in A \\times \\Theta \\times \\Omega} v_{(p, \\sigma)}(a, \\theta, \\omega) u(a, \\theta, \\omega)=\\int_{\\Delta(\\Omega)}\\left[\\sum_{\\theta \\in \\Theta} f(\\theta) \\sum_{\\omega \\in \\Omega} \\mu(\\omega) \\sum_{a \\in A} \\alpha(a \\mid \\theta, \\mu) u(a, \\theta, \\omega)\\right] \\tau_{(p, \\sigma)}(d \\mu) .\n$$\n\nSecond, we show that the allocation rule lacks profitable undetectable deviations and is ex ante individually rational.\n\nThe occupation measure satisfies Equation C. 27 To prove that Equation C. 27 holds, we first show that for all $g \\in C_{b}(A \\times \\Theta \\times \\Omega \\times \\Delta(\\Omega))$ and all $T \\in \\mathbb{N}$,\n\n$$\n\\int_{A \\times \\Theta \\times \\Omega \\times \\Delta(\\Omega)} g(a, \\theta, \\omega, \\mu) d \\bar{v}_{(p, \\sigma)}^{T, 2}=\\int_{A \\times \\Theta \\times \\Delta(\\Omega)} \\mathbb{E}_{\\mu}[g(a, \\theta, \\omega, \\mu)] d \\bar{v}_{(p, \\sigma), A \\Theta \\Delta(\\Omega)}^{T, 2}\n$$\n\nand for all $q \\in C_{b}(\\Theta \\times \\Delta(\\Omega))$ and all $T \\in \\mathbb{N}$,\n\n$$\n\\int_{\\Theta \\times \\Delta(\\Omega)} q(\\theta, \\mu) d \\bar{v}_{(p, \\sigma), \\Theta \\Delta}^{T, 1}=\\int_{\\Delta(\\Omega)} \\sum_{\\theta \\in \\Theta} f(\\theta) q(\\theta, \\mu) d \\bar{v}_{(p, \\sigma), \\Delta}^{T, 1}\n$$\n\nwhere the subscripts on $\\bar{v}$ next to ( $p, \\sigma$ ) are the spaces over which we take the marginals, and $\\Delta$ is shorthand notation for $\\Delta(\\Omega)$. We skip the proof of this step as it basically repeats the proof of the analogous step in Appendix C.1.\n\nBecause $\\Delta(A \\times \\Theta \\times \\Omega \\times \\Delta(\\Omega))$ is compact (Aliprantis and Border, 2006, Theorem 15.11), $\\bar{v}_{(p, \\sigma)}^{T, 1}$ has a convergent subsequence $\\left(\\bar{v}_{(p, \\sigma)}^{T_{n}, 1}\\right)_{n \\in \\mathbb{N}}$, which by Lemma C. 2 is also a convergent subsequence of $\\bar{v}_{(p, \\sigma)}^{T, 2}$. Let $\\bar{v}_{(p, \\sigma)}$ denote the weak* limit along $T_{n}$. The continuity of the projection implies that $v_{(p, \\sigma)}$ is the marginal of $\\bar{v}_{(p, \\sigma)}$ on $A \\times \\Theta \\times \\Omega$, and $\\tau_{(p, \\sigma)} \\equiv \\mathbb{P}_{\\sigma} \\circ \\mu_{\\infty}^{-1}$ is the marginal on $\\Delta(\\Omega)$. Moreover, Equation C. 29 and Equation C. 30 together imply that $v_{(p, \\sigma)}$ admits the decomposition on the right hand side of Equation C.27, and the result follows.\n\nThe allocation rule lacks profitable undetectable deviations We now show the allocation rule $\\alpha$ admits no profitable undetectable deviations. An undetectable deviation is a transition probability $\\sigma^{\\prime}$ from $\\Theta \\times \\Delta(\\Omega)$ to $\\Delta(\\Theta)$ such that for all $\\mu \\in \\Delta(\\Omega)$ and $\\theta^{\\prime} \\in \\Theta$\n\n$$\n\\sum_{\\theta \\in \\Theta} f(\\theta) \\sigma^{\\prime}\\left(\\theta^{\\prime} \\mid \\theta, \\mu\\right)=f\\left(\\theta^{\\prime}\\right) .\n$$\n\nConsider a deviation by the agent to $\\left(p, \\sigma^{\\prime}\\right)$ instead of $(p, \\sigma)$. That is, when his type is $\\theta$ and belief is $\\mu$, the agent chooses type $\\theta^{\\prime}$ with probability $\\sigma^{\\prime}\\left(\\theta^{\\prime} \\mid \\theta, \\mu\\right)$. In what follows, we index the induced distributions over histories only by $\\sigma$ and $\\sigma^{\\prime}$ as we are only changing the agent's reporting strategy. In particular, denote by $\\mathbb{P}_{\\sigma^{\\prime}}$ the induced probability distribution over terminal histories when the agent uses $\\left(p, \\sigma^{\\prime}\\right)$ instead of $(p, \\sigma)$.\n\nWe first claim that for every $t$ the marginal of $\\mathbb{P}_{\\sigma^{\\prime}}^{t}$ over $\\Omega \\times \\hat{H}^{t}$ coincides with that of $\\mathbb{P}_{\\sigma}^{t}$. Recall that for every $t$ we have that\n\n$$\n\\mathbb{P}_{\\sigma^{\\prime}}^{t+1}\\left(\\omega, h^{t}, \\theta, \\theta^{\\prime}, a\\right)=\\mathbb{P}_{\\sigma^{\\prime}}^{t}\\left(\\omega, h^{t}\\right) f(\\theta) \\sigma^{\\prime}\\left(\\theta^{\\prime} \\mid \\mu_{t}\\left(h^{t}\\right), \\theta\\right) \\varphi_{t}\\left(a \\mid \\omega, \\hat{h}^{t}, \\theta^{\\prime}\\right) .\n$$\n\nAdding up over $\\theta$ on both sides and using Equation C.31, we get:\n\n$$\n\\sum_{\\theta \\in \\Theta} \\mathbb{P}_{\\sigma^{\\prime}}^{t+1}\\left(\\omega, h^{t}, \\theta, \\theta^{\\prime}, a\\right)=\\mathbb{P}_{\\sigma^{\\prime}}^{t}\\left(\\omega, h^{t}\\right) f\\left(\\theta^{\\prime}\\right) \\varphi_{t}\\left(a \\mid \\omega, \\hat{h}^{t}, \\theta^{\\prime}\\right) .\n$$","text_sha256":"19d00a924d3fb547fb63ab2cb6f11ca465018ba66b6a70c154312c16dc242aa6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0042","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.2.1 Proof of Theorem 4 (necessity)","text":"Now, note that $h^{t}=\\left(\\hat{h}^{t}, \\tilde{\\theta}^{t-1}\\right)$ for some sequence $\\tilde{\\theta}^{t-1} \\in \\Theta^{t-1}$. If we add up on both sides over all such sequences we get\n\n$$\n\\sum_{\\theta \\in \\Theta, \\tilde{\\theta}^{t-1} \\in \\Theta^{t-1}} \\mathbb{P}_{\\sigma^{\\prime}}^{t+1}\\left(\\omega, \\hat{h}^{t}, \\tilde{\\theta}^{t-1}, \\theta, \\theta^{\\prime}, a\\right)=\\sum_{\\tilde{\\theta}^{t-1} \\in \\Theta^{t-1}} \\mathbb{P}_{\\sigma^{\\prime}}^{t}\\left(\\omega, \\hat{h}^{t}, \\tilde{\\theta}^{t-1}\\right) f\\left(\\theta^{\\prime}\\right) \\varphi_{t}\\left(a \\mid \\omega, \\hat{h}^{t}, \\theta^{\\prime}\\right) .\n$$\n\nNote that if the distribution over $\\Omega \\times \\hat{H}^{t}$ induced by $\\sigma^{\\prime}$ up to period $t$ is the same as that induced by $\\sigma$, we get that the right-hand side equals:\n\n$$\n\\mathbb{P}_{\\sigma, \\hat{\\mathcal{H}^{t}}}^{t}\\left(\\omega, \\hat{h}^{t}\\right) f\\left(\\theta^{\\prime}\\right) \\varphi_{t}\\left(a \\mid \\omega, \\hat{h}^{t}, \\theta^{\\prime}\\right),\n$$\n\nand hence $\\mathbb{P}_{\\sigma^{\\prime}, \\hat{\\mathcal{H}}^{t+1}}^{t+1}\\left(\\omega, \\hat{h}^{t}, \\theta^{\\prime}, a\\right)=\\mathbb{P}_{\\sigma, \\hat{\\mathcal{H}}^{t}}^{t}\\left(\\omega, \\hat{h}^{t}\\right) f\\left(\\theta^{\\prime}\\right) \\varphi_{t}\\left(a \\mid \\omega, \\hat{h}^{t}, \\theta^{\\prime}\\right)=\\mathbb{P}_{\\sigma, \\hat{\\mathcal{H}}^{t+1}}^{t+1}\\left(\\omega, \\hat{h}^{t}, \\theta^{\\prime}, a\\right)$. By definition of $\\mathbb{P}_{\\sigma^{\\prime}}$, we conclude that $\\mathbb{P}_{\\sigma^{\\prime}, \\hat{\\mathcal{H}}^{\\infty}}=\\mathbb{P}_{\\sigma, \\hat{\\mathcal{H}}^{\\infty}}$. Hence, the joint distribution over states, reports, and allocations is the same under $\\sigma$ and $\\sigma^{\\prime}$.\n\nLet $\\bar{v}_{\\sigma^{\\prime}}^{T, 1}, \\bar{v}_{\\sigma^{\\prime}}^{T, 2} \\in \\Delta(A \\times \\Theta \\times \\hat{\\Theta} \\times \\Omega \\times \\Delta(\\Omega))$ denote the analogue of the occupation measures in Equations C. 25 and C. 26 corresponding to $\\sigma^{\\prime}$, extended to account for the agent's reports. Below, the notation $\\hat{\\Theta}$ signifies those are the agent's reports. In what follows, recalling that the belief system depends only on the reported history and not the type history is useful. Equation C. 31 implies that for all measurable\nsubsets $\\tilde{\\Delta}$ of $\\Delta(\\Omega)$,\n\n$$\n\\begin{aligned}\n& \\sum_{\\theta \\in \\Theta} \\bar{v}_{\\sigma^{\\prime}}^{T, 1}\\left(\\left\\{\\left(a, \\theta, \\theta^{\\prime}, \\omega\\right)\\right\\} \\times \\tilde{\\Delta}\\right)=\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{h^{t} \\in H^{t}} \\mathbb{P}_{\\sigma^{\\prime}}^{t}\\left(\\omega, h^{t}\\right) f\\left(\\theta^{\\prime}\\right) \\varphi_{t}\\left(\\omega, \\hat{h}^{t}, \\theta^{\\prime}\\right)(a) \\mathbb{1}\\left[\\mu_{t}\\left(h^{t}\\right) \\in \\tilde{\\Delta}\\right] \\\\\n& =\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{\\hat{h}^{t} \\in \\hat{H}^{t}} \\mathbb{P}_{\\sigma^{\\prime}}^{t}\\left(\\omega, \\hat{h}^{t}\\right) f\\left(\\theta^{\\prime}\\right) \\varphi_{t}\\left(\\omega, \\hat{h}^{t}, \\theta^{\\prime}\\right)(a) \\mathbb{1}\\left[\\mu_{t}\\left(\\hat{h}^{t}\\right) \\in \\tilde{\\Delta}\\right] \\\\\n& =\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{\\hat{h}^{t} \\in \\hat{H}^{t}} \\mathbb{P}_{\\sigma}^{t}\\left(\\omega, \\hat{h}^{t}\\right) f\\left(\\theta^{\\prime}\\right) \\varphi_{t}\\left(\\omega, \\hat{h}^{t}, \\theta^{\\prime}\\right)(a) \\mathbb{1}\\left[\\mu_{t}\\left(\\hat{h}^{t}\\right) \\in \\tilde{\\Delta}\\right]=\\bar{v}_{\\sigma}^{T, 1}\\left(\\left\\{\\left(a, \\theta^{\\prime}, \\omega\\right)\\right\\} \\times \\tilde{\\Delta}\\right) .\n\\end{aligned}\n$$\n\nThe first equality uses the definition of undetectability, the second uses that all the terms depend only on the reported history, the third uses that $\\sigma$ and $\\sigma^{\\prime}$ induce the same distribution over states, reports, and allocations, and the last is the definition of the occupation measure induced by $\\sigma$. In words, the marginal of $\\bar{v}_{\\sigma^{\\prime}}^{T, 1}$ over allocations, reports, states, and beliefs, $\\bar{v}_{\\sigma^{\\prime}, A \\hat{\\Theta} \\Omega \\Delta}^{T, 1}$ coincides with $\\bar{v}_{\\sigma}^{T, 1}$.\n\nWe now show that $\\bar{v}_{\\sigma^{\\prime}}^{T, 1}$ and $\\bar{v}_{\\sigma^{\\prime}}^{T, 2}$ have a convergent subsequence with limit $\\bar{v}_{\\sigma^{\\prime}} \\in \\Delta(A \\times \\Theta \\times \\hat{\\Theta} \\times \\Omega \\times \\Delta(\\Omega))$ that admits the following decomposition:\n\n$$\n\\mathbb{E}_{\\bar{v}_{\\sigma^{\\prime}}}[u(a, \\theta, \\omega)]=\\int_{\\Delta(\\Omega)}\\left[\\sum_{\\theta \\in \\Theta} f(\\theta) \\sum_{\\theta^{\\prime} \\in \\Theta} \\sigma^{\\prime}(\\theta, \\mu)\\left(\\theta^{\\prime}\\right) \\sum_{a} \\alpha\\left(a \\mid \\theta^{\\prime}, \\mu\\right) u(a, \\theta, \\mu)\\right] d \\tau_{(p, \\sigma)},\n$$\n\nwhere $u(a, \\theta, \\mu)$ is the linear extension of $u(a, \\theta, \\cdot)$.\nWe proceed as follows: First, we show that for each $T$, under $v_{\\sigma^{\\prime}}^{T, 1}$, the allocation is independent of the true type conditional on the period- $t$ belief and the reported type. Indeed,\n\n$$\n\\begin{aligned}\n\\sum_{\\omega \\in \\Omega} v_{\\sigma^{\\prime}}^{T, 1}\\left(a, \\theta, \\theta^{\\prime}, \\omega, \\mu\\right) & =\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{h^{t} \\in H^{t}} \\mathbb{P}_{\\sigma^{\\prime}}\\left(h^{t}\\right)\\left(\\sum_{\\omega \\in \\Omega} \\mathbb{P}_{\\sigma^{\\prime}}\\left(\\omega \\mid h^{t}\\right) \\varphi_{t}\\left(\\omega, \\hat{h}^{t}, \\theta^{\\prime}\\right)(a)\\right) f(\\theta) \\sigma^{\\prime}\\left(\\mu_{t}\\left(h^{t}\\right), \\theta\\right)\\left(\\theta^{\\prime}\\right) \\mathbb{1}\\left[\\mu_{t}\\left(h^{t}\\right)=\\mu\\right] \\\\\n& =\\frac{f(\\theta) \\sigma^{\\prime}(\\theta, \\mu)\\left(\\theta^{\\prime}\\right)}{f\\left(\\theta^{\\prime}\\right)}\\left(\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{\\hat{h}^{t}: \\mu_{t}\\left(\\hat{h}^{t}\\right)=\\mu} \\mathbb{P}_{\\sigma^{\\prime}}^{t}\\left(\\hat{h}^{t}\\right) \\sum_{\\omega \\in \\Omega} \\mathbb{P}_{\\sigma^{\\prime}}\\left(\\omega \\mid \\hat{h}^{t}\\right) f\\left(\\theta^{\\prime}\\right) \\varphi_{t}\\left(\\hat{h}^{t}, \\theta^{\\prime}\\right)(a)\\right) \\\\\n& =\\frac{f(\\theta) \\sigma^{\\prime}(\\theta, \\mu)\\left(\\theta^{\\prime}\\right)}{f\\left(\\theta^{\\prime}\\right)} v_{\\sigma^{\\prime}, A \\hat{\\Theta} \\Delta}^{T, 1}\\left(a, \\theta^{\\prime}, \\mu\\right)=\\frac{f(\\theta) \\sigma^{\\prime}(\\theta, \\mu)\\left(\\theta^{\\prime}\\right)}{f\\left(\\theta^{\\prime}\\right)} v_{\\sigma, A \\Theta \\Delta}^{T, 1}\\left(a, \\theta^{\\prime}, \\mu\\right),\n\\end{aligned}\n$$\n\nwhere the third and fourth equalities use Equation C.25. Moreover, the same analysis as that under $\\sigma$ implies the agent's true type is independent of the belief.","text_sha256":"093536a694b0e0c5337312afd4afee6cc3529df3050b980a9904537fe6490b58"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0043","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.2.1 Proof of Theorem 4 (necessity)","text":"Therefore, $\\bar{v}_{\\sigma^{\\prime}, A \\Theta \\Theta \\leq}^{T, 1}$ admits decomposition:\n\n$$\n\\bar{v}_{\\sigma^{\\prime}, A \\Theta \\hat{\\Theta} \\Delta}^{T, 1}\\left(a, \\theta, \\theta^{\\prime}, \\mu\\right)=\\frac{f(\\theta) \\sigma^{\\prime}\\left(\\theta^{\\prime} \\mid \\theta, \\mu\\right)}{f\\left(\\theta^{\\prime}\\right)} \\bar{v}_{\\sigma^{\\prime}, A \\hat{\\Theta} \\Delta}^{T, 1}\\left(a, \\theta^{\\prime}, \\mu\\right)=\\frac{f(\\theta) \\sigma^{\\prime}\\left(\\theta^{\\prime} \\mid \\theta, \\mu\\right)}{f\\left(\\theta^{\\prime}\\right)} \\bar{v}_{\\sigma, A \\Theta \\Delta}^{T, 1}\\left(a, \\theta^{\\prime}, \\mu\\right),\n$$\n\nwhere the second equality follows from Equation C.32.\nSecond, by the same arguments as in Appendix C.1, $\\bar{v}_{\\sigma^{\\prime}}^{T, 2}\\left(a, \\theta, \\theta^{\\prime}, \\omega, \\mu\\right)$ admits decomposition $\\mu(\\omega) \\bar{v}_{\\sigma^{\\prime}, A \\Theta \\Theta \\leq}^{T, 2}\\left(a, \\theta, \\theta^{\\prime}, \\mu\\right)$ for each $T$.\n\nThird, convergent subsequences $v_{\\sigma^{\\prime}}^{T_{n_{m}}, 1}$ and $v_{\\sigma^{\\prime}}^{T_{n_{m}}, 2}$ exist with limit $\\bar{v}_{\\sigma^{\\prime}}$ (cf. Lemma C.2). ${ }^{45}$ We note two\n\n[^29]things. On the one hand, because our previous arguments show that the set of measures admitting the above decompositions is closed, the limit $\\bar{v}_{\\sigma^{\\prime}}$ admits the decomposition. That is,\n$$\n\\int_{A \\times \\Theta \\times \\hat{\\Theta} \\times \\Omega \\times \\Delta(\\Omega)} u(a, \\theta, \\omega) \\bar{v}_{\\sigma^{\\prime}}\\left(d\\left(a, \\theta, \\theta^{\\prime}, \\omega, \\mu\\right)\\right)=\\int_{A \\times \\hat{\\Theta} \\times \\Delta(\\Omega)} \\sum_{\\theta \\in \\Theta} \\frac{f(\\theta) \\sigma^{\\prime}\\left(\\theta^{\\prime} \\mid \\theta, \\mu\\right)}{f\\left(\\theta^{\\prime}\\right)}\\left(\\sum_{\\omega \\in \\Omega} \\mu(\\omega) u(a, \\theta, \\omega)\\right) d \\bar{v}_{\\sigma^{\\prime}, A \\hat{\\Theta} \\Delta}\n$$\nOn the other hand, because $T_{n_{m}}$ is a subsequence of $T_{n}$ and $\\bar{v}_{\\sigma^{\\prime}, A \\hat{\\Theta} \\Omega \\Delta}^{T, 1}=\\bar{v}_{\\sigma, A \\Theta \\Omega \\Delta}^{T, 1}$ and $\\bar{v}_{\\sigma, A \\Theta \\Omega \\Delta}^{T_{n}, 1} \\xrightarrow{w^{*}} \\bar{v}_{\\sigma}$, we can conclude that $\\bar{v}_{\\sigma^{\\prime}, A \\hat{\\Theta} \\Delta}=\\bar{v}_{\\sigma, A \\Theta \\Delta}$ and admits the same decomposition as $\\bar{v}_{\\sigma}$. We conclude that\n$$\n\\mathbb{E}_{\\bar{v}_{\\sigma^{\\prime}}}[u(a, \\theta, \\omega)]=\\int_{\\Delta(\\Omega)}\\left[\\sum_{\\theta, \\theta^{\\prime} \\in \\Theta} f(\\theta) \\sigma^{\\prime}\\left(\\theta^{\\prime} \\mid \\theta, \\mu\\right) \\sum_{a} \\alpha\\left(a \\mid \\theta^{\\prime}, \\mu\\right)\\left(\\sum_{\\omega \\in \\Omega} \\mu(\\omega) u(a, \\theta, \\omega)\\right)\\right] d \\tau_{(p, \\sigma)}\n$$\nConsequently,\n$$\n\\begin{aligned}\n\\lim \\sup _{T \\rightarrow \\infty} \\mathbb{E}_{\\sigma^{\\prime}}\\left[U_{T}\\right] \\geq \\lim _{m \\rightarrow \\infty} \\mathbb{E}_{\\sigma^{\\prime}}\\left[U_{T_{n_{m}}}\\right] & =\\mathbb{E}_{\\bar{v}_{\\sigma^{\\prime}}}[u(a, \\theta, \\omega)] \\\\\n& =\\mathbb{E}_{\\tau_{(p, \\sigma)}}\\left[\\sum_{\\theta, \\theta^{\\prime}, a} f(\\theta) \\sigma^{\\prime}(\\theta, \\mu)\\left(\\theta^{\\prime}\\right) \\alpha\\left(a \\mid \\theta^{\\prime}, \\mu\\right) u(a, \\theta, \\mu)\\right],\n\\end{aligned}\n$$\nwhere $u(a, \\theta, \\mu)$ is the linear extension of $u(a, \\theta, \\cdot)$. Because $\\sigma$ is a best response, we have that\n$$\n\\mathbb{E}_{\\tau_{(p, \\sigma)}}\\left[\\sum_{\\theta, a} f(\\theta) \\alpha(a \\mid \\theta, \\mu) u(a, \\theta, \\mu)\\right] \\geq \\mathbb{E}_{\\tau_{(p, \\sigma)}}\\left[\\sum_{\\theta, \\theta^{\\prime}, a} f(\\theta) \\sigma^{\\prime}(\\theta, \\mu)\\left(\\theta^{\\prime}\\right) \\alpha\\left(a \\mid \\theta^{\\prime}, \\mu\\right) u(a, \\theta, \\mu)\\right],\n$$\nwhich implies the two-stage mechanism lacks profitable undetectable deviations.\n\nThe allocation rule is ex ante individually rational Define\n\n$$\nU_{\\mathrm{net}}(\\mu)=\\sum_{\\theta \\in \\Theta} f(\\theta) \\sum_{\\omega \\in \\Omega} \\mu(\\omega)\\left[\\sum_{a \\in A} \\alpha(a \\mid \\theta, \\mu) u(a, \\theta, \\omega)-u\\left(a_{\\varnothing}, \\theta, \\omega\\right)\\right],\n$$\n\nto be the agent's (ex ante) payoff net of the outside option at belief $\\mu$. Ex ante individual rationality of $\\alpha$ is equivalent to $U_{\\text {net }}(\\mu) \\geq 0$ for all $\\mu$ in the support of $\\tau_{(p, \\sigma)}$.\n\nToward a contradiction, assume that $U_{\\text {net }}(\\mu)<0$ with positive probability under $\\tau_{(p, \\sigma)}$. By Lemma D. 1\n\n[^30]in Appendix D, a set $B \\subset \\Delta(\\Omega)$ open relative to $\\Delta(\\Omega)$ exists such that ${ }^{46}$\n$$\n\\int_{B} U_{\\text {net }}(\\mu) \\tau_{(p, \\sigma)}(d \\mu)<0\n$$\nMoreover, we can pick $B$ such that $\\tau_{(p, \\sigma)}(\\partial B)=0$, where $\\partial B$ denotes the boundary of $B$ relative to $\\Delta(\\Omega) .{ }^{47}$ Lastly, let $\\delta>0$ be such that\n$$\n\\int_{B} U_{\\text {net }}(\\mu) \\tau_{(p, \\sigma)}(d \\mu) \\leq-2 \\delta\n$$\nLet $\\left(\\mu_{t}\\left(h^{t}\\right)\\right)_{t \\in \\mathbb{N}, h^{t} \\in H^{t}}$ denote the belief process under $(p, \\sigma)$. For $L \\in \\mathbb{N}$, define a strategy $\\left(p^{L}, \\sigma^{L}\\right)$ as follows:\n\n1. $\\left(p_{t}^{L}\\left(h^{t}, \\cdot\\right), \\sigma_{t}^{L}\\left(h^{t}, \\cdot\\right)\\right)=\\left(p_{t}\\left(h^{t}, \\cdot\\right), \\sigma_{t}\\left(h^{t}, \\cdot\\right)\\right)$ if either $t<L$ OR $\\left(t \\geq L\\right.$ and $\\left.\\mu_{L}\\left(h^{L}\\right) \\notin B\\right)$, where $h^{L}$ precedes $h^{t}$,\n2. Otherwise, $\\left(p_{t}^{L}\\left(h^{t}, \\cdot\\right), \\sigma_{t}^{L}\\left(h^{t}, \\cdot\\right)\\right)=\\left(0, \\sigma_{t}\\left(h^{t}, \\cdot\\right)\\right)$ (note that when the agent quits the strategy can be specified arbitrarily.)","text_sha256":"f8cafc4fadbe3830c02e6b974400de77887ccc96994a6d6080243ba3ba9d7107"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0044","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.2.1 Proof of Theorem 4 (necessity)","text":"Note the agent's average payoff through period $T$ under ( $p^{L}, \\sigma^{L}$ ) can be written as follows:\n$$\n\\mathbb{E}_{\\left(p^{L}, \\sigma^{L}\\right)}\\left[U_{T}\\right]=\\mathbb{E}_{(p, \\sigma)}\\left[U_{T}\\right]-\\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=1}^{T}\\left(u\\left(a_{t}, \\theta_{t}, \\omega\\right)-u\\left(a_{\\varnothing}, \\theta_{t}, \\omega\\right)\\right) \\mathbb{1}\\left[t \\geq L \\text { and } \\mu_{L} \\in B\\right]\\right] .\n$$\nWe show that for sufficiently large $L$, $\\left(p^{L}, \\sigma^{L}\\right)$ is a profitable deviation. For $T \\geq L$, write\n$$\n\\begin{aligned}\n& \\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=L}^{T}\\left(u\\left(a_{t}, \\theta_{t}, \\omega\\right)-u\\left(a_{\\varnothing}, \\theta_{t}, \\omega\\right)\\right) \\mathbb{1}\\left[\\mu_{L} \\in B\\right]\\right]-\\int_{B} U_{\\text {net }}(\\mu) \\tau_{(p, \\sigma)}(d \\mu)= \\\\\n& =\\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=1}^{T}\\left(u\\left(a_{t}, \\theta_{t}, \\omega\\right)-u\\left(a_{\\varnothing}, \\theta_{t}, \\omega\\right)\\right) \\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right]-\\int_{B} U_{\\text {net }}(\\mu) \\tau_{(p, \\sigma)}(d \\mu) \\\\\n& -\\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=1}^{L-1}\\left(u\\left(a_{t}, \\theta_{t}, \\omega\\right)-u\\left(a_{\\varnothing}, \\theta_{t}, \\omega\\right)\\right) \\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right] \\\\\n& +\\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=L}^{T}\\left(u\\left(a_{t}, \\theta_{t}, \\omega\\right)-u\\left(a_{\\varnothing}, \\theta_{t}, \\omega\\right)\\right)\\left(\\mathbb{1}\\left[\\mu_{L} \\in B\\right]-\\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right)\\right] .\n\\end{aligned}\n$$\nLet $K=\\max _{\\theta, \\omega, a}\\left|\\left(u(a, \\theta, \\omega)-u\\left(a_{\\varnothing}, \\theta, \\omega\\right)\\right)\\right|$, and note that we can bound the term in the last line of Equation C. 35 as follows:\n$$\n\\left|\\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=L}^{T}\\left(u\\left(a_{t}, \\theta_{t}, \\omega\\right)-u\\left(a_{\\varnothing}, \\theta_{t}, \\omega\\right)\\right)\\left(\\mathbb{1}\\left[\\mu_{L} \\in B\\right]-\\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right)\\right]\\right| \\leq K \\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=L}^{T}\\left|\\mathbb{1}\\left[\\mu_{L} \\in B\\right]-\\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right| .\\right]\n$$\n\n[^31]Because $\\mu_{t} \\xrightarrow{w^{*}} \\mu_{\\infty} \\mathbb{P}_{(p, \\sigma)}$-a.s. (Lemma C.1) and $\\tau_{(p, \\sigma)}(\\partial B)=0$, we conclude: ${ }^{48}$\n\n$$\n\\lim _{L \\rightarrow \\infty} \\sup _{t \\geq L}\\left|\\mathbb{1}\\left[\\mu_{L} \\in B\\right]-\\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right|=0 \\mathbb{P}_{(p, \\sigma)} \\text {-a.s. }\n$$\n\nThen,\n\n$$\n\\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=L}^{T}\\left|\\mathbb{1}\\left[\\mu_{L} \\in B\\right]-\\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right|\\right] \\leq \\mathbb{E}_{(p, \\sigma)}\\left[\\sup _{t \\geq L}\\left|\\mathbb{1}\\left[\\mu_{L} \\in B\\right]-\\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right|\\right]\n$$\n\nand choose $\\bar{L}$ large enough so that for all $L \\geq \\bar{L}$, we have that:\n\n$$\n\\mathbb{E}_{(p, \\sigma)}\\left[\\sup _{t \\geq L}\\left|\\mathbb{1}\\left[\\mu_{L} \\in B\\right]-\\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right|\\right] \\leq \\delta / K .\n$$\n\nConsider now the term in the third line of Equation C. 35 and note that it is bounded in absolute value by $K(L-1) / T$, which tends to 0 as $T \\rightarrow \\infty$. Similarly, the term in the second line of Equation C. 35 vanishes as $T \\rightarrow \\infty$. ${ }^{49}$ Thus, for $L \\geq \\bar{L}$, we can find $\\bar{T}$ such that for all $T \\geq \\bar{T}^{50}$\n\n$$\n\\left|\\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=L}^{T}\\left(u\\left(a_{t}, \\theta_{t}, \\omega\\right)-u\\left(a_{\\varnothing}, \\theta_{t}, \\omega\\right)\\right) \\mathbb{1}\\left[\\mu_{L} \\in B\\right]\\right]-\\int_{B} U_{\\mathrm{net}}(\\mu) \\tau_{(p, \\sigma)}(d \\mu)\\right| \\leq \\frac{3}{2} \\delta,\n$$\n\nand hence\n\n$$\n\\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=L}^{T}\\left(u\\left(a_{t}, \\theta_{t}, \\omega\\right)-u\\left(a_{\\varnothing}, \\theta_{t}, \\omega\\right)\\right) \\mathbb{1}\\left[\\mu_{L} \\in B\\right]\\right] \\leq-\\frac{1}{2} \\delta .\n$$\n\nWe conclude that\n\n$$\n\\lim \\sup _{T \\rightarrow \\infty} \\mathbb{E}_{\\left(p^{L}, \\sigma^{L}\\right)}\\left[U_{T}\\right] \\geq \\lim _{T \\rightarrow \\infty} \\mathbb{E}_{(p, \\sigma)}\\left[U_{T}\\right]+\\frac{1}{2} \\delta,\n$$\n\na contradiction.","text_sha256":"6eef3eee76b8594847b5a8527e30ed6dcf014b0de23cfb12067f776726d91a9a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0045","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.2.2 Proof of Theorem 4 (sufficiency)","text":"## C.2.2 Proof of Theorem 4 (sufficiency)\n\nWe now show that all outcome distributions $\\vartheta \\in \\Delta(A \\times \\Theta \\times \\Omega)$ that admit the decomposition in Theorem 4 can be implemented via a dynamic mechanism. To this end, let $\\tau$ and $\\alpha: \\Theta \\times \\Delta(\\Omega) \\rightarrow \\Delta(A)$\n\n[^32]denote the belief distribution and the ex ante individually rational allocation rule without profitable undetectable deviations corresponding to $\\vartheta$. That is,\n$$\n\\vartheta(a, \\theta, \\omega)=\\int_{\\Delta(\\Omega)} \\mu(\\omega) f(\\theta) \\alpha(a \\mid \\theta, \\mu) \\tau(d \\mu)\n$$\nThe proof proceeds as follows:\n\n1. We first consider a fictitious setting in which there is no state uncertainty and we are given an allocation rule $\\alpha^{\\prime}: \\Theta \\rightarrow \\Delta(A)$ that is ex ante individually rational and lacks profitable undetectable deviations for some utility function $u^{\\prime}: A \\times \\Theta \\rightarrow \\mathbb{R}$. Proposition C. 1 shows that a dynamic mechanism exists that implements $\\alpha^{\\prime}$.\n2. We then show that if $\\vartheta$ satisfies Equation C.39, then a finite support belief distribution $\\tau^{\\prime}$ exists such that $\\vartheta$ and $\\alpha$ satisfies Equation C. 39 with $\\tau^{\\prime}$ instead of $\\tau$.\n3. Lastly, we use this result to construct a dynamic game that implements $\\vartheta$.\n\nStep 1 For this step, we consider a fictitious setting in which there is no state uncertainty and the designer faces a privately informed agent with payoffs $u^{\\prime}: A \\times \\Theta \\rightarrow \\mathbb{R}$, where $\\theta \\sim f \\in \\Delta(\\Theta) .{ }^{51}$\n\nSuppose we are given an allocation rule $\\alpha^{\\prime}: \\Theta \\rightarrow \\Delta(A)$ that admits no profitable undetectable deviations relative to $u^{\\prime}$ as in Definition 7 and is individually rational as in Definition 8. We have the following result:\n\nProposition C.1. Let $\\vartheta^{\\prime}=f(\\theta) \\alpha^{\\prime}(a \\mid \\theta) \\in \\Delta(A \\times \\Theta)$ such that $\\alpha^{\\prime}$ lacks profitable undetectable deviations and is ex ante individually rational. Then, a dynamic mechanism exists that implements $\\vartheta^{\\prime}$.\n\nProof of Proposition C.1. The proof is constructive. We build on the analysis of Margaria and Smolin (2018) and present a dynamic mechanism that alternates between communication and adjustment phases. In all phases, the mechanism selects allocations using reports $\\theta^{\\prime}$ according to $\\alpha^{\\prime}$. In a communication phase, the reports are those sent by the agent. In an adjustment phase, the agent's reports are disregarded; instead, the mechanism simulates reports to guarantee that the occupation measure over reports coincides with $f$ and these simulated reports are used to determine the allocation. The mechanism ensures that under any agent's strategy, the occupation measure over reports and allocations exists and equals $\\vartheta^{\\prime}$; thus, any strategy corresponds to an undetectable deviation. The length of communication phases grows in time. Thus, under truthtelling the relative length of adjustment phases vanishes in time, and the expected occupation measure over types and allocations exists and equals $\\vartheta^{\\prime}$. Because $\\alpha^{\\prime}$ lacks profitable undetectable deviations, it follows that truthtelling is optimal for the agent. Because $\\alpha^{\\prime}$ is ex ante individually rational, it follows that the participation constraints are satisfied.\n\nFormally, the mechanism consists of sequential blocks, each block starting with a communication phase followed by an adjustment phase. The lengths of communication phases are fixed at $L_{1}, L_{2}, \\ldots$\n\n[^33]such that $L_{n} \\rightarrow \\infty$ and $L_{n} / \\sum_{k \\leq n} L_{k} \\rightarrow 0$, e.g., $L_{n}=n$. The length of adjustment phase $N_{n}$ depends on the agent's reports in the communication phase in block $n$. Denote by $T_{n}$ the first period of block $n$, which is the first period of the corresponding communication phase. The first period of the corresponding adjustment phase is $T_{n}+L_{n}+1$. Denote by freq ${ }_{n}^{1}$ the average report frequencies in this block at the beginning of the adjustment stage:\n$$\n\\operatorname{freq}_{n}^{1}(\\hat{\\theta}) \\triangleq \\frac{1}{L_{n}} \\sum_{t=T_{n}}^{T_{n}+L_{n}-1} 1\\left(\\hat{\\theta}_{t}=\\hat{\\theta}\\right) .\n$$\nIf $\\operatorname{freq}_{n}^{1}=f$, then the adjustment phase is empty, and the mechanism proceeds to the next block. Otherwise, in the adjustment phase, the mechanism generates reports over $N_{n}$ periods to guarantee that at the end of the adjustment phase the expected frequency of reports in this block equals $f$ that is,\n$$\n\\mathbb{E}\\left[\\text { freq }_{n}^{2} \\mid \\text { freq }_{n}^{1}\\right]=f\n$$\nwhere\n$$\n\\operatorname{freq}_{n}^{2}(\\hat{\\theta}) \\triangleq \\frac{1}{L_{n}+N_{n}} \\sum_{t=T_{n}}^{T_{n}+L_{n}+N_{n}-1} 1\\left(\\hat{\\theta}_{t}=\\hat{\\theta}\\right) .\n$$\nTo do so, denote by $\\eta \\triangleq \\min _{\\theta} f(\\theta)$ and observe that $f \\in \\Delta(\\Theta)$ can be surrounded by a ball of radius $\\eta$ within the simplex $\\Delta(\\Theta)$. The adjustment phase lasts for $N_{n}$ periods where: ${ }^{52}$\n$$\nN_{n}=\\left\\lceil L_{n} \\frac{\\left\\|\\mathrm{freq}_{n}^{1}-f\\right\\|_{\\infty}}{\\eta}\\right\\rceil,\n$$\nand in each period of the adjustment phase the mechanism generates the reports i.i.d. according to $\\tilde{f}_{n}^{a}$ :\n$$\nf_{n}^{a}=f-\\left(\\text { freq }_{n}^{1}-f\\right) \\frac{L_{n}}{N_{n}} .\n$$\nThe construction ensures that $f_{n}^{a} \\in \\Delta(\\Theta)$, because $\\left\\|f_{n}^{a}-f\\right\\|_{\\infty} \\leq \\eta$, and that (C.41) holds, because\n$$\n\\mathbb{E}\\left[\\operatorname{freq}_{n}^{2} \\mid \\operatorname{freq}_{n}^{1}\\right]=\\frac{1}{L_{n}+N_{n}}\\left(L_{n} \\operatorname{freq}_{n}^{1}+N_{n} f_{n}^{a}\\right)=f\n$$\nThis in turn guarantees that the long-run distribution of reports (generated jointly by the agent and the mechanism) exists and equals $f$ irrespectively of the agent's strategy. Intuitively, the fact that each block becomes negligible relative to past history over time ensures the agent's reports in each block have less and less effect on the long run frequency of reports, whereas the adjustment phase ensures that the frequency of reports converges to $f$. Formally, for any history and $T$ denote by $n^{\\text {last }}(T)$ the number of the block to which $T$ belongs and by $T^{\\text {last }}(T)$ the first period of that block. Observe that for","text_sha256":"5d72534cd9bfb05353619adc4f6960b79ceaa0061fc5f90a009c63cebbcb1d3f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0046","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.2.2 Proof of Theorem 4 (sufficiency)","text":"[^34]any agent's strategy:\n$$\nN_{n} \\leq L_{n}\\left(\\max _{f^{\\prime}} \\frac{\\left\\|f^{\\prime}-f\\right\\|_{\\infty}}{\\eta}+1\\right) \\triangleq L_{n} \\bar{\\rho}\n$$\nTherefore,\n$$\n\\frac{\\left|T-T^{\\text {last }}(T)\\right|}{T^{\\text {last }}(T)} \\leq \\frac{L_{n^{\\text {last }}(T)}(1+\\bar{\\rho})}{\\sum_{k<n^{\\text {last }}(T)} L_{k}} \\xrightarrow[T \\rightarrow \\infty]{\\text { a.s. }} 0,\n$$\nwhere the limit result holds because $n^{\\text {last }}(T) \\xrightarrow[T \\rightarrow \\infty]{\\text { a.s. }} \\infty$ and $L_{n} / \\sum_{k \\leq n} L_{k} \\xrightarrow[n \\rightarrow \\infty]{ } 0$.\nThen, for any agent's strategy, for any $\\hat{\\theta} \\in \\Theta$,\n$$\n\\begin{aligned}\n\\lim _{T \\rightarrow \\infty} \\frac{1}{T} \\sum_{t=1}^{T} \\operatorname{Pr}\\left(\\hat{\\theta}_{t}=\\hat{\\theta}\\right) & =\\lim _{T \\rightarrow \\infty} \\mathbb{E}\\left[\\frac{f(\\hat{\\theta}) T^{\\text {last }}(T)+f^{\\text {last }}(T)\\left(T-T^{\\text {last }}(T)\\right)}{T^{\\text {last }}(T)+T-T^{\\text {last }}(T)}\\right] \\\\\n& =\\lim _{T \\rightarrow \\infty} \\mathbb{E}\\left[\\frac{f(\\hat{\\theta})+f^{\\text {last }}(T)\\left(T-T^{\\text {last }}(T) / T^{\\text {last }}(T)\\right.}{1+\\left(T-T^{\\text {last }}(T)\\right) / T^{\\text {last }}(T)}\\right]=f(\\hat{\\theta}),\n\\end{aligned}\n$$\nwhere $f^{\\text {last }}(T) \\in \\Delta(\\Theta)$ is the report frequency in the last block up to period $T$, and the last line follows from Equation C.46.\n\nSince the mechanism chooses allocations in all periods according to $\\alpha^{\\prime}$, it follows that for any agent's strategy $\\sigma^{\\prime}$ the induced occupation measure over allocations and type reports satisfies:\n\n$$\n\\lim _{T \\rightarrow \\infty} \\frac{1}{T} \\mathbb{E}_{\\sigma^{\\prime}}\\left[\\sum_{t=1}^{T} \\mathbb{1}\\left[\\left(a_{t}, \\hat{\\theta}_{t}\\right)=(a, \\hat{\\theta})\\right]\\right]=f(\\hat{\\theta}) \\alpha^{\\prime}(a \\mid \\hat{\\theta})=\\vartheta^{\\prime}(a, \\hat{\\theta}) .\n$$\n\nIn other words, for any reporting strategy the occupation measure over allocations and reports exists.\nWe now show that under truthtelling the occupation measure over types and allocations exists and equals $f(\\theta) \\alpha^{\\prime}(a \\mid \\theta)=\\vartheta(a, \\theta)$. To this end, assume that the agent always reports her true type. For any $T$, denote by $\\tilde{L}^{\\text {total }}(T)$ the total number of periods spent in communication phases before $T$ and by $\\tilde{N}^{\\text {total }}(T)$ the total number of periods spent in adjustment phases before $T$. Observe that by the strong law of large numbers, because $L_{n} \\rightarrow \\infty$,\n\n$$\n\\frac{N_{n}}{L_{n}} \\leq \\frac{\\left\\|\\operatorname{freq}_{n}^{1}-f\\right\\|_{\\infty}}{\\eta}+\\frac{1}{L_{n}} \\xrightarrow[n \\rightarrow \\infty]{\\text { a.s. }} 0 .\n$$\n\nTherefore,\n\n$$\n\\frac{\\tilde{N}^{\\text {total }}(T)}{\\tilde{N}^{\\text {total }}(T)+\\tilde{L}^{\\text {total }}(T)} \\underset{T \\rightarrow \\infty}{\\stackrel{\\text { a.s. }}{\\longrightarrow}} 0,\n$$\n\nbecause whenever $N_{n} / L_{n} \\rightarrow 0, \\lim _{T \\rightarrow \\infty} N^{\\text {total }(T)} /\\left(N^{\\text {total }}(T)+L^{\\text {total }}(T)\\right)=\\lim _{n \\rightarrow \\infty} N_{n} /\\left(L_{n}+N_{n}\\right)=0$.\n\nIt follows that\n\n$$\n\\begin{aligned}\n& \\lim _{T \\rightarrow \\infty} \\frac{1}{T} \\sum_{t=1}^{T} \\operatorname{Pr}\\left(\\left(\\theta_{t}, \\hat{\\theta}_{t}, a_{t}\\right)=(\\theta, \\hat{\\theta}, a)\\right) \\\\\n& =\\lim _{T \\rightarrow \\infty} \\mathbb{E}\\left[\\frac{\\tilde{L}^{\\text {total }}(T) 1(\\theta=\\hat{\\theta}) f(\\hat{\\theta}) \\alpha^{\\prime}(a \\mid \\hat{\\theta})+\\tilde{N}^{\\text {total }}(T) f^{\\text {adj }}(T)(\\theta, \\hat{\\theta}, a)}{\\tilde{N}^{\\text {total }}(T)+\\tilde{L}^{\\text {total }}(T)}\\right] \\\\\n& =1(\\theta=\\hat{\\theta}) f(\\hat{\\theta}) \\alpha^{\\prime}(a \\mid \\hat{\\theta})\n\\end{aligned}\n$$\n\nwhere $f^{\\text {adj }}(T) \\in \\Delta(\\Theta \\times \\Theta \\times A)$ is the average frequency of types, reports, and allocations in the adjustment phases before $T$. Therefore, under truthtelling, the occupation measure over allocations and types equals\n\n$$\n\\vartheta^{\\prime}(a, \\theta)=f(\\theta) \\alpha^{\\prime}(a \\mid \\theta) .\n$$\n\nHence, the agent's payoff in the dynamic mechanism under truthtelling is:\n\n$$\nU^{\\mathrm{truth}}=\\sum_{(a, \\theta)} f(\\theta) \\alpha^{\\prime}(a \\mid \\theta) u^{\\prime}(a, \\theta) .\n$$\n\nIt remains to show that the agent cannot achieve more than $U^{\\text {truth }}$ under any other strategy. To this end, fix and alternative strategy $\\sigma$, and denote by $U(\\sigma)=\\limsup _{T \\rightarrow \\infty} U_{T}(\\sigma)$ where:\n\n$$\nU_{T}(\\sigma)=\\frac{1}{T} \\sum_{t=1}^{T} \\sum_{a, \\theta} \\operatorname{Pr}\\left(\\left(a_{t}, \\theta_{t}\\right)=(a, \\theta)\\right) u^{\\prime}(a, \\theta) .\n$$\n\nConsider any convergent subsequence $\\left(U_{T_{n}}\\right)_{n=1}^{\\infty}$ along times $\\left\\{T_{n}\\right\\}_{n=1}^{\\infty}$. Because $\\Delta(A \\times \\Theta \\times \\Theta)$ is compact (Aliprantis and Border, 2006, Theorem 15.11), a convergent (sub)subsequence at times $\\left\\{T_{k}\\right\\}_{k=1}^{\\infty} \\subseteq$ $\\left\\{T_{n}\\right\\}_{n=1}^{\\infty}$ exists along which the occupation measure induced by $\\sigma$\n\n$$\nv_{\\sigma}^{T_{k}} \\xrightarrow{w^{*}} v_{\\sigma},\n$$\n\nfor some $v_{\\sigma} \\in \\Delta(A \\times \\Theta \\times \\Theta)$, which by (C.48) satisfies $v_{\\sigma}(a, \\hat{\\theta})=f(\\hat{\\theta}) \\alpha^{\\prime}(a \\mid \\hat{\\theta})$. It follows that for some undetectable deviation $v_{\\sigma}(\\hat{\\theta} \\mid \\theta)$ :\n\n$$\n\\lim _{n \\rightarrow \\infty} U_{T_{n}}=\\lim _{k \\rightarrow \\infty} U_{T_{k}}=\\sum_{\\theta, \\hat{\\theta}, a} f(\\theta) v_{\\sigma}(\\hat{\\theta} \\mid \\theta) \\alpha^{\\prime}(a \\mid \\hat{\\theta}) u^{\\prime}(a, \\theta) \\leq U^{\\mathrm{truth}},\n$$\n\nwhere the inequality follows because $\\alpha^{\\prime}(a \\mid \\hat{\\theta})$ lacks profitable undetectable deviations. Because this inequality holds for any convergent subsequence $\\left(U_{T_{n}}\\right)_{n=1}^{\\infty}$,\n\n$$\nU(\\sigma)=\\limsup _{T \\rightarrow \\infty} U_{T}(\\sigma) \\leq U^{\\text {truth }}\n$$\n\nFinally, observe that the construction ensures that after every history, truthtelling from there on delivers the continuation payoff $U^{\\text {truth }}$. Since $\\alpha^{\\prime}$ is ex ante individually rational, $U^{\\text {truth }} \\geq \\sum_{\\theta} f(\\theta) u^{\\prime}\\left(a_{\\varnothing}, \\theta\\right)$, and thus the participation constraints are satisfied. This concludes the proof. $\\square$","text_sha256":"591b2ab50599ea8c705fbd1b90d17765fe87140eff4ad50825b07c72e971d4ff"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0047","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C.2.2 Proof of Theorem 4 (sufficiency)","text":"Step 2 Consider now the outcome distribution $\\vartheta \\in \\Delta(A \\times \\Theta \\times \\Omega)$ satisfying Equation C.39. As we argue in the proof of Theorem 2, a finite $K \\leq|A||\\Theta||\\Omega|,\\left\\{\\mu_{1}, \\ldots, \\mu_{K}\\right\\} \\in \\Delta(\\Omega)$, and $\\tau^{\\prime} \\in \\Delta(\\Delta(\\Omega))$ exists such that\n\n$$\n\\vartheta(a, \\theta, \\omega)=f(\\theta) \\sum_{k=1}^{K} \\tau^{\\prime}\\left(\\mu_{k}\\right) \\mu_{k}(\\omega) \\alpha\\left(a \\mid \\theta, \\mu_{k}\\right) .\n$$\n\nStep 3 We now use steps 1 and 2 to complete the proof of Theorem 4, so in what follows we use the finite support representation of $\\vartheta$ in the previous step. By Bayes plausibility, a dynamic mechanism can generate the belief split $\\tau^{\\prime}$ in $T$ periods with $T \\leq\\left\\lceil\\log _{|A|}(|\\Omega||\\Theta||A|)\\right\\rceil$, by treating each sequence of allocations of length $T$ as a message. This can be achieved by making the mechanism constant on the agent's type reports during the first $T$ periods. Since each $\\alpha\\left(\\cdot \\mid \\cdot, \\mu_{k}\\right)$ for $k \\in\\{1, \\ldots, K\\}$ lacks profitable undetectable deviations and is individually rational, Proposition C. 1 implies that a dynamic mechanism exists that implements $\\vartheta$ by first generating the belief split $\\tau^{\\prime}$ and then implementing $\\alpha\\left(\\cdot \\mid \\cdot, \\mu_{k}\\right)$ in the corresponding continuation play.","text_sha256":"7663d223c647b7b49fd2efc55ab388df3235b803d4da22b3d2df08d629c005f6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0048","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D Proof of auxiliary results","text":"## D Proof of auxiliary results","text_sha256":"b3c80557923ae40d3f6a6206e82ae79c1b95e05b934f44a6fd4f07063911f6cb"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0049","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D. 1 Revelation principle for calibrated mechanism design","text":"## D. 1 Revelation principle for calibrated mechanism design\n\nIn the main text, we restricted attention to incentive compatible and individually rational calibrated mechanisms. We show in this appendix that this restriction is without loss of generality by considering mechanisms with arbitrary message spaces and participation and reporting decisions by the agents that constitute an equilibrium of the game induced by the mechanism and its calibrated information structure.\n\nMechanisms Let $2^{[N]} \\backslash \\varnothing$ denote the nonempty subsets of agents. Then, we can define a mechanism as a collection $\\left\\{\\left(M_{J}, \\phi_{J}\\right): J \\in 2^{[N]} \\backslash \\varnothing\\right\\}$, where\n\n$$\n\\phi_{J}: M_{J} \\times \\Omega \\times[0,1] \\rightarrow \\Delta\\left(A_{J}\\right),\n$$\n\nis the mechanism when agents in $J$ participate, where $M_{J}=\\times_{i \\in J} M_{i}$ and $A_{J}=\\times_{i \\in J} A_{i}$.\n\nInformation Structure Let $\\hat{S}_{i}=\\Delta\\left(A_{i}\\right)^{M_{i}}$ denote the collection of menus of lotteries with labels $M_{i}$, and let $\\hat{S}=\\times_{i \\in[N]} \\hat{S}_{i}$. An information structure is $(\\pi, \\hat{S})$, where $\\pi: \\Omega \\times[0,1] \\rightarrow \\hat{S}$.\n\nParticipation and reporting strategies It is notationally convenient to allow each agent to have her own randomization device $\\varepsilon_{i} \\sim U[0,1]$ and write agents' strategies as mappings $\\left(p_{i}, \\sigma_{i}\\right): \\Theta_{i} \\times \\hat{S}_{i} \\times$ $[0,1] \\rightarrow\\{0,1\\} \\times M_{i}$, where $p_{i}$ denotes agent $i$ 's participation decision, and $\\sigma_{i}$ her reporting strategy, conditional on participating. To distinguish the agents' randomization from that of the original mechanism, we reserve $\\varepsilon_{0}$ for the realization of the mechanism's randomization device.\n\nGiven $\\left(p_{i}, \\sigma_{i}\\right)_{i \\in[N]}$ and a mechanism $(\\phi, M)$, fix a profile $(\\theta, \\hat{s}, \\bar{\\varepsilon}) \\equiv\\left(\\theta_{i}, \\hat{s}, \\varepsilon_{i}\\right)_{i \\in[N]}$. This determines a set of agents that participate,\n\n$$\nJ(\\theta, \\hat{s}, \\bar{\\varepsilon})=\\left\\{j \\in[N]: p_{j}\\left(\\theta_{j}, \\hat{s}_{j}, \\varepsilon_{j}\\right)=1\\right\\},\n$$\n\nand let $J_{-i}(\\theta, \\hat{s}, \\bar{\\varepsilon})$ denote the projection of $J(\\theta, \\hat{s}, \\bar{\\varepsilon})$ on $J \\backslash\\{i\\}$. Note that $J_{-i}$ only depends on $\\left(\\theta_{-i}, \\hat{s}_{-i}, \\bar{\\varepsilon}_{-i}\\right)$. Lastly, write $\\phi_{J_{-i}\\left(\\theta_{-i}, \\hat{s}_{-i}, \\varepsilon_{-i}\\right) \\cup\\{i\\}}\\left(m_{i}, \\sigma_{J_{-i}\\left(\\theta_{-i}, \\hat{s}_{-i}, \\varepsilon_{-i}\\right)}, \\omega, \\varepsilon_{0}\\right) \\in \\Delta\\left(A_{J_{-i}\\left(\\theta_{-i}, \\hat{s}_{-i}, \\varepsilon_{-i}\\right) \\cup\\{i\\}}\\right)$ for\n\n$$\n\\sum_{m_{J_{-i}\\left(\\theta_{-i}, \\hat{s}_{-i}, \\varepsilon_{-i}\\right)}}\\left(\\prod_{j \\in J_{-i}\\left(\\theta_{-i}, \\hat{s}_{-i}, \\varepsilon_{-i}\\right)} \\sigma_{j}\\left(\\theta_{j}, \\hat{s}_{j}\\right)\\left(m_{j}\\right)\\right) \\phi_{J_{-i}\\left(\\theta_{-i}, \\hat{s}_{-i}, \\varepsilon_{-i}\\right) \\cup\\{i\\}}\\left(m_{i}, m_{J_{-i}\\left(\\theta_{-i}, \\hat{s}_{-i}, \\varepsilon_{-i}\\right)}, \\omega, \\varepsilon_{0}\\right)\n$$\n\nCalibrated information structures Given $\\left(p_{i}, \\sigma_{i}\\right)_{i \\in[N]}$ and a mechanism $(\\phi, M)$, the information structure $(\\pi, \\hat{S})$ is calibrated with the mechanism and the agents' strategies if whenever $\\pi\\left(\\omega, \\varepsilon_{0}\\right)=$ $\\left(\\hat{s}_{1}, \\ldots, \\hat{s}_{N}\\right)$, then for all $i, m_{i}$\n\n$$\n\\hat{s}_{i}\\left(\\cdot \\mid m_{i}\\right)=\\mathbb{E}_{\\tilde{\\theta}_{-i} \\sim f_{-i}(\\cdot \\mid \\omega), \\epsilon_{-i}}\\left[\\sum_{a_{-i} \\in A_{-i}} \\phi_{J_{-i}\\left(\\theta_{-i}, \\hat{s}_{-i}, \\varepsilon_{-i}\\right) \\cup\\{i\\}}\\left(m_{i}, \\sigma_{J_{-i}\\left(\\theta_{-i}, \\hat{s}_{-i}, \\varepsilon_{-i}\\right)}, \\omega, \\epsilon_{0}\\right)\\left(\\cdot, a_{-i}\\right)\\right] .\n$$\n\nBelow, to keep the presentation simple, we focus on the case in which the calibrated information structure has finite support.\n\nEquilibrium Given $\\left(p_{i}, \\sigma_{i}\\right)_{i \\in[N]}$, a mechanism $(\\phi, M)$ and an information structure $(\\pi, \\hat{S})$ calibrated with the mechanism and the agents' strategies, $\\left(p_{i}, \\sigma_{i}\\right)_{i \\in[N]}$ is an equilibrium if for all $i \\in[N]$, all $\\theta_{i} \\in \\Theta_{i}$, all $\\hat{s}_{i} \\in \\hat{S}_{i}$, and $\\varepsilon_{i} \\in[0,1]$, the following hold:\n\n$$\n\\begin{aligned}\n& \\sigma_{i}\\left(\\theta_{i}, \\hat{s}_{i}, \\varepsilon_{i}\\right) \\in \\arg \\max _{m_{i} \\in M_{i}} \\sum_{a_{i} \\in A_{i}} \\hat{s}_{i}\\left(a_{i} \\mid m_{i}\\right) \\mathbb{E}_{\\omega \\sim \\mu_{i}\\left(\\cdot \\mid \\theta_{i}, \\hat{s}_{i}\\right)}\\left[u_{i}\\left(a_{i}, \\theta_{i}, \\omega\\right)\\right], \\\\\n& p_{i}\\left(\\theta_{i}, \\hat{s}_{i}, \\varepsilon_{i}\\right) \\in \\arg \\max _{p \\in\\{0,1\\}} p \\sum_{a_{i} \\in A_{i}} \\hat{s}_{i}\\left(a_{i} \\mid \\sigma_{i}\\left(\\theta_{i}, \\hat{s}_{i}, \\varepsilon_{i}\\right)\\right) \\mathbb{E}_{\\omega \\sim \\mu_{i}\\left(\\cdot \\mid \\theta_{i}, \\hat{s}_{i}\\right)}\\left[u_{i}\\left(a_{i}, \\theta_{i}, \\omega\\right)\\right]+(1-p) \\mathbb{E}_{\\omega \\sim \\mu_{i}\\left(\\cdot \\mid \\theta_{i}, \\hat{s}_{i}\\right)}\\left[u_{i}\\left(a_{i \\varnothing}, \\theta_{i}, \\omega\\right)\\right],\n\\end{aligned}\n$$\n\nwhere $\\mu_{i}\\left(\\theta_{i}, \\hat{s}_{i}\\right) \\in \\Delta(\\Omega)$ denotes agent $i$ 's updated beliefs about the state when her type is $\\theta_{i}$ conditional on receiving signal $\\hat{s}_{i}$.\n\nRevelation Principle Fix $\\left(p_{i}, \\sigma_{i}\\right)_{i \\in[N]}$, a mechanism $(\\phi, M)$ and an information structure $(\\pi, \\hat{S})$ calibrated with the mechanism such that $\\left(p_{i}, \\sigma_{i}\\right)_{i \\in[N]}$ is an equilibrium. We construct a direct mechanism $\\left(\\phi^{*}, \\Theta\\right)$ and a calibrated information structure $\\left(\\pi^{*}, S^{*}\\right)$ calibrated with the mechanism under truthtelling and full participation such that truthtelling and full participation is an equilibrium.\n\nFirst, note that we can extend each $\\phi_{J}(\\cdot) \\in \\Delta\\left(A_{J}\\right)$ to a mechanism $\\bar{\\phi}_{J}(\\cdot) \\in \\Delta(A)$ as follows: for all $m \\in M_{J}, \\omega \\in \\Omega, \\varepsilon_{0} \\in[0,1]$, and $a_{J} \\in A_{J}$,\n\n$$\n\\bar{\\phi}_{J}\\left(m_{J}, \\omega, \\varepsilon_{0}\\right)(a)=\\phi_{J}\\left(m_{J}, \\omega, \\varepsilon_{0}\\right)\\left(a_{J}\\right) \\times \\delta_{a_{-J, \\phi}} .\n$$\n\nDefine a \"pseudo\"-mechanism as follows:","text_sha256":"a0e75c690280c032eb360a45d7dd6edbc40b2a3fcb618b9151a874e27dbc3f48"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0050","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D. 1 Revelation principle for calibrated mechanism design","text":"$$\n\\hat{\\phi}_{N}\\left(\\theta, \\omega, \\varepsilon_{0}, \\bar{\\varepsilon}\\right)=\\bar{\\phi}_{J\\left(\\theta, \\pi\\left(\\omega, \\varepsilon_{0}\\right), \\bar{\\varepsilon}\\right)}\\left(\\sigma_{J(\\theta, \\hat{s}, \\bar{\\varepsilon})}, \\omega, \\varepsilon_{0}\\right) .\n$$\n\nwhere $\\sigma_{J(\\theta, \\hat{s}, \\bar{\\varepsilon})}$ is the message vector generated by the strategies. Define the full participation mechanism $\\phi_{N}^{*}: \\Theta \\times \\Omega \\times[0,1] \\mapsto \\Delta(A)$ to be\n\n$$\n\\phi_{N}^{*}\\left(\\theta, \\omega, \\varepsilon_{0}\\right)(a)=\\int_{[0,1]^{N}} \\hat{\\phi}_{N}\\left(\\theta, \\omega, \\varepsilon_{0}, \\bar{\\varepsilon}\\right)(a) \\lambda^{N}(d \\bar{\\varepsilon})\n$$\n\nLet $S_{i}^{*}=\\Delta\\left(A_{i}\\right)^{\\Theta_{i}}$ and define $\\pi^{*}\\left(\\omega, \\varepsilon_{0}\\right)=\\left(s_{1}^{*}, \\ldots, s_{N}^{*}\\right) \\in \\times_{i \\in[N]} S_{i}^{*}$, where\n\n$$\ns_{i}^{*}\\left(\\cdot \\mid \\hat{\\theta_{i}}\\right)=\\mathbb{E}_{\\theta_{-i} \\sim f_{-i}(\\cdot \\mid \\omega)}\\left[\\sum_{a_{-i}} \\phi_{N}^{*}\\left(\\hat{\\theta_{i}}, \\theta_{-i}, \\omega, \\varepsilon_{0}\\right)\\left(\\cdot, a_{-i}\\right)\\right] .\n$$\n\nBy definition, the information structure is calibrated relative to full participation and truthful reporting.\nWe now show that full participation and truthful reporting is a best response to others participating and truthfully reporting into the mechanism. To this end, consider agent $i$ 's payoff from submitting report $\\theta_{i}^{\\prime}$ when observing $s_{i}^{*}$. Denoting by $\\Sigma\\left(\\omega, s_{i}^{*}\\right)$ the set of $\\varepsilon_{0}$ such that $\\pi_{i}^{*}=s_{i}^{*}$, this payoff is given by: ${ }^{53}$\n\n$$\n\\begin{aligned}\n& \\sum_{\\omega \\in \\Omega} \\frac{\\mu_{0}(\\omega) f_{i}\\left(\\theta_{i} \\mid \\omega\\right)}{\\operatorname{Pr}\\left(s_{i}^{*} \\mid \\theta_{i}\\right)} \\sum_{\\theta_{-i}} f_{-i}\\left(\\theta_{-i} \\mid \\omega\\right) \\int_{\\Sigma\\left(\\omega, s_{i}^{*}\\right)} \\sum_{a} \\phi_{N}^{*}\\left(\\theta_{i}^{\\prime}, \\theta_{-i}, \\omega, \\varepsilon_{0}\\right)\\left(a_{i}, a_{-i}\\right) \\lambda\\left(d \\varepsilon_{0}\\right) u_{i}\\left(a_{i}, \\theta_{i}, \\omega\\right)= \\\\\n& \\sum_{a_{i} \\in A_{i}} \\sum_{\\omega \\in \\Omega} \\frac{\\mu_{0}(\\omega) f_{i}\\left(\\theta_{i} \\mid \\omega\\right)}{\\operatorname{Pr}\\left(s_{i}^{*} \\mid \\theta_{i}\\right)} u_{i}\\left(a_{i}, \\theta_{i}, \\omega\\right) \\sum_{\\theta_{-i}} f_{-i}\\left(\\theta_{-i} \\mid \\omega\\right) \\int_{\\Sigma\\left(\\omega, s_{i}^{*}\\right)} \\sum_{a_{-i}} \\phi_{N}^{*}\\left(\\theta_{i}^{\\prime}, \\theta_{-i}, \\omega, \\varepsilon_{0}\\right)\\left(a_{i}, a_{-i}\\right) \\lambda\\left(d \\varepsilon_{0}\\right) \\\\\n& =\\int_{0}^{1}\\left[\\sum_{a_{i}} \\sum_{\\omega} \\frac{\\mu_{0}(\\omega) f_{i}\\left(\\theta_{i} \\mid \\omega\\right)}{\\operatorname{Pr}\\left(s_{i}^{*} \\mid \\theta_{i}\\right)} u_{i}\\left(a_{i}, \\theta_{i}, \\omega\\right) \\int_{\\Sigma\\left(\\omega, s_{i}^{*}\\right)}(\\star) \\lambda\\left(d \\varepsilon_{0}\\right)\\right] \\lambda\\left(d \\varepsilon_{i}\\right)\n\\end{aligned}\n$$\n\nwhere\n\n$$\n\\begin{aligned}\n& \\star=\\mathbb{E}_{\\theta_{-i} \\mid \\omega, \\bar{\\varepsilon}_{-i}}\\left[\\sum_{a_{-i}} \\hat{\\phi}_{N}\\left(\\theta_{i}^{\\prime}, \\theta_{-i}, \\varepsilon_{0}, \\bar{\\varepsilon}_{-i}\\right)\\left(a_{i}, a_{-i}\\right)\\right] \\\\\n& =\\mathbb{E}_{\\theta_{-i} \\mid \\omega, \\bar{\\varepsilon}_{-i}}\\left[\\sum_{a_{-i}} \\bar{\\phi}_{J\\left(\\theta_{i}^{\\prime}, \\theta_{-i}, \\hat{s}\\left(\\omega, \\varepsilon_{0}\\right), \\bar{\\varepsilon}\\right)}\\left(\\sigma_{J\\left(\\theta_{i}^{\\prime}, \\theta_{-i}, \\hat{s}\\left(\\omega, \\varepsilon_{0}\\right), \\bar{\\varepsilon}\\right)}, \\omega, \\varepsilon_{0}\\right)\\left(a_{i}, a_{-i}\\right)\\right] \\\\\n& =\\mathbb{1}\\left[p_{i}\\left(\\theta_{i}^{\\prime}, \\hat{s}_{i}\\left(\\omega, \\varepsilon_{0}\\right), \\varepsilon_{i}\\right)=1\\right] \\hat{s}_{i}\\left(a_{i} \\mid \\sigma_{i}\\left(\\theta_{i}^{\\prime}, \\hat{s}_{i}, \\varepsilon_{i}\\right)\\right)+\\left(1-\\mathbb{1}\\left[p_{i}\\left(\\theta_{i}^{\\prime}, \\hat{s}_{i}\\left(\\omega, \\varepsilon_{0}\\right), \\varepsilon_{i}\\right)=1\\right]\\right) \\delta_{a_{i, \\phi}}\\left(a_{i}\\right)\n\\end{aligned}\n$$\n\nBecause agent $i$ of type $\\theta_{i}$ could have imitated type $\\theta_{i}^{\\prime}$, reporting $\\theta_{i}$ dominates. By the same logic, when the agent reports $\\theta_{i}$, she obtains at least the payoff from participating in the mechanism.","text_sha256":"55d7877f3e270e4381b5d929c753bb5776cd9d107056024d3a2d3fc854f1357c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0051","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D. 2 Optimal Calibrated Auction","text":"## D. 2 Optimal Calibrated Auction\n\nProof of Proposition 5. The pointwise solution to the Myersonian problem allocates the good to agents in $N^{*}(\\theta, \\omega)=\\arg \\max _{i \\in[N] \\cup\\{0\\}}\\left[w_{i}(\\theta, \\omega)+J_{i}\\left(\\theta_{i}\\right) \\omega_{i}+\\omega_{0 i}\\right]$ where $i=0$ corresponds to an outside option with $w_{0} \\equiv J_{0} \\equiv \\omega_{00} \\equiv 0$. The conditions of the proposition ensure that for all $i \\in N$ and $j \\in[N] \\cup\\{0\\}$,\n\n$$\n\\frac{d}{d \\theta_{i}}\\left(w_{i}(\\theta, \\omega)+J_{i}\\left(\\theta_{i}, F_{i}\\right) \\omega_{i}+\\omega_{0 i}\\right) \\geq \\frac{d}{d \\theta_{i}}\\left(w_{j}(\\theta, \\omega)+J_{j}\\left(\\theta_{j}, F_{j}\\right) \\omega_{j}+\\omega_{0 j}\\right) .\n$$\n\nThus, an optimal selection $q^{*}(\\theta, \\omega)$ exists such that for each $i, \\theta_{-i}$, and $\\omega, q^{*}\\left(\\theta_{i}, \\theta_{-i}, \\omega\\right)$ is nondecreasing in $\\theta_{i}$ (e.g., one that uniformly randomizes over $N^{*}(\\theta, \\omega)$ ).\n\nDenote by $Q_{\\text {full }}$ the set of allocation rules implementable under full state disclosure. These are the rules such that for all $i$ and $\\omega, \\mathbb{E}_{F_{-i}}\\left[q_{i}\\left(\\theta_{i}, \\theta_{-i}, \\omega\\right)\\right]$ is non-decreasing in $\\theta_{i}$. It follows that $q^{*} \\in Q_{\\mathrm{full}}$, and hence $q^{*} \\in Q_{\\mathrm{My}}$. Thus, $q^{*}$ solves the Myersonian problem and can also be implemented by fully disclosing the state to the agents and conducting an optimal mechanism state-by-state. By revenue\n\n[^35]equivalence, the expected revenue of such implementation is the same as under no disclosure, and thus the designer obtains payoff $W_{\\text {My }}$. $\\square$","text_sha256":"2ee96f81e4132c275e5da9e751d322b841e7cfedcd7181bd7f89a1c70ede3895"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0052","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D. 3 Technical results from Appendix C","text":"## D. 3 Technical results from Appendix C\n\nProof of Lemma C.1. The set of continuous bounded functions on $\\tilde{\\Omega}$ is separable and hence it has a countable dense subset $\\left\\{g_{k}\\right\\}_{k \\in \\mathbb{N}} \\subset C_{b}(\\tilde{\\Omega})$. It is immediate to see that $\\mu_{n} \\xrightarrow{w^{*}} \\mu$ if and only if for all $k \\in \\mathbb{N}$ $\\int g_{k} d \\mu_{n} \\rightarrow \\int g_{k} d \\mu$.\n\nFor each $k \\in \\mathbb{N}$ define a real-valued, bounded, martingale on ( $\\mathcal{H}^{\\infty}, \\mathcal{B}_{\\mathcal{H}^{\\infty}}, \\mathbb{P}_{\\sigma}$ ) as follows:\n\n$$\nM_{t}^{k}\\left(\\tilde{\\omega}, h^{\\infty}\\right)=\\int_{\\tilde{\\Omega}} g_{k}\\left(\\omega^{\\prime}, \\varepsilon\\right) d \\mu_{t}\\left(\\tilde{\\omega}, h^{\\infty}\\right)\\left(\\omega^{\\prime}, \\varepsilon\\right)\n$$\n\nDoob's martingale convergence theorem implies that $M_{t}^{k}\\left(\\tilde{\\omega}, h^{\\infty}\\right)=\\mathbb{E}\\left[g_{k} \\mid h^{t}\\right] \\rightarrow M_{\\infty}^{k}\\left(\\tilde{\\omega}, h^{\\infty}\\right)=\\mathbb{E}\\left[g_{k} \\mid h^{\\infty}\\right]$ $\\mathbb{P}_{\\sigma}$-a.s. Let $E_{k}$ denote the subset of $\\mathcal{H}^{\\infty}$ where convergence happens, and note that $\\mathbb{P}_{\\sigma}\\left(E_{k}\\right)=1$.\n\nLet $E=\\cap_{k} E_{k}$ and note that $\\mathbb{P}_{\\sigma}(E)=1$. Then, on $E$, we have that for all $k \\in \\mathbb{N}$,\n\n$$\n\\int_{\\tilde{\\Omega}} g_{k}\\left(\\omega^{\\prime}, \\varepsilon\\right) d \\mu_{t}\\left(\\tilde{\\omega}, h^{\\infty}\\right)\\left(\\omega^{\\prime}, \\varepsilon\\right) \\rightarrow M_{\\infty}^{k}\\left(\\tilde{\\omega}, h^{\\infty}\\right)\n$$\n\nFix now a terminal history ( $\\tilde{\\omega}, h^{\\infty}$ ). Because $\\Delta(\\tilde{\\Omega})$ is compact (Aliprantis and Border, 2006, Theorem 15.11), the sequence $\\left(\\mu_{t}\\left(\\tilde{\\omega}, h^{\\infty}\\right)\\right)_{t \\in \\mathbb{N}}$ has a convergent subsequence $\\mu_{t_{j}}\\left(\\tilde{\\omega}, h^{\\infty}\\right) \\xrightarrow{w^{*}} \\tilde{\\mu}$. Passing the limit along $t_{j}$ in Equation D. 1 we have that for all $k \\in \\mathbb{N}$\n\n$$\n\\int_{\\tilde{\\Omega}} g_{k}\\left(\\omega^{\\prime}, \\varepsilon\\right) d \\mu_{t}\\left(\\tilde{\\omega}, h^{\\infty}\\right)\\left(\\omega^{\\prime}, \\varepsilon\\right) \\rightarrow \\int_{\\tilde{\\Omega}} g_{k}\\left(\\omega^{\\prime}, \\varepsilon\\right) d \\tilde{\\mu}\n$$\n\nBecause the set $\\left\\{g_{k}: k \\in \\mathbb{N}\\right\\}$ determines the convergent subsequences, any subsequential limits must be equal. Hence $\\mu_{t}(\\cdot)$ converges on $E$ and call this limit $\\tilde{\\mu}_{\\infty}$. Hence on $E$ we have that $\\mu_{t}\\left(h^{\\infty}\\right) \\xrightarrow{w^{*}} \\tilde{\\mu}_{\\infty}\\left(h^{\\infty}\\right)$.\n\nNow, for each $k, \\int g_{k} d \\mu_{\\infty}=\\mathbb{E}\\left[g_{k} \\mid h^{\\infty}\\right]$ almost surely. Hence, $\\tilde{\\mu}_{\\infty}$ is a version of the law of $\\tilde{\\Omega}$ conditional on $h^{\\infty}$. This is $\\mathbb{P}_{\\sigma}\\left(\\cdot \\mid h^{\\infty}\\right)$, completing the proof. $\\square$\n\nProof of Lemma C.2. Fix a continuous and bounded function $g \\in C_{b}(A \\times \\Theta \\times \\Omega \\times \\Delta(\\Omega))$ and define for each terminal history $\\left(\\omega, h^{\\infty}\\right)$\n\n$$\n\\Delta_{t}\\left(\\omega, h^{\\infty}\\right)=g\\left(a_{t}\\left(h^{\\infty}\\right), \\theta_{t}\\left(h^{\\infty}\\right), \\omega, \\mu_{t}\\left(h^{\\infty}\\right)\\right)-g\\left(a_{t}\\left(h^{\\infty}\\right), \\theta_{t}\\left(h^{\\infty}\\right), \\omega, \\mu_{t+1}\\left(h^{\\infty}\\right)\\right) .\n$$\n\nAs in Lemma C.1, let $E$ denote the probability-1 subset of $\\mathcal{H}^{\\infty}$ on which $\\mu_{t} \\xrightarrow{w^{*}} \\mu_{\\infty} .{ }^{54}$ Then, on $E$, $\\Delta_{t}\\left(\\omega, h^{\\infty}\\right) \\rightarrow 0$, and hence,\n\n$$\n\\mathbb{E}_{\\sigma}\\left[\\Delta_{t}\\left(\\omega, h^{\\infty}\\right)\\right] \\rightarrow 0,\n$$\n\n[^36]as $t \\rightarrow \\infty$. Now, for every $T$,\n$$\nD_{T}(g) \\equiv \\mathbb{E}_{\\bar{v}_{\\sigma}^{T, 1}}[g]-\\mathbb{E}_{\\bar{v}_{\\sigma}^{T, 2}}[g]=\\frac{1}{T} \\sum_{t=1}^{T} \\mathbb{E}_{\\sigma}\\left[\\Delta_{t}\\right]\n$$\nand hence the left-hand side goes to 0 as $T \\rightarrow \\infty$.\nNow, let $T_{n}$ be such that $\\bar{v}_{\\sigma}^{T_{n}, 2} \\xrightarrow{w^{*}} \\bar{v}$ for some $\\bar{v} \\in \\Delta(A \\times \\Theta \\times \\Omega \\times \\Delta(\\Omega))$. Note that\n$$\n\\mathbb{E}_{\\bar{v}_{\\sigma}^{T n, 1}}[g]=\\mathbb{E}_{\\bar{v}_{\\sigma}^{T n, 2}}[g]+D_{T_{n}}[g] \\rightarrow \\mathbb{E}_{\\bar{v}}[g]+0,\n$$\nso a subsequential limit of $\\bar{v}_{\\sigma}^{T_{n}, 2}$ is a subsequential limit of $\\bar{v}_{\\sigma}^{T_{n}, 1}$. Switching the role of 1 and 2, we obtain the opposite set inclusion and the result follows. $\\square$\n\nProof of Lemma C.3. Fix a continuous function $g$ on $\\Delta(\\tilde{\\Omega})$. Then, we want to show that\n\n$$\n\\mathbb{E}_{\\tau_{T}}[g] \\rightarrow \\mathbb{E}_{\\mathbb{P}_{\\sigma} \\circ \\mu_{\\infty}^{-1}}[g] .\n$$\n\nBy Lemma C.1, $\\mu_{t} \\xrightarrow{w^{*}} \\mu_{\\infty} \\mathbb{P}_{\\sigma}$-almost surely and $g$ is continuous, we have that $\\mathbb{E}_{\\sigma}\\left[g\\left(\\mu_{t}\\right)\\right] \\rightarrow \\mathbb{E}_{\\sigma}\\left[g\\left(\\mu_{\\infty}\\right)\\right]$ by dominated convergence theorem. ${ }^{55}$ Because eventually constant sequences have Cesàro limits, we have that\n\n$$\n\\frac{1}{T} \\sum_{t=1}^{T} \\mathbb{E}_{\\sigma}\\left[g\\left(\\mu_{t}\\right)\\right] \\rightarrow \\mathbb{E}_{\\sigma}\\left[g\\left(\\mu_{\\infty}\\right)\\right] .\n$$\n\nAnd now we are basically done, because\n\n$$\n\\mathbb{E}_{\\tau_{T}}[g]=\\int_{H^{\\infty}} \\frac{1}{T} \\sum_{t=1}^{T} g\\left(\\mu_{t}\\left(h^{\\infty}\\right)\\right) \\mathbb{P}_{\\sigma}\\left(d h^{\\infty}\\right)=\\frac{1}{T} \\sum_{t=1}^{T} \\mathbb{E}_{\\sigma}\\left[g\\left(\\mu_{t}\\right)\\right] \\rightarrow \\mathbb{E}_{\\sigma}\\left[g\\left(\\mu_{\\infty}\\right)\\right]=\\int g d\\left(\\mathbb{P}_{\\sigma} \\circ \\mu_{\\infty}^{-1}\\right) .\n$$\n\nIn other words, the occupation measure on beliefs induced by the strategy (and the prior, the type distribution, and the mechanism) is the push-forward measure $\\left(\\mathbb{P}_{\\sigma} \\circ \\mu_{\\infty}^{-1}\\right)$. In particular, that $\\mathbb{P}_{\\sigma}$ is a measure implies that $\\left(\\mathbb{P}_{\\sigma} \\circ \\mu_{\\infty}^{-1}\\right)$ is a measure itself (Bogachev, 2007, Chapter 3.6). $\\square$","text_sha256":"bb17fc2163c5ecb431d11c4d310fe4e5e6c2c3860a41d4a3e3848ed49d49e282"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0053","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D. 3 Technical results from Appendix C","text":"Proof of Lemma C.4. We now show the agent can ensure the payoff $U\\left(\\tau_{\\phi}\\right)$ in Equation C.21, which corresponds to the agent's maximum payoff under the calibrated information structure $\\pi_{\\phi}$. Recall that this information structure is the one that corresponds to the partition of $\\tilde{\\Omega}, \\mathscr{P}$, defined as follows: $\\tilde{\\omega}, \\tilde{\\omega}^{\\prime}$ in the same cell $P$ of $\\mathscr{P}$ if for all $(a, m), \\phi(a \\mid m, \\tilde{\\omega})=\\phi\\left(a \\mid m, \\tilde{\\omega}^{\\prime}\\right)$. Conditional on cell $P$, the associated posterior is $\\mu(\\mid P) \\in \\Delta(\\tilde{\\Omega})$. Let $\\tau_{\\phi} \\in \\Delta(\\Delta(\\tilde{\\Omega}))$ denote the induced belief distribution (with mean $\\mu_{0} \\otimes \\eta$ ).\n\nTo prove the result, we consider the strategy $\\sigma_{N}^{\\prime}$ parameterized by a number $N$ and defined as follows. In the exploration phase of $\\sigma_{N}^{\\prime}$, the agent plays each message $m$ for $N$ rounds. Let $\\mu_{N|M|}\\left(h^{N|M|}\\right)$ denote the agent's beliefs as a function of the realized sequence of allocations implied by $h^{N|M|}$. For each\n\n[^37]history that succeeds $h^{N|M|}$, the agent of type $\\theta$ plays the message $m$ that solves\n$$\n\\max _{m \\in M} \\sum_{\\tilde{\\omega}} \\mu_{N|M|}\\left(h^{N|M|}\\right)(\\tilde{\\omega}) \\sum_{a \\in A} \\phi(a \\mid m, \\tilde{\\omega}) u(a, \\theta, \\omega) .\n$$\nIt is immediate to verify that the agent's (limit) average payoff under $\\sigma_{N}^{\\prime}$ is given by:\n$$\n\\begin{aligned}\nU\\left(\\sigma_{N}^{\\prime}\\right) & =\\mathbb{E}_{\\sigma_{N}^{\\prime}}\\left[\\sum_{\\theta \\in \\Theta} f(\\theta) \\max _{m \\in M} \\sum_{\\tilde{\\omega}} \\mu_{N|M|}\\left(h^{N|M|}\\right)(\\tilde{\\omega}) \\sum_{a \\in A} \\phi(a \\mid m, \\tilde{\\omega}) u(a, \\theta, \\omega)\\right]= \\\\\n& =\\sum_{h^{N|M|} \\in H^{N|M|}} \\mathbb{P}_{\\sigma_{N}^{\\prime}}^{N|M|}\\left(h^{N|M|}\\right) u^{*}\\left(\\mu_{N|M|}\\left(h^{N|M|}\\right)\\right),\n\\end{aligned}\n$$\nwhere $u^{*}$ is as in Equation C.20.\nBelow, we show that the distribution of beliefs under $\\sigma_{N}^{\\prime}, \\mathbb{P}_{\\sigma_{N}^{\\prime}} \\circ \\mu_{N|M|}^{-1} \\xrightarrow{w^{*}} \\tau_{\\phi}$. Consequently, as $u^{*}$ is continuous and bounded on $\\Delta(\\tilde{\\Omega})$, for any $\\delta>0$, we can choose $N_{\\delta}$ so that for all $N \\geq N_{\\delta}$,\n$$\n\\left|U\\left(\\sigma_{N}^{\\prime}\\right)-U\\left(\\tau_{\\phi}\\right)\\right| \\leq \\delta .\n$$\nConsequently, the agent's payoff under $\\sigma$ must be $U\\left(\\tau_{\\phi}\\right)$ because by definition for all $\\delta>0^{56}$\n$$\n\\liminf _{T \\rightarrow \\infty} \\mathbb{E}_{\\sigma}\\left[U_{T}\\right] \\geq \\limsup _{T \\rightarrow \\infty} \\mathbb{E}_{\\sigma_{N_{\\delta}}^{\\prime}}\\left[U_{T}\\right]=U\\left(\\sigma_{N_{\\delta}}^{\\prime}\\right) \\geq U\\left(\\tau_{\\phi}\\right)-\\delta,\n$$\nand hence,\n$$\n\\liminf _{T \\rightarrow \\infty} \\mathbb{E}_{\\sigma}\\left[U_{T}\\right] \\geq \\lim _{\\delta \\rightarrow 0} U\\left(\\tau_{\\phi}\\right)-\\delta=U\\left(\\tau_{\\phi}\\right) .\n$$\nwhich completes the proof.\nWe now complete the missing step:\n\nThe law of $\\mu_{N|M|}$ converges to $\\tau_{\\phi}$ It is useful to write the bottom line of Equation D. 2 as follows:\n\n$$\n\\sum_{\\tilde{\\omega} \\in \\tilde{\\Omega}}\\left(\\mu_{0} \\otimes \\eta\\right)(\\tilde{\\omega}) \\mathbb{E}_{\\mathbb{P}_{\\sigma_{N}^{\\prime}}^{N|M|}(\\cdot \\mid \\tilde{\\omega})}\\left[u^{*}\\left(\\mu_{N|M|}\\right)\\right] .\n$$\n\nWe show that the conditional law of $\\mu_{N|M|}, \\mathbb{P}_{\\sigma_{N}^{\\prime}}(\\cdot \\mid \\tilde{\\omega})$, converges to the Dirac measure on $\\mu(\\cdot \\mid P(\\tilde{\\omega}))$. Noting that $\\tau_{\\phi}=\\sum_{\\tilde{\\omega} \\in \\tilde{\\Omega}}\\left(\\mu_{0} \\otimes \\eta\\right)(\\tilde{\\omega}) \\delta_{\\mu(\\cdot \\mid P(\\tilde{\\omega}))}$ completes the proof.\n\nFor the exploration block, define for each $m \\in M$, the empirical frequency $\\hat{\\phi}_{N, m}:(\\Theta \\times M \\times A)^{N|M|} \\rightarrow$ $\\Delta(A)$, as follows\n\n$$\n\\hat{\\phi}_{N, m}\\left(h^{N|M|}\\right)(a)=\\frac{1}{N} \\sum_{n=1}^{N} \\mathbb{1}\\left[A_{m, n}=a\\right],\n$$\n\nwhere $A_{m, n}$ is the $n^{\\text {th }}$ draw from $A$ when the message is $m$. Let $\\hat{\\phi}_{N}: H^{N|M|} \\rightarrow \\Delta(A)^{M}$ denote the vector\n\n[^38]of empirical frequencies. The agent's belief at history $h^{N|M|}$ is given by:\n$$\n\\mu_{N|M|}\\left(h^{N|M|}\\right)(\\tilde{\\omega})=\\frac{\\left(\\mu_{0} \\otimes \\eta\\right)(\\tilde{\\omega}) \\prod_{m \\in M} \\prod_{a \\in A} \\phi(a \\mid m, \\tilde{\\omega})^{N \\hat{\\phi}_{N, m}\\left(h^{N|M|}\\right)(a)}}{\\sum_{\\tilde{\\omega}^{\\prime}}\\left(\\mu_{0} \\otimes \\eta\\right)\\left(\\tilde{\\omega}^{\\prime}\\right) \\prod_{m \\in M} \\prod_{a \\in A} \\phi\\left(a \\mid m, \\tilde{\\omega}^{\\prime}\\right)^{N \\hat{\\phi}_{N, m}\\left(h^{N|M|}\\right)(a)}} .\n$$\nDenote by $\\tau_{N|M|, \\tilde{\\omega}} \\in \\Delta(\\Delta(\\tilde{\\Omega}))$ the law of $\\mu_{N|M|}$ conditional on $\\tilde{\\omega}$, i.e., $\\tau_{N|M|, \\tilde{\\omega}}=\\mathbb{P}_{\\sigma_{N}^{\\prime}}(\\cdot \\mid \\tilde{\\omega}) \\circ \\mu_{N|M|}^{-1}$. Below, we show that $\\tau_{N|M|, \\tilde{\\omega}}$ converges weakly to $\\delta_{\\mu(\\cdot \\mid P(\\tilde{\\omega}))}$.\n\nSuppose the true state is $\\tilde{\\omega}^{\\star}$. Then, $\\left(A_{m, 1}, \\ldots, A_{m, N}\\right)$ are drawn i.i.d. from distribution $\\phi\\left(\\cdot \\mid m, \\tilde{\\omega}^{\\star}\\right)$. Fix a continuous and bounded function $g$ on $A$. Then, almost surely, ${ }^{57}$\n\n$$\n\\int_{A} g d \\hat{\\phi}_{N}(\\cdot \\mid m)=\\frac{1}{N} \\sum_{n=1}^{N} g\\left(A_{m, n}\\right) \\rightarrow \\mathbb{E}_{\\phi}\\left[g\\left(A_{m, 1}\\right)\\right]=\\sum_{a \\in A} g(a) \\phi\\left(a \\mid m, \\tilde{\\omega}^{\\star}\\right),\n$$\n\nby the strong law of large numbers applied to the i.i.d random variables $\\left(g\\left(A_{m, 1}\\right), \\ldots, g\\left(A_{m, N}\\right)\\right)$. Because this holds for all $g$, then $\\hat{\\phi}_{N}(\\cdot \\mid m) \\xrightarrow{w^{*}} \\phi\\left(\\cdot \\mid m, \\tilde{\\omega}^{\\star}\\right)$ almost surely when the true state is $\\tilde{\\omega}^{\\star}$.\n\nFix an arbitrary state $\\tilde{\\omega}$ and consider the ratio of the right-hand side of Equation D. 3 at $\\tilde{\\omega}$ and $\\tilde{\\omega}^{\\star}$ :","text_sha256":"72103e9d140d5853a15d65c78fc162f71cf9c97071443475c6707182ecc5da20"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0054","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D. 3 Technical results from Appendix C","text":"$$\n\\frac{\\mu_{N|M|}\\left(h^{N|M|}\\right)(\\tilde{\\omega})}{\\mu_{N|M|}\\left(h^{N|M|}\\right)\\left(\\tilde{\\omega}^{\\star}\\right)}=\\frac{\\left(\\mu_{0} \\otimes \\eta\\right)(\\tilde{\\omega})}{\\left(\\mu_{0} \\otimes \\eta\\right)\\left(\\tilde{\\omega}^{\\star}\\right)} \\prod_{a \\in A} \\prod_{m \\in M}\\left(\\frac{\\phi(a \\mid m, \\tilde{\\omega})}{\\phi\\left(a \\mid m, \\tilde{\\omega}^{\\star}\\right)}\\right)^{N \\hat{\\phi}_{N, m}\\left(h^{N|M|}\\right)(a)}\n$$\n\nSuppose $\\tilde{\\omega} \\notin P\\left(\\tilde{\\omega}^{\\star}\\right)$. By definition of the partition $\\mathscr{P}$, a message $m \\in M$ and allocation $a \\in A$ exist such that $\\phi(a \\mid m, \\tilde{\\omega}) \\neq \\phi\\left(a \\mid m, \\tilde{\\omega}^{\\star}\\right)$. Taking logarithm on both sides of Equation D. 4 and dividing by $N$,\n\n$$\n\\frac{1}{N} \\log \\left(\\frac{\\mu_{N|M|}\\left(h^{N|M|}\\right)(\\tilde{\\omega})}{\\mu_{N|M|}\\left(h^{N|M|}\\right)\\left(\\tilde{\\omega}^{\\star}\\right)}\\right)=\\frac{1}{N} \\log \\left(\\frac{\\left(\\mu_{0} \\otimes \\eta\\right)(\\tilde{\\omega})}{\\left(\\mu_{0} \\otimes \\eta\\right)\\left(\\tilde{\\omega}^{\\star}\\right)}\\right)+\\sum_{a^{\\prime} \\in A} \\sum_{m^{\\prime} \\in M} \\hat{\\phi}_{N, m^{\\prime}}\\left(h^{N|M|}\\right)\\left(a^{\\prime}\\right) \\log \\left(\\frac{\\phi\\left(a^{\\prime} \\mid m^{\\prime}, \\tilde{\\omega}\\right)}{\\phi\\left(a^{\\prime} \\mid m^{\\prime}, \\tilde{\\omega}^{\\star}\\right)}\\right) .\n$$\n\nBecause $\\hat{\\phi}_{N}(\\cdot \\mid m) \\xrightarrow{w^{*}} \\phi\\left(\\cdot \\mid m, \\tilde{\\omega}^{\\star}\\right)$ almost surely when the true state is $\\tilde{\\omega}^{\\star}$,\n\n$$\n\\lim _{N \\rightarrow \\infty} \\frac{1}{N} \\log \\left(\\frac{\\mu_{N|M|}\\left(h^{N|M|}\\right)(\\tilde{\\omega})}{\\mu_{N|M|}\\left(h^{N|M|}\\right)\\left(\\tilde{\\omega}^{\\star}\\right)}\\right)=-\\sum_{m \\in M} \\mathrm{D}_{\\mathrm{KL}}\\left(\\phi\\left(\\cdot \\mid m, \\tilde{\\omega}^{\\star}\\right) \\mid \\phi(\\cdot \\mid m, \\tilde{\\omega})\\right),\n$$\n\nwhere $\\mathrm{D}_{\\mathrm{KL}}$ is the Kullback-Leibler divergence. Note that at least one of the terms in the KL-divergence is positive as $\\phi(\\cdot \\mid m, \\tilde{\\omega}) \\neq \\phi\\left(\\cdot \\mid m, \\tilde{\\omega}^{\\star}\\right)$. Hence,\n\n$$\n\\lim _{N \\rightarrow \\infty} \\log \\left(\\frac{\\mu_{N|M|}\\left(h^{N|M|}\\right)(\\tilde{\\omega})}{\\mu_{N|M|}\\left(h^{N|M|}\\right)\\left(\\tilde{\\omega}^{\\star}\\right)}\\right)=-\\infty,\n$$\n\nmeaning that $\\mu_{N|M|}\\left(h^{N|M|}\\right)(\\tilde{\\omega}) / \\mu_{N|M|}\\left(h^{N|M|}\\right)\\left(\\tilde{\\omega}^{\\star}\\right) \\rightarrow 0$.\nSuppose now that $\\tilde{\\omega} \\in P\\left(\\tilde{\\omega}^{\\star}\\right)$. Then, Equation D. 4 reduces to\n\n$$\n\\frac{\\mu_{N|M|}\\left(h^{N|M|}\\right)(\\tilde{\\omega})}{\\mu_{N|M|}\\left(h^{N|M|}\\right)\\left(\\tilde{\\omega}^{\\star}\\right)}=\\frac{\\left(\\mu_{0} \\otimes \\eta\\right)(\\tilde{\\omega})}{\\left(\\mu_{0} \\otimes \\eta\\right)\\left(\\tilde{\\omega}^{\\star}\\right)},\n$$\n\nfor all $N$.\n\n[^39]Collecting both cases, we conclude that conditional on the true state being $\\tilde{\\omega}^{\\star}$,\n\n$$\n\\sum_{\\tilde{\\omega} \\in P\\left(\\tilde{\\omega}^{\\star}\\right)} \\mu_{N|M|}(\\tilde{\\omega}) \\rightarrow_{N \\rightarrow \\infty} 1,\n$$\n\nand moreover, within the cell, the fixed-ratio property implies the law $\\tau_{N|M|, \\tilde{\\omega}^{\\star}} \\xrightarrow{w^{*}} \\delta_{\\mu\\left(\\cdot \\mid P\\left(\\tilde{\\omega}^{\\star}\\right)\\right)}$. We conclude that the unconditional belief distribution, $\\sum_{\\tilde{\\omega} \\in \\tilde{\\Omega}}\\left(\\mu_{0} \\otimes \\eta\\right)(\\tilde{\\omega}) \\tau_{N|M|, \\tilde{\\omega}} \\xrightarrow{w^{*}} \\sum_{\\tilde{\\omega} \\in \\tilde{\\Omega}}\\left(\\mu_{0} \\otimes \\eta\\right)(\\tilde{\\omega}) \\delta_{\\mu(P(\\tilde{\\omega}))}=$ $\\tau_{\\phi}$. In particular,\n\n$$\nU\\left(\\sigma_{N}^{\\prime}\\right)=\\mathbb{E}_{\\mathbb{P}_{\\sigma_{N}^{\\prime}} \\circ \\mu_{N|M|}^{-1}}\\left[u^{*}(\\mu)\\right] \\rightarrow \\mathbb{E}_{\\tau_{\\phi}}\\left[u^{*}(\\mu)\\right]\n$$\n\ncompleting the proof. $\\square$\n\nLemma D.1. Suppose $u: \\Delta(\\Omega) \\rightarrow \\mathbb{R}$ satisfies that\n\n$$\n\\int_{\\Delta(\\Omega)} u(\\mu) \\tau(d \\mu)<\\int_{\\Delta(\\Omega)} \\max \\{u(\\mu), 0\\} \\tau(d \\mu)\n$$\n\nthen a set $B \\subseteq \\Delta(\\Omega)$ open relative to $\\Delta(\\Omega)$ exists such that\n\n$$\n\\int_{B} u(\\mu) \\tau(d \\mu)<0\n$$\n\nProof. Define the positive and negative parts of $u$ :\n\n$$\nu_{+}(\\mu):=\\max \\{u(\\mu), 0\\}, \\quad u_{-}(\\mu):=\\max \\{-u(\\mu), 0\\} .\n$$\n\nThen $u=u_{+}-u_{-}$pointwise. Integrating and using the assumed strict inequality,\n\n$$\n\\int u d \\tau=\\int u_{+} d \\tau-\\int u_{-} d \\tau<\\int u_{+} d \\tau \\Rightarrow \\int u_{-} d \\tau>0 .\n$$\n\nHence the set $N:=\\{\\mu \\in \\Delta(\\Omega): u(\\mu)<0\\}$ has strictly positive mass under $\\tau$.\nEmbed $\\Delta(\\Omega) \\subset \\mathbb{R}^{|\\Omega|}$. Extend $\\tau$ to a finite Borel measure $\\tilde{\\tau}$ on $\\mathbb{R}^{d}$ by\n\n$$\n\\tilde{\\tau}(A):=\\tau(A \\cap \\Delta(\\Omega)) \\quad\\left(A \\subseteq \\mathbb{R}^{d} \\text { Borel }\\right),\n$$\n\nand extend $u$ to $\\tilde{u}: \\mathbb{R}^{d} \\rightarrow[-1,1]$ by $\\tilde{u}=u$ on $\\Delta(\\Omega)$ and $\\tilde{u}=0$ on $\\mathbb{R}^{d} \\backslash \\Delta(\\Omega)$. Then $\\tilde{u} \\in L^{1}(\\tilde{\\tau})$ and $\\tilde{\\tau}(N)=\\tau(N)>0$.\n\nLet $B_{\\mathbb{R}|\\Omega|}(x, r)$ denote the ball in $\\mathbb{R}^{|\\Omega|}$ with center $x$ and radius $r$. By the Lebesgue differentiation theorem for finite Borel measures on $\\mathbb{R}^{d}$, there is a $\\tilde{\\tau}$-full-measure set $D \\subseteq \\mathbb{R}^{d}$ such that for every $x \\in D$,\n\n$$\n\\lim _{r \\downarrow 0} \\frac{1}{\\tilde{\\tau}\\left(B_{\\mathbb{R}^{d}}(x, r)\\right)} \\int_{B_{\\mathbb{R}^{d}}(x, r)} \\tilde{u} d \\tilde{\\tau}=\\tilde{u}(x),\n$$\n\nwhenever $\\tilde{\\tau}\\left(B_{\\mathbb{R}^{d}}(x, r)\\right)>0$ (and this positivity holds for all sufficiently small $r$ for $\\tilde{\\tau}$-a.e. $x$ ). Since $\\tilde{\\tau}(N \\cap D)>0$, choose $\\mu_{0} \\in N \\cap D$. Then $\\tilde{u}\\left(\\mu_{0}\\right)=u\\left(\\mu_{0}\\right)<0$. Therefore the above limit is strictly negative,\nso there exists $r_{0}>0$ such that for all $0<r<r_{0}$,\n\n$$\n\\int_{B_{\\mathbb{R}^{d}}\\left(\\mu_{0}, r\\right)} \\tilde{u} d \\tilde{\\tau}<0\n$$\n\nFor such an $r$, let $B:=B_{\\Delta}\\left(\\mu_{0}, r\\right)=\\Delta(\\Omega) \\cap B_{\\mathbb{R}^{d}}\\left(\\mu_{0}, r\\right)$, which is an open ball in $\\Delta(\\Omega)$. Using the definitions of $\\tilde{\\tau}$ and $\\tilde{u}$,\n\n$$\n\\int_{B} u d \\tau=\\int_{B_{\\mathbb{R}^{d}}\\left(\\mu_{0}, r\\right)} \\tilde{u} d \\tilde{\\tau}<0\n$$\n\nThis proves the claim. $\\square$","text_sha256":"23edc7708479a713f2ea5cda1243a06a9ac2463efe09a1da05444389b691b7cb"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0055","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D.3.1 Revelation principle for limit of means preferences","text":"## D.3.1 Revelation principle for limit of means preferences\n\nWe show in this section that when the designer uses dynamic mechanisms, it is without loss of generality for the designer to employ direct dynamic mechanisms that (i) implement the outside option at all histories after the agent first exercises her option not to participate in the mechanism, and (ii) for which the agent's best response is to always participate and truthfully report her type. This justifies the class of mechanisms we employ in the analysis of Section 5.2.\n\nHistories, mechanisms, and strategies As in the main text, to simplify notation, we do not include the agent's decision to participate in the mechanism in the histories of the game. Instead, we follow the convention that if the agent does not participate, it is as if she reported $\\varnothing$ and the allocation is $a_{\\varnothing}$. Formally, let $M A_{\\varnothing}=(M \\times A) \\cup\\left\\{\\left(\\varnothing, a_{\\varnothing}\\right)\\right\\}$. With this notation, a history through period $t$ is an element of $\\hat{H}_{M}^{t} \\equiv\\left(M A_{\\varnothing}\\right)^{t-1}$ and let $\\hat{\\mathcal{H}}_{M}^{t}=\\Omega \\times \\hat{H}_{M}^{t} .{ }^{58}$\n\nA mechanism is a collection $\\varphi \\equiv\\left(\\varphi_{t}\\right)_{t=1}^{\\infty}$ such that the mechanism in period $t$ is a mapping $\\varphi_{t}$ : $\\hat{\\mathcal{H}}_{M}^{t} \\times M \\rightarrow \\Delta(A)$.\n\nLet $H_{M}^{t}=\\left(\\Theta \\times M A_{\\varnothing}\\right)^{t-1}=\\Theta^{t-1} \\times \\hat{H}_{M}^{t}$. The agent's strategy, ( $p, \\sigma$ ), is given by her participation strategy $p_{t}: H_{M}^{t} \\times \\Theta \\rightarrow[0,1]$, and conditional on participating, her reporting strategy $\\sigma_{t}: H_{M}^{t} \\times \\Theta \\rightarrow \\Delta(M)$.\n\nThe distribution over terminal histories To obtain the complete description of the paths on the tree we need to append $\\Omega$ to $H_{M}^{t}$; hence the paths through period $t-1$ are $\\Omega \\times H_{M}^{t} \\equiv \\mathcal{H}_{M}^{t}$. The distributions over states and agent's types, the agent's strategy, and the mechanism induce a distribution over the terminal histories $\\mathcal{H}_{M}^{\\infty} \\equiv \\Omega \\times H_{M}^{\\infty}$, which we denote by $\\mathbb{P}_{\\varphi,(p, \\sigma)} \\in \\Delta\\left(\\Omega \\times H_{M}^{\\infty}\\right)$, as it is now useful to keep track of the mechanism. We denote by $\\mathbb{E}_{(p, \\sigma)}$ the expectation under this measure. The distribution $\\mathbb{P}_{\\varphi,(p, \\sigma)} \\in \\Delta\\left(\\Omega \\times H_{M}^{\\infty}\\right)$ is the unique distribution that satisfies that for all $t \\in \\mathbb{N}, \\tilde{\\mathcal{H}}_{M}^{t} \\subset \\Omega \\times H_{M}^{t}$,\n\n$$\n\\mathbb{P}_{\\varphi,(p, \\sigma)}\\left(\\tilde{\\mathcal{H}}_{M}^{t} \\times \\prod_{s=t+1}^{\\infty}\\left(\\Theta \\times M A_{\\varnothing}\\right)\\right)=\\mathbb{P}_{\\varphi,(p, \\sigma)}^{t}\\left(\\tilde{\\mathcal{H}}_{M}^{t}\\right),\n$$\n\n[^40]where the distributions $\\left(\\mathbb{P}_{\\varphi,(p, \\sigma)}^{t}\\right)_{t \\in \\mathbb{N}}$ satisfy\n$$\n\\begin{aligned}\n\\mathbb{P}_{\\varphi,(p, \\sigma)}^{t+1}\\left(\\omega, h_{M}^{t}, \\theta, m, a\\right) & =\\mathbb{P}_{\\varphi,(p, \\sigma)}^{t}\\left(\\omega, h_{M}^{t}\\right) f(\\theta) p_{t}\\left(h_{M}^{t}, \\theta\\right) \\sigma_{t}\\left(h_{M}^{t}, \\theta\\right)(m) \\varphi_{t}\\left(\\omega, \\hat{h}_{M}^{t}, m\\right)(a), \\\\\n\\mathbb{P}_{\\varphi,(p, \\sigma)}^{t+1}\\left(\\omega, h_{M}^{t}, \\theta, \\varnothing, a\\right) & =\\mathbb{P}_{\\varphi,(p, \\sigma)}^{t}\\left(\\omega, h_{M}^{t}\\right) f(\\theta)\\left(1-p_{t}\\left(h_{M}^{t}, \\theta\\right)\\right) \\mathbb{1}\\left[a=a_{\\varnothing}\\right] .\n\\end{aligned}\n$$\n\nOutcome distribution Our interest is in the distribution over payoff-relevant outcomes, $\\Omega \\times(\\Theta \\times A)^{\\infty}$, and hence on the marginal of $\\mathbb{P}_{\\varphi,(p, \\sigma)}$ on $\\Omega \\times(\\Theta \\times A)^{\\infty}$, which we denote by $\\overline{\\mathbb{P}}_{\\varphi,(p, \\sigma)}$.\n\nBest response We say that strategy $(p, \\sigma)$ is a best response for the agent if for all alternative strategies $\\left(p^{\\prime}, \\sigma^{\\prime}\\right)$, we have that\n\n$$\n\\liminf _{T \\rightarrow \\infty} \\mathbb{E}_{(p, \\sigma)}\\left[U_{T}\\right] \\geq \\limsup _{T \\rightarrow \\infty} \\mathbb{E}_{\\left(p^{\\prime}, \\sigma^{\\prime}\\right)}\\left[U_{T}\\right],\n$$\n\nwhere recall $U_{T}$ is the agent's average payoff until period $T$.\n\nDirect and full participation mechanisms A special case of the above game is that in which $M=\\Theta$, and whenever the agent does not participate, the mechanism chooses $a_{\\varnothing}$ with probability 1 for any message in all continuation histories. We call these mechanisms direct and full participation mechanisms. Below, when $M=\\Theta$, we drop the dependence of the set histories on $M$.\n\nFormally, let $\\hat{\\mathcal{H}}_{\\varnothing}^{t}$ denote the subset of $\\hat{\\mathcal{H}}^{t}$ such that at some point the sequence ( $\\varnothing, a_{\\varnothing}$ ) appears. We define mechanisms\n\n$$\n\\tilde{\\varphi}_{t}: \\hat{\\mathcal{H}}^{t} \\times \\Theta \\rightarrow \\Delta(A),\n$$\n\nsuch that $\\tilde{\\varphi}_{t}\\left(\\omega, \\hat{h}^{t}, \\cdot\\right)=\\mathbb{1}\\left[a=a_{\\varnothing}\\right]$ whenever $\\left(\\omega, \\hat{h}^{t}\\right) \\in \\hat{\\mathcal{H}}_{\\varnothing}^{t}$.\nTheorem D.1. Suppose that $(p, \\sigma)$ is a best response to mechanism $\\varphi$. Then, a direct and full participation mechanism $\\tilde{\\varphi}$ exists such that\n\n1. Participation with probability 1 and truthtelling are a best response for the agent,\n2. The distribution over $\\Omega \\times(\\Theta \\times A)^{\\infty}$ induced by $(\\varphi,(p, \\sigma))$ is the same as that induced by $\\tilde{\\varphi}$ under participation and truthtelling.\n\nProof. Fix a mechanism $\\varphi=\\left(\\varphi^{t}\\right)_{t \\geq 1}$ and a best response ( $p, \\sigma$ ) for the agent in the sense of Equation 13. Let $\\mathbb{P}_{\\varphi,(p, \\sigma)}$ denote the induced distribution over $\\mathcal{H}^{\\infty}$. We write $\\left(\\omega,\\left(\\theta_{t}, m_{t}, a_{t}\\right)_{t \\geq 1}\\right)$ for a generic realization, where $m_{t}=\\varnothing \\Rightarrow a_{t}=a_{\\varnothing}$.","text_sha256":"350e4e110f6e5f250b9c8ca8114aa87a8f1c69761f9c754f47ddde54420c65b4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0056","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D.3.1 Revelation principle for limit of means preferences","text":"The proof proceeds in three steps. In the first step, we construct a direct (but not full participation) mechanism $\\tilde{\\varphi}$, which under participation and truthtelling after every history on path implements the same outcome distribution as $(\\varphi,(p, \\sigma))$. In the second step, we verify that participation and truthtelling after every history on path is a best response to $\\tilde{\\varphi}$. In the third step, we construct a direct and full participation mechanism from $\\tilde{\\varphi}$. That the agent can always quit the mechanism at each step and obtain $a_{\\varnothing}$ and Step 3 implies that participation and truthtelling after every history is also a best response to the full participation mechanism obtained from $\\tilde{\\varphi}$.\n\nStep 1: We first construct the direct mechanism $\\tilde{\\varphi}_{t}: \\hat{\\mathcal{H}}^{t} \\times \\Theta \\rightarrow \\Delta(A)$. Define a collection of transition probabilities $\\kappa_{t}: \\mathcal{H}_{M}^{t} \\times \\Theta \\rightarrow \\Delta(M \\cup\\{\\varnothing\\})$ as follows:\n\n$$\n\\kappa_{t}\\left(m_{t} \\mid h_{M}^{t}, \\theta_{t}\\right)=\\left(1-p_{t}\\left(h_{M}^{t}, \\theta_{t}\\right)\\right) \\mathbb{1}\\left[m_{t}=\\varnothing\\right]+p_{t}\\left(h_{M}^{t}, \\theta_{t}\\right) \\sigma_{t}\\left(h_{M}^{t}, \\theta_{t}\\right)\\left(m_{t}\\right) .\n$$\n\nWe construct $\\tilde{\\varphi}$ recursively. In period 1, if the state is $\\omega$ and the report is $\\theta_{1}$, the designer draws fictitious $m_{1}$ from $\\kappa_{1}\\left(\\cdot \\mid \\theta_{1}\\right)$, and implements $a_{1}=a_{\\varnothing}$ if $m_{1}=\\varnothing$, and otherwise draws $a_{1} \\sim \\varphi_{1}\\left(\\omega, m_{1}\\right)$.\n\nRecursively, for $t \\geq 2$, if the sequence of reports, fictitious messages, and allocations is $\\left(\\theta^{\\prime t-1}, m^{t-1}, a^{t-1}\\right)=$ $\\left(\\theta_{s}^{\\prime}, m_{s}, a_{s}\\right)_{s=1}^{t-1}$ and the agent reports $\\theta_{t}$, the designer draws $m_{t}$ from $\\kappa_{t}\\left(\\cdot \\mid \\theta^{\\prime t-1}, m^{t-1}, a^{t-1}, \\theta_{t}\\right)$ and implements $a_{t}=a_{\\varnothing}$ if $m_{t}=\\varnothing$, and otherwise draws $a_{t} \\sim \\varphi_{t}\\left(\\omega, m^{t-1}, a^{t-1}, m_{t}\\right) .{ }^{59}$\n\nIt is immediate that under truthtelling and participation the mechanism $\\tilde{\\varphi}$ implements the same distribution over $\\Omega \\times(A \\times \\Theta)^{\\infty} .{ }^{60}$\n\nStep 2: Let $\\left(p^{*}, \\sigma^{*}\\right)$ denote the agent's strategy that participates and truthfully reports after every history. We now show that $\\left(p^{*}, \\sigma^{*}\\right)$ is a best response to $\\tilde{\\varphi}$ in the sense of Equation D.8.\n\nTo do so, we show that for any strategy ( $\\tilde{p}, \\tilde{\\sigma}$ ) in the game induced by the direct mechanism $\\tilde{\\varphi}$, a strategy $\\left(p^{\\prime}, \\sigma^{\\prime}\\right)$ exists such that\n\n$$\n\\mathbb{E}_{\\tilde{\\varphi},(\\tilde{p}, \\tilde{\\sigma})}\\left[U_{T}\\right]=\\mathbb{E}_{\\varphi,\\left(p^{\\prime}, \\sigma^{\\prime}\\right)}\\left[U_{T}\\right] \\quad \\text { for all } T,\n$$\n\nwhere $U_{T}$ is the average payoff through period $T$. Given Equation D. 9 and the best-response property of $(p, \\sigma)$ to $\\varphi$,\n\n$$\n\\liminf _{T \\rightarrow \\infty} \\mathbb{E}_{\\varphi,(p, \\sigma)}\\left[U_{T}\\right] \\geq \\limsup _{T \\rightarrow \\infty} \\mathbb{E}_{\\varphi,\\left(p^{\\prime}, \\sigma^{\\prime}\\right)}\\left[U_{T}\\right] \\quad \\text { for all }\\left(p^{\\prime}, \\sigma^{\\prime}\\right) .\n$$\n\nUsing Step 1, we have $\\mathbb{E}_{\\varphi,(p, \\sigma)}\\left[U_{T}\\right]=\\mathbb{E}_{\\tilde{\\varphi},\\left(p^{*}, \\sigma^{*}\\right)}\\left[U_{T}\\right]$ for all $T$, and by Equation D. 9 we have $\\mathbb{E}_{\\varphi,\\left(p^{\\prime}, \\sigma^{\\prime}\\right)}\\left[U_{T}\\right]=$ $\\mathbb{E}_{\\tilde{\\varphi},(\\tilde{p}, \\tilde{\\sigma})}\\left[U_{T}\\right]$ for all $T$. Hence\n\n$$\n\\liminf _{T \\rightarrow \\infty} \\mathbb{E}_{\\tilde{\\varphi},\\left(p^{*}, \\sigma^{*}\\right)}\\left[U_{T}\\right] \\geq \\limsup _{T \\rightarrow \\infty} \\mathbb{E}_{\\tilde{\\varphi},(\\tilde{p}, \\tilde{\\sigma})}\\left[U_{T}\\right] \\quad \\text { for all }(\\tilde{p}, \\tilde{\\sigma}),\n$$\n\nwhich is exactly the definition of $\\left(p^{*}, \\sigma^{*}\\right)$ being a best response to $\\tilde{\\varphi}$. It remains to construct $\\left(p^{\\prime}, \\sigma^{\\prime}\\right)$ and verify (D.9).\n\nWe show how the agent can emulate the strategy ( $\\tilde{p}, \\tilde{\\sigma}$ ) in the game induced by the indirect mechanism via strategy $\\left(p^{\\prime}, \\sigma^{\\prime}\\right)$. Define $\\tilde{\\kappa}_{t}: H^{t} \\times \\Theta \\rightarrow \\triangle(\\Theta \\cup\\{\\varnothing\\})$ as follows\n\n$$\n\\tilde{\\kappa}_{t}\\left(\\theta^{\\prime} \\mid h^{t}, \\theta_{t}\\right)=\\left(1-\\tilde{p}\\left(h^{t}, \\theta_{t}\\right)\\right) \\mathbb{1}\\left[\\theta^{\\prime}=\\varnothing\\right]+\\tilde{p}\\left(h^{t}, \\theta_{t}\\right) \\tilde{\\sigma}\\left(h^{t}, \\theta_{t}\\right)\\left(\\theta^{\\prime}\\right) .\n$$\n\nThe strategy ( $p^{\\prime}, \\sigma^{\\prime}$ ) privately simulates the report process induced by ( $\\tilde{p}, \\tilde{\\sigma}$ ) in the direct mechanism, and conditional on the fictitious reports, generates the actual message in the indirect mechanism $\\varphi$ using the kernel $\\kappa_{t}$ in Step 1, evaluated at the fictitious type history.\n\n[^41]Formally, for $t \\geq 1$ given the history of types, messages, and allocations through period $t,\\left(\\theta^{t-1}, m^{t-1}, a^{t-1}\\right)$, and the privately tracked fictitious type reports $\\theta^{\\prime-1}$, ${ }^{61}$ the agent of type $\\theta_{t}$ draws a fictitious report $\\theta_{t}^{\\prime} \\sim \\tilde{\\kappa}_{t}\\left(\\cdot \\mid \\theta^{t-1}, \\theta^{\\prime \\text { t }-1}, a^{t-1}, \\theta_{t}\\right)$. If $\\theta_{t}^{\\prime}=\\varnothing$, then $m_{t}=\\varnothing$ (the agent does not participate in period $t$ ). Otherwise, $\\theta_{t}^{\\prime} \\in \\Theta$ and $m_{t}$ is drawn from $\\kappa_{t}\\left(\\cdot \\mid \\theta^{\\prime-1}, m^{t-1}, a^{t-1}, \\theta_{t}^{\\prime}\\right)$. The allocation is $a_{\\varnothing}$ upon rejection, and $a_{t} \\sim \\varphi_{t}\\left(\\omega, m^{t-1}, a^{t-1}, m_{t}\\right)$, otherwise.","text_sha256":"78f6e7a00cb1785c3f373447e348600e91321e1010583bed75608353e7e29867"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0057","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D.3.1 Revelation principle for limit of means preferences","text":"It is immediate that Equation D. 9 holds and hence $\\left(p^{*}, \\sigma^{*}\\right)$ is a best response to $\\tilde{\\varphi}:{ }^{62}$ In the extensive form game induced by $\\tilde{\\varphi}$, the designer simulates the agent's participation and reporting strategies using $(p, \\sigma)$ based on the agent's type reports and determines allocations in the mechanism. When the agent's strategy is given by $(\\tilde{p}, \\tilde{\\sigma})$, the process described in the above paragraph correspond to the designer's simulated participation and reporting strategies, and allocations continued to be determined by $\\varphi$. Hence, in the extensive form game induced by $\\tilde{\\varphi}$, when the agent plays ( $\\tilde{p}, \\tilde{\\sigma}$ ), it is as if she faces mechanism $\\varphi$ and plays strategy ( $p^{\\prime}, \\sigma^{\\prime}$ ). The best response property of ( $p, \\sigma$ ) implies that playing $\\left(p^{\\prime}, \\sigma^{\\prime}\\right)$ yields a weakly worse payoff, and hence $(\\tilde{p}, \\tilde{\\sigma})$ is not a profitable deviation from $\\left(p^{*}, \\sigma^{*}\\right)$ in the direct mechanism $\\tilde{\\varphi}$.\n\nStep 3: Modify $\\tilde{\\varphi}$ at all histories that include at least one non-participation decision, so that the mechanism implements the outside option $a_{\\varnothing}$. With this modification, the mechanism satisfies the full participation property. It is immediate that participation and truthtelling after every history remains a best response. $\\square$\n\n[^42]\n[^0]:    *The latest version of the paper can be found here. We thank Dirk Bergemann, James Best, Elliot Lipnowski, Emir Kamenica, Navin Kartik, Stephen Morris , Phil Reny, Marzena Rostek, and Refine.ink for valuable comments and suggestions, as well as seminar audiences at EARIE 2024, WUSTL Economic Theory Conference 2024, MIT IDSS Distinguished Speaker Series, the Workshop in Market Design, Bonn, Berlin HU, CERGE-EI, Gerzensee, UCL, Naples, Stanford University, MIT Sloan, and the CEPR Workshop on Contracts, Incentives and Information at Collegio Carlo Alberto. Laura Doval gratefully acknowledges financial support from the Sloan Foundation. Alex Smolin gratefully acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (grant ANR-17-EURE-0010) and the AI Interdisciplinary Institute ANITI (grant ANR-23-IACL-0002). This paper was partly written while the first author was visiting Stanford University and the second author was visiting Northwestern University and Columbia Business School; we are grateful for their hospitality.\n    ${ }^{\\dagger}$ Columbia Business School and CEPR. E-mail: laura.doval@columbia.edu.\n    ${ }^{\\ddagger}$ Toulouse School of Economics and CEPR. E-mail: alexey.v.smolin@gmail.com.\n\n[^1]:    ${ }^{1}$ In (generalized) two-stage mechanisms, the designer communicates with the agents before the agents communicate with the mechanism. Whereas this communication is a restriction on the set of implementable outcomes relative to the single-designer Myersonian benchmark, Attar et al. (2025) show that allowing competing principals to first communicate with agents expands the set of implementable outcomes.\n\n[^2]:    ${ }^{2}$ We assume the agent has limit-of-means preferences, so we can pass to the $\\delta \\rightarrow 1$ limit without approximation.\n    ${ }^{3}$ Theorem 3 holds when the agent's type is drawn once at the beginning without further assumptions on the distribution. As we explain in Section 5, we choose the i.i.d. specification for the evolution of the agent's private information to put repeated and dynamic mechanisms on a more equal footing.\n\n[^3]:    ${ }^{4}$ There is also a literature that studies a designer's disclosure of information that must be elicited from the agents (Eső and Szentes, 2007; Bergemann and Pesendorfer, 2007; Li and Shi, 2017; Krähmer, 2020; Bergemann et al., 2022a,b; Smolin, 2023). By contrast, the designer knows the realization of the state and also what information the two-stage mechanism discloses to the agents, so he need not elicit this information.\n\n[^4]:    ${ }^{5}$ Assuming the allocation space is a product space is without loss of generality. Any restriction on the allocations, such as all agents must receive the same allocation, can be incorporated as restrictions on the support of the mechanism.\n    ${ }^{6}$ Because we allow for lotteries over allocations, that the allocation space is finite does not preclude the case of transferable utility. Indeed, we could let each $A_{i}=\\tilde{A_{i}} \\times\\{-K, K\\}$ for some large enough $K>0$.\n\n[^5]:    ${ }^{7}$ When the agent is infinitely patient as in Section 5, we can exhibit a sequence of strategies under which the agent (approximately) learns this mapping. See the proof of Lemma C. 4 in Appendix D.\n    ${ }^{8}$ Gentzkow and Kamenica (2017) highlight that the language in Green and Stokey (2022) allows one to describe the correlation across signal structures. This is exactly what we need to allow the designer to obfuscate the agents' ability to learn. It is again instructive to consider the single-agent case. For each type report $\\theta \\in \\Theta$, the mechanism can be seen as an information structure $\\phi(\\theta, \\cdot): \\Omega \\times[0,1] \\rightarrow \\Delta(A)$. Thus, the randomization device allows the designer to control the correlation across these different signal structures, which in turn disciplines what the agent stands to learn when experimenting with different reports.\n    ${ }^{9}$ The terminology is by analogy to reduced form auctions where the map from own types to own probabilities of being allocated the good are referred to as the interim allocation.\n\n[^6]:    ${ }^{10}$ When we consider mechanisms with arbitrary message spaces in Appendix D.1, the requirement of calibration is relative to both the mechanism and agents' equilibrium participation and reporting strategies.\n    ${ }^{11}$ Thus, we are assuming that $a_{\\varnothing}=\\left(a_{i \\varnothing}\\right)_{i \\in N}$ is an element of $A$. In Appendix D.1, we consider more general participation decisions, allowing the mechanism to condition on the set of participating agents, but even with this extra generality, it is still without loss to restrict attention to mechanisms that induce full participation.","text_sha256":"bba5e5354f5f6019339b5a253c07e103ad61140c08e1d2c7e8744b5e23cd7bfb"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0058","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D.3.1 Revelation principle for limit of means preferences","text":"[^7]:    ${ }^{12}$ It is common for online platforms to inform agents of the consequences of their choices: marketplaces inform sellers of their probability of sale at different posted prices, and transportation providers inform riders of their probability of receiving a seat upgrade at different bid levels. Even insurance companies provide consumers with projected expenditures under different plan choices.\n\n[^8]:    ${ }^{13}$ Indeed, in the analysis of the dynamic interaction in Section 5, we assume the agent only observes her type and her allocation, but not her payoffs.\n    ${ }^{14}$ This assumption is routinely made in dynamic settings. See Pavan et al. (2014) and Cesa-Bianchi et al. (2024) for two examples in the context of agents' behavior within mechanisms.\n    ${ }^{15}$ In Section 3, we provide an example in which when agents have access to the calibrated information structure the optimal Myersonian mechanism fails to be incentive compatible.\n\n[^9]:    ${ }^{16}$ A tempting comparison is Maskin and Tirole (1990, Prop. 11): with private values and quasilinear utilities, the informed principal's unique equilibrium payoff coincides with the state-by-state optimum. The authors show this conclusion depends on quasilinearity: absent this assumption, an informed principal can benefit from concealing his information in the case of private values. Instead, Theorem 1 relies neither on quasilinearity nor on the designer's lack of commitment.\n\n[^10]:    ${ }^{17}$ Formally,\n\n    $$\n    \\mu_{0}(\\omega \\mid \\theta)=\\frac{\\mu_{0}(\\omega) f(\\theta \\mid \\omega)}{\\sum_{\\omega^{\\prime} \\in \\Omega} \\mu_{0}\\left(\\omega^{\\prime}\\right) f\\left(\\theta \\mid \\omega^{\\prime}\\right)}\n    $$\n\n    ${ }^{18}$ We refer the reader to the appendix for our mathematical conventions, in particular, the definition of the corresponding $\\sigma$-algebras.\n\n[^11]:    ${ }^{19}$ Formally, define the belief distribution induced by $\\beta, \\tau_{\\beta}=\\mu_{0} \\otimes \\beta$. The claim is that $\\mathbb{E}_{\\tau_{\\beta}}[\\mu]=\\mu_{0}$.\n    ${ }^{20}$ This is a consequence of Bayes rule: beliefs are a sufficient statistic for $\\omega$. Hence, conditional on $(\\theta, \\mu)$, the allocation rule carries no more information about the state.\n\n[^12]:    ${ }^{21}$ In contrast to the model of Section 2, we are assuming the set of types and allocations to be intervals in the real line. The results in the previous sections go through with richer type and allocation spaces, at the cost of more notation.\n\n[^13]:    ${ }^{22}$ This assumption ensures the Lipschitz continuity of the agent's indirect utility function when the allocation space is infinite. Because $\\Omega$ is finite, requiring the condition to hold state-by-state suffices.\n\n[^14]:    ${ }^{23}$ The calibrated mechanism which state-by-state implements the optimal direct mechanism under common knowledge of the state may reveal less than full information about the state, e.g., because at $\\omega$ and $\\omega^{\\prime}$ the same direct mechanism is optimal. The point is that pooling those states does not weaken the incentive constraints of the agent, so it is as if the designer were forced to reveal the state.\n\n[^15]:    ${ }^{24}$ Restricting attention to deterministic mechanisms, Ottaviani and Prat (2001) obtain the optimality of full disclosure without such a linearity assumption. Their model, however, is different from ours and that of the aforementioned papers: the agent's type is not payoff relevant and the agent's type and the state are affiliated. Thus, while related in spirit, Proposition 2 is distinct from their result.\n    ${ }^{25}$ See Szabadi (2018) for a similar observation.\n\n[^16]:    ${ }^{26}$ Because payoffs are quasilinear, considering mechanisms that do not randomize on transfers is without loss of generality.\n    ${ }^{27}$ The equi-Lipschitz condition on $u$ ensures we can take the derivative inside the integral.\n\n[^17]:    ${ }^{28}$ The designer can be viewed as an online advertising platform and the agent as an advertiser. State $\\omega$ represents the click-through rate of an ad slot, and higher $\\theta$ corresponds to a larger advertiser willing to pay more for exposure. The designer's payoff captures both the value created by advertising and the disutility from showing ads of large advertisers, e.g., due to user brand fatigue.\n\n[^18]:    ${ }^{29}$ In the statement, $w_{k \\theta_{i}}$ denotes the derivative of $w_{k}$ with respect to $\\theta_{i}$, for $k \\in\\{1, \\ldots, N\\}$.\n    ${ }^{30}$ To be sure, Theorem 3 extends to the case in which the agent's type is fully persistent and correlated with the state.\n\n[^19]:    ${ }^{31}$ The Ionescu-Tulcea extension theorem implies this measure is always well-defined for any mechanism and any agent's\n\n[^20]:    strategy. See Appendix C. 1 for details.\n    ${ }^{32}$ Throughout this section, limits of measures should always be understood in the weak* sense.\n    ${ }^{33}$ Note that in mechanism design one always focuses on mechanisms that have well-defined best responses in single-agent settings, and equilibria in multi-agent ones.\n\n[^21]:    ${ }^{34}$ By assumption, the marginal of $\\bar{v}_{\\sigma}^{T}$ on $A \\times \\Theta \\times M \\times \\tilde{\\Omega}$ converges to $v_{\\sigma}$. Moreover, we show that the marginal of $\\bar{v}_{\\sigma}^{T}$ on $\\Delta(\\tilde{\\Omega})$ also converges (Lemma C.3). However, this is not enough to ensure the convergence of $\\bar{v}_{\\sigma}^{T}$.\n    ${ }^{35}$ Whereas the above two-stage mechanism is described in terms of beliefs over $\\Omega \\times \\mathcal{E}$, we show in the appendix how to derive from it a two-stage mechanism in terms of beliefs over $\\Omega$.\n\n[^22]:    ${ }^{36}$ This notation allows us to keep the definitions of the histories when the agent participates and does not participate symmetric, and saves us on including the agent's participation strategy in the histories.\n    ${ }^{37}$ That is, starting from a dynamic mechanism $\\varphi$ and a best response strategy ( $p, \\sigma$ ), one can construct an alternative mechanism $\\varphi^{\\prime}$ such that participation and truthtelling are a best response for the agent and preserves the distribution over $(\\Omega \\times \\Theta \\times A)^{\\infty}$ induced by $(p, \\sigma)$ and $\\varphi$.","text_sha256":"881bb2708135a0c06c537f335f3dc99ebc66f80a55a70271dd3e7921d317f965"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0059","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D.3.1 Revelation principle for limit of means preferences","text":"[^23]:    ${ }^{38}$ This is easily seen in the example after Definition 7. Note the mechanism that allocates the good to the agent if and only if her type is $\\theta_{2}$ satisfies monotonicity. Hence, a transfer scheme exists that implements this allocation rule with transfers.\n    ${ }^{39}$ In a repeated principal-agent game with communication, Meng (2021) shows that the principal can guarantee in the patient limit his complete information payoff subject to the constraint that her actions satisfy the cyclic monotonicity condition in Rochet (1987). We view the results as complementary: We focus on implementable outcome distributions, instead of payoffs, when the agent has limit of the means preferences, which makes our notion of implementation exact.\n\n[^24]:    ${ }^{40}$ A standard argument implies that if $\\vartheta$ satisfies Equation 16, then a finite support $\\beta^{\\prime}$ exists such that $\\vartheta$ satisfies Equation 16 with $\\beta^{\\prime}$ in place of $\\beta$.\n\n[^25]:    ${ }^{41}$ Let $\\mu_{0}\\left(\\cdot \\mid \\theta_{i}\\right) \\in \\Delta(\\Omega)$ denote the prior of the agent with type $\\theta_{i}$ and $\\mu\\left(\\cdot \\mid s_{i}\\right)$ denote the updated belief of an agent with prior\n\n[^26]:    $\\mu_{0}$ upon observing signal $s_{i}^{*}$. When the signal is $s_{i}^{*}$, the agent with type $\\theta_{i}$ updates her beliefs to:\n\n    $$\n    \\mu_{i}\\left(\\cdot \\mid \\theta_{i}, s_{i}^{*}\\right)=\\frac{\\mu_{0}\\left(\\cdot \\mid \\theta_{i}\\right) \\cdot \\frac{\\mu\\left(\\cdot \\mid s_{i}^{*}\\right)}{\\mu_{0}(\\cdot)}}{\\left\\|\\mu_{0}\\left(\\cdot \\mid \\theta_{i}\\right) \\cdot \\frac{\\mu\\left(\\cdot \\mid s_{i}^{*}\\right)}{\\mu_{0}(\\cdot)}\\right\\|},\n    $$\n\n    where the ⋅ and / operations are meant componentwise, and $\\|\\cdot\\|$ is the $l^{1}$-norm.\n    ${ }^{42}$ Namely, Equation B. 5 implies that for all $(\\theta, \\omega), \\vartheta(\\cdot \\mid \\theta, \\omega) \\in \\operatorname{clco}\\{\\alpha(\\cdot \\mid \\theta, \\mu): \\mu \\in \\Delta(\\Omega)\\}$. Rubin and Wesler (1958) implies that our under assumptions $\\operatorname{clco}\\{\\alpha(\\cdot \\mid \\theta, \\mu): \\mu \\in \\Delta(\\Omega)\\}=\\operatorname{co}\\{\\alpha(\\cdot \\mid \\theta, \\mu): \\mu \\in \\Delta(\\Omega)\\}$, and the rest of the claim follows from Carathéodory's theorem.\n    ${ }^{43}$ To extend the result to the case in which the agent's type is correlated with the state, note the following. Knowing $\\mu_{0}$\n\n[^27]:    updates to $\\mu_{k}$ conditional on $s$ is enough to pin down the agent's belief $\\mu(\\cdot \\mid \\theta, s)$, with respect to which the agent's incentive compatibility and individual rationality constraints are defined (see footnote 41).\n\n[^28]:    ${ }^{44}$ We could expand the mechanism by allowing the agent to have a message which triggers the outside option, but this is not necessary as $\\alpha$ is individually rational.\n\n[^29]:    ${ }^{45}$ Lemma C. 2 implies that $\\bar{v}_{\\sigma^{\\prime}}^{T, 1}$ and $\\bar{v}_{\\sigma^{\\prime}}^{T, 2}$ have the same set of subsequential limits. Indeed, let $g$\n\n[^30]:    denote any continuous bounded function on $\\Omega \\times \\Delta(\\Omega) \\times \\Theta \\times \\Theta \\times A$. Let $D_{T}(g)=\\mathbb{E}_{\\bar{v}_{\\sigma^{\\prime}}^{T, 2}}[g]-\\mathbb{E}_{\\bar{v}_{\\sigma^{\\prime}}^{T, 1}}[g]=$ $\\frac{1}{T} \\sum_{t=1}^{T} \\mathbb{E}_{\\sigma^{\\prime}}\\left[g\\left(a_{t}, \\theta_{t}, \\theta_{t}^{\\prime}, \\omega, \\mu_{t+1}\\right)-g\\left(a_{t}, \\theta_{t}, \\theta_{t}^{\\prime}, \\omega, \\mu_{t}\\right)\\right]$. The argument in Lemma C. 2 implies that $D_{T}(g) \\rightarrow 0$ as $T \\rightarrow \\infty$ (this does not rely on the existence of a limit, just the convergence of beliefs and the continuity of $g$ ). Now, let $T_{n}$ be such that $\\bar{v}_{\\sigma^{\\prime}}^{T_{n}, 1} \\xrightarrow{w^{*}} \\tilde{v}$. Note that\n\n    $$\n    \\int g d \\bar{v}_{\\sigma^{\\prime}}^{T_{n}, 2}=\\int g d \\bar{v}_{\\sigma^{\\prime}}^{T_{n}, 1}+D_{T_{n}}(g) \\rightarrow \\int g d \\tilde{v}+0,\n    $$\n\n    so a subsequential limit of $\\bar{v}_{\\sigma^{\\prime}}^{T, 1}$ is a subsequential limit of $\\bar{v}_{\\sigma^{\\prime}}^{T, 2}$. Switching the role of 1 and 2, we obtain the opposite set inclusion.\n\n[^31]:    ${ }^{46}$ A set $X \\subseteq \\Delta(\\Omega)$ is open relative to $\\Delta(\\Omega)$ if an open set $Y \\subseteq \\mathbb{R}^{|\\Omega|}$ exists such that $X=Y \\cap \\Delta(\\Omega)$. The boundary relative to $\\Delta(\\Omega)$ is analogously defined via open sets relative to $\\Delta(\\Omega)$.\n    ${ }^{47}$ Lemma D. 1 provides an interval of radii $r \\in\\left(0, r_{0}\\right)$ such that $\\int_{B(\\hat{\\mu}, r)} U_{\\text {net }}(\\mu) \\tau_{(p, \\sigma)}(d \\mu)<0$. Note that only countable many such $r$ can have $\\tau_{(p, \\sigma)}(\\partial B(\\hat{\\mu}, r))>0$ (the boundaries for different radii are disjoint), so we can always pick $r$ such that $\\tau_{(p, \\sigma)}(\\partial B(\\hat{\\mu}, r))=0$ and preserve the negative sign.\n\n[^32]:    ${ }^{48}$ The property that $\\tau_{(p, \\sigma)}(\\partial B)=0$ ensures that $\\mathbb{1}\\left[\\mu_{t}\\left(h^{\\infty}\\right) \\in B\\right]$ is eventually constant almost surely. Let $E=\\left\\{h^{\\infty}: \\mu_{\\infty}\\left(h^{\\infty}\\right) \\notin\\right.$ $\\partial B\\}$. On $E$, either $\\mu_{\\infty}$ is in the interior of $B$ (relative to $\\Delta(\\Omega)$ ) or in the interior of $B^{\\complement}$ (relative to $\\Delta(\\Omega)$ ), that is, an $\\epsilon>0$ exists such that $\\left(B\\left(\\mu_{\\infty}, \\epsilon\\right) \\cap \\Delta(\\Omega)\\right) \\subset B$ or $B^{\\complement}$. In either case, for each $h^{\\infty}$, there exists $N\\left(h^{\\infty}\\right)$ such that for all $t \\geq N\\left(h^{\\infty}\\right)$, $\\mu_{t}\\left(h^{\\infty}\\right) \\in\\left(B\\left(\\mu_{\\infty}, \\epsilon\\right) \\cap \\Delta(\\Omega)\\right)$ and hence $\\mathbb{1}\\left[\\mu_{t}\\left(h^{\\infty}\\right) \\in B\\right]$ is eventually constant. When $\\tau_{(p, \\sigma)}(\\partial B)=0$, we have that $E$ has probability 1 under $\\mathbb{P}_{(p, \\sigma)}$.\n    ${ }^{49}$ Indeed, $\\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=1}^{T}\\left(u\\left(a_{t}, \\theta_{t}, \\omega\\right)-u\\left(a_{\\varnothing}, \\theta_{t}, \\omega\\right)\\right) \\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right]=\\mathbb{E}_{\\bar{v}_{(p, \\sigma)}^{T, 1}}\\left[\\left(u(a, \\theta, \\omega)-u\\left(a_{\\varnothing}, \\theta, \\omega\\right)\\right) \\mathbb{1}[\\mu \\in B]\\right]$ and our previous analysis implies it converges to $\\int_{B} U_{\\text {net }}(\\mu) \\tau_{(p, \\sigma)}(d \\mu)$.\n    ${ }^{50}$ Choose $\\bar{T}$ so that for all $T \\geq \\bar{T}$ :","text_sha256":"7e7b41a2c20a9202ec69fa1303568267a511b2366076b99f271177d8a9d88c04"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0060","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D.3.1 Revelation principle for limit of means preferences","text":"$$\n    \\begin{aligned}\n    & \\left|\\mathbb{E}_{(p, \\sigma)}\\left[\\frac{1}{T} \\sum_{t=1}^{T}\\left(u\\left(a_{t}, \\theta_{t}, \\omega\\right)-u\\left(a_{\\varnothing}, \\theta_{t}, \\omega\\right)\\right) \\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right]-\\int_{B} U_{\\text {net }}(\\mu) \\tau_{(p, \\sigma)}(d \\mu)\\right| \\\\\n    & +\\left|\\mathbb{E}\\left[-\\frac{1}{T} \\sum_{t=1}^{L-1}\\left(u\\left(a_{t}, \\theta_{t}, \\omega\\right)-u\\left(a_{\\varnothing}, \\theta_{t}, \\omega\\right)\\right) \\mathbb{1}\\left[\\mu_{t} \\in B\\right]\\right]\\right| \\leq \\delta / 2 .\n    \\end{aligned}\n    $$\n\n[^33]:    ${ }^{51}$ Anticipating our construction in item 3, for each belief $\\mu$ in the support of $\\tau$, the allocation rule $\\alpha^{\\prime}(\\cdot \\mid \\cdot, \\mu)$ has no profitable undetectable deviations relative to payoff function $u^{\\prime}(a, \\theta)=\\sum_{\\omega \\in \\Omega} \\mu(\\omega) u(a, \\theta, \\omega)$.\n\n[^34]:    ${ }^{52}$ Any $\\|.\\|_{p}$ would work, but larger $p$ results in weakly shorter adjustment phases.\n\n[^35]:    ${ }^{53}$ In the expressions that follow, recall the full participation mechanism $\\phi_{N}^{*}$ already averages over the agents' own randomization devices.\n\n[^36]:    ${ }^{54}$ To be sure, the proof of Lemma C. 1 is written in the context of repeated mechanisms but it extends verbatim to dynamic mechanisms with simple notational adjustments.\n\n[^37]:    ${ }^{55}$ To be sure,\n\n    $$\n    \\mathbb{E}_{\\sigma}\\left[g\\left(\\mu_{t}\\right)\\right]=\\int_{H^{\\infty}} g\\left(\\mu_{t}\\left(h^{\\infty}\\right)\\right) \\mathbb{P}_{\\sigma}\\left(d h^{\\infty}\\right)\n    $$\n\n[^38]:    ${ }^{56}$ The argument shows that the agent's equilibrium payoff is at least $U\\left(\\tau_{\\phi}\\right)$. However, it is immediate that $U\\left(\\tau_{\\phi}\\right)$ is the most the agent can make in the game as $\\tau_{\\phi}$ extracts all information from the mechanism.\n\n[^39]:    ${ }^{57}$ This almost surely is under the law of $A$ under $\\phi\\left(\\cdot \\mid m, \\tilde{\\omega}^{\\star}\\right)$.\n\n[^40]:    ${ }^{58}$ We index histories by the messages to distinguish these histories from those when the designer uses direct mechanisms.\n\n[^41]:    ${ }^{59}$ Recall the fictitious reports encode the agent's participation decisions in the original mechanism.\n    ${ }^{60}$ In fact, if we kept track of the designer's draws of fictitious messages, the new mechanism implements the same distribution over terminal histories $\\mathcal{H}_{M}^{\\infty}$, and a fortiori, its marginal over $\\Omega \\times(A \\times \\Theta)^{\\infty}$ is the same.\n\n[^42]:    ${ }^{61}$ Recall that to minimize notation and make history lengths symmetric across participation and nonparticipation, we record the agent's rejection of the mechanism as the empty message $\\varnothing$.\n    ${ }^{62}$ In fact, the construction ensures the stronger property that $\\mathbb{P}_{\\tilde{\\varphi},(\\tilde{p}, \\tilde{\\sigma})}^{T}=\\mathbb{P}_{\\varphi,\\left(p^{\\prime}, \\sigma^{\\prime}\\right)}^{T}$, when in a slight abuse of notation we keep track of the fictitious messages in $\\mathbb{P}_{\\tilde{\\varphi},(\\tilde{p}, \\tilde{\\sigma})}^{T}$.","text_sha256":"f7accc918e2f1b5514be8d5e0dd8e6bce8d31adca0e4b244c6ac9309cfdf3574"}
{"schema_version":"1.0","chunk_id":"alex-smolin:calibrated-mechanism-design:2026-02-18:0061","work_id":"alex-smolin:calibrated-mechanism-design","paper_id":"alex-smolin:calibrated-mechanism-design:2026-02-18","title":"Calibrated Mechanism Design","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2026-02-18","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md","source_record":"https://alexsmolin.com/#research","citation":"Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Laura Doval; Alex Smolin\n\n**Canonical citation:** Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md\n\n**Source record:** https://alexsmolin.com/#research\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"2e58794c7b91f630da642153d2f098693f95f55d5b33f9c5146c9b251a63bb47"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0001","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Shota Ichihashi; Alex Smolin.\n> Canonical citation: Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"e30e47303c98db5b60b663db222f7be475138be7acbb009bc85d0df56723142f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0002","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Buyer-Optimal Algorithmic Recommendations","text":"# Buyer-Optimal Algorithmic Recommendations\n\n**Authors:** Shota Ichihashi; Alex Smolin\n\n**Manuscript date:** 2025-02-15\n\n#### Abstract\n\nIn markets where algorithmic data processing is increasingly prevalent, recommendation algorithms can substantially affect trade and welfare. We consider a setting in which an algorithm recommends a product based on its value to the buyer and its price. We characterize an algorithm that maximizes the buyer's expected payoff and show that it strategically biases recommendations to induce lower prices. Revealing the buyer's value to the seller leaves overall payoffs unchanged while leading to more dispersed prices and a more equitable distribution of surplus across buyer types. These results extend to all Pareto-optimal algorithms and to multiseller markets, with implications for AI assistants and e-commerce ranking systems.\n\n[^0]","text_sha256":"dd4f80ede9eac4c7427a3da21b1f1720c3d3ccc308df5db74d2ffc6bdd551694"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0003","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nIn today's digital economy, algorithmic decision-making is transforming how consumers navigate markets-from price trackers that seek and pinpoint lower-priced products, to robo-advisors that propose financial securities, to e-commerce platforms where ranking algorithms shape which products capture buyers' attention, and to AI-powered personal assistants that advise individuals on their purchasing decisions. ${ }^{1}$ Algorithms are not merely processing information but are actively influencing consumer and seller behavior.\n\nIn this paper, we formalize this dynamic by developing a model of algorithmic recommendations and examining how they can reshape market outcomes and redistribute welfare. Our model emphasizes three key features of recommendation algorithms. First, they operate according to preprogrammed rules. Second, they are capable of locating and processing critical information about a product's existence, inherent value, and price. Third, by influencing when and how recommendations are made, these algorithms can affect the purchasing decisions by the buyers and pricing decisions by the sellers.\n\nWe begin by developing a baseline model of bilateral trade. In this setting, a single product is traded between a buyer and a seller in the presence of uncertainty about both trade costs and values. The seller privately observes her production cost-her type- while the buyer is initially uninformed about both the product's existence and its value. An algorithm, however, can discover the product's value or, equivalently, produce its best estimate and issue a recommendation based on this information combined with the seller's posted price. Upon receiving a recommendation, the buyer updates her belief and decides whether to purchase at the offered price; if no recommendation is received, no trade occurs- that is, the algorithm can serve as a gatekeeper. The seller, fully aware of the algorithm's design, sets her price strategically to maximize profit.\n\nThe algorithm shapes equilibrium trade and welfare. We abstract away from the source of the algorithm and instead analyze all algorithms with desirable welfare properties. Motivated by the fact that currently most revenue for AI assistants comes from\n\n[^1]their users, we present most of the analysis with the objective being the buyer's expected surplus, thus characterizing a buyer-optimal algorithm. However, our characterization and key results extend, with minimal changes, to any Pareto-optimal algorithm in the space of buyer and seller surpluses. Thus, our findings speak to a variety of market structures that include, for example, two-sided platforms aiming to accommodate both of their sides.\n\nA buyer-optimal algorithm must navigate a trade-off: It needs to reward sellers for offering lower prices by increasing the frequency of recommendations, while ensuring that beneficial trades are not foregone. We show that this balance is optimally achieved by an algorithm that uses a pseudo value threshold-recommending the product when this adjusted metric exceeds the price, even if the true value might not. The equilibrium we characterize (Proposition 1) reveals three key insights. First, although the algorithm's design and the equilibrium prices depend on both the cost and value distributions, the final product allocation is driven solely by the cost distribution. Second, by deliberately deviating from a simple ex post optimal rule-eschewing recommendations at high prices even when the true value is favorable and endorsing products at low prices even when the value falls short-the algorithm increases the buyer's price sensitivity, thereby pressuring the seller to offer lower prices. ${ }^{2}$ Third, compared to standard monopoly pricing with ex post optimal trade, buyer-optimal algorithmic recommendations substitute high-value, high-cost trades with low-value, low-cost trades (Proposition 2).\n\nThe complete characterization of an optimal algorithm and the equilibrium strategies allows us to uncover major changes in the welfare implications of third-degree price discrimination compared to standard monopoly pricing. We show that if the seller can distinguish and price discriminate among different buyer segments, the algorithm adapts accordingly and fully mitigates the average effects of market segmentation. As a result, price discrimination does not affect the average price, buyer's total surplus, seller profits, or product allocation (Proposition 3). At the same time, finer segmentation enables the seller to correlate prices with values, and within a class of monotone segmentations, it\n\n[^2]leads to more dispersed equilibrium prices and a more concentrated-and thus more equitable-distribution of consumer surplus (Proposition 4). These results hold broadly and demonstrate that algorithmic recommendations can serve as a powerful consumer protection tool, complementing existing regulatory methods (e.g., Scott Morton et al. (2019)).\n\nOur analysis extends in several dimensions. First, our characterization and market segmentation insights extend to any Pareto-optimal algorithm by simply incorporating Pareto weights into the definition of a pseudo-value (Proposition 5). Second, most of our key results and characterizations generalize to settings with multiple competing sellers (Proposition 6). Notably, despite strategic competition among sellers, the specifics of market segmentation do not affect the average price, total buyer surplus, or seller profits (Proposition 7), and finer segmentation leads to more dispersed prices (Proposition 8). Third, we expand our analysis by allowing the buyer to be informed in advance about either the product's existence or its value, showing how this information can potentially harm the buyer and discussing how algorithm design can mitigate this harm (Proposition 9 and Proposition 10).","text_sha256":"0e618ea445e2ce170088e1e8bd88a3034dd44477a2e123c136dfbd61c083c293"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0004","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"Related literature.- Our paper is closest to the recent strand of economic literature that examines methods of empowering buyers in monopolistic settings via information control. Roesler and Szentes (2017) analyze buyer-optimal learning in a bilateral trade setting. Like us, they show that the buyer benefits from ex post imperfect decisions to influence the seller's pricing; that is, full learning about the value is not optimal. Unlike us, they require learning to occur before the price is set, which limits its impact (see further Section 3.1 and Section 5.3). Deb and Roesler (forth.) extend this analysis to the case of a multiproduct monopoly and Bergemann et al. (2023) to auctions; Condorelli and Szentes (2020) analyze the buyer-optimal distribution of values within a given interval. We contribute to this literature by allowing the buyer's information to depend on the price and by allowing the seller to have private information. ${ }^{3}$\n\n[^3]In our study of algorithm design under sellers' private information, we integrate Bayesian persuasion (e.g., Kamenica and Gentzkow (2011)) with mechanism design (Baron and Myerson (1982)). Several works have combined these frameworks in trade settings. Among these, the most closely related studies are Yang (2022), Bergemann and Bonatti (2024), Xu and Yang (2024), and Bergemann et al. (forth.). ${ }^{4}$ We share the perspective that new information technologies enable powerful third parties to process data, extract valuable insights, and shape market outcomes by serving as informational gatekeepers. However, our focus differs in that those studies allow the intermediary to charge transfers and study revenue maximization, whereas we focus on achieving Pareto efficiency. We further differ from Bergemann and Bonatti (2024) and Bergemann et al. (forth.) in the nature of the friction faced by the designer: Whereas they contend with buyers' off-site purchase opportunities, we focus on the private information held by sellers. In this respect, we are closer to Yang (2022) and Xu and Yang (2024), sharing some features of the solution with them (Remark 2).\n\nOur analysis in Section 4 offers a novel perspective on the classic question of the impact of price discrimination based on consumer information, as studied in the market segmentation literature (e.g., Bergemann et al. (2015) and Haghpanah and Siegel (2023)). We show that consumer use of algorithms may introduce a new welfare implication whereby price discrimination results in a more equal distribution of consumer surplus without affecting average welfare outcomes. This finding contributes to the recent literature that explores ways to promote equality and fairness through mechanism design (Kleinberg et al. (2018), Dworczak et al. (2021), Akbarpour et al. (2024)) or information design (Doval and Smolin (2024)).\n\nFinally, our work contributes broadly to recent research strands in the economics of algorithmic decision-making. A significant body of work has examined algorithmic pricing, largely focusing on how such tools bolster seller power. Empirical studies-such as those by Calvano et al. (2020), Asker et al. (2022), and Assad et al. (2024)-document how algorithms facilitate collusion and dynamic pricing, while theoretical work (e.g., Sal-\n\n[^4]cedo (2015), Lamba and Zhuk (2023)) and combined empirical-theoretical approaches (e.g., Brown and MacKay (2023), Johnson et al. (2023)) investigate their strategic impact. In contrast, we explore how algorithmic recommendations can empower buyers by mitigating information asymmetries and counteracting traditional price discrimination. This buyer-centric perspective resonates with early insights from Galbraith (1952), offering an alternative approach to balancing market power. ${ }^{5,6}$","text_sha256":"7987b018c82181fcbbfcf328980e24dd8108a9ff4c28c59b20659dcec9d68689"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0005","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Baseline Model","text":"## 2 Baseline Model\n\nThere is a buyer and a seller. The buyer may purchase a product but initially knows neither the existence nor the value of the product: The value is denoted by $v$ and is distributed according to distribution $G$ with positive density $g$ over its support [0, 1]. A recommendation algorithm or simply algorithm provides the buyer with information about the product by means of recommendations. ${ }^{7}$ The algorithm is characterized by a function $r:[0,1] \\times \\mathbb{R}_{+} \\rightarrow[0,1]$ such that for any pair $(v, p)$ of a realized value $v \\in[0,1]$ and a product price $p \\in \\mathbb{R}_{+}$, the algorithm recommends that the buyer purchase the product with probability $r(v, p) .{ }^{8}$\n\nThe seller can produce one unit of a product at cost $c$, which is her private type. The type distribution $F$ has support $[0,1]$ with positive density $f$.\n\n[^5]Timing For any given algorithm, which is commonly known to both players, the timing of the game is as follows. First, nature draws the seller's type $c$ and the buyer's value $v$. Second, the seller privately observes her type $c$ but not value $v$, and posts a price, $p$. With probability $1-r(v, p)$, the algorithm does not recommend the product, in which case trade does not occur and the game ends. With probability $r(v, p)$, the algorithm recommends the product to the buyer, in which case the buyer observes the recommendation and the price, and then decides whether to buy the product. If trade occurs, the buyer and seller obtain ex post payoffs $v-p$ and $p-c$, respectively. Otherwise, both players obtain zero payoffs.\n\nGiven the algorithm, the solution concept is a perfect Bayesian equilibrium. If the product is recommended, the buyer updates the expected value of the product to\n\n$$\n\\mathbb{E}[v \\mid \\text { recommended, } p]=\\frac{\\int_{0}^{1} x r(x, p) g(x) \\mathrm{d} x}{\\int_{0}^{1} r(x, p) g(x) \\mathrm{d} x},\n$$\n\nand then purchases the product whenever this value weakly exceeds the price. A pair of an algorithm and a buyer's strategy induces a demand curve, which maps each price to a probability of trade. In equilibrium, each seller type takes this demand curve as given and chooses a price that maximizes her expected profit.\n\nObjective We call the buyer's ex ante expected payoff buyer surplus and the ex ante seller's expected payoff seller profit. An algorithm attains a given buyer surplus if this buyer surplus arises in an equilibrium under this algorithm. For most of the analysis, we focus on the recommendation algorithms that maximize buyer surplus. However, in Section 5.1, we show that the same analysis readily applies to the whole class of Pareto-optimal algorithms.\n\nDefinition 1 (Buyer-Optimal Algorithm). A recommendation algorithm is buyeroptimal if it attains a greater buyer surplus than any other recommendation algorithm.\n\nIn what follows, it will be useful to distinguish between seller types who trade and those who do not under a given algorithm and their posted prices. Given an algorithm\nand an equilibrium, we say that a price is active if it results in a strictly positive trade probability and is inactive otherwise. Similarly, we say that a type is active if she posts an active price with a strictly positive probability and is inactive otherwise.","text_sha256":"bd9ca5eb1a2def982c4e5f6e530b6a37b85354d4c4e80ef134ccba3296727c1e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0006","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Buyer-Optimal Algorithm","text":"## 3 Buyer-Optimal Algorithm\n\nIn this section, we characterize the buyer-optimal algorithm. We say that an algorithm $r$ is a threshold algorithm if there exists a threshold function $\\hat{v}: \\mathbb{R}_{+} \\rightarrow[0,1]$ such that $r(v, p)=\\mathbb{1}(v \\geq \\hat{v}(p))$, i.e., the algorithm recommends the product with probability 1 if the value exceeds a price-dependent threshold and with probability 0 otherwise.\n\nLemma 1 (Threshold Algorithms). For any algorithm $r$, there exists a threshold algorithm under which the buyer follows the recommendations and that yields a weakly greater buyer surplus than $r$ and the same seller profit as $r$.\n\nThe proofs of this and all other results are in the Appendix. Lemma 1 shows that threshold algorithms span a Pareto frontier in the space of buyer surplus and seller profit. Intuitively, the buyer can be set to follow the recommendations because the algorithm can anticipate and mimic the buyer's response. In turn, the Pareto efficiency of threshold algorithms follows from the observation that each seller type is concerned solely with trade volume whereas the buyer surplus is maximized when the higher values are prioritized. Consequently, if a buyer-optimal algorithm exists, then it can be found in the class of threshold algorithms, and in what follows, we focus on threshold algorithms.\n\nThe optimal choice of a threshold function must balance the trade-off between maximizing the trade surplus and incentivizing the seller to lower the price. One natural option is to set $\\hat{v}(p)=p$ so that the product is recommended if and only if the value exceeds the price. This ex post optimal algorithm maximizes the buyer's payoff given fixed prices. However, the algorithm fails to maximize buyer surplus because it underuses the opportunity to dampen equilibrium prices.","text_sha256":"9bd16d1be61fde1a6de4949063d9ebf941d266b4ee82a20a67301b5947aef57b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0007","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.1 Known Seller Cost","text":"### 3.1 Known Seller Cost\n\nThe simplest case to illustrate the power of algorithmic recommendations is to consider a limit case of our model in which the seller cost is commonly known.\n\nClaim 1 (Known Cost). Suppose the seller cost is commonly known to be $c \\in[0,1]$. Then an algorithm that employs a threshold function $\\hat{v}$ defined by $\\hat{v}(c)=c$ and $\\hat{v}(p)>1$ for $p \\neq c$ is optimal and extracts the full surplus from the seller.\n\nThe condition $\\hat{v}(p)>1$ for $p \\neq c$ effectively forces the seller to set the price equal to the marginal cost, thereby eliminating any seller rents, while $\\hat{v}(c)=c$ ensures that trade is efficient and generates maximal surplus. In equilibrium, the entire surplus is generated and appropriated by the buyer, which confirms the algorithm's optimality.\n\nClaim 1 highlights a key distinction between our setting and Roesler and Szentes (2017). In their model, the cost is commonly known, but the algorithm cannot condition information on price nor exclude the seller from trade. Consequently, even though equilibrium trade is efficient, the seller earns positive rents. In contrast, in our model, the algorithm can condition information on price and exclude the seller from trade by not informing the buyer of the existence of the product. The combination of these properties enables the buyer to extract full surplus when the seller's cost is known.\n\nIn Section 5.3.1, we elaborate on the roles of price dependence and gatekeeping: When cost uncertainty is low, gatekeeping is particularly valuable, while price dependence is not, because the algorithm effectively provides information at only a single price. Conversely, when cost uncertainty is high-and we provide precise distributional conditions under which this holds-price dependence is as effective whether or not the buyer could purchase the product without the recommendation.\n\nWhile the optimal algorithm in Claim 1 is simple, it relies on precise knowledge of seller cost and is fragile to small perturbations: Even an arbitrarily small increase in cost would lead to a complete collapse of trade. In practice, the algorithm may lack the information necessary to identify seller cost. Moreover, this asymmetry can be exacerbated by idiosyncratic cost shocks arising from supply disruptions or alternative\nsale opportunities. These observations motivate our subsequent analysis of private seller information.","text_sha256":"35d61a8530d0aebaded8846a4940be21b45bf62e9ca33b665d7d238781be244d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0008","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.2 General Case","text":"### 3.2 General Case\n\nTo find an optimal algorithm in the general case of an unknown seller cost, we build on Lemma 1 and frame the designer's problem as a nonlinear screening problem in which the recommendation threshold responds to the price. The choice of a threshold at any given price simultaneously determines the expected trade surplus, which is valued by the buyer, and the expected trade volume, which is valued by the seller. We recover the optimal threshold function by adapting the seminal analysis of Baron and Myerson (1982) and confirm that, with this algorithm, the buyer indeed finds it optimal to follow the recommendations.\n\nThe optimal algorithm and equilibrium pricing are easier to describe not in terms of price-dependent thresholds for values but in terms of value-dependent thresholds for prices. Specifically, denote the virtual cost function by $\\gamma(c) \\triangleq c+F(c) / f(c)$, and assume that it is continuous and strictly increasing on $[0,1] .{ }^{9}$ For each $v \\in[0,1]$, define the buyer's pseudo value as\n\n$$\ny(v) \\triangleq \\mathbb{E}_{\\tilde{v} \\sim G}\\left[\\gamma^{-1}(\\tilde{v}) \\mid \\tilde{v} \\geq v\\right] .\n$$\n\nThe pseudo value $y(v)$ is an increasing function of $v$. When the true value is sufficiently low, close to 0, the pseudo value is higher than the true value, $y(v)>v$, because $y(0)=\\mathbb{E}_{\\tilde{v} \\sim G}\\left[\\gamma^{-1}(\\tilde{v})\\right]>0$. When the true value is sufficiently high, close to 1 , the pseudo value is below the true value, $y(v)<v$, because $y(1)=\\gamma^{-1}(1)<1$. Define $\\bar{c} \\triangleq \\gamma^{-1}(1) .^{10}$\n\n[^6]Proposition 1 (Buyer-Optimal Algorithm). A buyer-optimal algorithm recommends the product if and only if $y(v) \\geq p$. Under this algorithm, the seller of type $c \\leq \\bar{c}$ posts price $p^{*}(c)=y(\\gamma(c))$, and the seller of type $c>\\bar{c}$ is inactive. Under this algorithm and pricing, the trade occurs if and only if $v \\geq \\gamma(c)$.\n\nProposition 1 reveals two notable features. First, the impact of the value and cost distributions can be decoupled: Even though the optimal algorithm and the equilibrium prices depend both on the cost and value distributions, the optimal product allocation depends only on the cost distribution because the product is traded if and only if the value exceeds the virtual cost. This feature is crucial for the market segmentation results in Section 4.\n\nSecond, the buyer-optimal algorithm makes two types of ex post errors: When the true value is sufficiently high, the pseudo value is below the true value, and the optimal algorithm never recommends the product when the value is below the price. At the same time, for seller types that satisfy $y(v)<y(\\gamma(c))<v$, the algorithm does not recommend the product even though the value exceeds the price. In contrast, when the true value is sufficiently low, the pseudo value is higher than the true value and the algorithm always recommends the product when the value exceeds the price. However, for seller types that satisfy $v<y(\\gamma(c))<y(v)$, the algorithm recommends the product even though the value is below the price. As a result, compared with the ex post optimal algorithm, the buyer-optimal algorithm overrecommends at low prices and underrecommends at high prices. These distortions benefit the buyer because they incentivize the seller to set lower prices.\n\nExample 1 (Uniform). We illustrate the buyer-optimal algorithm when $c$ and $v$ are uniformly distributed on [0, 1]. In this case, the virtual cost is $\\gamma(c)=2 c$; the pseudo value is $y(v)=\\mathbb{E}_{\\tilde{v} \\sim U[0,1]}\\left[\\left.\\frac{\\tilde{v}}{2} \\right\\rvert\\, \\tilde{v} \\geq v\\right]=(1+v) / 4$; and the associated threshold function $\\hat{v}(p)$ is equal to 0 for $p<1 / 4$, to $4 p-1$ for $p \\in[1 / 4,1 / 2]$ and to 1 for $p>1 / 2$. The equilibrium price posted by active type $c$ is $p^{*}(c)=y(\\gamma(c))=(1+2 c) / 4$ for $c \\in[0,1 / 2]$. Types $c>1 / 2$ are inactive and post, for example, $p^{*}(c)=1 / 2$. A buyer who receives a recommendation to purchase at price $p \\in[1 / 4,1 / 2]$ infers that the product's expected\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: Optimal recommendation algorithm (left) and the resulting equilibrium pricing strategy and trade region (right). $v \\sim U[0,1], c \\sim U[0,1]$.\n\nvalue is $(4 p-1+1) / 2=2 p>p$ and is thus strictly willing to purchase it.\nThe left side of Figure 1 depicts the optimal recommendation threshold (solid line) along with the ex post optimal recommendation threshold (dashed line). As we discussed above, the ex ante optimal algorithm is suboptimal ex post in two ways: If the product price is low, i.e., $p<1 / 3$, it recommends the product even when the value is below the price; if the product price is high, i.e., $p>1 / 3$, the algorithm does not recommend the product even when the value is above the price.\n\nThe right side of Figure 1 depicts the resulting equilibrium pricing and trade: The price $p^{*}(c)$ posted by the seller of type $c$, the region of values and types in which the trade occurs (filled area), and the efficient trade region (area encircled by dashed lines). In accordance with Proposition 1, under an optimal algorithm, trade occurs whenever the buyer value is greater than the seller's virtual cost. Type $c=0$ always trades; all higher types post progressively higher prices and serve progressively fewer buyers. Types $c>1 / 2$ never trade. Equilibrium active prices span the interval [1/4, 1/2].\n\nWe compare the equilibrium outcomes under the buyer-optimal algorithm and the ex post optimal algorithm. Under the ex post optimal algorithm, the seller's problem is a standard monopoly problem with the optimal price $p^{m}(c)=\\frac{1+c}{2}$, and trade occurs if\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 2: Comparison of the equilibrium outcomes under the ex ante optimal algorithm and the ex post optimal algorithm in terms of the seller's pricing strategies (left) and trade regions (right).","text_sha256":"4cfc0727609d593b27e385eefa038b4f3e2f18bc274b22098f20ee3c9116538d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0009","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.2 General Case","text":"and only if $v \\geq p^{m}(c)$. As indicated by the left panel of Figure 2, as we move from the ex post optimal algorithm to the buyer-optimal algorithm, each seller type $c \\leq 0.5$-who transacts with a positive probability under both algorithms-decreases its price by 0.25; thus, the buyer-optimal algorithm induces a uniform downward shift of the equilibrium prices across all seller types. (In this example, the price change is associated with the uniform decrease of the seller profits: The interim profit for seller type $c$ under the ex post optimal algorithm is $\\frac{(1-c)^{2}}{4}$, whereas that under the buyer-optimal algorithm is $\\frac{(1-2 c)^{2}}{4}$ if $c \\leq \\frac{1}{2}$ and 0 otherwise. However, in general, some seller types may prefer the buyer-optimal algorithm due to the overrecommendations.)\n\nComplementary, the right panel of Figure 2 depicts the sets of pairs of value $v$ and cost $c$ under which transaction occurs. Under the ex post optimal algorithm, transaction occurs when $v \\geq p^{m}(c)$. The figure indicates that compared to the ex post optimal algorithm, the buyer-optimal algorithm reduces transactions between high-value buyers and high-cost sellers (the red region) and increases transactions between low-value buyers and low-cost sellers (the blue region). $\\square$\n\nThe last observation of the previous example turns out to be more general. To describe the result, let $\\Delta^{*}$ denote the set of $(c, v)$ pairs such that trade occurs under the buyer-optimal algorithm, and let $\\Delta^{m}$ denote the set of $(c, v)$ pairs such that trade occurs under the ex post optimal algorithm.\n\nProposition 2 (Allocation Substitution). If $F$ has a strictly decreasing reversed hazard rate $\\frac{f}{F}$ and $G$ has a strictly increasing hazard rate $\\frac{g}{1-G}$, then for any $(c, v) \\in$ $\\Delta^{*} \\backslash \\Delta^{m}$ and $\\left(c^{\\prime}, v^{\\prime}\\right) \\in \\Delta^{m} \\backslash \\Delta^{*}$ such that $(c, v) \\neq\\left(c^{\\prime}, v^{\\prime}\\right)$, we have $c<c^{\\prime}$ and $v<v^{\\prime}$.\n\nProposition 2 shows that in regular environments, switching from monopoly pricing to algorithmic recommendations leads to a systematic shift in equilibrium trade: Highvalue, high-cost transactions are replaced by low-value, low-cost ones. Intuitively, algorithmic recommendations push prices down by under-recommending high-value products and over-recommending low-value ones. This, in turn, reduces trade with high-cost sellers while increasing trade with low-cost sellers.\n\nRemark 1 (Comparison to Full Commitment Benchmark). As our proof reveals, the optimal algorithm in Proposition 1 attains the same outcome as when the buyer is aware of the product's existence, can observe the product's value, and has full commitment power over purchasing decisions and monetary transfers at different product values. As a result, even though the algorithm serves only information, it effectively transfers market power from the seller to the buyer. This observation has two consequences. First, the same algorithm remains optimal in the case of fully automated trade, i.e., if it could execute transactions without having the buyer in a loop, or if the algorithm could charge monetary transfers to the seller, e.g., referral or commission fees. Second, the same outcome would be optimal even if the seller could employ more general trade protocols than a posted price. In that case, a buyer-optimal algorithm would recommend products sold via posted prices according to the characterization in Proposition 1 and would never recommend products sold via alternative protocols.\n\nRemark 2 (Optimal Product Allocation). The findings in Proposition 1 are related to those of Yang (2022) and Xu and Yang (2024). They study revenue-maximizing\nintermediation by a platform that can assess consumer value and can charge payments to both sides but is uncertain about seller costs. Our paper and their papers differ in the designer's objective, the proof techniques, and the optimal mechanisms. At the same time, the equilibrium product allocations coincide-the good is traded if and only if the value exceeds virtual costs. This outcome is not obvious a priori and suggests that a revenue-maximizing platform would be more aligned, in terms of the resulting product allocation, with the side about which it has more information.","text_sha256":"e32adaad19e671d75426356909a592dc96ce08b72ad6e23d4fe99dd878331e12"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0010","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Algorithm Design and Market Segmentation","text":"## 4 Algorithm Design and Market Segmentation\n\nAlgorithmic recommendations have major consequences for third-degree price discrimination. To demonstrate this, we allow the seller to observe and base prices on signal $\\mathcal{I}=(S, \\pi)$ informative about the buyer's value. The signal consists of a set $S$ of signal realizations $s$ and a family of probability distributions $\\{\\pi(\\cdot \\mid v)\\}_{v \\in[0,1]}$ over $S$. We write $\\hat{\\pi}(s)$ for the marginal probability of signal realization $s \\in S$ and $G_{s}$ for the posterior value distribution conditional on $s$. Each signal can be viewed as a market segmentation, with $\\hat{\\pi}$ capturing the relative frequency of buyer segments and $G_{s}$ capturing the distribution of buyer values within each segment (cf. Bergemann et al. (2015)). The signal is exogenous, and the signal realization is independent of the seller's type.\n\nWe assume that the algorithm can perfectly distinguish different market segments so that the cost information remains the only private information of the seller. ${ }^{11}$ As the seller can set different prices in different segments, the optimal algorithm's recommendations should depend on the segment as well. In fact, the buyer-optimal segmentdependent algorithm must be buyer-optimal in each segment and is thus characterized in each segment $s$ by Proposition 1, with the value distribution being $G_{s}:{ }^{12}$ For each $s \\in S$, the optimal algorithm recommends the product if and only if the corresponding\n\n[^7]pseudo value exceeds the price, i.e.,\n$$\ny_{s}(v) \\triangleq \\mathbb{E}_{\\tilde{v} \\sim G_{s}}\\left[\\gamma^{-1}(\\tilde{v}) \\mid \\tilde{v} \\geq v\\right] \\geq p .\n$$\nIn equilibrium, the seller with type $c$ posts a price of\n$$\np_{s}^{*}(c) \\triangleq y_{s}(\\gamma(c))=\\mathbb{E}_{\\tilde{v} \\sim G_{s}}\\left[\\gamma^{-1}(\\tilde{v}) \\mid \\tilde{v} \\geq \\gamma(c)\\right]\n$$\nand transacts whenever $v \\geq \\gamma(c)$.\nImportantly, in contrast to the recommendation function or the seller's pricing, the product allocation, when viewed as a function of value and cost, is the same across all segments. This enables us to derive sharp implications of finer market segmentation. For any active type $c$, we define the distribution of prices of type $c$ as the distribution of $p_{s}^{*}(c)$ when we fix $c$ and draw $s$ from distribution $\\hat{\\pi}$. The corresponding profit of type $c$ is\n$$\n\\pi(c) \\triangleq \\mathbb{E}_{s, v}\\left[\\left(p_{s}^{*}(c)-c\\right) \\mathbb{1}\\left(y_{s}(v) \\geq p_{s}^{*}(c)\\right) \\mid c\\right],\n$$\nwhere the expectation is taken with respect to $v \\sim G$ and $s \\sim \\pi(\\cdot \\mid v)$.\nProposition 3 (Segmentation Neutrality). For any signal $\\mathcal{I}$ available to the seller, the buyer-optimal algorithm induces the same ex post product allocation, the same expected price and profit of each seller type, and the same ex ante buyer surplus.\n\nProof Outline. Consider the buyer-optimal algorithms with and without the seller's signal $\\mathcal{I}$. Under either algorithm, in equilibrium, the trade occurs if and only if $v \\geq \\gamma(c)$, and the highest type, $c=1$, earns zero profit as she never trades. We can view the two algorithms as indirect mechanisms that induce the same allocation rule and yield the same profit for type $c=1$ when the seller prices in an incentive-compatible way. A version of the revenue equivalence theorem (Lemma 2 in the appendix) implies that the individual profit of each seller type with and without the seller's signal $\\mathcal{I}$ must coincide. Consequently, the buyer surplus, which is the total surplus minus the seller's ex ante\nprofit, is also identical in the two settings. $\\square$\n\nProposition 3 establishes in a stark manner that no seller types benefit from having more information about the buyer value, and the buyer neither benefits from nor is harmed by the release of such information on average, as long as this release is accounted for in the algorithm design.\n\nDespite the neutral aspects highlighted by Proposition 3, a change in market segmentation does affect the optimal algorithm, the equilibrium pricing, and the distribution of payoffs across buyers with different valuations. To analyze the redistribution effect, we define the buyer surplus at value $v$ and type $c, w(v, c)$, as the equilibrium expected payoff of the buyer conditional on his value being $v$ and the seller's type being $c$ :\n\n$$\n\\left.w(v, c) \\triangleq \\mathbb{E}_{s}\\left[\\left(v-p_{s}^{*}(c)\\right) \\mathbb{1}\\left(y_{s}(v)\\right) \\geq p_{s}^{*}(c)\\right) \\mid v\\right],\n$$\n\nwhere the expectation is taken with respect to signal realization $s \\sim \\pi(\\cdot \\mid v)$ conditional on value $v$. Similarly, we define the distribution of buyer surplus at type $c$ as the distribution of $w(v, c)$ with $v \\sim G$. Furthermore, we will obtain a cleaner characterization and stronger results for the natural class of monotone partitional signals.\n\nDefinition 2 (Monotone Partitional Signal). A signal $\\mathcal{I}$ is monotone partitional if there exists a finite partition of $[0,1]$ into intervals $\\left\\{I_{1}, \\ldots, I_{n}\\right\\}$ such that for each interval $I_{k}$ and each $v \\in I_{k}, \\pi(\\cdot \\mid v)$ assigns probability 1 to either (i) $s=v$ or (ii) $s=k$.\n\nUnder a monotone partitional signal, each market segment is either a singleton or an interval. Furthermore, different buyer values belong to different segments; thus, the segment dependency in the description of an algorithm, which already conditions on the value, is redundant. The buyer-optimal algorithm and equilibrium pricing can be described segment by segment. Consider a segment $[\\underline{v}, \\bar{v}]$, which either equals $I_{k}$ for some $k$ or satisfies $\\underline{v}=\\bar{v}$. By Proposition 1, for all $v \\in[\\underline{v}, \\bar{v}]$, the buyer-optimal algorithm recommends the product if and only if the pseudo value $y(v)=\\mathbb{E}_{\\tilde{v} \\sim G}\\left[\\gamma^{-1}(\\tilde{v}) \\mid \\bar{v} \\geq \\tilde{v} \\geq v\\right]$ is above the price. Given this algorithm, in the segment $[\\underline{v}, \\bar{v}]$, the seller of type $c$ posts\na price","text_sha256":"abfab39b293de2dd113e36e44e10c4e7cd3a9fae2ea61a7e66f068dc94f357b0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0011","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Algorithm Design and Market Segmentation","text":"$$\np_{[\\underline{v}, \\bar{v}]}^{*}(c)= \\begin{cases}\\mathbb{E}_{\\tilde{v} \\sim G}\\left[\\gamma^{-1}(\\tilde{v}) \\mid \\bar{v} \\geq \\tilde{v} \\geq \\gamma(c)\\right] & \\text { if } \\quad \\underline{v} \\leq \\gamma(c) \\leq \\bar{v}, \\\\ \\mathbb{E}_{\\tilde{v} \\sim G}\\left[\\gamma^{-1}(\\tilde{v}) \\mid \\bar{v} \\geq \\tilde{v} \\geq \\underline{v}\\right] & \\text { if } \\quad \\gamma(c)<\\underline{v} .\\end{cases}\n$$\n\nIn particular, if the buyer value is revealed to be $v$ (i.e., $\\underline{v}=\\bar{v}=v$ ), the algorithm recommends the product if and only if the price is below $\\gamma^{-1}(v)$, and any seller with a cost below this value posts price $\\gamma^{-1}(v)$.\n\nProposition 4 (Segmentation Redistribution). If signal $\\mathcal{I}_{H}$ is Blackwell more informative than $\\mathcal{I}_{L}$, then the distribution of prices set by each seller type under $\\mathcal{I}_{H}$ is a mean-preserving spread of that under $\\mathcal{I}_{L}$. Furthermore, if signals $\\mathcal{I}_{H}$ and $\\mathcal{I}_{L}$ are monotone partitional, then the distribution of individual buyer surplus at any seller type under $\\mathcal{I}_{H}$ is a mean-preserving contraction of that under $\\mathcal{I}_{L}$.\n\nWhen the seller faces a more informative signal, the posterior beliefs on values conditional on signal realizations and the event $v \\geq \\gamma(c)$ become a mean-preserving spread of the posterior beliefs when the seller faces a less informative signal. By Equation 2, the equilibrium prices are linear in these beliefs; thus, the prices undergo a mean-preserving spread as the signal becomes more informative.\n\nTo gain intuition about the buyer surplus at different values, compare the seller who has no information and the seller who perfectly observes the value. When the seller has no information, the seller of type $c$ posts a price of $\\mathbb{E}_{\\tilde{v} \\sim G}\\left[\\gamma^{-1}(\\tilde{v}) \\mid \\tilde{v} \\geq \\gamma(c)\\right]$ regardless of the value, and any buyer with value $v \\geq \\gamma(c)$ trades at that price. When the seller has full information, the price depends on the segment, and for the buyer with value $v$, the seller of type $c$ posts a price of $\\gamma^{-1}(v)$. The buyer trades at this price, which is increasing in value. As a result, the seller's information increases the prices set for buyers with higher values, whereas the average remains the same by the first part of the argument. Consequently, the seller's information decreases the individual surplus of buyers with high values and increases the individual surplus of buyers with low values, leading to a more equalized surplus distribution.\n\nThe intuition behind the general monotone partitional signals is similar. In fact, the proof of Proposition 4 establishes an additional result: For any monotone partitional signal $\\mathcal{I}$ and type $c$, a cutoff $v(c, \\mathcal{I})$ exists such that the buyer surplus at value $v$ and type $c$ is greater with signal $\\mathcal{I}$ than under no information if and only if $v \\leq v(c, \\mathcal{I}) .{ }^{13}$\n\nExample 1 (Continued). Let $v$ and $c$ be uniformly distributed on [ 0,1 ]. Suppose that the seller has access to a monotone partitional signal with a uniform grid, i.e., each $I_{k}$ is an interval between $\\frac{k-1}{n}$ and $\\frac{k}{n}$. All values in interval $I_{k}$ are pooled into signal $k$. Let $I(v)$ denote the interval to which value $v$ belongs.\n\nWe fix any type $c<\\frac{1}{2}$ and examine how the equilibrium price and buyer surplus at $v$ and $c$ vary across $v \\geq 2 c$. Let $k=1, \\ldots, n$ satisfy $2 c \\in I_{k}$. If $v \\in I_{m}$, the seller will observe signal realization $m$ and infer that the value is uniformly distributed between $\\frac{m-1}{n}$ and $\\frac{m}{n}$. As a result, the price posted by type $c$ against value $v$ is\n\n$$\np(c, v)=\\mathbb{E}_{\\tilde{v} \\sim G}\\left[\\left.\\frac{\\tilde{v}}{2} \\right\\rvert\\, \\tilde{v} \\geq 2 c, \\tilde{v} \\in I(v)\\right]= \\begin{cases}\\frac{2 c n+k}{4 n}, & \\text { if } v \\in I_{k}, \\\\ \\frac{2 m-1}{4 n}, & \\text { if } v \\in I_{m}, m>k .\\end{cases}\n$$\n\nThe buyer's surplus at $v \\geq 2 c$ is $w(v, c)=v-p(c, v)$.\nFigure 3 depicts the optimal algorithm thresholds and the buyer surplus $\\mathbb{E}_{c \\sim F}[w(\\cdot, c)]$ at different values for the uninformative signal, the signal represented by binary partition $\\{[0,0.5],(0.5,1]\\}$ and the fully informative signal. As the market segmentation becomes finer, the equilibrium price responds more to the buyer value, which redistributes the surplus from higher to lower values. As a result, the finer market segmentation, or more buyer information provided to the seller, makes the distribution of buyer surplus more equalized across different values.\n\nProposition 4 generalizes this observation for Blackwell-comparable market segmen-\n\n[^8]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 3: Buyer-optimal algorithm thresholds (left) and buyer surplus at different values (right). Computed at no segmentation (solid), binary partition (dashed), and full segmentation (dotted). $v \\sim U[0,1], c \\sim U[0,1]$.\n\ntations, which, in the case of uniform monotone partitions, correspond to partitions with $n_{H}$ and $n_{L}$ elements such that $n_{H} / n_{L} \\in \\mathbb{N}$. $\\square$\n\nAn important takeaway from our analysis in this section is that consumers can win the technological race against sellers. Advances in information technology not only drive the proliferation of algorithmic recommendations but also enable sellers to engage in third-degree price discrimination. Our neutrality results show that the benefits of buyer-optimal algorithmic consumption can fully offset the harm caused by price discrimination. However, the choice of the algorithm is crucial for this, e.g., an ex-post optimal algorithm could erode all buyer welfare under perfect price discrimination.","text_sha256":"9d5e327de2d3f4aeae7f2a7819ac185097f539f8d62af3da3372bdcb2de2b7f5"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0012","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Extensions","text":"## 5 Extensions","text_sha256":"fc0645123651211059b3c23002fdc6f773f706f6659046b0fd5667fd1c584177"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0013","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.1 Pareto-Optimal Algorithmic Recommendations","text":"### 5.1 Pareto-Optimal Algorithmic Recommendations\n\nThus far, we have focused on buyer-optimal algorithmic recommendations. A natural and broader question is what algorithmic recommendations are Pareto optimal, i.e., which\nbuyer surplus and seller profit cannot be simultaneously improved upon. In Lemma 1, we showed that such recommendations are induced by threshold algorithms. In this section, we show that the characterization of Pareto-optimal algorithms is fully analogous to the characterization of a buyer-optimal algorithm.\n\nTo this end, assume that the designer's objective is a weighted average of buyer surplus and seller profit, with weights of $\\alpha$ and $1-\\alpha$, respectively. Call an algorithm $\\alpha$-optimal if it maximizes this objective for weight $\\alpha$. As $\\alpha$ spans [0,1], the $\\alpha$-optimal algorithms span the range from seller-optimal to socially-optimal to buyer-optimal. Define an $\\alpha$-virtual cost as follows:\n\n$$\n\\gamma_{\\alpha}(c) \\triangleq c+\\max \\left\\{\\frac{2 \\alpha-1}{\\alpha}, 0\\right\\} \\frac{F(c)}{f(c)},\n$$\n\nand assume that for all $\\alpha \\in[0,1], \\gamma_{\\alpha}(c)$ is strictly increasing. ${ }^{14}$ Define an $\\alpha$-pseudo value as:\n\n$$\ny_{\\alpha}(v) \\triangleq \\mathbb{E}_{\\tilde{v} \\sim G}\\left[\\gamma_{\\alpha}^{-1}(\\tilde{v}) \\mid \\tilde{v} \\geq v\\right] .\n$$\n\nProposition 5 ( $\\alpha$-Optimal Algorithm). An $\\alpha$-optimal algorithm recommends the product if and only if $y_{\\alpha}(v) \\geq p$. Under this algorithm, type $c$ posts $p^{*}(c)=y_{\\alpha}\\left(\\gamma_{\\alpha}(c)\\right)$. Under this algorithm and pricing, the trade occurs if and only if $v \\geq \\gamma_{\\alpha}(c)$.\n\nProposition 5 shows that the characterization of an $\\alpha$-optimal algorithm follows verbatim the characterization of a buyer-optimal algorithm, with the virtual cost and the pseudo value replaced by their $\\alpha$-analogs. Importantly, the product allocation continues to be independent of value distribution. Consequently, all the results on market segmentation in Section 4 apply to any Pareto-optimal algorithmic recommendations.\n\nAs the weight $\\alpha$ attached to buyer surplus decreases, the difference between an $\\alpha$ virtual cost and a true cost decreases. By Proposition 5, this translates into the trade occurring over a broader range of costs and values, thus generating more total surplus.\n\n[^9]Moreover, for all $\\alpha \\leq 1 / 2$, the $\\alpha$-virtual cost coincides with the true cost. Therefore, a seller-optimal algorithm and a socially-efficient algorithm coincide and, as shown below, feature a simple recommendation structure:\n\nCorollary 1 (Seller-Optimal Algorithm). A threshold algorithm with $\\hat{v}(p)$ such that $\\mathbb{E}[v \\mid v \\geq \\hat{v}(p)]=p$ for all $p \\in[\\mathbb{E}[v], 1]$ simultaneously maximizes the seller profit and total surplus and, moreover, achieves efficient trade.\n\nThe seller-optimal algorithm maximizes efficiency at the expense of the buyer. For any price, the algorithm maximally pools products of different values to the extent that the buyer is still willing to purchase when recommended. This results in a threshold recommendation, and given the full support assumption on $G$, a threshold is uniquely defined for all $p \\in[\\mathbb{E}[v], 1]$. Under this algorithm, the buyer is guaranteed a zero expected payoff irrespective of the posted price. Thus, the seller, regardless of cost, understands that she captures all the surplus generated, and her goal of maximizing profit aligns perfectly with efficiency. As a result, the seller of type $c$ will post a price $p(c)$ that leads to an efficient trade, i.e., $p(c)=\\mathbb{E}[v \\mid v \\geq c]$. The resulting product allocation is efficient, the seller obtains the maximal feasible surplus, and the buyer is left with no rent. ${ }^{15}$\n\nCorollary 1 entails a notable feature: The persuasion constraint of the buyer, i.e., the requirement that the buyer is always willing to follow recommendations, does not constrain the designer at all points of the Pareto frontier except the seller-optimal one. Intuitively, by Corollary 1, $\\alpha$-optimal algorithms for $\\alpha \\in[0,1 / 2]$ induce the same, efficient outcome, where the persuasion constraint holds with equality but does not constrain the designer. In turn, for $\\alpha>1 / 2$, as the weight given to buyer surplus increases, the recommendations become even more favorable to the buyer, and the persuasion constraint becomes slack.\n\n[^10]","text_sha256":"720d5d689c474720faf4388daf865bbd2025a05e1111a022de7f0db4117e5eaf"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0014","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.2 Competing Sellers","text":"### 5.2 Competing Sellers\n\nThus far, we have focused on a monopoly setting in which a given product can be supplied only by a single seller. Applied seller-by-seller, this analysis also covers cases where multiple sellers offer noncompeting products. However, a natural alternative is a market in which sellers compete for the buyer so that the recommendation algorithm directs the buyer to one out of many alternatives (e.g., Hagiu and Jullien (2011), Hagiu et al. (2022), Elliott et al. (2022), Bar-Isaac and Shelegia (2022)). In this section, we show that our characterization of optimal algorithms and the main welfare implications extend to that setting.\n\nFormally, we consider the following extension of the main setting. There is a single buyer with unit demand. There are $J$ sellers indexed by $j=1, \\ldots, J$, each offering a single product. The buyer values for the products, $\\left(v_{1}, \\ldots, v_{J}\\right) \\in \\mathbb{R}^{J}$, are drawn from a joint distribution $G \\in \\Delta\\left([0,1]^{J}\\right)$ and can be arbitrarily correlated. The cost of each seller $j$ is drawn from $F_{j}$, independent of other costs or the value profile. For notational convenience, we introduce a dummy seller indexed by $j=0$ with $v_{0}=0$ and $c_{0}=0$ that corresponds to the buyer's decision not to buy anything.\n\nAn algorithm is a function $r:[0,1]^{J} \\times \\mathbb{R}_{+}^{J} \\rightarrow \\Delta^{J+1}$, where $\\Delta^{J+1}$ is a $J+1$ dimensional simplex, so that for any profiles of realized values $v=\\left(v_{1}, \\ldots, v_{J}\\right)$ and prices $p=\\left(p_{1}, \\ldots, p_{J}\\right)$, the algorithm recommends that the buyer purchase one of the products or none according to $r(v, p) \\in \\Delta^{J+1}$. The algorithm is commonly known to the buyer and sellers. Given an algorithm, nature draws the seller types $c_{j}$ and the buyer values $v_{j}$. All sellers privately observe their types but not the buyer values or the types of other sellers and simultaneously post their prices $p_{j}$. The algorithm makes recommendations according to $r$. If no product is recommended, trade does not occur. If a product of seller $j$ is recommended, the buyer observes the recommendation and the price and then decides whether to buy the product. If product $j$ is purchased, the buyer and seller $j$ obtain ex post payoffs $v_{j}-p_{j}$ and $p_{j}-c_{j}$, respectively. Otherwise, the players obtain zero payoffs. The solution concept is perfect Bayesian equilibrium.\n\nDespite featuring strategic interaction between the sellers, this setting can be ana-\nlyzed analogously to the single seller case. The main idea is that each seller's private information and thus the incentive constraints are similar in both cases: From the perspective of each seller, the value and cost uncertainty, as well as the strategic behavior of other sellers, matter only insofar as they affect her demand curve, which can be encoded in a single variable.\n\nSpecifically, for each $j$, denote by $\\gamma_{j}\\left(c_{j}\\right)$ her virtual cost. For the dummy seller 0 , set $\\gamma_{0}\\left(c_{0}\\right)=0$. Assume that for $j=1, \\ldots, J, \\gamma_{j}$ is strictly increasing and continuous in $c_{j}$. Define $\\bar{c}_{j} \\triangleq \\gamma_{j}^{-1}(1)$. Define an auxiliary random variable\n\n$$\n\\theta_{j}=v_{j}-\\max _{k \\in\\{0,1, \\ldots, J\\} \\backslash j}\\left\\{v_{k}-\\gamma_{k}\\left(c_{k}\\right)\\right\\},\n$$\n\ni.e., $\\theta_{j}$ is the value of seller $j$ 's product minus the highest virtual surplus among all other sellers as long as the latter is positive. Define\n\n$$\np_{j}^{*}\\left(c_{j}\\right) \\triangleq \\mathbb{E}_{\\theta_{j}}\\left[\\gamma_{j}^{-1}\\left(\\theta_{j}\\right) \\mid \\gamma_{j}^{-1}\\left(\\theta_{j}\\right) \\geq c_{j}\\right]\n$$\n\nand observe that $p_{j}^{*}\\left(c_{j}\\right)$ is a strictly increasing function. Define the inverse function of $p_{j}^{*}$ as $p_{j}^{*-1}$ with the (nonstandard) convention that $p_{j}^{*-1}(p)=0$ for $p<p_{j}^{*}(0)$ and $p_{j}^{*-1}(p)=1$ for $p>p_{j}^{*}(1)$.\n\nProposition 6 (Buyer-Optimal Algorithm with Competing Sellers). A buyeroptimal algorithm recommends the product of seller $j^{*}(v, p)$ such that\n\n$$\nj^{*}(v, p) \\in \\underset{j \\in\\{0,1, \\ldots, J\\}}{\\operatorname{argmax}} v_{j}-\\gamma_{j}\\left(p_{j}^{*-1}\\left(p_{j}\\right)\\right),\n$$\n\nwith ties being broken arbitrarily. Under this algorithm, seller $j$ of type $c_{j} \\leq \\bar{c}_{j}$ posts price $p_{j}^{*}\\left(c_{j}\\right)$ and seller $j$ of type $c_{j}>\\bar{c}_{j}$ is inactive. Under this algorithm and pricing, for any realized profile of $v$ and $c$, the buyer trades with seller $j^{*} \\in \\operatorname{argmax}_{j \\in\\{0,1, \\ldots, J\\}} v_{j}-\\gamma_{j}\\left(c_{j}\\right)$.\n\nProposition 6 directly extends Proposition 1. To see this, observe that in the case of $J=1$, the condition $v_{1}-\\gamma_{1}\\left(p_{1}^{*-1}\\left(p_{1}\\right)\\right) \\geq v_{0}-\\gamma_{0}\\left(p_{0}^{*-1}\\left(p_{0}\\right)\\right)=0$ is equivalent to the condition $y\\left(v_{1}\\right) \\geq p_{1}$; thus, the two propositions describe the same algorithm albeit\nin different terms. In the case of many sellers, the condition $v_{i}-\\gamma_{i}\\left(p_{i}^{*-1}\\left(p_{i}\\right)\\right) \\geq v_{j}-$ $\\gamma_{j}\\left(p_{j}^{*-1}\\left(p_{j}\\right)\\right)$ for $i, j \\neq 0$ cannot be easily translated into the language of pseudo values, so we present the buyer-optimal algorithm as in (9).\n\nImportantly, as in the case of a single seller, Proposition 6 establishes that the equilibrium product allocation does not depend on the distribution of product values. Similarly, it allows us to succinctly analyze the impact of market segmentation. Formally, market segmentation is defined by an information structure $\\mathcal{I}=(S, \\pi)$ that consists of a set $S=\\times_{j} S_{j}$ of signal realizations $s_{j}$ privately observed by each seller, and a family of probability distributions $\\{\\pi(\\cdot \\mid v)\\}_{v \\in[0,1]^{J}}$ over $S$. The signal is commonly known and exogenous, the signal realizations are independent of the seller types but can be arbitrarily correlated across sellers, and the algorithm can base recommendations on the realized signals, the values, and the product prices. ${ }^{16}$\n\nProposition 7 (Segmentation Neutrality with Competing Sellers). For any market segmentation, the buyer-optimal algorithm induces the same ex post product allocation, the same expected price and profit of each seller type, and the same ex ante buyer surplus.","text_sha256":"04f9e41794d715353b40703db6a86fb3b10111b96345ff25a532588e464bc0f8"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0015","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.2 Competing Sellers","text":"Proposition 7 implies the remarkable neutrality of market segmentation if the buyer uses an algorithm to guide consumption choices. As in Section 4, leaking buyer data not only does not harm the buyer but also cannot benefit him, despite competition among sellers. Intuitively, because the algorithm can assess the buyer's value and is designed prior to pricing decisions, it shifts the bargaining power to the buyer, and informing the sellers can only reduce the attainable buyer surplus. Furthermore, by adapting to the specifics of market segmentation, the optimal algorithm design can perfectly absorb the impact of information leakage on total buyer surplus, seller profits, and product allocation.\n\nAt the same time, market segmentation does affect equilibrium pricing and the re-\n\n[^11]distribution of buyer surplus across different value profiles. Our analysis behind Proposition 7 reveals that equilibrium pricing can be decomposed across sellers, with each seller's pricing strategy depending only on her beliefs about the value profile, and being indifferent to information observed by other sellers. This immediately allows us to claim an impact of finer market segmentation on prices. Specifically, say that $\\mathcal{I}_{H}$ is Blackwell more informative than $\\mathcal{I}_{L}$ for seller $j$, if the corresponding marginal $\\left(S_{j}, \\pi_{j}\\right)$ with $\\pi_{j}:[0,1]^{J} \\rightarrow \\Delta\\left(S_{j}\\right)$ derived from $(S, \\pi)$ is Blackwell more informative about $v$. Then, we have:\n\nProposition 8 (Segmentation Redistribution with Competing Sellers). If signal $\\mathcal{I}_{H}$ is Blackwell more informative than $\\mathcal{I}_{L}$ for seller $j$, then for any given type of seller $j$, the distribution of prices set by seller $j$ under $\\mathcal{I}_{H}$ is a mean-preserving spread of that under $\\mathcal{I}_{L}$.\n\nBy Proposition 8, any finer market segmentation-regardless of how the additional information is correlated across sellers-results in a clear pattern of more dispersed prices, just as under a monopoly. However, the impact on surplus distribution is subtler than that under a monopoly. With multiple sellers, finer segmentation may group lower values for one seller's product with higher values for another seller's product, leading to a higher trade price and lower surplus for some low-value buyers, thus violating the mean-preserving contraction property.","text_sha256":"a714748ca80374428b8e41e80df0914fd0c57661580164d97853f24508c9e582"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0016","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.3 Informed Buyer","text":"### 5.3 Informed Buyer\n\nUp to this point, we have deliberately assumed that the algorithm has full control over the buyer's information both about the product's existence and about the product's value. This assumption offered a clear benchmark for studying algorithmic recommendations, provided the algorithm with maximal information to control and thus established the upper bound on achievable buyer surplus. In this section, we relax this assumption and show how our analysis remains relevant even if the buyer is partially informed.","text_sha256":"ac2f09a2edd959b6b22a5a46a2f023253a65f217edbe5be388332de6d80abef0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0017","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.3.1 Information about Product Existence","text":"### 5.3.1 Information about Product Existence\n\nWe have assumed that the buyer cannot purchase the product if it is not recommended. This assumption is relevant in online settings where recommendation systems are used primarily to discover and bring products to the buyer's attention, and thus effectively serve as gatekeepers. ${ }^{17}$ A natural alternative setting is one in which the buyer already knows the product exists and where to purchase it but may still be unsure about the match value. To address that setting, in this subsection, we allow the buyer to purchase the product even if it is not recommended, and impose a constraint that the buyer prefers not to purchase the nonrecommended product at each price.\n\nFirst, consider the simplest case in which the seller's cost is commonly known to be $c_{0}$. This case is closest to the paper by Roesler and Szentes (2017) and differs only in the timing of the recommendations. In their setting, recommendations come before the price is posted and thus cannot condition on the price; in our setting, the recommendations can condition on the price.\n\nWhen the costs are known to be $c_{0}$, a natural candidate for a buyer-optimal algorithm is to recommend the product if and only if $p=c_{0}$ and $v \\geq c_{0}$. If this algorithm suffices to incentivize the seller to set $p=c_{0}$, then it is buyer-optimal because the outcome is efficient and leaves the seller with zero profit.\n\nIf $\\mathbb{E}[v] \\leq c_{0}$, an algorithm can attain this outcome by revealing no information whenever $p>c_{0}$ and thus dissuading the buyer from purchasing at a price above $c_{0}$. In contrast, if $\\mathbb{E}[v]>c_{0}$, the seller can secure a positive profit by charging a price in $\\left(c_{0}, \\mathbb{E}[v]\\right)$ because no algorithm can make the buyer believe that the expected value of the product is always below $p<\\mathbb{E}[v]$. In this case, the buyer-optimal algorithm deters the seller from setting a higher price via adversarial persuasion: At each price, the algorithm provides the buyer with information that minimizes the probability of trade.\n\n[^12]Specifically, the algorithm reveals whether $v<\\hat{v}(p)$, where $\\hat{v}(p)$ is such that\n\n$$\n\\mathbb{E}[\\tilde{v} \\mid \\tilde{v}<\\hat{v}(p)]=\\min \\{p, \\mathbb{E}[v]\\},\n$$\n\nand persuades the buyer to purchase only when $v \\geq \\hat{v}(p)$. The maximum profit the seller can guarantee against adversarial persuasion is\n\n$$\n\\underline{\\pi} \\triangleq \\max _{p \\geq c_{0}}\\left(p-c_{0}\\right)[1-G(\\hat{v}(p)] .\n$$\n\nThe buyer-optimal algorithm induces an efficient trade while leaving the seller with this profit:\n\nProposition 9 (Known Product, Known Cost). Suppose that $F$ is concentrated at $c_{0} \\in[0,1)$ and that the buyer can purchase the product even when not recommended. The buyer-optimal algorithm recommends the product if $p^{*}=c_{0}+\\frac{\\pi}{1-G\\left(c_{0}\\right)}$ and $v \\geq c_{0}$ and follows all other prices with adversarial persuasion. Under this algorithm, the seller posts a price $p^{*}$, and the equilibrium trade is efficient.\n\nLike in the setting of Roesler and Szentes (2017), a buyer-optimal algorithm leads to efficient trade. Unlike the setting of Roesler and Szentes (2017), the seller's rent is driven by adversarial persuasion price-by-price and thus is lower, reaching zero when $\\mathbb{E}[v]<c_{0}$, e.g., when the product can be counterfeit or harmful with a high probability.\n\nRecall from Claim 1 that if the seller's cost were known to be $c_{0}$ in our original setup, the buyer-optimal algorithm would recommend the product if and only if $p=c_{0}$ and $v \\geq c_{0}$. This algorithm always attains the efficient outcome and leaves the seller with zero profit. Therefore, when $c_{0}<\\mathbb{E}[v]$, the design and consequences of the optimal algorithm depend on whether the buyer can purchase the nonrecommended product, whereas if $c_{0}>\\mathbb{E}[v]$ they do not depend on it.\n\nFurthermore, when the seller's costs are uncertain, the buyer-optimal algorithms in both cases can coincide even when the cost could fall below $\\mathbb{E}[v]$. This happens, for example, in the uniform setting of Example 1: Under the buyer-optimal algorithm, the\nlack of recommendation is a sufficiently negative signal at any price to dissuade the buyer from the purchase. More generally:\n\nProposition 10 (Known Product. Unknown Cost). If $\\int_{0}^{\\gamma(c)}[v-c] \\mathrm{d} G(v) \\leq 0$ for each $c \\in[0, \\bar{c}]$, then even if the buyer can purchase the product when not recommended, the algorithm in Proposition 1 is buyer-optimal.\n\nThe condition of Proposition 10 ensures that whenever the product is not recommended under the algorithm of Proposition 1, the buyer infers that the expected value of the product is below the price. This holds for many classes of distributions, for example, when (i) $F(c)=c^{\\alpha}$ and $G(v)=v^{\\beta}$ with $0<\\alpha \\leq \\beta$ or (ii) $G$ is uniform and $F(c) / c$ is increasing. ${ }^{18}$ To see the intuition behind this sufficient condition, suppose that the seller posts a price of $p^{*}(c)$, and the algorithm recommends that the buyer not purchase the product, which by Proposition 1, reveals that $v \\leq \\gamma(c)$. If the buyer purchases the product, his payoff must decrease because the seller's profit increases but total surplus decreases because $\\int_{0}^{\\gamma(c)}[v-c] \\mathrm{d} G(v) \\leq 0$. Thus, the buyer is willing to follow the recommendation not to buy.","text_sha256":"07eee70cad9901fe682e7f0fed217478d4f3aecb47e1b26fde2a637f1cd00c64"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0018","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.3.2 Information about Product Value","text":"### 5.3.2 Information about Product Value\n\nThus far, we have assumed that the buyer does not obtain any product information beyond what is provided by the algorithm. This stylized assumption is intended to capture the context of experience goods, which can be difficult to judge based on appearance and for which individual taste shocks are sufficiently variable yet can be estimated by a welltrained recommendation system. In this section, we allow for the possibility that the buyer observes additional information about the value when a product is recommended while maintaining the original ignorance of the product's existence. ${ }^{19}$\n\nWe show how our previous analysis informs this setting. First, observe that the buyer's incentives in the buyer-optimal algorithm of Proposition 1 are generally slack.\n\n[^13]That is, whenever a product is recommended, the buyer strictly prefers to follow the recommendation. Therefore, a small amount of extraneous information that does not significantly lower the posterior expectation will not interfere with the algorithm's design.\n\nSecond, our market segmentation analysis implies that even if the buyer can perfectly assess the value of the product upon seeing it, the algorithm can still achieve the same total buyer surplus, seller surplus, and product allocation, although this would require informing both the seller and the buyer. Specifically, suppose that the buyer observes the value $v$ of the product whenever it is recommended. When the seller does not know $v$, the algorithm in Proposition 1 is not incentive compatible because the buyer will ignore the recommendation when $p$ and $v$ are such that $v<p<y(v)$, which occurs for low values. However, if the algorithm perfectly informs the seller about $v$, then the buyer-optimal algorithm, as characterized in Section 4, recommends the product if and only if $p \\leq \\gamma^{-1}(v)<v$. Under this algorithm, buyers with all values $v$ are willing to follow the recommendations because, intuitively, informed sellers lower the prices offered to low-value buyers. By Proposition 3, this algorithm implements the same total buyer surplus, seller profits, and product allocation. Remarkably, in the case of consumption driven by algorithmic recommendations, third-degree price discrimination not only does not harm the total buyer surplus but also may be beneficial if the algorithm cannot fully control the buyer value information.","text_sha256":"aa38a70265cfb747092e1284bd07373df3d6b8e98b2bc93bd3a0c64439640aa6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0019","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Conclusion","text":"## 6 Conclusion\n\nIn this paper, we studied the question of optimal algorithmic recommendations in the presence of strategic pricing. We showed that optimal recommendations must strike a balance between increasing the trade surplus and inducing low prices. Algorithmic recommendations drastically change the predictions of third-degree price discrimination, whereby finer market segmentations by the sellers do not affect the total consumer surplus or seller profits but result in larger price spreads and a more equitable surplus distribution.\n\nWe view our work as a stepping stone toward a better understanding of algorithmic design in strategic settings, an area of growing importance at the intersection of economics and computer science (e.g., Goktas et al. (2025)). First, our model of algorithms is deliberately stylized to analyze strategic motives in a clear and tractable way. A practical implementation would ideally incorporate many engineering concerns from which we abstracted away, such as value estimation details, computational complexity, and robustness. Second, it would be interesting to study market structures for algorithm providers and understand which of the algorithms that we characterize are favored by one or another market structure. Third, the developed ideas of algorithmic decisions can be exported beyond consumption settings, such as to algorithmic matching or algorithmic negotiations. All this further research can be built upon the analytical framework proposed in this paper.","text_sha256":"a2ee9252eaf8479cf9ba1f704b8091ee65805d0379e29836ba243a7b89ebed1a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0020","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAkbarpour, M., P. Dworczak, and S. D. Kominers (2024): \"Redistributive Allocation Mechanisms,\" Journal of Political Economy, 132, 1831-1875.\n\nAsker, J., C. Fershtman, and A. Pakes (2022): \"Artificial Intelligence, Algorithm Design, and Pricing,\" AEA Papers and Proceedings, 112, 452-456.\n\nAssad, S., R. Clark, D. Ershov, and L. Xu (2024): \"Algorithmic Pricing and Competition: Empirical Evidence from the German Retail Gasoline Market,\" Journal of Political Economy, 132, 723-771.\n\nBar-Isaac, H. and S. Shelegia (2022): \"Monetizing Steering,\" Working paper.\n\nBaron, D. P. and R. B. Myerson (1982): \"Regulating a Monopolist with Unknown Costs,\" Econometrica, 911-930.\n\nBergemann, D. and A. Bonatti (2024): \"Data, Competition, and Digital Platforms,\" American Economic Review, 114, 2553-2595.\n\nBergemann, D., A. Bonatti, and N. Wu (forth.): \"How Do Digital Advertising Auctions Impact Product Prices?\" Review of Economic Studies.\n\nBergemann, D., B. Brooks, and S. 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Sappington (1994): \"Supplying Information to Facilitate Price Discrimination,\" International Economic Review, 309-327.\n\nLibgober, J. and X. Mu (2021): \"Informational Robustness in Intertemporal Pricing,\" Review of Economic Studies, 88, 1224-1252.\n\nLiu, Q., K. Mierendorff, X. Shi, and W. Zhong (2019): \"Auctions with Limited Commitment,\" American Economic Review, 109, 876-910.\n\nLoertscher, S. and L. M. Marx (2022): \"Incomplete Information Bargaining with Applications to Mergers, Investment, and Vertical Integration,\" American Economic Review, 112, 616-649.\n\nLuca, M. and O. Reshef (2021): \"The Effect of Price on Firm Reputation,\" Management Science, 67, 4408-4419.\n\nMyerson, R. B. (1981): \"Optimal Auction Design,\" Mathematics of Operations Research, 6, 58-73.\n\nMyerson, R. B. and M. A. Satterthwaite (1983): \"Efficient Mechanisms for Bilateral Trading,\" Journal of Economic Theory, 29, 265-281.\n\nMylovanov, T. and T. Tröger (2014): \"Mechanism Design by an Informed Principal: Private Values with Transferable Utility,\" Review of Economic Studies, 81, 1668-1707.\n\nRoesler, A.-K. and B. Szentes (2017): \"Buyer-Optimal Learning and Monopoly Pricing,\" American Economic Review, 107, 2072-80.\n\nSalcedo, B. (2015): \"Pricing Algorithms and Tacit Collusion,\" Working paper.\n\nScott Morton, F., P. Bouvier, A. Ezrachi, B. Jullien, R. Katz, G. Kimmelman, A. D. Melamed, and J. Morgenstern (2019): \"Committee for the Study of Digital Platforms: Market Structure and Antitrust Subcommittee Report,\" Stigler Center for the Study of the Economy and the State, University of Chicago Booth School of Business, 36.\n\nShaked, M. and J. G. Shanthikumar (2007): Stochastic Orders, Springer.\n\nSmolin, A. (2023): \"Disclosure and Pricing of Attributes,\" Rand Journal of Economics, 54, 570-597.","text_sha256":"11f8a052569fda8be267bb3a7d46a18c1dc9aab4ac03d45a1b8de2ad2730a326"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0021","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"White House (2023): \"Executive Order on the Safe, Secure, and Trustworthy Development and Use of Artificial Intelligence,\" https://www.whitehouse.gov/ briefing-room/presidential-actions/2023/10/30/executive-order-onthe-safe-secure-and-trustworthy-development-and-use-of-artificialintelligence/, accessed: September 05, 2024.\n\nXu, W. and K. H. Yang (2024): \"Equivalent Mechanisms for Information Intermediation,\" Working paper.\n\nYang, K. H. (2022): \"Selling Consumer Data for Profit: Optimal Market-Segmentation Design and Its Consequences,\" American Economic Review, 112, 1364-93.","text_sha256":"780a654a290b74ce1934894629e7f006567f17f6652d7acc173ae493e01d4f81"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0022","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Appendix: Ommited Proofs","text":"## Appendix: Ommited Proofs","text_sha256":"de1789263c8858216065fe7f4be6da72866bceb17e928a9b8d2833531f25fdcb"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0023","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Proof of Lemma 1","text":"## A Proof of Lemma 1\n\nTake any algorithm $r$. For each $p \\geq 0$, let $q_{r}(p) \\triangleq \\int_{0}^{1} r(v, p) d G(v)$ denote the probability with which the product is recommended, and thus purchased, under $r$. We define a new algorithm $\\hat{r}$ as $\\hat{r}(v, p) \\triangleq \\mathbb{1}\\left(v>G^{-1}\\left(1-q_{r}(p)\\right)\\right)$. At each price $p$, this algorithm recommends the product with the same probability as $r, 1-G\\left(G^{-1}\\left(1-q_{r}(p)\\right)\\right)=q_{r}(p)$. Moreover, the expected value of the product, conditional on the recommendation, is greater under $\\hat{r}$ than under $r$. As a result, the buyer will purchase the product whenever it is recommended by $\\hat{r}$, and at each price $p$, the seller will earn the same profit under both $r$ and $\\hat{r}$. Therefore, $\\hat{r}$ has an equilibrium that attains a greater buyer surplus than $r$ with the same seller profit as $r$. $\\square$","text_sha256":"a717b975960a32c481cfa5b95123497cf5025eea7a5b7f4ee0adbc9f56f833d5"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0024","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B Proof of Proposition 1","text":"## B Proof of Proposition 1\n\nBy the revelation principle, we can study algorithm design by analyzing direct mechanisms in which the seller reports the type to the designer and the designer chooses which valuations to allocate to the seller and at which price. Furthermore, by Lemma 1, we can focus on threshold allocations. The designer's problem can thus be stated as follows:\n\n$$\n\\begin{array}{rlr}\n\\max _{\\hat{v}:[0,1] \\rightarrow[0,1], p:[0,1] \\rightarrow \\mathbb{R}_{+}} & \\int_{0}^{1} \\int_{\\hat{v}(c)}^{1}(v-p(c)) \\mathrm{d} G \\mathrm{~d} F & \\\\\n\\text { s.t. } & \\int_{\\hat{v}(c)}^{1}(p(c)-c) \\mathrm{d} G \\geq \\int_{\\hat{v}\\left(c^{\\prime}\\right)}^{1}\\left(p\\left(c^{\\prime}\\right)-c\\right) \\mathrm{d} G & \\forall c, c^{\\prime} \\in[0,1] \\\\\n& \\int_{\\hat{v}(c)}^{1}(p(c)-c) \\mathrm{d} G \\geq 0 & \\forall c \\in[0,1]\n\\end{array}\n$$\n\nOne way to solve this problem is to reformulate it in familiar terms. Because the value is continuously distributed, the expected trade probability $q \\triangleq \\int_{\\hat{v}}^{1} \\mathrm{~d} G$ is strictly\ndecreasing in $\\hat{v}$, spanning [ 0,1 ] as $\\hat{v}$ spans [ 0,1 ]. Hence, $q$ and $v$ are in a one-to-one relationship, and instead of maximizing over $\\hat{v}(c)$, we can maximize over $q(c)$. With a small abuse of notation, denote by $\\hat{v}(q)$ the threshold that results in a given $q$ and by $V(q) \\triangleq \\int_{\\hat{v}(q)}^{1} v \\mathrm{~d} G$ the corresponding trade surplus. The trade surplus is strictly increasing in $q$ with $V(0)=0$ and $V(1)=\\mathbb{E}[v]$. Moreover,\n\n$$\n\\frac{\\mathrm{d} V}{\\mathrm{~d} q}=\\frac{\\partial V / \\partial \\hat{v}}{\\partial q / \\partial \\hat{v}}=\\frac{-\\hat{v} g(\\hat{v})}{-g(\\hat{v})}=\\hat{v}(q) .\n$$\n\nAs such, $V(q)$ is a concave function with $V^{\\prime}(0)=1$ and $V^{\\prime}(1)=0$. Finally, we denote the expected revenue by $t(c) \\triangleq p(c) \\int_{\\hat{v}(c)}^{1} \\mathrm{~d} G$. In terms of these variables, we can restate problem (12) as follows:\n\n$$\n\\begin{array}{rlr}\n\\max _{q:[0,1] \\rightarrow[0,1], t:[0,1] \\rightarrow \\mathbb{R}_{+}} \\int_{0}^{1}(V(q(c))-t(c)) \\mathrm{d} F & \\\\\n\\text { s.t. } t(c)-c q(c) \\geq t\\left(c^{\\prime}\\right)-c q\\left(c^{\\prime}\\right) & \\forall c, c^{\\prime} \\in[0,1] \\\\\nt(c)-c q(c) \\geq 0 & \\forall c \\in[0,1] .\n\\end{array}\n$$\n\nProblem (14) is analogous to the problem analyzed by Baron and Myerson (1982) if $q$ is interpreted as a quantity produced and $V$ is interpreted as the welfare generated by producing quantity $q$. Its celebrated solution sets the optimal quantity to equalize marginal welfare benefits with virtual costs and the optimal transfer to guarantee the incentive-compatible profit distribution:\n\n$$\n\\begin{aligned}\nV^{\\prime}(q(c)) & =\\gamma(c) \\\\\nt(c)-q(c) c & =\\int_{c}^{1} q(x) \\mathrm{d} x=\\int_{c}^{1} 1-G(\\gamma(x)) \\mathrm{d} x\n\\end{aligned}\n$$\n\nBy Equation 13, we can translate this solution back to problem (12) as\n\n$$\n\\begin{aligned}\n\\hat{v}(c) & =\\gamma(c), \\\\\np(c) & =c+\\frac{\\int_{c}^{1} 1-G(\\gamma(x)) \\mathrm{d} x}{1-G(\\gamma(c))} \\\\\n& =c+\\frac{\\int_{c}^{1}(x-c) g(\\gamma(x)) \\gamma^{\\prime}(x) \\mathrm{d} x}{1-G(\\gamma(c))} \\quad \\text { (integration by parts) } \\\\\n& \\left.=c+\\frac{\\int_{\\gamma(c)}^{\\gamma(1)}\\left(\\gamma^{-1}(v)-c\\right) g(v) \\mathrm{d} v}{1-G(\\gamma(c))} \\quad \\text { (change of variable with } v=\\gamma(x)\\right) \\\\\n& =\\frac{\\int_{\\gamma(c)}^{\\gamma(1)} \\gamma^{-1}(x) g(x) \\mathrm{d} x}{1-G(\\gamma(c))} \\\\\n& =\\mathbb{E}\\left[\\gamma^{-1}(v) \\mid v \\geq \\gamma(c)\\right] \\\\\n& =y(\\gamma(c)) .\n\\end{aligned}\n$$\n\nWe now show that the algorithm and the equilibrium in the statement attain the same outcome as above. First, the buyer is willing to purchase the product when recommended because\n\n$$\n\\mathbb{E}[v \\mid y(v) \\geq p(c)]=\\mathbb{E}[v \\mid y(v) \\geq y(\\gamma(c))]=\\mathbb{E}[v \\mid v \\geq \\gamma(c)] \\geq \\mathbb{E}\\left[\\gamma^{-1}(v) \\mid v \\geq \\gamma(c)\\right]=p(c) .\n$$\n\nThus, the expected value of the product conditional on each possible price exceeds the price.\n\nSecond, the seller with each type $c$ is willing to set price $y(\\gamma(c))$. Deviating to another price in $[y(\\gamma(0)), y(\\gamma(\\bar{c}))]$ is not profitable because of the incentive compatibility constraints of the mechanism. Deviating to a price below $y(\\gamma(0))$ or above $y(\\gamma(\\bar{c}))$ is not profitable either because it results in a lower profit than $p=y(\\gamma(0))$ or no trade.\n\nFinally, if the buyer follows the recommendation and each type $c$ sets price $y(\\gamma(c))$, the trade occurs if and only if $y(v) \\geq y(\\gamma(c))$, or equivalently, if $v \\geq \\hat{v}(c)=\\gamma(c)$.\n\nIn summary, the algorithm described in the statement implements the solution to problem (12) in equilibrium. Therefore, it is a buyer-optimal algorithm. $\\square$","text_sha256":"1927fff74fcf146e994507f440fc074968d966143ea22ad5b2763689fb69ad81"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0025","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"C Proof of Proposition 2","text":"## C Proof of Proposition 2\n\nUnder the buyer-optimal algorithm, trade occurs if and only if $v \\geq \\gamma(c)$. If $F$ has a decreasing reversed hazard rate, then $\\gamma(c)=c+\\frac{F(c)}{f(c)}$ implies that $\\gamma^{\\prime}(c) \\geq 1$. Under the ex post optimal algorithm, trade occurs if and only if $v \\geq r^{-1}(c)$, where $r(v)=v-\\frac{1-G(v)}{g(v)}$ is the virtual valuation function. If $G$ has an increasing hazard rate, then $r^{\\prime}(v)>1$, which implies that\n\n$$\n\\frac{d}{d c} r^{-1}(c)=\\frac{1}{r^{\\prime}\\left(r^{-1}(c)\\right)}<1 .\n$$\n\nThus, $\\gamma(c)$ is steeper than $r^{-1}(c)$, meaning that in the $(c, v)$-space, the curve $v=\\gamma(c)$ crosses the curve $v=r^{-1}(c)$ at most once and from below.\n\nSince $\\gamma(c)$ starts at 0 when $c=0$ and ends above 1 when $c=1$, while $r^{-1}(c)$ starts above 0 at $c=0$ and ends at 1 at $c=1$, it follows that for low values of $c, \\gamma(c)<r^{-1}(c)$, whereas for high values of $c, \\gamma(c)>r^{-1}(c)$. Consequently, there exists a unique crossing point $c^{*}$ such that $\\gamma\\left(c^{*}\\right)=r^{-1}\\left(c^{*}\\right)$. We then obtain qualitatively the same picture as the right panel of Figure 2. This establishes the result. $\\square$","text_sha256":"c56d0b4687ece0b1f7b20a2465c4cafb922b3cbe5e6963cede9565d1dc34f796"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0026","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"D Proofs of Proposition 3 and Proposition 4","text":"## D Proofs of Proposition 3 and Proposition 4\n\nTo prove Proposition 3, we establish a version of the payoff equivalence theorem for our model (cf. Myerson (1981) and Krishna (2009)).\n\nLemma 2 (Payoff Equivalence). For each $i \\in\\{1,2\\}$, take an algorithm $r_{i}$, market segmentation $\\mathcal{I}_{i}$, and equilibrium $\\mathcal{E}_{i}$. Suppose that $\\mathcal{E}_{1}$ and $\\mathcal{E}_{2}$ have the same allocation rule in terms of $v$ and $c$ and the same profit of the seller at type $c=1$. Then, the seller's profit of any type and the buyer surplus are identical between $\\mathcal{E}_{1}$ and $\\mathcal{E}_{2}$.\n\nProof. Let $q(v, c)$ denote the probability of a trade when the value is $v$ and the type is $c$, and let $q(c)=\\int_{0}^{1} q(v, c) \\mathrm{d} G(v)$ denote the expected probability of a trade for type $c$. Upon calculating these objects, we take expectation with respect to the possible segments. Let $\\bar{\\pi}$ denote the profit of the seller with the highest type, $c=1$. By assumption, $q(\\cdot, \\cdot)$ and $\\bar{\\pi}$ are the same between $\\mathcal{E}_{1}$ and $\\mathcal{E}_{2}$. Additionally, let $\\pi_{i}(c)$ and $t_{i}(c)$ denote the profit and the expected monetary transfer, respectively, at type $c$ in equilibrium $\\mathcal{E}_{i}$.\n\nIn equilibrium $\\mathcal{E}_{i}$, type $c=1$ cannot earn a strictly higher profit by imitating the pricing strategy of type $c^{\\prime}$ in every segment in $\\mathcal{I}_{i}$. This incentive compatibility constraint is\n\n$$\nt_{i}(c)-c q(c) \\geq t_{i}\\left(c^{\\prime}\\right)-c q\\left(c^{\\prime}\\right), \\forall c, c^{\\prime} \\in[0,1] .\n$$\n\nThe envelope theorem implies that\n\n$$\n\\pi_{i}(c)=\\bar{\\pi}+\\int_{c}^{1} q(x) \\mathrm{d} x\n$$\n\nThe right-hand side does not depend on $i$. Thus, the seller's profit is the same between $\\mathcal{E}_{1}$ and $\\mathcal{E}_{2}$ for every seller type. The buyer surplus is the same between $\\mathcal{E}_{1}$ and $\\mathcal{E}_{2}$ because it is the total surplus from allocation rule $q(\\cdot, \\cdot)$ minus the seller's profit, neither of which depends on $i$. $\\square$","text_sha256":"b6f5ecf1e4c7564b390d84e4e34b46f4bc5e80247dea9d5066d90e1e3628448d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0027","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proof of Proposition 3","text":"## Proof of Proposition 3\n\nTake any signal, $\\mathcal{I}$. For any signal realization, the optimal algorithm induces a trade if and only if $v \\geq \\gamma(c)$. Hence, the ex post allocation of the product is independent of the signal, as is the total surplus. Furthermore, the highest seller type $c=1$ always earns zero profits. Lemma 2 then implies that the seller profit of all types and the buyer surplus are independent of the signal. $\\square$","text_sha256":"98780925621f7501bcba58d73d33b9582950a89bc6dc925a09414cd5d1a15b4a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0028","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proof of Proposition 4","text":"## Proof of Proposition 4\n\nFirst, we show that as the signal becomes more informative, the distribution of prices set by each active seller type undergoes a mean-preserving spread. To see this, consider any signals $\\mathcal{I}_{H}$ and $\\mathcal{I}_{L}$ such that $\\mathcal{I}_{H}$ is more informative than $\\mathcal{I}_{L}$. Recall that $\\hat{\\pi}_{H}$ and $\\hat{\\pi}_{L}$ denote the respective ex ante distributions of the signal realizations.\n\nFix any active type $c$. For each $\\alpha \\in\\{L, H\\}$, let $G_{\\alpha, s}^{c} \\in \\Delta[0,1]$ denote the posterior distribution of value $v$ conditional on (i) signal $s$ being realized under signal $\\mathcal{I}_{\\alpha}$ and (ii) $v \\geq \\gamma(c)$. The equilibrium price of type $c$ after observing signal realization $s$ under\nsignal $\\mathcal{I}_{\\alpha}$ is\n\n$$\np(c \\mid s, \\alpha) \\triangleq \\int_{0}^{1} \\gamma^{-1}(v) \\mathrm{d} G_{\\alpha, s}^{c}(v) .\n$$\n\nLet $\\mathcal{G}_{\\alpha}^{c} \\in \\Delta \\Delta[0,1]$ denote the distribution of posteriors $G_{\\alpha, s}^{c}$ for a fixed $c$. Specifically, $\\mathcal{G}_{\\alpha}^{c}$ is the distribution of random variable $G_{\\alpha, s}^{c}$ with $s \\sim \\hat{\\pi}_{\\alpha}$. Because signal $\\mathcal{I}_{H}$ is more informative than signal $\\mathcal{I}_{L}, \\mathcal{G}_{H}^{c}$ is a mean-preserving spread of $\\mathcal{G}_{L}^{c} .{ }^{20}$ As the price is linear in posterior $G_{\\alpha, s}^{c}$, the mean-preserving spread relation between the distributions of posteriors, $\\mathcal{G}_{H}^{c}$ and $\\mathcal{G}_{L}^{c}$, imply the mean-preserving spread relation between real-valued random variables $p(c \\mid s, H)$ and $p(c \\mid s, L)$. Therefore, we conclude that $p(c \\mid s, H)$ with $s \\sim \\hat{\\pi}_{H}$ is a mean-preserving spread of $p(c \\mid s, L)$ with $s \\sim \\hat{\\pi}_{L}$. Therefore, the distribution of prices set by each active type under signal $\\mathcal{I}_{H}$ is a mean-preserving spread of the price distribution under signal $\\mathcal{I}_{L}$.\n\nSecond, we establish the results on monotone partitions. In what follows, we view a monotone partitional signal as a partition of [0, 1] and use an \"interval\" to mean an interval with a positive length, excluding a singleton set.\n\nTake any monotone partitional signals, $\\mathcal{I}_{H}$ and $\\mathcal{I}_{L}$, such that $\\mathcal{I}_{H}$ is finer than $\\mathcal{I}_{L}$. We can create partition $\\mathcal{I}_{H}$ by applying the following operations finitely many times to partition $\\mathcal{I}_{L}$ : (i) taking an interval from $\\mathcal{I}_{L}$ and dividing it into two subintervals or (ii) taking an interval from $\\mathcal{I}_{L}$ and fully revealing the values within it. The latter operation means partitioning interval $[a, b]$ into $\\{\\{v\\}\\}_{v \\in[a, b]}$. To obtain our result, it suffices to show that applying (i) or (ii) to any given monotone partitional signal leads to a meanpreserving contraction of the buyer surplus at any seller type. We consider these two operations in turn.\n\nOperation (i). Fix any monotone partitional signal $\\mathcal{I}$ that is different from the fully informative signal. Suppose that we take interval $\\left[v_{i}, v_{i+1}\\right]$ from $\\mathcal{I}$ and split it into $\\left[v_{i}, \\hat{v}\\right]$ and $\\left[\\hat{v}, v_{i+1}\\right]$ for some $\\hat{v} \\in\\left(v_{i}, v_{i+1}\\right)$. We show that after this partitioning, the\n\n[^14]buyer surplus $w(v, c)$, when we fix $c$ but draw $v$ from $G$, undergoes a mean-preserving contraction.\n\nFirst, we consider the values and types that are affected by the operation, i.e., $(v, c)$ such that $\\gamma(c) \\leq v<v_{i+1}$. Before the operation, the buyer's ex post payoff is\n\n$$\nw_{0}(v, c) \\triangleq v-\\mathbb{E}_{\\tilde{v} \\sim G}\\left[\\gamma^{-1}(\\tilde{v}) \\mid \\tilde{v} \\in\\left[\\gamma(c), v_{i+1}\\right]\\right], \\forall v \\in\\left[\\gamma(c), v_{i+1}\\right] .\n$$\n\nAfter the operation, the buyer's ex post payoff is\n\n$$\nw_{1}(v, c) \\triangleq \\begin{cases}v-\\mathbb{E}_{\\tilde{v} \\sim G}\\left[\\gamma^{-1}(\\tilde{v}) \\mid \\tilde{v} \\in[\\gamma(c), \\hat{v}]\\right] & \\text { if } \\quad v \\in[\\gamma(c), \\hat{v}] \\\\ v-\\mathbb{E}_{\\tilde{v} \\sim G}\\left[\\gamma^{-1}(\\tilde{v}) \\mid \\tilde{v} \\in\\left[\\hat{v}, v_{i+1}\\right]\\right] & \\text { if } \\quad v \\in\\left[\\hat{v}, v_{i+1}\\right] .\\end{cases}\n$$\n\nNote that by applying Operation (i), the ex post payoff of the buyer with value $v \\in$ $[\\gamma(c), \\hat{v}]$ increases because of the lower price and that of $v \\in\\left[\\hat{v}, v_{i+1}\\right]$ decreases because of the higher price.\n\nFor each $k \\in\\{0,1\\}$, consider the distribution of $w_{k}(v, c)$ when $v \\sim G\\left(\\cdot \\mid \\tilde{v} \\in\\left[\\gamma(c), v_{i+1}\\right]\\right)$. First, they have the same mean because the expected price remains the same before and after the operation. Second, because $w_{1}(\\cdot, c)$ crosses $w_{0}(\\cdot, c)$ once from above, the CDF of $w_{1}(v, c)$ crosses the CDF of $w_{0}(v, c)$ once from above. The equal mean property and the single-crossing property imply, by Theorem 3.A. 44 (Condition 3.A.59) of Shaked and Shanthikumar (2007), that $w_{0}(v, c)$ is a mean-preserving spread of $w_{1}(v, c)$ when $v \\sim G\\left(\\cdot \\mid \\tilde{v} \\in\\left[\\gamma(c), v_{i+1}\\right]\\right)$.\n\nTherefore, for a fixed $c$, the buyer's ex post surplus conditional on $v \\in\\left[\\gamma(c), v_{i+1}\\right]$ under signal $\\mathcal{I}_{L}$ is a mean-preserving spread of that under signal $\\mathcal{I}_{H}$. The same relationship trivially holds for the ex post surpluses of value $v<\\gamma(c)$ or $v>v_{i+1}$ because those types do not trade or continue to face the same price. In summary, for any fixed $c$, the buyer's ex post surplus under signal $\\mathcal{I}_{L}$ is a mean-preserving spread of that under signal $\\mathcal{I}_{H}$ conditional on each of the three cases, $v \\in\\left[\\gamma(c), v_{i+1}\\right], v<\\gamma(c)$, and $v>v_{i+1}$. The mean-preserving spread relationship is closed under mixtures (e.g., Theorem 3.A.12(b) of Shaked and Shanthikumar (2007)). Thus, for any fixed $c$, the distribution of the\nbuyer's surplus under signal $\\mathcal{I}_{L}$ is a mean-preserving spread of the distribution of the buyer's surplus under signal $\\mathcal{I}_{H}$.\n\nOperation (ii). We can apply the same logic as the case of Operation (i) by defining $w_{1}(v, c)$ as\n\n$$\nw_{1}(v, c) \\triangleq v-\\gamma^{-1}(v), \\forall v \\in\\left[v_{i}, v_{i+1}\\right] .\n$$ $\\square$","text_sha256":"f032ebea2b3ea665685930b37ccbc4f9ad438906f145d9dfbcd550d5d48f0e33"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0029","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"E Proof of Proposition 5","text":"## E Proof of Proposition 5\n\nThe proof follows that of Proposition 1. The designer's problem is stated as follows:\n\n$$\n\\begin{array}{rlr}\n\\max _{\\hat{v}:[0,1] \\rightarrow[0,1], p:[0,1] \\rightarrow \\mathbb{R}_{+}} & \\int_{0}^{1} \\int_{\\hat{v}(c)}^{1} \\alpha(v-p(c))+(1-\\alpha)(p(c)-c) \\mathrm{d} G \\mathrm{~d} F & \\\\\n\\text { s.t. } & \\int_{\\hat{v}(c)}^{1}(p(c)-c) \\mathrm{d} G \\geq \\int_{\\hat{v}\\left(c^{\\prime}\\right)}^{1}\\left(p\\left(c^{\\prime}\\right)-c\\right) \\mathrm{d} G & \\forall c, c^{\\prime} \\in[0,1] \\\\\n& \\int_{\\hat{v}(c)}^{1}(p(c)-c) \\mathrm{d} G \\geq 0 & \\forall c \\in[0,1]\n\\end{array}\n$$\n\nFor $\\alpha>0$, the designer's objective can also be written as\n\n$$\n\\int_{0}^{1} \\int_{\\hat{v}(c)}^{1} v-p(c)+\\frac{1-\\alpha}{\\alpha}(p(c)-c) \\mathrm{d} G \\mathrm{~d} F .\n$$\n\nRepeat the same step to rewrite $\\int_{\\hat{v}(c)}^{1} v \\mathrm{~d} G$ as in the proof of Proposition 1. We can then write the designer's problem as\n\n$$\n\\begin{array}{cl}\n\\max _{q:[0,1] \\rightarrow[0,1], t:[0,1] \\rightarrow \\mathbb{R}_{+}} \\int_{0}^{1}\\left[V(q(c))-t(c)+\\frac{1-\\alpha}{\\alpha}(t(c)-c q(c))\\right] \\mathrm{d} F, & \\\\\n\\text { s.t. } t(c)-c q(c) \\geq t\\left(c^{\\prime}\\right)-c q\\left(c^{\\prime}\\right) & \\forall c, c^{\\prime} \\in[0,1], \\\\\nt(c)-c q(c) \\geq 0 & \\forall c \\in[0,1] .\n\\end{array}\n$$\n\nWe consider two cases.\n\nCase 1: $\\alpha \\geq \\frac{1}{2}$ When the designer places a weakly higher weight on buyer surplus than seller profit (i.e., when $\\alpha \\geq \\frac{1}{2}$ ), Problem (17) is analogous to the problem analyzed by Baron and Myerson (1982), where the weight \" $\\alpha$ \" in their Lemma 2 is replaced by $\\frac{1-\\alpha}{\\alpha}$ for our proof. Thus, its solution sets the optimal quantity to equalize marginal welfare benefits with $\\alpha$-virtual costs and the optimal transfer to guarantee the incentivecompatible profit distribution:\n\n$$\n\\begin{aligned}\nV^{\\prime}(q(c)) & =\\gamma_{\\alpha}(c) \\\\\nt(c)-q(c) c & =\\int_{c}^{1} q(x) \\mathrm{d} x=\\int_{c}^{1} 1-G\\left(\\gamma_{\\alpha}(x)\\right) \\mathrm{d} x\n\\end{aligned}\n$$\n\nThe rest of of the proof is identical with that of Proposition 1, where we replace $\\gamma(c)$ in that proof with $\\gamma_{\\alpha}(c)$.\n\nWe use the following observation for the next case: At $\\alpha=\\frac{1}{2}$, the $\\alpha$-virtual costs equals the seller's true cost. Consequently, under the $\\frac{1}{2}$-optimal algorithm, type $c$ sets a price of\n\n$$\ny_{\\frac{1}{2}}(c) \\triangleq \\mathbb{E}_{\\tilde{v} \\sim G}[\\tilde{v} \\mid \\tilde{v} \\geq c],\n$$\n\nand the algorithm recommends the product if and only if $v \\geq c$. As a result, the product allocation is efficient and buyer surplus is 0.\n\nCase 2: $\\alpha<\\frac{1}{2}$ In this case, the designer places a strictly higher weight on seller profit than buyer surplus. We show that the solution to the designer's problem with weight $\\alpha^{\\prime}<\\frac{1}{2}$ is equal to the solution with weight $\\alpha=\\frac{1}{2}$. Suppose to the contrary that between the solution with weight $\\alpha^{\\prime}$ and the solution with weight $\\alpha$, either buyer surplus or seller profit is different. Then, both buyer surplus and seller profit must be different. For example, if the two solutions have the same buyer surplus but the solution with weight $\\alpha^{\\prime}$ has a strictly higher seller profit, then the designer with $\\alpha=\\frac{1}{2}$ would mimic the designer with weight $\\alpha^{\\prime}$.\n\nIf both buyer surplus and seller profit are different, then it must be the case that\nunder the solution with a higher weight on seller profit, $\\alpha^{\\prime}$, the seller profit is strictly higher and the buyer surplus is strictly lower than under $\\alpha=\\frac{1}{2}$. However, it would mean that buyer surplus at $\\alpha^{\\prime}$ would be negative because buyer surplus at $\\alpha=\\frac{1}{2}$ is 0. This is a contradiction, because in equilibrium, the buyer can secure zero payoffs by not purchasing anything. Therefore, the solution to the designer's problem with weight $\\alpha^{\\prime}<\\frac{1}{2}$ is equal to the solution at weight $\\alpha=\\frac{1}{2}$. In terms of the $\\alpha$-virtual costs, this means that we can use (6) for any $\\alpha \\in[0,1]$. $\\square$","text_sha256":"036801e6bff7d7aef08b103a97a4ed612ccdf906b2dc26053d074ddf40391cda"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0030","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"F Proofs of Proposition 6 and Proposition 7","text":"## F Proofs of Proposition 6 and Proposition 7\n\nWe prove a result that implies both Proposition 6 and Proposition 7 as corollaries. Assume that the sellers face an information structure $\\mathcal{I}=(S, \\pi)$ that consists of a set $S=\\times_{j} S_{j}$ of signal realizations $s_{j} \\in S_{j}$ privately observed by each seller and a family of probability distributions $\\{\\pi(\\cdot \\mid v)\\}_{v \\in[0,1]^{J}}$ over $S$. We write $\\tilde{s}_{j}$ for seller $j$ 's signal as a random variable and $s_{j} \\in S_{j}$ for a generic realization.\n\nDenote the set of real sellers by $\\mathcal{J} \\triangleq\\{1, \\ldots, J\\}$ and the set of all sellers, together with a dummy seller, by $\\mathcal{J}_{0} \\triangleq\\{0,1, \\ldots, J\\}$. When we say that the buyer purchases from (or transacts with) seller 0, it means that the buyer does not purchase from any seller $j=1, \\ldots, J$. The profiles of the signal realizations, values, types, and prices are denoted as $\\mathbf{s}, \\mathbf{v}, \\mathbf{c}$, and $\\mathbf{p}$, respectively. When we refer to a profile that excludes seller $j$, we use notations such as $\\mathbf{s}_{-j}$ and $\\mathbf{v}_{-j}$. With a slight abuse of notation, we write $F, G, F_{-j}$, and $G_{-j}$, for the distributions of $\\mathbf{c}, \\mathbf{v}, \\mathbf{c}_{-j}$, and $\\mathbf{v}_{-j}$, respectively. Unless otherwise stated, these vectors and joint distributions exclude the dummy seller.\n\nRecall that we defined an auxiliary random variable\n\n$$\n\\theta_{j}=v_{j}-\\max _{k \\in \\mathcal{J}_{0} \\backslash\\{j\\}}\\left\\{v_{k}-\\gamma_{k}\\left(c_{k}\\right)\\right\\} .\n$$\n\nFor each $j$, let $\\bar{v}_{j}\\left(s_{j}\\right)$ be the supremum of the support of the posterior distribution of $v_{j}$ conditional on $\\tilde{s}_{j}=s_{j}$. Define $\\bar{c}_{j}\\left(s_{j}\\right)=\\gamma_{j}^{-1}\\left(\\bar{v}_{j}\\left(s_{j}\\right)\\right)$. For each $j \\in \\mathcal{J}, s_{j} \\in S_{j}$, and\n$c_{j} \\in\\left[0, \\bar{c}_{j}\\left(s_{j}\\right)\\right]$, define\n\n$$\np_{j}^{*}\\left(c_{j}, s_{j}\\right) \\triangleq \\mathbb{E}_{\\theta_{j}}\\left[\\gamma_{j}^{-1}\\left(\\theta_{j}\\right) \\mid \\theta_{j} \\geq \\gamma_{j}\\left(c_{j}\\right), \\tilde{s}_{j}=s_{j}\\right] .\n$$\n\nThe conditional expectation is well-defined for any $c_{j} \\leq \\bar{c}_{j}\\left(s_{j}\\right)$ or equivalently $\\gamma_{j}\\left(c_{j}\\right) \\leq$ $\\bar{v}_{j}\\left(s_{j}\\right)$ because $\\theta_{j}=\\bar{v}_{j}\\left(s_{j}\\right)$ is in the support of the posterior distribution of $v_{j}$ conditional on $\\tilde{s}_{j}=s_{j}$. This is because $v_{j}=\\bar{v}_{j}\\left(s_{j}\\right)$ is in the support, and $v_{k} \\leq \\gamma_{k}\\left(c_{k}\\right)$ for all $k \\neq j$ could occur with a positive probability. Given signal realizations $\\mathbf{s}$, we say that seller $j$ 's type $c_{j}$ is active if $c_{j} \\leq \\bar{c}_{j}\\left(s_{j}\\right)$. Otherwise, the type is inactive. Seller $j$ 's price $p_{j}$ is said to be active if $p_{j}$ is in the range of $p_{j}^{*}\\left(\\cdot, s_{j}\\right)$; otherwise, the price is called inactive. Note that any active type sets an active price.\n\nIn this appendix, for simplicity, we focus on the case in which for each seller $j$ and active price $p_{j}$, there exists a unique type, denoted by $p_{j}^{*-1}\\left(p_{j}, s_{j}\\right)$, that solves $p_{j}^{*}\\left(c_{j}, s_{j}\\right)=p_{j}$. This is the case, for example, if value $v_{j}$ has a full support on [0, 1] conditional on each signal realization $s_{j}$. The proof for the case in which multiple types may set the same price is relegated to the Supplementary Material. ${ }^{21}$ For any inactive price, we set $p_{j}^{*-1}\\left(p_{j}, s_{j}\\right)=1$; for the dummy seller $j=0$, we set $p_{j}^{*-1}\\left(p_{j}, s_{j}\\right)=0$.\n\nWe now define an algorithm that we will prove to be optimal for the buyer.\nDefinition 3. Define the candidate algorithm as follows: At each profile of signal realizations $\\mathbf{s}=\\left(s_{1}, \\ldots, s_{J}\\right)$, values $\\mathbf{v}=\\left(v_{1}, \\ldots, v_{J}\\right)$, and prices $\\mathbf{p}=\\left(p_{1}, \\ldots, p_{J}\\right)$, the candidate algorithm recommends trading with seller $j^{*}(\\mathbf{v}, \\mathbf{p}, \\mathbf{s})$ such that\n\n$$\nj^{*}(\\mathbf{v}, \\mathbf{p}, \\mathbf{s}) \\in \\underset{j \\in \\mathcal{J}_{0}}{\\operatorname{argmax}} v_{j}-\\gamma_{j}\\left(p_{j}^{*-1}\\left(p_{j}, s_{j}\\right)\\right) .\n$$\n\nIf multiple sellers attain the maximized value in Equation 20, the algorithm breaks ties in favor of sellers such that $p_{j}^{*-1}\\left(p_{j}, s_{j}\\right)>0$. Other than this restriction, ties are broken arbitrarily.\n\nThe following result characterizes the buyer-optimal algorithm and equilibrium under\n\n[^15]any information structure.\n\nProposition 11 (Market Segmentation with Competing Sellers). For any information structure $\\mathcal{I}$, the corresponding candidate algorithm is a buyer-optimal algorithm. In equilibrium, seller $j$ of type $c_{j} \\leq \\bar{c}_{j}\\left(s_{j}\\right)$ posts price $p_{j}^{*}\\left(c_{j}, s_{j}\\right)$, and any type $c_{j}>\\bar{c}_{j}\\left(s_{j}\\right)$ sets some inactive price above 1. Under this algorithm and pricing, for any realized profile of values and costs, the buyer trades with seller $j^{*} \\in \\operatorname{argmax}_{j \\in \\mathcal{J}_{0}} v_{j}-\\gamma_{j}\\left(c_{j}\\right)$. Moreover, the profit of any seller of any type and the total buyer surplus are independent of $\\mathcal{I}$.\n\nProof. The proof consists of three steps.\nStep 1: Characterizing a buyer-optimal direct mechanism. First, we derive a buyeroptimal direct mechanism, where the direct mechanism is in the sense of Myerson (1981), i.e., a mechanism that fully controls product allocation and transfers across all players. By the revelation principle, since any information structure combined with an algorithm can be viewed as an indirect mechanism, a buyer-optimal mechanism must achieve a weakly higher buyer surplus than a buyer-optimal algorithm under any information structure.\n\nGiven a profile of values $\\mathbf{v}$ and reported types $\\mathbf{c}$, let $q_{j}(\\mathbf{v}, \\mathbf{c})$ be the probability of allocating seller $j$ 's product to the buyer, and let $t_{j}(\\mathbf{v}, \\mathbf{c})$ be the monetary transfer from the buyer to seller $j$. The direct mechanism design problem can be written as:","text_sha256":"fcd36fec5e0430a88a19c84ae5961343241958a938879a28757b2f9681872577"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0031","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"F Proofs of Proposition 6 and Proposition 7","text":"$$\n\\begin{aligned}\n\\max _{q:[0,1]^{2 J} \\rightarrow[0,1], t:[0,1]^{2 J} \\rightarrow \\mathbb{R}} \\sum_{j=1}^{J} \\int_{[0,1]^{J}} \\int_{[0,1]^{J}}\\left(v_{j} q_{j}(\\mathbf{v}, \\mathbf{c})-t_{j}(\\mathbf{v}, \\mathbf{c})\\right) \\mathrm{d} F \\mathrm{~d} G & \\\\\n\\text { s.t. } & T_{j}\\left(c_{j}\\right)-c_{j} Q_{j}\\left(c_{j}\\right) \\geq T_{j}\\left(c_{j}^{\\prime}\\right)-c_{j} Q_{j}\\left(c_{j}^{\\prime}\\right), \\\\\n& T_{j}\\left(c_{j}\\right)-c_{j} Q_{j}\\left(c_{j}\\right) \\geq 0, \\\\\n& Q_{j}\\left(c_{j}\\right)=\\int_{[0,1]^{J}} \\int_{[0,1]^{J-1}} q_{j}\\left(\\mathbf{v}, c_{j}, \\mathbf{c}_{-j}\\right) \\mathrm{d} F_{-j} \\mathrm{~d} G, \\quad \\forall j \\in \\mathcal{J}, c_{j}, c_{j}^{\\prime} \\in[0,1], \\\\\n& T_{j}\\left(c_{j}\\right)=\\int_{[0,1]^{J}} \\int_{[0,1]^{J-1}} t_{j}\\left(\\mathbf{v}, c_{j}, \\mathbf{c}_{-j}\\right) \\mathrm{d} F_{-j} \\mathrm{~d} G, \\quad \\forall j \\in \\mathcal{J}, c_{j} \\in[0,1], \\\\\n& \\sum_{j \\in \\mathcal{J}} q_{j}(\\mathbf{v}, \\mathbf{c}) \\leq 1,\n\\end{aligned}\n$$\n\nThe standard mechanism design arguments imply that only the participation constraint of $c_{j}=1$ for each seller $j$ binds at the optimum, and that the IC constraints are equivalent to the local IC constraints (see Equation 22 below) with $Q_{j}(\\cdot)$ being weakly decreasing for each $j$ (cf. Baron and Myerson (1982)). Using the local IC constraints, we can rewrite the expected transfer as follows:\n\n$$\n\\begin{aligned}\n\\sum_{j=1}^{J} \\int_{[0,1]^{J}} \\int_{[0,1]^{J}} t_{j}(\\mathbf{v}, \\mathbf{c}) \\mathrm{d} F \\mathrm{~d} G & =\\sum_{j=1}^{J} \\int_{0}^{1} T_{j}(x) f_{j}(x) \\mathrm{d} x \\\\\n& =\\sum_{j=1}^{J} \\int_{0}^{1}\\left(x+\\frac{F_{j}(x)}{f_{j}(x)}\\right) Q_{j}(x) f(x) \\mathrm{d} x \\\\\n& =\\sum_{j=1}^{J} \\int_{0}^{1} \\gamma_{j}(x) Q_{j}(x) f(x) \\mathrm{d} x\n\\end{aligned}\n$$\n\nPlugging this into the objective and using $Q_{j}(x)=\\int_{[0,1]^{J}} \\int_{[0,1]^{J-1}} q_{j}\\left(\\mathbf{v}, x, \\mathbf{c}_{-j}\\right) \\mathrm{d} F_{-j} \\mathrm{~d} G$, we can rewrite the designer's problem as the choice of a product allocation rule to maximize virtual surplus:\n\n$$\n\\int_{[0,1]^{J}} \\int_{[0,1]^{J}} \\sum_{j=1}^{J}\\left(v_{j}-\\gamma_{j}\\left(c_{j}\\right)\\right) q_{j}(\\mathbf{v}, \\mathbf{c}) f(\\mathbf{c}) \\mathrm{d} \\mathbf{c} \\mathrm{~d} G .\n$$\n\nWe can maximize the virtual surplus by choosing $\\left\\{q_{j}(\\mathbf{v}, \\mathbf{c})\\right\\}_{j \\in \\mathcal{J}}$ to maximize the integrand for each $(\\mathbf{v}, \\mathbf{c})$. The optimal mechanism allocates seller $j$ 's product to the buyer, $q_{j}(\\mathbf{v}, \\mathbf{c})=1$, if seller $j$ has the highest virtual surplus $v_{j}-\\gamma_{j}\\left(c_{j}\\right)$ and it is nonnegative; otherwise, $q_{j}(\\mathbf{v}, \\mathbf{c})=0$. Let $q^{D}$ be this optimal product allocation rule and $Q_{j}^{D}\\left(c_{j}\\right)$ be the interim allocation probability for seller $j$ with type $c_{j}$. Under the optimal mechanism, the monetary transfer $T^{D}$ must satisfy\n\n$$\nT_{j}^{D}\\left(c_{j}\\right)=Q^{D}\\left(c_{j}\\right) c_{j}+\\int_{c_{j}}^{1} Q_{j}^{D}(x) \\mathrm{d} x, \\forall c_{j} \\in[0,1]\n$$\n\nUnder the optimal mechanism, the participation constraints for the highest types bind and thus each seller $j$ with the type $c_{j}=1$ earns zero profit.\n\nStep 2: Connecting with the candidate algorithm. In this step, we show that the candi-\ndate algorithm has an equilibrium in which the product allocation rule and the profits of the highest types are the same as those in the optimal direct mechanism.\n\nFirst, suppose that each seller follows the pricing strategy described in the proposition. Take any profile of signal realizations $\\mathbf{s}$, values $\\mathbf{v}$, and types $\\left(\\hat{c}_{1}, \\ldots, \\hat{c}_{J}\\right)$. Let $\\mathbf{p}$ be the resulting price profile posted by the sellers. For each seller that posts an active price $p_{j}$, the candidate algorithm calculates the unique type that sets price $p_{j}$ according to Equation 19 and recommends a seller that maximizes virtual surplus. Also, the candidate algorithm never recommends a seller that sets an inactive price, because their corresponding virtual surplus is always negative. Thus if all sellers use the pricing rule in Equation 19 and the buyer always follows the recommendations, then for any profile of prices that can arise, the candidate algorithm recommends the product of the seller with the highest virtual surplus and thus induces the same product allocation as the buyer-optimal mechanism. In particular, the buyer never purchases the product from a seller who has a negative virtual surplus, which means that seller $j$ with $c_{j}=1$ earns zero profit.\n\nWe now show that the pricing rule in Equation 19 is indeed an equilibrium if the buyer always purchases the recommended product. Take any active seller $j \\in \\mathcal{J}$ with type $c_{j}$. The seller cannot profit from setting an inactive price, because the candidate algorithm never recommends a seller at an inactive price. Alternatively, suppose that the seller has type $c_{j}$ but deviates to an active price which would be chosen by type $c_{j}^{\\prime}$. Let $H$ be the distribution of $\\gamma_{j}^{-1}\\left(\\theta_{j}\\right)$ conditional on $\\tilde{s}_{j}=s_{j}$. We can compare the profits without and with the deviation as follows:\n\n$$\n\\begin{aligned}\n& \\operatorname{Pr}\\left(\\gamma_{j}^{-1}\\left(\\theta_{j}\\right)-c_{j} \\geq 0 \\mid \\tilde{s}_{j}=s_{j}\\right) \\cdot \\mathbb{E}_{\\theta_{j}}\\left[\\gamma_{j}^{-1}\\left(\\theta_{j}\\right)-c_{j} \\mid \\gamma_{j}^{-1}\\left(\\theta_{j}\\right)-c_{j} \\geq 0, \\tilde{s}_{j}=s_{j}\\right] \\\\\n= & \\int_{c_{j}}^{1}\\left(x-c_{j}\\right) d H(x) \\\\\n\\geq & \\int_{c_{j}^{\\prime}}^{1}\\left(x-c_{j}\\right) d H(x) \\\\\n= & \\operatorname{Pr}\\left(\\gamma_{j}^{-1}\\left(\\theta_{j}\\right)-c_{j}^{\\prime} \\geq 0 \\mid \\tilde{s}_{j}=s_{j}\\right) \\cdot \\mathbb{E}_{\\theta_{j}}\\left[\\gamma_{j}^{-1}\\left(\\theta_{j}\\right)-c_{j} \\mid \\gamma_{j}^{-1}\\left(\\theta_{j}\\right)-c_{j}^{\\prime} \\geq 0, \\tilde{s}_{j}=s_{j}\\right]\n\\end{aligned}\n$$\n\nHere, the first line is the profit from following the candidate strategy, and the last line is the profit from deviation.\n\nThe other case is when a deviating seller $j$ has an inactive type $c_{j}$. In this case, conditional on $s_{j}$, any possible realization of $\\theta_{j}$ satisfies $\\gamma_{j}\\left(c_{j}\\right)>\\theta_{j}$. Thus, the profit from the deviation to active type $c_{j}^{\\prime}$, which is given by the last line of the above inequalities, will be negative. We conclude that each seller has no profitable deviation.","text_sha256":"fca4a3f5bf62364ed71bccceb131a523b6cf4716f56578271e2e70befb96a4ac"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0032","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"F Proofs of Proposition 6 and Proposition 7","text":"The last part of this step is to show that the buyer is willing to purchase the product whenever recommended, i.e., conditional on knowing the identity and the price of the recommended product. We present a substantially stronger statement: The buyer is willing to follow recommendations even if she additionally observes the realized signal $s_{j}$ and the type $c_{j}$ of the recommended seller $j$. For each $c_{j}$, we have:\n\n$$\n\\begin{aligned}\n& \\mathbb{E}\\left[v_{j} \\mid v_{j}-\\gamma_{j}\\left(c_{j}\\right) \\geq \\max _{k \\in \\mathcal{J}_{0} \\backslash\\{j\\}} v_{k}-\\gamma_{k}\\left(c_{k}\\right), \\tilde{s}_{j}=s_{j}\\right] \\\\\n= & \\mathbb{E}\\left[v_{j} \\mid \\gamma^{-1}\\left(\\theta_{j}\\right) \\geq c_{j}, \\tilde{s}_{j}=s_{j}\\right] \\\\\n\\geq & \\mathbb{E}\\left[\\gamma_{j}^{-1}\\left(\\theta_{j}\\right) \\mid \\gamma^{-1}\\left(\\theta_{j}\\right) \\geq c_{j}, \\tilde{s}_{j}=s_{j}\\right] \\\\\n= & p_{j}^{*}\\left(c_{j}, s_{j}\\right),\n\\end{aligned}\n$$\n\nwhere the inequality holds because:\n\n$$\nv_{j} \\geq \\theta_{j}=v_{j}-\\max _{k \\in \\mathcal{J}_{0} \\backslash\\{j\\}}\\left\\{v_{k}-\\gamma_{k}\\left(c_{k}\\right)\\right\\} \\geq \\gamma_{j}^{-1}\\left(\\theta_{j}\\right) .\n$$\n\nStep 3: Establishing the \"payoff equivalence.\" Let $\\left\\{\\left(Q_{j}\\left(c_{j}\\right), T_{j}\\left(c_{j}\\right)\\right\\}_{j \\in \\mathcal{J}, c_{j} \\in[0,1]}\\right.$ be the interim allocation probability $Q_{j}\\left(c_{j}\\right)$ and expected revenue $T_{j}\\left(c_{j}\\right)$ for each seller $j$ and type $c_{j}$ under the candidate algorithm. Recall that $\\left\\{\\left(Q_{j}^{D}\\left(c_{j}\\right), T_{j}^{D}\\left(c_{j}\\right)\\right\\}_{j \\in \\mathcal{J}, c_{j} \\in[0,1]}\\right.$ denote the corresponding objects in the optimal direct mechanism.\n\nWe have shown that (i) $\\left\\{\\left(Q_{j}\\left(c_{j}\\right), T_{j}\\left(c_{j}\\right)\\right\\}_{j \\in \\mathcal{J}, c_{j} \\in[0,1]}\\right.$ is an equilibrium object and thus satisfies the first two constraints of Equation 21, i.e., the incentive compatibility and participation constraints; (ii) in the candidate algorithm, the profits of the highest seller types are 0; and (iii) $Q_{j}=Q_{j}^{D}$ for each seller $j$ because they come from the same ex\npost product allocation rule. Thus, the interim expected revenue of each seller $j$ under the candidate algorithm must satisfy\n\n$$\n\\begin{aligned}\nT_{j}\\left(c_{j}\\right) & =Q\\left(c_{j}\\right) c_{j}+\\int_{c_{j}}^{1} Q_{j}(x) d x \\\\\n& =Q^{D}\\left(c_{j}\\right) c_{j}+\\int_{c_{j}}^{1} Q_{j}^{D}(x) d x \\\\\n& =T_{j}^{D}\\left(c_{j}\\right)\n\\end{aligned}\n$$\n\nwhere the first equality comes from (i) and (ii), the second from (iii), and the third from Equation 22. Therefore, the interim profit of each seller, $T_{j}\\left(c_{j}\\right)-Q\\left(c_{j}\\right) c_{j}$, is the same between the candidate algorithm and the optimal mechanism. As a result, the buyer surplus, which is the total surplus (uniquely determined by $q^{D}$ ) minus the seller profit, is the same between the algorithm described in the statement and the optimal mechanism in Step 1. $\\square$\n\nProofs of Proposition 6 and Proposition 7 Proposition 6 holds by setting information structure $\\mathcal{I}$ to the uninformative structure, e.g., $S_{j}=\\{\\emptyset\\}$ for each seller $j$. Proposition 7 is a direct corollary of Proposition 11. $\\square$","text_sha256":"5aa90b3dbe3decd936153be4595bd4c411a775a64700388509ca0631445f12bb"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0033","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"G Proof of Proposition 8","text":"## G Proof of Proposition 8\n\nThe proof follows that of Proposition 4. Consider any seller $j$ with type $c_{j}$. Consider a fictitious situation in which seller $j$ 's prior belief over $\\left(\\mathbf{v}, \\mathbf{c}_{-j}\\right)$ is given by the conditional distribution of $\\left(\\mathbf{v}, \\mathbf{c}_{-j}\\right)$ given $\\theta_{j} \\geq c_{j}$ (calculated starting from the true prior). We call seller $j$ 's prior in this situation as the fictitious prior and any posterior belief updated from the fictitious prior a fictitious posterior.\n\nUnder any information structure, the price posted by seller $j$ with type $c_{j}$, which is (19), is linear in seller $j$ 's fictitious posterior belief over $\\left(\\mathbf{v}, \\mathbf{c}_{-j}\\right)$ (which pins down $\\theta_{j}$ ) conditional on signal realization $s_{j}$. If seller $j$ obtains more information about v in the sense stated in the proposition, the seller also gains more information about $\\left(\\mathbf{v}, \\mathbf{c}_{-j}\\right)$. Thus, the distribution of seller $j$ 's fictitious posterior belief over $\\left(\\mathbf{v}, \\mathbf{c}_{-j}\\right)$ undergoes a\nmean-preserving spread. The same logic as in the proof of Proposition 4 implies that seller $j$ 's price undergoes a mean-preserving spread. $\\square$","text_sha256":"3161f70f25e2e333e4a05892262798addf6efcd26c03d51acd01f09b3e84f58f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0034","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"H Proof of Proposition 10","text":"## H Proof of Proposition 10\n\nWe borrow the notation from the proof of Proposition 1 and let $\\mu=\\mathbb{E}_{v \\sim G}[v]$. Suppose that the buyer faces the optimal algorithm of Proposition 1. Because the buyer is willing to follow the algorithm's recommendation to purchase, it suffices to show that the buyer is also willing to follow the recommendation to not purchase. This constraint is equivalent to the condition that the buyer's ex ante payoff from following the recommendation weakly exceeds the payoff from always buying the product regardless of the recommendation. For any active price $p \\in[0, p(\\bar{c}))$, the condition is written as\n\n$$\nV(q(c))-t(c) \\geq \\mu-p(c)\n$$\n\nor\n\n$$\nV(q(c))-c q(c)-\\int_{c}^{1} q(x) d x \\geq \\mu-c-\\frac{\\int_{c}^{1} q(x) \\mathrm{d} x}{q(c)} .\n$$\n\nBecause $q(c) \\leq 1$, a sufficient condition for inequality (23) is\n\n$$\nV(q(c))-c q(c) \\geq \\mu-c .\n$$\n\nWe can rewrite this inequality as\n\n$$\n\\int_{\\gamma(c)}^{1} v \\mathrm{~d} G(v)-c \\int_{\\gamma(c)}^{1} 1 \\mathrm{~d} G(v) \\geq \\int_{0}^{1} v \\mathrm{~d} G(v)-c \\int_{0}^{1} 1 \\mathrm{~d} G(v),\n$$\n\nor, equivalently,\n\n$$\n\\int_{0}^{\\gamma(c)}[v-c] \\mathrm{d} G(v) \\leq 0 .\n$$\n\nFinally, the buyer follows the recommendation to not buy the product at any price $p$ that is not active, i.e., $p \\geq p^{*}(\\bar{c})$. Recall that the buyer-optimal algorithm provides no information about $v$ at price $p>p^{*}(\\bar{c})$. Plugging $c=\\bar{c}$ into $\\int_{0}^{\\gamma(c)}[v-c] \\mathrm{d} G(v) \\leq 0$, we\nobtain $\\mu-\\bar{c} \\leq 0$. Thus, if $p>p^{*}(\\bar{c})$, we have $\\mu-p \\leq \\mu-p^{*}(\\bar{c})=\\mu-\\bar{c} \\leq 0$. $\\square$","text_sha256":"24c17f5549b6c9d31f91e6584efec3e8ff52c764edc21a40aed5ba89f9902364"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0035","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Supplementary Material for Appendix F","text":"## Supplementary Material for Appendix F\n\nIn the appendix, we assumed that each active price is posted by a unique type. In this Supplementary Material, we drop this assumption and prove that the candidate algorithm continues to maximize virtual surplus.\n\nRecall the pricing equation (Equation 19). For each $j \\in \\mathcal{J}, s_{j} \\in S_{j}$, and active price $p_{j} \\in \\mathbb{R}$, define\n\n$$\nC_{j}\\left(p_{j}, s_{j}\\right) \\triangleq\\left\\{c_{j} \\in[0,1]: p_{j}^{*}\\left(c_{j}, s_{j}\\right)=p_{j}\\right\\}\n$$\n\nas the set of the types of seller $j$ that choose price $p_{j}$. For each $p_{j}$, we define function $p_{j}^{*-1}\\left(p_{j}, s_{j}\\right)$ as follows: For each $j \\in \\mathcal{J}$,\n\n$$\np_{j}^{*-1}\\left(p_{j}, s_{j}\\right)= \\begin{cases}\\max C_{j}\\left(p_{j}, s_{j}\\right) & \\text { if } \\quad C_{j}\\left(p_{j}, s_{j}\\right) \\neq \\emptyset \\\\ 1 & \\text { if } \\quad C_{j}\\left(p_{j}, s_{j}\\right)=\\emptyset .\\end{cases}\n$$\n\nFor the dummy seller $j=0$, we set $p_{j}^{*-1}\\left(p_{j}, s_{j}\\right)=0$.\nSuppose that each seller follows the pricing strategy described in the proposition. Take any profile of signal realizations $\\mathbf{s}$, values $\\mathbf{v}$, and types $\\left(\\hat{c}_{1}, \\ldots, \\hat{c}_{J}\\right)$. Let $\\mathbf{p}$ be the resulting price profile posted by the sellers. Suppose that the candidate algorithm recommends seller $j^{*} \\in \\mathcal{J}$. Without loss, assume $j^{*}=1$. We show that seller 1 has the highest, nonnegative virtual surplus. For each seller $j \\in \\mathcal{J}$, there exists some $c_{j}$ such that\n\n$$\nv_{j}-\\gamma_{j}\\left(p_{j}^{*-1}\\left(p_{j}, s_{j}\\right)\\right)=v_{j}-\\gamma_{j}\\left(c_{j}\\right) .\n$$\n\nIf $p_{j}$ is an active price, $c_{j}=\\max C_{j}\\left(p_{j}, s_{j}\\right)$. If $p_{j}$ is an inactive price, $c_{j}=1$ by construction. Let $c_{j}\\left(p_{j}\\right)$ be the type that satisfies Equation 24. Then Equation 20 implies that\n\n$$\nv_{1}-\\gamma_{1}\\left(c_{1}\\left(p_{1}\\right)\\right) \\geq \\max _{k \\in \\mathcal{J}_{0} \\backslash\\{1\\}} v_{k}-\\gamma_{k}\\left(c_{k}\\left(p_{k}\\right)\\right),\n$$\n\nso that seller 1 has the largest virtual surplus under type profile $\\left(c_{1}\\left(p_{1}\\right), \\ldots, c_{J}\\left(p_{J}\\right)\\right)$. The rest of the proof is devoted to showing that seller 1 has the largest virtual surplus under\ntype profile $\\left(\\hat{c}_{1}, \\ldots, \\hat{c}_{J}\\right)$ as well, i.e.,\n\n$$\nv_{1}-\\gamma_{1}\\left(\\hat{c}_{1}\\right) \\geq \\max _{k \\in \\mathcal{J}_{0} \\backslash\\{1\\}} v_{k}-\\gamma_{k}\\left(\\hat{c}_{k}\\right) .\n$$\n\nNote that if $p_{k}$ is inactive for some non-recommended seller $k$, any type $\\hat{c}_{k}$ that sets $p_{k}$ satisfies $\\hat{c}_{k}>\\bar{c}_{k}\\left(s_{k}\\right)$, which implies $v_{k}-\\gamma_{k}\\left(\\hat{c}_{k}\\right)<0$. Thus, replacing one inactive type $c_{k}\\left(p_{k}\\right)$ with another inactive type $\\hat{c}_{k}$ in the RHS of Equation 26 does not affect the inequality (and it does not affect the argument below). ${ }^{22}$ Thus, to simplify exposition, we assume that all sellers set active prices, or equivalently, $c_{k}\\left(p_{k}\\right) \\leq \\bar{c}_{k}\\left(s_{k}\\right)$ for each $k \\in \\mathcal{J}$. We change the type of each seller from $c_{j}\\left(p_{j}\\right)$ to $\\hat{c}_{j}$ one by one and show that seller 1 continues to maximize virtual surplus at each step.\n\nFirst, suppose that we change the type of seller 1 from $c_{1}\\left(p_{1}\\right)$ to $\\hat{c}_{1}$. Because $c_{1}\\left(p_{1}\\right)=$ $\\max C_{1}\\left(p_{1}, s_{1}\\right)$, we have $c_{1}\\left(p_{1}\\right) \\geq \\hat{c}_{1}$. Therefore, we obtain\n\n$$\nv_{1}-\\gamma_{1}\\left(\\hat{c}_{1}\\right) \\geq \\max _{k \\in \\mathcal{J}_{0} \\backslash\\{1\\}} v_{k}-\\gamma_{k}\\left(c_{k}\\left(p_{k}\\right)\\right) .\n$$\n\nNext, for each seller $\\ell=2, \\ldots, J$, we replace $c_{\\ell}\\left(p_{\\ell}\\right)$ in the RHS of Equation 27 with another active type $\\hat{c}_{\\ell} \\neq c_{\\ell}\\left(p_{\\ell}\\right)$ that sets the same price $p_{\\ell}$, and show that the inequality is preserved. To begin with, for a given $\\ell \\geq 2$, we consider the following inequalities:\n\n$$\nv_{1}-\\gamma_{1}\\left(\\hat{c}_{1}\\right) \\geq \\max \\left\\{\\max _{k \\in\\{2,3 \\ldots, \\ell-1\\}} v_{k}-\\gamma_{k}\\left(\\hat{c}_{k}\\right), v_{\\ell}-\\gamma_{\\ell}\\left(c_{\\ell}\\left(p_{\\ell}\\right)\\right), \\max _{k \\in\\{\\ell+1, \\ldots, J, 0\\}} v_{k}-\\gamma_{k}\\left(c_{k}\\left(p_{k}\\right)\\right)\\right\\},\n$$\n\nwhere we ignore $\\max _{k \\in\\{2,3 \\ldots, \\ell-1\\}} v_{k}-\\gamma_{k}\\left(\\hat{c}_{k}\\right)$ when $\\ell=2$, and\n\n$$\n\\max \\left\\{\\max _{k \\in\\{1, \\ldots, \\ell-1\\}} v_{k}-\\gamma_{k}\\left(\\hat{c}_{k}\\right), \\max _{k \\in\\{\\ell+1, \\ldots, J, 0\\}} v_{k}-\\gamma_{k}\\left(c_{k}\\left(p_{k}\\right)\\right)\\right\\}<v_{\\ell}-\\gamma_{\\ell}\\left(\\hat{c}_{\\ell}\\right) .\n$$\n\nEquation 28 means that seller 1 continues to maximize virtual surplus after we change the type of each seller $k \\leq \\ell-1$ from $c_{k}\\left(p_{k}\\right)$ to $\\hat{c}_{k}$. Equation 29 means that the inequality is\n\n[^16]reversed after we replace the type of seller $\\ell$. When we consider a version of Equation 28 in which we replace $\\ell$ with $k^{\\prime}$, we refer to the inequality as Equation $28\\left(k^{\\prime}\\right)$. Note that Equation 28( $\\ell$ ) is Equation 28, and we have already shown that Equation 28(2) holds.\n\nWe also consider the following inequalities:\n\n$$\n\\max _{k \\in \\mathcal{J}_{0} \\backslash\\{\\ell\\}} v_{k}-\\gamma_{k}\\left(c_{k}\\right)>v_{\\ell}-\\gamma_{\\ell}\\left(c_{\\ell}\\left(p_{\\ell}\\right)\\right)\n$$\n\nand\n\n$$\n\\max _{k \\in \\mathcal{J}_{0} \\backslash\\{\\ell\\}} v_{k}-\\gamma_{k}\\left(c_{k}\\right)<v_{\\ell}-\\gamma_{\\ell}\\left(\\hat{c}_{\\ell}\\right) .\n$$\n\nWe will show that if Equation 28 and Equation 29 hold, then Equation 30 and Equation 31 hold for a positive measure of type profiles $c_{-\\ell}$, which, as we show, leads to a contradiction.\n\nFor each $\\ell \\geq 2$, assume that Equation $28\\left(k^{\\prime}\\right)$ for $k^{\\prime}=2, \\ldots ., \\ell$ and Equation 29 hold. Note that Equation 29 implies $c_{\\ell}\\left(p_{\\ell}\\right)>\\hat{c}_{\\ell} \\geq 0$. Thus, we have $c_{\\ell}\\left(p_{\\ell}\\right)>0$. We consider four cases.","text_sha256":"2bc7ea55bee0dbe4c23277f45da8252151d2d3c602f0dcc9c38c68c26f3fa61a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0036","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Supplementary Material for Appendix F","text":"Case 1. First, suppose that (i) Equation 28 holds with strict inequality or (ii) $\\hat{c}_{1}>0$. In either case, we can find a positive measure of type profiles $c_{-\\ell}$ such that Equation 30 and Equation 31 hold. For example, if $\\hat{c}_{1}>0$, then replacing $\\hat{c}_{1}$ with a slightly lower $\\hat{c}_{1}-\\epsilon$ satisfies Equation 28 and Equation 29 with strict inequalities, because the virtual cost functions are assumed to be continuous and strictly increasing. We can then find a positive measure of type profiles $c_{-\\ell}$ that maintain these strict inequalities, leading to Equation 30 and Equation 31 for a positive measure of $c_{-\\ell}$ (conditional on $s_{\\ell}$ ). This is a contradiction, because the existence of such $c_{-\\ell}$ 's implies that there is a positive measure of $\\theta_{\\ell}$ 's such that $\\theta_{\\ell}>\\gamma_{\\ell}\\left(c_{\\ell}\\left(p_{\\ell}\\right)\\right)$ and $\\theta_{\\ell}<\\gamma_{\\ell}\\left(\\hat{c}_{\\ell}\\right)$, i.e., types $c_{\\ell}\\left(p_{\\ell}\\right)$ and $\\hat{c}_{\\ell}$ set different prices (see Equation 19).\n\nCase 2. Suppose that (i) Equation $28\\left(k^{\\prime}\\right)$ holds with equality at every step $k^{\\prime} \\leq \\ell$, (ii) $\\hat{c}_{1}=0$, (iii) and the RHS of Equation 28 is positive. Note that Points (i) and (ii) imply that $c_{1}\\left(p_{1}\\right)=\\hat{c}_{1}=0$, so seller 1 sets a price for which there is a unique active type (to see this, note that if $c_{1}\\left(p_{1}\\right)>\\hat{c}_{1}$ and Equation 28(2) holds with equality, then\n\nEquation 25 would fail). This means that seller $\\ell$ does not attain the maximized value of the RHS in Equation 28, because if both sellers 1 and $\\ell$ maximize virtual surplus and $\\hat{c}_{1}=0<c_{\\ell}\\left(p_{\\ell}\\right)$, the candidate algorithm would recommend seller $\\ell$ instead of seller 1 (see the tie-breaking rule described in Definition 3). Because the RHS of Equation 28 is not determined by seller $\\ell$ but both sides are positive, we can slightly increase the cost of each seller $k \\neq 1, \\ell$ to make Equation 28 and Equation 29 strict. We can then find a positive measure of type profiles $c_{-\\ell}$ such that Equation 30 and Equation 31 hold, which leads to a contradiction by the same argument as in Case 1.\n\nCase 3. Suppose that (i) Equation 28 holds with equality but there is some step $k^{\\prime}<\\ell$ at which Equation 28( $k^{\\prime}$ ) is strict, (ii) $\\hat{c}_{1}=0$, (iii) and the RHS of Equation 28 is positive. Point (i) implies that there is some $k^{\\prime} \\in\\{2, \\ldots, \\ell-1\\}$ such that Equation $28\\left(k^{\\prime}\\right)$ is strict but Equation $28\\left(k^{\\prime}+1\\right)$ holds with equality, which occurs only when the RHS of the inequality increases as we move from Equation $28\\left(k^{\\prime}\\right)$ to Equation $28\\left(k^{\\prime}+1\\right)$. It means that the RHS of Equation 28 is not determined by seller $\\ell$, whose type did not change in earlier steps. By the same argument as Case 2, we can find type profiles $c_{-\\ell}$ where Equation 28 and Equation 29 hold with strict inequalities, which leads to a contradiction.\n\nCase 4. Suppose that both sides of Equation 28 are 0, and $\\hat{c}_{1}=0$. Both sides of Equation $28\\left(k^{\\prime}\\right)$ are 0 for every step $k^{\\prime}=2, \\ldots, \\ell$, because after each replacement, the LHS of Equation $28\\left(k^{\\prime}\\right)$ remains 0 and the RHS is weakly greater than 0 . By the same argument as in Case 2, we conclude that that seller $\\ell$ does not attain the maximized value of the RHS in Equation 28 (otherwise, seller $\\ell$ would be recommended according to the tie-breaking rule of the candidate algorithm). But it means that seller $\\ell$ has a negative virtual surplus, which is a contradiction.\n\nIn summary, we have shown that for any information structure, if all sellers use the pricing rule in Equation 19 and the buyer always follows the recommendations, for any profile of prices that can arise, the candidate algorithm recommends the product of the seller with the highest virtual surplus. The rest of the proof follows the same argument as in the appendix.\n\n[^0]:    *Ichihashi: Queen's University, Department of Economics, shotaichihashi@gmail.com. Smolin: Toulouse School of Economics, University of Toulouse Capitole and CEPR, alexey.v.smolin@gmail.com. We would like to thank Nemanja Antic, Heski Bar-Isaac, Dirk Bergemann, Alessandro Bonatti, Daniele Condorelli, Piero Gottardi, Emir Kamenica, Caio Lorecchio, Matthew Mitchell, Alessandro Pavan, Nicola Persico, and Udayan Vaidya, and audiences at ESSET Gerzensee 2024, Berkeley-Columbia-Duke-Northwestern-MIT IO Theory Conference 2024, and other venues, for their valuable suggestions and discussions. The authors acknowledge support from the Economics of Digital Services initiative at the University of Pennsylvania and from the NET Institute. Smolin acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d'Avenir) program (grant ANR-17-EURE-0010) and through the Artificial and Natural Intelligence Toulouse Institute (ANITI). Javier Gonzalez-Morin provided excellent research assistance.\n\n[^1]:    ${ }^{1}$ As of February 2025, the top three productivity apps on the Apple Store are AI assistants: ChatGPT, DeepSeek, and Gemini.\n\n[^2]:    ${ }^{2}$ This finding highlights the importance of the strategic context for an algorithm assessment and AI regulation (cf. White House (2023); European Commission (2024)).\n\n[^3]:    ${ }^{3}$ The dependence of information on price may also arise from a worst-case analysis, as in the work of Libgober and Mu (2021), in which the buyer's information is chosen to minimize the seller's profits.\n\n[^4]:    ${ }^{4}$ See also Lee (2021), Bergemann et al. (2022), and Smolin (2023).","text_sha256":"d68c1ac40624445efe8f8664a1103ab4ea44b1f2a2ff8e72e56a34d69d1f419e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0037","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Supplementary Material for Appendix F","text":"[^5]:    ${ }^{5}$ Thus, algorithmic recommendations can be viewed as an effective alternative to the joint use of an intermediary (see Decarolis and Rovigatti (2021) for online advertising) or to a merger (see Loertscher and Marx (2022) for multifirm bargaining).\n    ${ }^{6}$ One can view our setting as enabling a buyer from the classic setting of Myerson and Satterthwaite (1983) to commit to values and prices at which she would be purchasing a product. In this sense, we proceed in the opposite direction from the literature on limited commitment, which investigates how the inability to commit, typically on the part of a seller or a mechanism designer, affects equilibrium trade outcomes (e.g., Mylovanov and Tröger (2014), Liu et al. (2019)).\n    ${ }^{7}$ In this setting, providing extra information about $v$ upon recommendation is unnecessary.\n    ${ }^{8}$ The dependence of recommendations on price can be requested by the buyer in a conversation with an AI assistant, programmed directly, as seen in Amazon's search ranking algorithms (Lee and Musolff (2023), Farronato et al. (2023)), or it can arise indirectly through consumer feedback technology (Luca and Reshef (2021), Chakraborty et al. (2022)), wherein higher prices, all else being equal, lead to lower consumer satisfaction and ratings.\n\n[^6]:    ${ }^{9}$ If $\\gamma(c)$ were not everywhere increasing, then one would simply use an ironed version of it.\n    ${ }^{10}$ Type $\\bar{c}$ exists and is unique because $\\gamma(\\cdot)$ is strictly increasing and continuous on [0, 1], and $\\gamma(0)=$ $0<1 \\leq \\gamma(1)$.\n\n[^7]:    ${ }^{11}$ As the algorithm knows the value, this assumption is trivially satisfied if the seller's signal is fully informative or, more generally, partitional.\n    ${ }^{12}$ Formally, in Section 3, we assumed that the value distribution has full support on an interval. However, our derivation of Proposition 1 did not rely on this assumption, and thus the result applies to any market segment.\n\n[^8]:    ${ }^{13}$ Monotone partitional signals are not the only class of signals under which the distribution of individual buyer surplus undergoes a mean-preserving contraction. For example, in the previous version of the draft, we established this result for the truth-or-noise signals (Lewis and Sappington, 1994). However, some signal restrictions are necessary: If the finer segmentation separates high-value buyers from medium-value buyers while pooling them with low-value buyers, the price charged to those buyers may decrease, exacerbating payoff inequality.\n\n[^9]:    ${ }^{14}$ As before, if for some $\\alpha \\in[0,1], \\gamma_{\\alpha}(c)$ were not everywhere increasing, then one would simply use an ironed version of it.\n\n[^10]:    ${ }^{15}$ Gottardi and Mezzetti (2024) use a similar argument to construct an efficient one-shot mediation mechanism.\n\n[^11]:    ${ }^{16}$ As before, the assumption that the algorithm can base recommendations on the realized signals captures the idea that the sellers have no information beyond that accessed by the algorithm. This assumption is automatically satisfied if the signals are deterministic functions of the value profile, i.e., partitional or fully informative signals.\n\n[^12]:    ${ }^{17}$ This assumption is consistent with the consideration set approach to model recommender systems. See, for example, Dinerstein et al. (2018) and Lee and Musolff (2023).\n\n[^13]:    ${ }^{18}$ Point (i) follows from direct calculation. For Point (ii), note that the condition $\\left(\\frac{F(c)}{c}\\right)^{\\prime} \\geq 0$ is written as $\\frac{\\gamma(c)}{c} \\leq 2$, which is equivalent to $\\int_{0}^{\\gamma(c)}[v-c] \\mathrm{d} G(v) \\leq 0$ when $G=U[0,1]$.\n    ${ }^{19}$ The remaining case of a buyer perfectly informed about the product's existence and value is a textbook monopoly setting.\n\n[^14]:    ${ }^{20}$ Distribution $\\mathcal{G}_{H}$ of posteriors being a mean-preserving spread of $\\mathcal{G}_{L}$ means that there exist $\\Delta[0,1]$ -valued random variables $Z_{H}$ and $Z_{L}$ such that $Z_{H} \\sim \\mathcal{G}_{H}, Z_{L} \\sim \\mathcal{G}_{L}$ and $\\mathbb{E}\\left(Z_{H} \\mid Z_{L}\\right)=Z_{L}$.\n\n[^15]:    ${ }^{21}$ Multiple active types may set the same price if, for example, the signal is such that the support of the posterior distribution for $v_{j}$ is non-convex for some signal realizations.\n\n[^16]:    ${ }^{22}$ This argument also implies that if the candidate algorithm does not recommend any seller at a given price profile, then under any type profile that is consistent with the price profile, all the sellers have non-positive virtual surplus.","text_sha256":"a3077a81e4c4a8e9901566f8f4f77520492c06e8878b744899657e6125f05d3b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15:0038","work_id":"alex-smolin:buyer-optimal-algorithmic-recommendations","paper_id":"alex-smolin:buyer-optimal-algorithmic-recommendations:2025-02-15","title":"Buyer-Optimal Algorithmic Recommendations","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025-02-15","language":"en","version_type":"working-paper","canonical_url":"https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md","source_record":"https://arxiv.org/abs/2309.12122","citation":"Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Shota Ichihashi; Alex Smolin\n\n**Canonical citation:** Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md\n\n**Source record:** https://arxiv.org/abs/2309.12122\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"08c2c84b91025820cf7a4d7c8c9187d4c49801d24c5983c22a77bf2d65affc65"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0001","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Alex Smolin; Takuro Yamashita.\n> Canonical citation: Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"0e555fa7d7908628bd74cc907cb4886bacb3fd8af50f4f34d8748c11e954444d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0002","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Information Design in Smooth Games","text":"# Information Design in Smooth Games\n\n**Authors:** Alex Smolin; Takuro Yamashita\n\n**Manuscript date:** 2026-03-11\n\n#### Abstract\n\nWe study information design in games where players choose from a continuum of actions and have continuously differentiable payoffs. We show that an information structure is optimal when the equilibrium it induces can also be implemented in a principal-agent contracting problem. Building on this result, we characterize optimal information structures in symmetric linear-quadratic games. With common values, targeted disclosure is robustly optimal across all priors. With interdependent and normally distributed values, linear disclosure is uniquely optimal. We illustrate our findings with applications in venture capital, Bayesian polarization, and price competition.\n\n[^0]Keywords: Bayesian persuasion, information design, dual certification, firstorder approach, linear-quadratic games, targeted disclosure, Gaussian coupling, linear disclosure.","text_sha256":"f175be4f97f04fc80065b75bbea39dad06bbf5cc7398f194e62bc3309513431a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0003","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nAs advances in IT infrastructure and artificial intelligence expand our capacity to collect, process, and generate data, an increasing number of firms, regulators, and individuals must decide which information to supply to strategically interacting agents. Which information should investors receive to allocate capital most efficiently? Which demand signals should be disclosed to improve market outcomes? What are the limits of Bayesian polarization? We show that these and related questions can be tractably addressed within a single framework, and that the optimal policies are simple, intuitive, and robust.\n\nOur main analysis focuses on concave games of incomplete information, i.e., games in which each player's action lies in a convex set and each player's payoff is weakly concave in that action, but our methods extend to any smooth game. Concave games are a staple of applied economic modeling with fixed information structures, because equilibria can be tractably characterized by first-order conditions. We show that the same considerations render tractable the task of designing the information structure itself.\n\nTo solve the information-design problem, we adopt a duality-based approach. The dual can be viewed as an adversarial contracting problem between a principal and an omniscient agent who both observes the state and controls all players' actions. The dual-certification theorem (Theorem 1) states that if a state-action distribution emerges in equilibrium under both an information structure (in the information-design problem) and a contract (in the adversarial-contracting problem), then that information structure and that contract are optimal for their respective problems. The contract thus serves as an optimality certificate. Furthermore, it is capable of certifying any optimal information structure (Proposition 1), which enables us to establish whether the optimal information structure is unique or, when several are optimal, to identify their common features.\n\nWe show that (certifiably) optimal information structures are prior-robust: once optimal under one prior, they remain optimal under any other prior as long as the implemented allocation rule's support is contained within the original support in every state (Proposition 2). This has two consequences. First, if an optimal information\nstructure is fully informative about the state, it remains optimal under all priors. Second, unless the problem is trivial, an optimal information structure cannot induce a fullsupport action distribution in every state, which argues against adding full-support, extraneous, independent noise to individual signals.\n\nWe apply the solution method in two broad, symmetric settings with linear-quadratic payoffs for both the players and the designer. In each case, the certifying contract is symmetric across players and affine in actions, yet it yields qualitatively different optimal information policies.\n\nIn the first setting, the state is one-dimensional (Section 5). We derive the parameters of the certifying contract and use them to establish the optimality of a novel information structure we term targeted disclosure: it fully reveals the state to a subset of players while leaving the rest completely uninformed. This structure is remarkably simple and distributionally robust-it remains optimal under any prior. In addition, when the state is normally distributed, Gaussian coupling, which adds normal noise to each player's signal that cancels out in aggregate, is optimal as well.\n\nIn the second setting, the state is multidimensional with jointly normal components (Section 6). We show that a certifying contract can be found within the class of symmetric affine contracts by solving a single-variable minimization problem. This contract certifies the unique optimality of linear disclosure: an information structure that gives each player some linear statistic of the state. The resulting information structure is noise-free and symmetric ex ante.\n\nIn Section 8, we illustrate these results in three concrete applications. In Section 8.1, we analyze optimal capital fundraising when several investors independently choose how much to invest in a project of uncertain quality. We show that the information structure that maximizes the project's expected return is exclusive disclosure, a targeted disclosure in which only one investor is informed. Relative to full or no disclosure, exclusive disclosure prevents the dissipation of returns as the investor pool grows, offering a possible rationale for a common venture-capital practice.\n\nIn Section 8.2, we examine the limits of Bayesian polarization when each player\nforms a forecast from private information. We show that the information structure that maximizes a natural polarization index is a targeted disclosure that informs exactly half of the players. Hence, maximal polarization occurs when the population splits into two cohorts that are internally uniform but sharply divergent from each other. This finding underscores how media segregation can intensify societal polarization.\n\nIn Section 8.3, we study optimal price recommendations in a differentiated-product duopoly with linear demand and stochastic demand shocks. We characterize the information structures that maximize any weighted average of consumer and producer surplus. The resulting recommended prices are linear functions of the demand shocks and exhibit discontinuous regime changes with respect to the weight on consumer surplus. This observation underscores the risk that algorithms generating price recommendations-able to shift objectives and adapt policies far faster than human decision-makers-can destabilize markets.","text_sha256":"ba90dff05601c2feb6ab803b36343c893c63bf9b532536e0b48f88f6073d432c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0004","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"Related Literature Our paper contributes to the recent and flourishing literature on information design. Much of this literature focuses on information design with a single player, called Bayesian persuasion (Rayo and Segal (2010), Kamenica and Gentzkow (2011)). Popular solution methods are belief based, i.e., they operate within the space of the receiver's belief distributions. ${ }^{1}$ A natural continuation of this research agenda is a study of information design in multiplayer games (Bergemann and Morris (2016), Taneva (2019)). In games, players' beliefs constitute infinite hierarchies, which renders the beliefbased approach less tractable (Mathevet, Perego, and Taneva (2020)). Instead, in games, an action-based approach rooted in the revelation principle is promising, as it frames\n\n[^1]the design problem as a linear program and enables the use of duality machinery. ${ }^{2,3}$\nGalperti and Perego (2018) and Galperti, Levkun, and Perego (2024) employ this action-based approach to study information design in games with finitely many actions, imposing minimal structure on payoffs. They develop an economic interpretation of the Lagrange multipliers associated with Bayes' plausibility as the value of data records and propose the idea of pooling externalities across records; all these observations apply to our setting.\n\nIn contrast, we study games with infinitely many actions and impose a concavity structure on payoffs, thus enabling us to rely on a first-order approach for incentives (Holmström (1979), Mirrlees (1999)) that leads to more succinct and tractable primal and dual problems. This approach was introduced in a Bayesian persuasion setting by Kolotilin $(2012,2018)$ and further refined by Kolotilin, Corrao, and Wolitzky (2025). These papers study an information-design problem with a single player and a onedimensional state, identifying when censorship or, respectively, assortative disclosures are optimal. We deepen and extend this approach, adapting it to settings with multiple players and a multidimensional state.\n\nMost of our current understanding of optimal information in games is drawn from the study of Gaussian signals in games with quadratic payoffs and a normally distributed state (e.g., Angeletos and Pavan $(2007,2009)$, Bergemann and Morris (2013), Bergemann, Heumann, and Morris (2015, 2021), Ui (2020)). In that literature, as well as in a vast body of work in macroeconomics and finance, the Gaussian form of players' signals is imposed ad hoc for analytical convenience.\n\nRecent arguments suggest that Gaussian signals are not only convenient but often optimal among all information structures. For example, Tamura $(2012,2018)$ estab-\n\n[^2]lished the optimality of a Gaussian signal in a setting with a single player by building on the statistical properties of a covariance matrix of posterior expectations. Bergemann, Heumann, and Morris (2017) and Miyashita and Ui (2023) extend this argument to games. ${ }^{4}$ Our results align with these findings and show that in many scenarios the optimal Gaussian structures are symmetric and noise-free; moreover, the certification method allows direct identification of the optimal informational parameters. Furthermore, we provide tools that establish when Gaussian information structures are uniquely optimal.\n\nAt the same time, we show that in many settings targeted disclosure is optimal. These information structures are simple to implement and distributionally robust, offering a strong positive message for the information design literature, where existing solutions often lack these properties.","text_sha256":"c97505c061ee9d1c8f7794482bd46ee62326ddf0168bcbe5dbc00281a05426f5"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0005","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Design Problem","text":"## 2 Design Problem\n\nWe study a standard information design problem as presented by Bergemann and Morris (2016), extended to accommodate a continuum of players' actions.\n\nPayoffs There are $N$ players indexed by $i, 1 \\leq N<\\infty$, and an information designer. Each player chooses an action $a_{i} \\in A_{i}=\\mathbb{R}$. We denote an action profile by $a \\in A=\\times_{i} A_{i}$ and write $\\left(a_{i}, a_{-i}\\right)$ when highlighting player $i$ 's action.\n\nA state $\\omega$ is distributed over a Polish set $\\Omega$, according to a full-support prior $\\mu_{0} \\in$ $\\Delta(\\Omega)$. The action profile and the state jointly determine payoffs through\n\n$$\n\\begin{aligned}\n& u_{i}: A \\times \\Omega \\rightarrow \\mathbb{R}, \\\\\n& v: A \\times \\Omega \\rightarrow \\mathbb{R},\n\\end{aligned}\n$$\n\nfor each player $i$ and for the designer, respectively. The tuple $\\left(\\left(A_{i}, u_{i}\\right)_{i=1}^{N}, \\mu_{0}\\right)$ constitutes\n\n[^3]the basic game.\n\nInformation The players and the designer start with a commonly known prior belief about the state $\\omega$ that coincides with the prior $\\mu_{0}$. The designer can provide additional information to players by choosing an information structure $\\mathcal{I}=(S, \\pi)$ that consists of a measurable signal set $S=\\times_{i} S_{i}$ and a likelihood function $\\pi \\in \\Delta(S \\times \\Omega)$ that has $\\mu_{0}$ as its state marginal distribution. This information structure determines the sets of private signals the players can observe and, through the likelihood function, their informational content.\n\nFirst, the designer chooses an information structure $\\mathcal{I}$. Second, the state $\\omega$ and the signal profile $s$ are realized according to $\\mathcal{I}$. Finally, each player privately observes his signal $s_{i}$ and chooses an action $a_{i}$.\n\nWe will regularly refer to two benchmark information structures: full disclosure, with $S_{i}=\\Omega$ and $s_{i} \\equiv \\omega$ for all $i$; and no disclosure, with $\\left|S_{i}\\right|=1$ for all $i$.\n\nEquilibrium The basic game together with the information structure chosen by the designer determine a Bayesian game of incomplete information. In that game, each player's behavior is described by a strategy that maps any received signal to a possibly random action, $\\sigma_{i}: S_{i} \\rightarrow \\Delta\\left(A_{i}\\right)$, and we consider as an equilibrium concept a Bayes Nash equilibrium:\n\nDefinition 1. (Bayes Nash Equilibrium) For a given information structure $\\mathcal{I}$, a strategy profile $\\sigma=\\left(\\sigma_{1}, \\ldots, \\sigma_{N}\\right)$ constitutes a Bayes Nash equilibrium if\n\n$$\n\\mathbb{E}_{\\mathcal{I}, \\sigma_{i}, \\sigma_{-i}}\\left[u_{i}\\left(a_{i}, a_{-i}, \\omega\\right)\\right] \\geq \\mathbb{E}_{\\mathcal{I}, \\sigma_{i}^{\\prime}, \\sigma_{-i}}\\left[u_{i}\\left(a_{i}^{\\prime}, a_{-i}, \\omega\\right)\\right]\n$$\n\nfor all $i$ and $\\sigma_{i}^{\\prime}: S_{i} \\rightarrow \\Delta\\left(A_{i}\\right) .{ }^{5}$\nAn information structure and a strategy profile determine a distribution over the action profiles in each state $\\alpha: \\Omega \\rightarrow \\Delta(A)$, which we call an allocation rule, and the corre-\n\n[^4]sponding designer's expected payoff $\\mathbb{E}_{\\mathcal{I}, \\sigma}[v(a, \\omega)]$. The value of an information structure is defined as the maximal designer's expected payoff that can arise in equilibrium of the induced game: if the game has multiple equilibria, the designer can choose the one she prefers, whereas if no equilibrium exists, the value is undefined. An information-design problem consists of finding an information structure with a maximal value without placing any additional restrictions on the sets of signals or the likelihood function, apart from a mild \"admissibility\" condition. (This condition, and other omitted formal details, are deferred to Appendix A.)\n\nDefinition 2. (Optimal Information Structure) An information structure is optimal if there does not exist an information structure with a strictly higher value.\n\nThe search for an optimal information structure is complicated by the scale of the basic game: multiple players, after receiving private signals, choose actions from a continuum while anticipating one another's behavior. Our central simplifying assumption is:\n\nAssumption 1. (Concave Payoffs) For all $i=1, \\ldots, N, \\omega \\in \\Omega$, and $a_{-i} \\in A_{-i}$, $u_{i}\\left(a_{i}, a_{-i}, \\omega\\right)$ is continuously differentiable in $a_{i}$, weakly concave in $a_{i}$, and obtains its maximum at some finite value.\n\nWe call a basic game in which Assumption 1 is satisfied a concave game. ${ }^{6}$ In a concave game, for any player $i$ and equilibrium belief $\\mu \\in \\Delta\\left(A_{-i} \\times \\Omega\\right)$ about the others' actions and the state, a best response $a_{i}^{*}$ exists and satisfies the first-order condition:\n\n$$\n\\left.\\frac{\\partial}{\\partial a_{i}} \\mathbb{E}_{\\mu}\\left[u_{i}\\left(a_{i}, a_{-i}, \\omega\\right)\\right]\\right|_{a_{i}=a_{i}^{*}}=\\left.\\mathbb{E}_{\\mu}\\left[\\frac{\\partial}{\\partial a_{i}} u_{i}\\left(a_{i}, a_{-i}, \\omega\\right)\\right]\\right|_{a_{i}=a_{i}^{*}} \\triangleq \\mathbb{E}_{\\mu}\\left[\\dot{u}_{i}\\left(a_{i}^{*}, a_{-i}, \\omega\\right)\\right]=0,\n$$\n\nwhere we denote the player's marginal payoff function by\n\n$$\n\\dot{u}_{i}(a, \\omega) \\triangleq \\frac{\\partial u_{i}(a, \\omega)}{\\partial a_{i}} .\n$$\n\n[^5]An equilibrium under any given information structure is characterized by a system of such conditions, one for each player's signal.\n\nThe information-design problem can be simplified by appealing to the revelation principle (Myerson (1982)): it is without loss of generality to focus on direct information structures that inform each player about a recommended action $S=A$ and are such that all players are obedient, i.e., are willing to follow the recommendations. Each direct information structure corresponds to a measure $\\pi \\in \\Delta(A \\times \\Omega)$, and the informationdesign problem can be formulated as a constrained maximization over these measures:","text_sha256":"7901ef38cec97672666c261800fd6937596e15f83a8b80329944541b2fd75e32"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0006","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Design Problem","text":"$$\n\\begin{aligned}\nV^{P} \\triangleq & \\sup _{\\pi \\in \\Delta(A \\times \\Omega)} \\int_{A \\times \\Omega} v(a, \\omega) \\mathrm{d} \\pi \\\\\n& \\quad \\text { s.t. } \\int_{A_{i}^{\\prime} \\times A_{-i} \\times \\Omega} \\dot{u}_{i}(a, \\omega) \\mathrm{d} \\pi=0 \\quad \\forall i=1, \\ldots, N, \\text { measurable } A_{i}^{\\prime} \\subseteq A_{i}, \\\\\n& \\int_{A \\times \\Omega^{\\prime}} \\mathrm{d} \\pi=\\int_{\\Omega^{\\prime}} \\mathrm{d} \\mu_{0} \\quad \\forall \\text { measurable } \\Omega^{\\prime} \\subseteq \\Omega .\n\\end{aligned}\n$$\n\nConstraints (6) are a proper formulation of first-order conditions (4) in light of a continuum of recommended actions. These constraints capture players' obedience and effectively require that for each player $i$, the marginal of $\\pi$ on $A_{i}$ weighted by the marginal utilities equals zero measure. Constraints (7) capture Bayes' plausibility and, likewise, require that the marginal of $\\pi$ on $\\Omega$ equals the prior $\\mu_{0}$.","text_sha256":"6c587fb78e653e3472de97ded62b0b9aabe7f636eac305f5019eaf0c2dd5a575"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0007","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Solution Method","text":"## 3 Solution Method\n\nProblem (5-7) is linear in $\\pi$, yet unwieldy to solve directly. In the spirit of linear programming, we view it as a primal problem and call any $\\pi \\in \\Delta(A \\times \\Omega)$ a primal measure. If a primal measure satisfies the constraints of the primal problem, then we call that measure implementable by information. We can construct a dual problem as\nfollows (e.g., Anderson and Nash (1987)):\n\n$$\n\\begin{aligned}\n& V^{D} \\triangleq \\inf _{\\lambda \\in \\times_{i} \\mathcal{M}\\left(A_{i}\\right), \\gamma \\in \\mathcal{M}(\\Omega)} \\int_{\\Omega} \\gamma(\\omega) \\mathrm{d} \\mu_{0} \\\\\n& \\quad \\text { s.t. } \\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega)+\\gamma(\\omega) \\geq v(a, \\omega) \\forall a \\in A, \\omega \\in \\Omega,\n\\end{aligned}\n$$\n\nwhere $\\mathcal{M}(X)$ denotes the space of measurable real-valued functions on $X$. The minimization arguments, the dual variables $(\\lambda, \\gamma)$, represent the Lagrange multipliers associated with the primal incentive constraints (6) and the feasibility constraints (7), respectively.\n\nThe problem (8) is a generalization of the dual problem of Kolotilin (2018) and Kolotilin et al. (2025) to multiple players. By the arguments analogous to those of Galperti et al. (2024), the optimal $\\gamma^{*}(\\omega)$ measures the marginal benefit of increasing the frequency of state $\\omega$. Importantly, the optimal $\\gamma^{*}(\\omega)$ can be solved away: the objective in (8) is additively separable in $\\gamma(\\omega)$, and the constraints at different states $\\omega$ are linked only through the variables $\\lambda$. Hence, for any $\\lambda$ and $\\omega$, the optimal choice of $\\gamma(\\omega)$ is the smallest value consistent with the dual constraints:\n\n$$\n\\gamma^{*}(\\omega ; \\lambda)=\\sup _{a \\in A} w(a, \\omega, \\lambda),\n$$\n\nwhere the dual payoff function $w$ is defined as\n\n$$\nw(a, \\omega, \\lambda) \\triangleq v(a, \\omega)-\\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega) .\n$$\n\nAs a result, the problem (8) can be restated as follows:\n\n$$\nV^{D}=\\inf _{\\lambda \\in \\times_{i} \\mathcal{M}\\left(A_{i}\\right)} \\mathbb{E}\\left[\\sup _{a \\in A} w(a, \\omega, \\lambda)\\right] .\n$$\n\nProblem (10) admits a simple economic interpretation as a contracting problem between a dual principal and a dual agent. First, the principal chooses an incentive contract $\\lambda$ that consists of $N$ functions $\\lambda_{i}\\left(a_{i}\\right)$ and determines the agent's payoff according to (9). Second, the state $\\omega$ is realized. Finally, the agent perfectly observes the state and chooses\nthe whole action profile $a \\in A$ to maximize his payoff. The contracting is adversarial in that the principal aims to minimize the agent's expected payoff.\n\nIf the best responses exist at all states and induce the joint action-state measure $\\pi(a, \\omega)$, then we say that $\\lambda$ implements $\\pi$ by incentives and that $\\pi$ is implementable by incentives, by contract $\\lambda$.\n\nTheorem 1. (Weak Duality. Dual Certification) If $\\pi \\in \\Delta(A \\times \\Omega)$ is implementable by information and is implementable by incentives by contract $\\lambda$, then (i) $\\pi$ solves the information-design problem, (ii) $\\lambda$ solves the adversarial-contracting problem, and (iii) $V^{P}=V^{D}$.\n\nTheorem 1 offers a solution method based on optimality certification. When the conditions of the theorem hold, we say that $\\lambda$ is a (dual) certificate of $\\pi$, that $\\lambda$ certifies the optimality of $\\pi$, and that $\\pi$ is certifiably optimal. Similarly, if an information structure induces a certifiably optimal measure, then we call that information structure certifiably optimal.\n\nWe highlight two general properties of certificates. First, every certificate $\\lambda^{*}$ has a clear economic meaning: it represents Lagrange multipliers associated with the obedience constraints (6). Accordingly, $\\lambda_{i}^{*}\\left(a_{i}\\right)$ measures the marginal value for the information designer from perturbing the obedience constraint of action $a_{i}$. Second, each certificate is \"universal\":\n\nProposition 1. (Universality) If $\\lambda$ certifies the optimality of measure $\\pi$, and $\\pi^{\\prime}$ is another optimal measure, then $\\lambda$ also certifies the optimality of measure $\\pi^{\\prime}$.\n\nBy Proposition 1, a certificate does more than certify a single information structure- it constrains the entire set of optima. Every optimal measure must be certified by the same certificate; equivalently, it must be the dual agent's best response to the same contract in the dual problem. We exploit this property in Section 5, where multiple optima exist, to isolate their common features, and in Section 6 to show that the optimal measure is unique.\n\nThe difference $V^{D}-V^{P}$ between the optimal values of primal and dual problems is nonnegative and constitutes a duality gap. The solution to the information-design problem can be certified if and only if (i) solutions to both primal and dual problems exist and (ii) the duality gap is equal to zero, $V^{D}=V^{P}$. Thus, either all optimal information structures can be certified or none of them can.\n\nAny constructed certificate, by its very existence, proves that the duality gap is zero and that both primal and dual solutions exist. However, it may be useful to know $a$ priori whether one can expect to find such a certificate in a given problem.\n\nClaim 1. (Strong Duality) If each $A_{i}$ and $\\Omega$ are compact, and $v$ and each $\\dot{u}_{i}$ are continuous on $\\Omega \\times A$, then $V^{D}=V^{P}$.\n\nClaim 1 follows from standard Fenchel-Rockafellar arguments. We record the result mainly as a benchmark, suggesting that strong duality is to be expected in regular environments. Many applications below feature noncompact action spaces. In those environments, a general strong-duality theorem would require additional coercivity and integrability assumptions to rule out mass escaping to infinity and to ensure that the dual certificates remain well behaved. Rather than impose such conditions abstractly, in the applications below we verify strong duality constructively by exhibiting explicit certificates.","text_sha256":"fb7423d8abe7ec17d1c388fdcceb5c8b868f87b65deca4bdc197dc6e7a3b4968"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0008","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Robustness of Optimal Information Structures","text":"## 4 Robustness of Optimal Information Structures\n\nBy Theorem 1, an allocation rule induced by a certifiably optimal information structure must be chosen freely by a fully informed dual agent. This observation has two consequences. First, the prior is irrelevant for the implementability of an allocation rule by incentives since the prescribed action profiles must be optimal state-by-state. Second, if the optimal allocation rule randomizes over several action profiles at a given state, the dual agent must be indifferent among those profiles and could therefore randomize over them with any probabilities. Hence, only the support of the action profiles matters for implementability by incentives:\n\nProposition 2. (Robustness) Let two concave information-design problems differ only in their priors, if at all. Let $\\pi_{1} \\in \\Delta(A \\times \\Omega)$ be certifiably optimal in the first problem. If $\\pi_{2} \\in \\Delta(A \\times \\Omega)$ is implementable by information in the second problem and $\\operatorname{supp} \\pi_{2}(\\cdot \\mid$ $\\omega) \\subseteq \\operatorname{supp} \\pi_{1}(\\cdot \\mid \\omega)$ for all $\\omega \\in \\Omega$, then $\\pi_{2}$ is certifiably optimal in the second problem.\n\nProposition 2 shows that certifiably optimal information structures are, to some extent, prior-robust: once optimal under one prior, an information structure remains optimal under any other prior, provided it still implements an allocation rule whose support is no larger in every state. It is specific to our setting, which features a continuum of actions and thus more flexible players' best responses, and does not hold in generic games with finitely many actions (cf. Kamenica and Gentzkow (2011)).\n\nGenerally, the larger the support is, the easier it is to construct multiple information structures that implement allocation rules within that support. In the extreme case, if the action support covers the whole action space, then the support condition of Proposition 2 has no bite, and any information structure can be certified to be optimal.\n\nCorollary 1. (Full-Support Noise) If measure $\\pi \\in \\Delta(A \\times \\Omega)$ is certifiably optimal and $\\operatorname{supp} \\pi(\\cdot \\mid \\omega)=A$ for all $\\omega \\in \\Omega$, then any information structure is certifiably optimal.\n\nCorollary 1 presents a case against using extraneous noises that induce full-support action profiles in concave games with finitely many players. The information structures that employ such noises can never be certifiably optimal, except in trivial cases in which the designer's expected payoff is invariant to the information provided. This finding resonates with the analysis of Taneva (2019), who studied a two-player binary setting and showed that sending conditionally independent signals is never strictly optimal, as well as with the analysis of Candogan and Strack (2023), who established optimality of partitional signals in a class of games. However, such extraneous independent noises may optimally appear in the limit information structure as the number of players grows to infinity, as we discuss in Section 7.\n\nFurthermore, Proposition 2 enables us to assess the optimality of full state transparency. We call an information structure fully informative about the state if each player\ncan deduce the state with certainty from her private signal. There could be many such structures, each differing in the state-by-state coordination of players' actions, including full disclosure. However, any such structure can implement the same allocation rule under all priors.\n\nCorollary 2. (Full State Information) If a certifiably optimal information structure is fully informative about the state, then it is certifiably optimal under all priors.\n\nCorollary 2 shows that, once full disclosure is optimal, it is extremely robust: it remains optimal under any prior (cf. Jehiel (2015)).","text_sha256":"84b6f3edcb29a7947d6299a23c6cc2348f34396db55b7f6bae2d0b26c50675ee"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0009","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Linear-Quadratic Games: Common Value","text":"## 5 Linear-Quadratic Games: Common Value\n\nThe certification approach developed in the previous section applies to any concave game (see Online Appendix C) and, more generally, to any smooth game (Section 7). In this section, we demonstrate the approach in symmetric linear-quadratic games with a common state. We first develop general results and then illustrate them through two applications: capital fundraising and expectation polarization.\n\nWe define a linear quadratic symmetric game with a common state as a game with $\\Omega \\subseteq \\mathbb{R}$, and the following payoff structure:\n\n$$\n\\begin{aligned}\nu_{i}(a, \\omega) & =\\left(\\omega+b_{1}\\right) a_{i}+\\frac{1}{2}\\left(q^{o}-\\frac{q^{c}}{N}\\right) a_{i}^{2}+q^{c} a_{i} \\bar{a}+f_{1}\\left(a_{-i}, \\omega\\right), \\\\\nv(a, \\omega) & =\\left(h \\omega+b_{2}\\right) \\bar{a}+p^{o} \\check{a}+p^{c} \\bar{a}^{2}+f_{2}(\\omega),\n\\end{aligned}\n$$\n\nwhere $\\bar{a} \\triangleq \\sum_{j} a_{j} / N, \\check{a} \\triangleq \\sum_{j} a_{j}^{2} / N, f_{1}$ and $f_{2}$ are arbitrary functions of the indicated arguments, $h \\geq 0$, and $q^{o}, q^{c}, p^{o}, p^{c}$ satisfy the concavity conditions:\n\n$$\n\\begin{gathered}\nq^{o}<0, q^{o}+q^{c}<0 \\\\\nq^{o} p^{c}>p^{o} q^{c}\n\\end{gathered}\n$$\n\nCondition (13) ensures that $u_{i}$ is strictly concave in $a_{i}$ for all $N \\geq 1$, whereas condition\n(14) ensures concavity of the dual payoff under conjectured certificates. This game is strategically equivalent to, and thus admits the same solution as, a simpler normalized game, in which $\\mathbb{E}[\\omega]=0$,\n\n$$\n\\begin{aligned}\nu_{i}(a, \\omega) & =\\omega a_{i}+\\frac{1}{2}\\left(q^{o}-\\frac{q^{c}}{N}\\right) a_{i}^{2}+q^{c} a_{i} \\bar{a}, \\\\\nv(a, \\omega) & =(h \\omega+b) \\bar{a}+p^{o} \\check{a}+p^{c} \\bar{a}^{2},\n\\end{aligned}\n$$\n\nwhere $h, q^{o}, q^{c}, p^{o}, p^{c}$ are as in the original game and $b=h \\mathbb{E}[\\omega]+b_{2}-2\\left(\\mathbb{E}[\\omega]+b_{1}\\right)\\left(p^{o}+\\right.$ $\\left.p^{c}\\right) /\\left(q^{o}+q^{c}\\right)$. Therefore, we can focus on an analysis of the normalized game.\n\nTo find optimality certificates, observe that the players' marginal payoffs are linear:\n\n$$\n\\dot{u}_{i}(a, \\omega)=\\omega+q^{o} a_{i}+q^{c} \\bar{a},\n$$\n\nand the dual payoff function takes the form\n\n$$\nw(a, \\omega, \\lambda)=(h \\omega+b) \\bar{a}+p^{o} \\check{a}+p^{c} \\bar{a}^{2}-\\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right)\\left(\\omega+q^{o} a_{i}+q^{c} \\bar{a}\\right) .\n$$\n\nGiven the linear-quadratic nature of the environment, we conjecture that an optimal allocation rule is linear in the state. Because the allocation rule must be a best response given the dual payoff (16), it is natural to posit a certificate that is affine and symmetric across players:\n\n$$\n\\lambda_{i}\\left(a_{i}\\right)=-\\frac{1}{N}\\left(x a_{i}+x_{0}\\right) .\n$$\n\nWith this certificate, the dual payoff function becomes\n\n$$\nw\\left(a, \\omega, x, x_{0}\\right)=x_{0} \\omega+\\left((h+x) \\omega+b+x_{0}\\left(q^{o}+q^{c}\\right)\\right) \\bar{a}+\\left(p^{c}+x q^{c}\\right) \\bar{a}^{2}+\\left(p^{o}+q^{o} x\\right) \\check{a} .\n$$\n\nIf the coefficient in front of $\\check{a}$ is nonzero, $p^{o}+q^{o} x \\neq 0$, then the dual best-response is uniquely defined, linear in state, and symmetric across players; thus, such $x$ could certify only the optimality of full disclosure or no disclosure. To certify the optimality of partial\ndisclosure, we must have\n\n$$\n\\begin{gathered}\nx^{*}=-\\frac{p^{o}}{q^{o}} \\\\\nw\\left(a, \\omega, x^{*}, x_{0}\\right)=x_{0} \\omega+\\left(\\left(h-\\frac{p^{o}}{q^{o}}\\right) \\omega+b+x_{0}\\left(q^{o}+q^{c}\\right)\\right) \\bar{a}+\\left(p^{c}-\\frac{p^{o} q^{c}}{q^{o}}\\right) \\bar{a}^{2} .\n\\end{gathered}\n$$\n\nWith this choice of $x$, the dual payoff function is a function solely of the average action $\\bar{a}$. Because of (15) and $\\mathbb{E}[\\omega]=0$, any incentive compatible expected average action must equal to 0; therefore we must have\n\n$$\n\\begin{gathered}\nx_{0}^{*}=-\\frac{b}{q^{o}+q^{c}} \\\\\n\\bar{a}^{*}(\\omega)=\\frac{p^{o}-h q^{o}}{2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)} \\omega\n\\end{gathered}\n$$\n\nAny allocation rule that results in this average action is implementable by incentives by the affine contract (17). By Theorem 1, if the same allocation rule can be implemented by information, then that allocation rule is optimal. ${ }^{7}$ A natural way to implement such a linear average action is to fully inform a subset of players:\n\nDefinition 3. (Targeted Disclosure) For $k \\in\\{0,1, \\ldots, N\\}$, a $k$-targeted disclosure is an information structure that fully reveals the state to $k$ players while providing no information to others.\n\nA 0-targeted disclosure is no disclosure and $N$-targeted disclosure is full disclosure. Under a $k$-targeted disclosure, there exists an equilibrium where each uninformed player plays $a_{i}^{N D} \\equiv 0$, whereas each informed player plays\n\n$$\na_{i}^{I}(\\omega)=-\\frac{N}{q^{c} k+q^{o} N} \\omega,\n$$\n\n[^6]resulting in the average action\n$$\n\\bar{a}(\\omega)=\\frac{k}{N} a_{i}^{I}(\\omega)=-\\frac{k}{q^{c} k+q^{o} N} \\omega .\n$$\n\nTheorem 2. (Optimality of Targeted Disclosure) In a normalized linear-quadratic symmetric game with a common state, if\n\n$$\nk^{*} \\triangleq \\frac{q^{o}\\left(h q^{o}-p^{o}\\right)}{q^{o}\\left(2 p^{c}-h q^{c}\\right)-p^{o} q^{c}} N\n$$\n\nis in $\\{0,1, \\ldots, N\\}$, then $k^{*}$-targeted disclosure is optimal. If $k^{*} \\notin[0, N],{ }^{8}$ then either (i) $p^{o}<h q^{o}$, in which case no disclosure is optimal, or (ii) $\\left(h q^{o}-p^{o}\\right)\\left(q^{o}+q^{c}\\right)>2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)$, in which case full disclosure is optimal.\n\nProof. If $k$ is in $\\{0,1, \\ldots, N\\}$, then by Theorem 1, comparing (18) and (19), a $k$-targeted disclosure is optimal if\n\n$$\n\\begin{aligned}\n& -\\frac{k}{q^{c} k+q^{o} N}=\\frac{p^{o}-h q^{o}}{2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)} \\\\\n& \\Leftrightarrow k=\\frac{q^{o}\\left(h q^{o}-p^{o}\\right)}{q^{o}\\left(2 p^{c}-h q^{c}\\right)-p^{o} q^{c}} N .\n\\end{aligned}\n$$\n\nIf $k \\notin[0, N]$, then given the conditions (13), either (i) or (ii) holds. ${ }^{9}$ In case of (i), the optimality of no disclosure can be certified by an affine contract with $\\left(x, x_{0}\\right)=$ $\\left(-h,-b /\\left(q^{o}+q^{c}\\right)\\right)$. In case of (ii), the optimality of full disclosure can be certified by an affine contract with $\\left(x, x_{0}\\right)=\\left(h-2\\left(p^{o}+p^{c}\\right) /\\left(q^{o}+q^{c}\\right),-b /\\left(q^{o}+q^{c}\\right)\\right)$. (See Appendix A. 3 for details.) $\\square$","text_sha256":"38f6bdf1718832fe052a90103b5071d4254e070cdbc59e527716681357784043"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0010","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Linear-Quadratic Games: Common Value","text":"Theorem 2 highlights the importance of targeted disclosure in linear-quadratic games with a common state. First, if $k^{*} \\notin(0, N)$, then either 0-targeted disclosure or $N$ targeted disclosure is optimal. Second, if $k^{*}$ is in $\\{0,1, \\ldots, N\\}$, then $k^{*}$-targeted disclosure is exactly optimal, and the parameter space that yields it spans a wide range of\n\n[^7]economically relevant settings. Third, if $k^{*} \\in(0, N)$ but is not an integer, a $\\left\\lceil k^{*}\\right\\rceil$-targeted disclosure is asymptotically optimal. Indeed, the affine contract above provides, by weak duality, an upper bound on the designer's value, and under any obedient allocation the contract term has expectation zero. Under this contract, the dual payoff depends only on the average action and is maximized at (18). Because the average action induced by $k$-targeted disclosure depends continuously on $k / N$, and $\\left\\lceil k^{*}\\right\\rceil / N \\rightarrow k^{*} / N$, the payoff under $\\left\\lceil k^{*}\\right\\rceil$-targeted disclosure converges to the optimum as $N$ grows.\n\nThe optimality of targeted disclosure is notable for three reasons. First, when $k^{*} \\in$ $[1, N-1]$, the policy is asymmetric even though the underlying environment is symmetric. Second, because the argument never relies on the prior, the same targeted-disclosure policy is optimal for all priors (cf. Proposition 2). Third, the policy is remarkably simple to implement. The latter two properties highlight the exceptional practical applicability of targeted disclosure.\n\nAt the same time, optimal information structures need not be unique. When targeted disclosure is optimal, the designer is free to choose which players receive information. That choice leaves aggregate equilibrium outcomes unchanged but shifts surplus among players. For example, an ex-ante symmetric variant of $k^{*}$-targeted disclosure first publicly draws a random subset of $k^{*}$ players and then applies $k^{*}$-targeted disclosure to that subset. Furthermore, qualitatively distinct information structures can also be optimal, specifically when the state is normally distributed.\n\nFor the rest of this section, assume $N \\geq 2$.\n\nDefinition 4. (Gaussian Coupling) For $\\beta \\in \\mathbb{R}$ and $\\sigma^{2}>0, a\\left(\\beta, \\sigma^{2}\\right)$-Gaussian coupling is an information structure such that for all $i$ and $\\omega$,\n\n$$\ns_{i}=\\beta \\omega+\\varepsilon_{i}-\\frac{1}{N-1} \\sum_{j \\neq i} \\varepsilon_{j},\n$$\n\nwhere each noise term $\\varepsilon_{i} \\sim N\\left(0, \\sigma^{2}\\right)$ is independent from $\\omega$ and $\\varepsilon_{j}$ for $j \\neq i$.\nA Gaussian coupling distorts the state by adding Gaussian noise to each player's signal, with the individual noises coupled in such a way that they vanish upon aggregation-\nso that the sum of all signals is a deterministic function of the state $\\omega$. When $\\beta \\neq 0$, knowledge of all signals perfectly reveals the state. However, observing only $s_{i}$ provides imperfect information about the state; in this sense, a Gaussian coupling splits the complete state information across players. When the state is normally distributed, a Gaussian coupling corresponds to a Gaussian information structure-that is, the signals and the state are jointly normally distributed.\n\nTheorem 3. (Optimality of Gaussian Coupling) In a normalized linear-quadratic symmetric game with a common state, if $\\omega \\sim N\\left(0, \\sigma_{\\omega}^{2}\\right)$ and $k^{*}$ as defined in (20) lies in $(0, N),{ }^{10}$ then a $\\left(\\beta, \\sigma^{2}\\right)$-Gaussian coupling is optimal, where\n\n$$\n\\begin{aligned}\n\\beta & =\\frac{p^{o}-h q^{o}}{2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)} \\\\\n\\sigma^{2} & =\\frac{N-1}{N} \\frac{\\beta\\left(1+\\left(q^{c}+q^{o}\\right) \\beta\\right)}{-q^{o}} \\sigma_{\\omega}^{2}\n\\end{aligned}\n$$\n\nIn the optimal equilibrium, $a_{i} \\equiv s_{i}$.\nProof. Consider $\\left(\\beta, \\sigma^{2}\\right)$-Gaussian coupling with $\\left(\\beta, \\sigma^{2}\\right)$ as defined by (21) and (22). If each player plays $a_{i} \\equiv s_{i}$, then $\\bar{a}(\\omega)=\\beta \\omega=\\bar{a}^{*}(\\omega)$, as defined in (18). By Theorem 1, if $a_{i} \\equiv s_{i}$ is incentive-compatible and $\\sigma^{2}>0$, then the $\\left(\\beta, \\sigma^{2}\\right)$-Gaussian coupling is optimal.\n\nTo show incentive compatibility, observe that $\\omega+q^{o} a_{i}+q^{c} \\bar{a}$ and $a_{i}$ are jointly normally distributed. Therefore, the incentive compatibility is equivalent to\n\n$$\n\\begin{aligned}\n\\mathbb{E}\\left[\\left(\\omega+q^{o} a_{i}+q^{c} \\bar{a}\\right) a_{i}\\right] & =0, \\\\\n\\mathbb{E}\\left[\\left(\\omega+q^{o} a_{i}+q^{c} \\beta \\omega\\right) a_{i}\\right] & =0, \\\\\n\\left(1+q^{c} \\beta\\right) \\beta \\sigma_{\\omega}^{2}+q^{o}\\left(\\beta^{2} \\sigma_{\\omega}^{2}+\\sigma^{2}+\\frac{1}{N-1} \\sigma^{2}\\right) & =0,\n\\end{aligned}\n$$\n\nwhich, after rearrangement, is equivalent to (22).\nFinally, it is straightforward to verify that if $k^{*} \\in(0, N)$, then $\\sigma^{2}>0$; see Appendix A. 3 for details. $\\square$\n\n[^8]Theorem 3 highlights an alternative way of providing information optimally-by carefully coupling the players' individual noises. In the case of a normally distributed state, this information structure comes closest to the existing Gaussian information design literature (e.g., Angeletos and Pavan (2007, 2009), Bergemann and Morris (2013), Bergemann, Heumann, and Morris (2015, 2021), Ui (2020)). However, the optimality of a Gaussian coupling depends crucially on the prior's Gaussian form, making this solution less robust than targeted disclosure. Accordingly, in the applications that are presented in Section 8.1 and Section 8.2, we focus on optimal targeted disclosure. Nonetheless, we encourage the reader to bear in mind the fact that if the state is normally distributed in those applications, then providing information via a Gaussian coupling is also optimal.","text_sha256":"2e3f2ccf06d2454b51b003ebe498357c0ab560b3ad0a365e5bcd654bbf9b18ae"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0011","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Linear-Quadratic Games: Interdependent Values","text":"## 6 Linear-Quadratic Games: Interdependent Values\n\nIn this section, we apply the certification approach to strategic environments with interdependent values and thus with a multidimensional state. For tractability, we focus on jointly normally distributed state components. As in the previous section, we first present general results and then illustrate them with an economic application: informing competitive pricing.\n\nFormally, we study symmetric games with $\\omega \\sim N\\left(\\mu \\mathbf{1}, \\sigma^{2} M(1, \\rho)\\right)$, where $M(x, y)$ denotes the $N \\times N$ matrix whose diagonal entries equal $x$ and off-diagonal entries equal $y$, and $\\rho \\in(-1 /(N-1), 1)$. The players' payoffs are\n\n$$\n\\begin{aligned}\nu_{i}(a, \\omega) & =\\left(\\omega_{i} p^{o}+\\breve{\\omega}_{-i} p^{c}+b_{1}\\right) a_{i}+\\frac{q^{o}}{2} a_{i}^{2}+q^{c} a_{i} \\breve{a}_{-i}+f_{1}\\left(a_{-i}, \\omega\\right), \\\\\nv(a, \\omega) & =\\sum_{i=1}^{N}\\left(\\omega_{i} \\tilde{p}^{o}+b_{2}\\right) a_{i}+\\left(\\sum_{i=1}^{N} \\sum_{j \\neq i} \\omega_{i} a_{j}\\right) \\tilde{p}^{c}+\\left(\\sum_{i=1}^{N} a_{i}^{2}\\right) \\tilde{q}^{o}+\\left(\\sum_{i=1}^{N} \\sum_{j \\neq i} a_{i} a_{j}\\right) \\tilde{q}^{c}+f_{2}(\\omega),\n\\end{aligned}\n$$\n\nwhere $\\breve{\\omega}_{-i} \\triangleq \\sum_{j \\neq i} \\omega_{j}, \\breve{a}_{-i} \\triangleq \\sum_{j \\neq i} a_{j}, f_{1}$ and $f_{2}$ are arbitrary functions of the indicated arguments, and $q^{o}, q^{c}, p^{o}, p^{c}, \\tilde{q}^{o}, \\tilde{q}^{c}, \\tilde{p}^{o}, \\tilde{p}^{c}$ are commonly known parameters that satisfy\nthe concavity and \"genericity\" conditions:\n\n$$\n\\begin{gathered}\nq^{o}-q^{c}<0, q^{o}+(N-1) q^{c}<0, p^{o}>\\left|p^{c}\\right|, \\\\\nq^{o} \\tilde{q}^{c} \\neq q^{c} \\tilde{q}^{o}, \\\\\n\\left(q^{o}-q^{c}\\right)\\left(\\tilde{p}^{o}-\\tilde{p}^{c}\\right) \\neq\\left(\\tilde{q}^{o}-\\tilde{q}^{c}\\right)\\left(p^{o}-p^{c}\\right), \\\\\n\\left(q^{o}+(N-1) q^{c}\\right)\\left(\\tilde{p}^{o}+(N-1) \\tilde{p}^{c}\\right) \\neq\\left(\\tilde{q}^{o}+(N-1) \\tilde{q}^{c}\\right)\\left(p^{o}+(N-1) p^{c}\\right) .\n\\end{gathered}\n$$\n\nCondition (25) ensures that $u_{i}$ is strictly concave in $a_{i}$ for all $N \\geq 1$, whereas other conditions avoid knife-edge cases (e.g., division by zero).\n\nThis game is strategically equivalent to-and thus admits the same solution as-a simpler normalized game, in which $\\omega \\sim N(0, M(1, \\rho))$,\n\n$$\n\\begin{aligned}\nu_{i}(a, \\omega) & =\\left(\\omega_{i} p^{o}+\\breve{\\omega}_{-i} p^{c}\\right) a_{i}+\\frac{q^{o}}{2} a_{i}^{2}+q^{c} a_{i} \\breve{a}_{-i} \\\\\nv(a, \\omega) & =\\sum_{i=1}^{N}\\left(\\omega_{i} \\tilde{p}^{o}+b\\right) a_{i}+\\left(\\sum_{i=1}^{N} \\sum_{j \\neq i} \\omega_{i} a_{j}\\right) \\tilde{p}^{c}+\\left(\\sum_{i=1}^{N} a_{i}^{2}\\right) \\tilde{q}^{o}+\\left(\\sum_{i=1}^{N} \\sum_{j \\neq i} a_{i} a_{j}\\right) \\tilde{q}^{c}\n\\end{aligned}\n$$\n\nand where $q^{o}, q^{c}, p^{o}, p^{c}, \\tilde{q}^{o}, \\tilde{q}^{c}, \\tilde{p}^{o}, \\tilde{p}^{c}$ are the same as in the original game. Therefore, we can focus on the analysis of the normalized game.\n\nGiven the greater complexity of the present setting, we proceed slightly differently from the previous section. We still rely on affine contracts for certification; however, instead of calculating the optimal contract parameters in closed form, we invoke an envelope theorem. We show that an optimal symmetric affine contract implements an obedient allocation rule, thereby certifying both its global optimality in the dual problem and the optimality of the corresponding measure in the primal problem.\n\nFormally, observe that the players' marginal payoffs are linear:\n\n$$\n\\dot{u}_{i}(a, \\omega)=\\omega_{i} p^{o}+\\breve{\\omega}_{-i} p^{c}+q^{o} a_{i}+q^{c} \\breve{a}_{-i},\n$$\n\nand the dual payoff function is\n\n$$\nw(a, \\omega, \\lambda)=v(a, \\omega)-\\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right)\\left(\\omega_{i} p^{o}+\\breve{\\omega}_{-i} p^{c}+q^{o} a_{i}+q^{c} \\breve{a}_{-i}\\right) .\n$$\n\nGiven the linear-quadratic nature of the environment, we conjecture that an optimal allocation rule is linear in the state. As this allocation rule must be a best-response given the dual payoff (30), we posit a certificate that is affine and symmetric across players:\n\n$$\n\\lambda_{i}\\left(a_{i}\\right)=-\\left(x a_{i}+x_{0}\\right) .\n$$\n\nWith this certificate, the dual payoff function can be written as\n\n$$\n\\begin{aligned}\nw\\left(a, \\omega, x, x_{0}\\right) & \\sim \\omega^{T} M\\left(\\tilde{p}^{o}+x p^{o}, \\tilde{p}^{c}+x p^{c}\\right) a+a^{T} M\\left(\\tilde{q}^{o}+x q^{o}, \\tilde{q}^{c}+x q^{c}\\right) a \\\\\n& +\\left(b+x_{0}\\left(q^{o}+(N-1) q^{c}\\right)\\right) \\breve{a},\n\\end{aligned}\n$$\n\nwhere we omitted an action-independent term $x_{0}\\left(p^{o}+(N-1) p^{c}\\right) \\sum_{i=1}^{N} \\omega_{i}$. Because of (29) and $\\mathbb{E}[\\omega]=0$, any incentive-compatible expected individual action must equal zero, and therefore we conjecture that the coefficient in front of $\\breve{a}$ equals zero and thus\n\n$$\n\\begin{aligned}\nx_{0}^{*} & =-\\frac{b}{q^{o}+(N-1) q^{c}}, \\\\\nw\\left(a, \\omega, x, x_{0}^{*}\\right) & \\sim \\omega^{T} M\\left(\\tilde{p}^{o}+x p^{o}, \\tilde{p}^{c}+x p^{c}\\right) a+a^{T} M\\left(\\tilde{q}^{o}+x q^{o}, \\tilde{q}^{c}+x q^{c}\\right) a \\\\\n& \\sim \\omega^{T} M\\left(\\bar{p}^{o}(x), \\bar{p}^{c}(x)\\right) a+a^{T} M\\left(\\bar{q}^{o}(x), \\bar{q}^{c}(x)\\right) a .\n\\end{aligned}\n$$\n\nwhere we defined\n\n$$\n\\begin{aligned}\n& \\bar{q}^{o}(x) \\triangleq \\tilde{q}^{o}+x q^{o}, \\bar{q}^{c}(x) \\triangleq \\tilde{q}^{c}+x q^{c} \\\\\n& \\bar{p}^{o}(x) \\triangleq \\tilde{p}^{o}+x p^{o}, \\bar{p}^{c}(x) \\triangleq \\tilde{p}^{c}+x p^{c}\n\\end{aligned}\n$$\n\nFor the optimal allocation rule $a^{*}(\\omega)$ to maximize (32), $M\\left(\\bar{q}^{o}(x), \\bar{q}^{c}(x)\\right)$ must be negative semi-definite; otherwise, the value of the dual payoff diverges to $+\\infty$.\n\nLemma 1. $M\\left(\\bar{q}^{o}(x), \\bar{q}^{c}(x)\\right)$ is negative semi-definite if and only if\n\n$$\nx \\geq \\underline{x} \\triangleq \\max \\left\\{-\\frac{\\tilde{q}^{o}-\\tilde{q}^{c}}{q^{o}-q^{c}},-\\frac{\\tilde{q}^{o}+(N-1) \\tilde{q}^{c}}{q^{o}+(N-1) q^{c}}\\right\\} .\n$$\n\n$M\\left(\\bar{q}^{o}(x), \\bar{q}^{c}(x)\\right)$ is negative definite if and only if $x>\\underline{x}$.\nIf $x>\\underline{x}, M\\left(\\bar{q}^{o}(x), \\bar{q}^{c}(x)\\right)$ is negative definite, and the optimal best-response is\n\n$$\na^{*}(\\omega, x)=-\\frac{1}{2} M^{-1}\\left(\\bar{q}^{o}(x), \\bar{q}^{c}(x)\\right) M\\left(\\bar{p}^{o}(x), \\bar{p}^{c}(x)\\right) \\omega .\n$$\n\nThe resulting expected value of the dual payoff as a function of $x$ is","text_sha256":"f580aea2cc528db416cede6ded7a788f7bcac43169f9da5a67cf43d982b277dc"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0012","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Linear-Quadratic Games: Interdependent Values","text":"$$\n\\begin{aligned}\nW(x) & \\triangleq \\mathbb{E}\\left[w\\left(a^{*}(\\omega, x), \\omega, x, x_{0}^{*}\\right)\\right] \\\\\n& =-\\frac{1}{4} \\mathbb{E}\\left[\\omega^{T} M\\left(\\bar{p}^{o}(x), \\bar{p}^{c}(x)\\right) M^{-1}\\left(\\bar{q}^{o}(x), \\bar{q}^{c}(x)\\right) M\\left(\\bar{p}^{o}(x), \\bar{p}^{c}(x)\\right) \\omega\\right] .\n\\end{aligned}\n$$\n\nLemma 2. $\\lim _{x \\downarrow \\underline{x}} W(x)=\\lim _{x \\uparrow+\\infty} W(x)=+\\infty$, and $\\min _{x \\geq \\underline{x}} W(x)$ exists.\nBy Lemma 2, the dual payoff admits a minimum at some $x^{*}>\\underline{x}$. We show that this $x^{*}$ certifies an optimal information structure.\n\nDefinition 5. (Linear Disclosure) For $R=\\left(r_{1}, \\ldots, r_{N}\\right) \\in \\mathbb{R}^{N \\times N}$, an $R$-linear disclosure is an information structure such that for all $i$ and $\\omega$,\n\n$$\ns_{i}=r_{i}^{T} \\omega=\\sum_{j=1}^{N} r_{i, j} \\omega_{j} .\n$$\n\nA linear disclosure informs player $i$ about a linear statistic of the state with the responsiveness weights $r_{i}$. When $N=1$, a linear disclosure corresponds to no disclosure if $r=0$, and to full disclosure otherwise. When $N \\geq 2$ and $R$ has full rank, each signal is imperfectly informative, yet the full signal profile perfectly reveals the state. In that case, like a Gaussian coupling, a linear disclosure spreads complete state information across players. When the state is normally distributed, a linear disclosure is a Gaussian information structure.\n\nTheorem 4. (Optimality of Linear Disclosure) In a normalized symmetric linearquadratic Gaussian game with a multidimensional state, an $R^{*}$-linear disclosure is optimal, where\n\n$$\nR^{*}=-\\frac{1}{2} M^{-1}\\left(\\bar{q}^{o}\\left(x^{*}\\right), \\bar{q}^{c}\\left(x^{*}\\right)\\right) M\\left(\\bar{p}^{o}\\left(x^{*}\\right), \\bar{p}^{c}\\left(x^{*}\\right)\\right),\n$$\n\nand $x^{*}$ is any point in $\\arg \\min _{x>\\underline{x}} W(x)$. In the optimal equilibrium, $a_{i} \\equiv s_{i}$. The induced measure $\\pi \\in \\Delta(A \\times \\Omega)$ is uniquely optimal.\n\nProof. By Lemma 2, the dual payoff admits a minimum at some $x^{*}>\\underline{x}$. Consider any such $x^{*}$. By (35), the dual best-response is $a^{*}\\left(\\omega, x^{*}\\right)=R^{*} \\omega$; hence, this allocation rule is implementable by incentives, by the contract $\\lambda_{i}\\left(a_{i}\\right)=-\\left(x^{*} a_{i}+x_{0}^{*}\\right)$. The allocation rule is symmetric in $i$ and linear in $\\omega$. Recall that\n\n$$\nW(x)=\\mathbb{E}\\left[v\\left(a^{*}(\\omega, x), \\omega\\right)+\\sum_{i=1}^{N}\\left(x a_{i}^{*}(\\omega, x)+x_{0}^{*}\\right) \\dot{u}_{i}\\left(a^{*}(\\omega, x), \\omega\\right)\\right] .\n$$\n\nSince $a^{*}(\\omega, x)$ is an interior maximizer and $x$ is an interior minimizer of the dual payoff, it follows by the Envelope Theorem that\n\n$$\n\\left.\\frac{d W}{d x}\\right|_{x=x^{*}}=\\mathbb{E}\\left[\\sum_{i=1}^{N} a_{i}^{*}\\left(\\omega, x^{*}\\right) \\dot{u}_{i}\\left(a^{*}\\left(\\omega, x^{*}\\right), \\omega\\right)\\right]=0 .\n$$\n\nBy symmetry in $i$, it follows that for all $i$,\n\n$$\n\\mathbb{E}\\left[a_{i}^{*}\\left(\\omega, x^{*}\\right) \\dot{u}_{i}\\left(a^{*}\\left(\\omega, x^{*}\\right), \\omega\\right)\\right]=0 .\n$$\n\nBecause $a_{i}^{*}\\left(\\omega, x^{*}\\right)$ is linear in $\\omega$ and the components of $\\omega$ are jointly normally distributed, $a_{i}^{*}\\left(\\omega, x^{*}\\right)$ and $\\dot{u}_{i}\\left(a^{*}\\left(\\omega, x^{*}\\right), \\omega\\right)$ are also jointly normally distributed. Furthermore, $\\mathbb{E}\\left[\\dot{u}_{i}\\left(a^{*}\\left(\\omega, x^{*}\\right), \\omega\\right)\\right]=0$, and, by (39), $a_{i}^{*}\\left(\\omega, x^{*}\\right)$ and $\\dot{u}_{i}\\left(a^{*}\\left(\\omega, x^{*}\\right), \\omega\\right)$ are uncorrelated, and thus independent. Therefore, for all $i$,\n\n$$\n\\mathbb{E}\\left[\\dot{u}_{i}\\left(a^{*}\\left(\\omega, x^{*}\\right), \\omega\\right) \\mid a_{i}^{*}\\left(\\omega, x^{*}\\right)\\right]=0 .\n$$\n\nTherefore, the allocation rule $a^{*}\\left(\\omega, x^{*}\\right)$ is implementable by information, by $R^{*}$-linear disclosure. By Theorem 1, the optimality follows.\n\nFinally, as the dual agent's best response to $\\lambda\\left(x^{*}, x_{0}^{*}\\right)$ is unique, it follows from Proposition 1 that the corresponding measure is uniquely optimal. $\\square$\n\nTheorem 4 shows that an optimal information structure can be found within a simple class of symmetric noise-free Gaussian information structures (cf. Bergemann et al. (2015)). Moreover, its parameters can be identified by solving a one-dimensional minimization of $W(x)$.\n\nIn contrast with the common-state setting of Section 5, the optimal measure is unique here. Formally, in that setting the optimal certificate leaves the dual agent indifferent among many action profiles, whereas in this interdependent-value environment it yields a single best response. Conceptually, having one state component per player introduces richer uncertainty, which in turn fully determines the optimal recommendations.","text_sha256":"458e2f44372a02fcb06ea0bd71df746557fc3ea72a526fa8546ce6b7a8ad6943"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0013","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"7 Extensions","text":"## 7 Extensions\n\nBounded Action Spaces We have assumed that action spaces are unbounded, to avoid corner solutions and simplify the exposition. However, the certification approach can be easily extended to games with upper and lower bounds on each player's action space, as demonstrated in Appendix B. The primal problem formulation remains the same, except that the boundary first-order conditions must be inequalities rather than equalities. The dual problem formulation is identical, with the added sign constraints on the contract functions evaluated at the boundary actions.\n\nInfinite Economies Our analysis considers games with a finite number of players. It does not directly cover games with a continuum of players, as considered in the literature on infinite economies, e.g., by Angeletos and Pavan (2007) and Bergemann and Morris (2013), even though we anticipate that an analogous certification approach, suitably extended, would work in those settings as well.\n\nNonetheless, our analysis enables determining an optimal information structure in a game with a fixed number of players and then examining its limit as the number of players approaches infinity. Our results suggest that in many quadratic economies with a normally distributed state, Gaussian information structures are certifiably optimal and an optimal aggregate behavior is a deterministic function of a state. With a finite number of players, the latter condition necessitates the designer to either not introduce extraneous noise at all or to ensure that the noises are carefully coupled. With an infinite number of players, this condition can be met by adding independent noises and relying on the law of large numbers; the optimal aggregate behavior can be ensured by the large population size rather than by noise correlation. However, our analysis suggests that even in infinite economies, alternative information structures such as targeted disclosure may also be optimal, and robustly so.\n\nGeneral Smooth Games The tractability of the certification approach in concave games hinges on the capacity of first-order conditions to succinctly represent players' incentives. In general smooth games, i.e., games in which each player's payoff is continuously differentiable in their own action, these first-order conditions may be insufficient, as they can select a suboptimal local maximum or even a minimum in a player's payoff. However, they remain necessary, which enables a straightforward extension of our analysis: in a smooth but not necessarily concave information-design problem, one constructs a relaxed primal problem that features only first-order conditions. This problem can be solved using the certification approach. In the final step, one verifies whether players obey the recommendation of the information structure found. If so, this information structure solves the original information-design problem.","text_sha256":"d124051d4d9ae27b3b09585d5e6654409804c957de409d9215878354a1cbb06a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0014","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"8 Applications","text":"## 8 Applications\n\nIn this section, we illustrate the developed machinery with three economic applications. ${ }^{11}$\n\n[^9]","text_sha256":"9f200c79e60a053b36f0168a60b65309cb07a1383399b5d1ece2f1e67cba3240"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0015","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"8.1 Persuading Investors","text":"### 8.1 Persuading Investors\n\nAn important question in securities regulation is how much disclosure should be required from firms seeking capital (e.g., Carvajal, Rostek, and Sublet (2018)). Although many factors matter, in this section we focus on two. On the benefit side, greater transparency helps investors reduce risk and make better decisions; on the cost side, it intensifies cream-skimming, as investors herd into the same projects and crowd out investment gains. ${ }^{12}$ We characterize the optimal balance between these forces within our framework and show that the resulting solution aligns with key venture-capital practices.\n\nFormally, we consider an investment game in the spirit of Angeletos and Pavan (2007) and Bergemann and Morris (2013). There are $N \\geq 2$ investors, who simultaneously decide how much to invest in a project, $A_{i}=\\mathbb{R}$. The profitability of the project is uncertain; it depends on the unknown project quality $\\theta \\in \\mathbb{R}$ and on the total amount of investment $\\breve{a} \\triangleq \\sum_{i=1}^{N} a_{i}$. The ex post payoff of player $i$ is\n\n$$\nu_{i}(a, \\theta)=(\\theta-r \\breve{a}) a_{i}-c a_{i},\n$$\n\nwhere $r>0$ is the congestion parameter and $c>0$ is the opportunity cost of investment. As $r>0$, the project features decreasing returns to scale, i.e., its average profitability decreases in the total investment. We can conveniently rewrite payoff (41) as\n\n$$\nu_{i}(a, \\omega)=r(\\omega-\\breve{a}) a_{i}\n$$\n\nwhere the state $\\omega \\in \\Omega$ is a normalized project quality defined as $\\omega \\triangleq(\\theta-c) / r$.\nGiven (42), for any belief $\\mu \\in \\Delta\\left(A_{-i} \\times \\Omega\\right)$, the player $i$ 's best response can be found via the first-order condition to equal\n\n$$\na_{i}^{*}(\\mu)=\\mathbb{E}_{\\mu}\\left[\\frac{\\omega-\\breve{a}_{-i}}{2}\\right],\n$$\n\n[^10]where $\\breve{a}_{-i} \\triangleq \\sum_{j \\neq i} a_{j}$, so that the best response linearly increases in the normalized quality expectation and linearly decreases in the expected amount of total investment made by other players. The players' actions are thus strategic substitutes.\n\nThe information designer has full control over the information regarding the project quality and can privately convey it to each player, thereby persuading that player to invest more or less. The designer aims to maximize the total profits generated by the project with her ex post payoff being ${ }^{13}$\n\n$$\nv(a, \\omega)=\\sum_{i=1}^{N}(\\omega-\\breve{a}) a_{i}=(\\omega-\\breve{a}) \\breve{a}=\\omega \\breve{a}-\\breve{a}^{2} .\n$$\n\nInvestment Control If the designer could control the individual investment directly, then she would set the total investment to respond to the state as $\\breve{a}^{F B}(\\omega)=\\omega / 2$. As we later show, this first-best allocation rule cannot be implemented via pure information control; however, it can be roughly approximated.\n\nNo Disclosure and Full Disclosure Two other natural benchmarks are the cases of no disclosure and full disclosure. In these cases, the designer's payoff can be easily computed as:\n\n$$\n\\begin{aligned}\nv^{N D} & =\\frac{N}{(N+1)^{2}} \\mathbb{E}^{2}[\\omega], \\\\\nv^{F D} & =\\frac{N}{(N+1)^{2}}\\left(\\mathbb{E}^{2}[\\omega]+\\mathbb{V}[\\omega]\\right) .\n\\end{aligned}\n$$\n\nComparing the designer's payoffs across the two benchmarks shows that full disclosure strictly dominates no disclosure. However, both information structures suffer from a scaling problem: as the number of players goes to infinity, the designer's payoff and thus the total project profit converges to zero. In the limit, the individual rent as well as the total profit are dissipated. We now show that this problem can be mitigated by adopting an optimal information structure.\n\n[^11]Optimal Information For any information structure, taking the ex-ante expectation of both sides of (43), applying the law of iterated expectations, and summing across players (cf. Bergemann et al. (2017)), we obtain that the expected total investment remains constant at\n\n$$\n\\mathbb{E}[\\breve{a}]=\\frac{N}{N+1} \\mathbb{E}[\\omega]=\\frac{N}{N+1} \\frac{\\mathbb{E}[\\theta]-c}{r}>\\frac{1}{2} \\mathbb{E}[\\omega]=\\mathbb{E}\\left[\\breve{a}^{F B}(\\omega)\\right] .\n$$\n\nConsequently, whenever $\\mathbb{E}[\\omega]>0$, information control can never achieve the full-control benchmark. However, while the designer cannot affect the ex-ante total amount of investment, she can direct investment toward more productive projects. An optimal way to do so is:\n\nProposition 3. (Persuading Investment by Exclusivity) For any number of players, a 1-targeted disclosure is optimal.\n\nProof. The setting is an instance of the general setting (11-12) with $h=N, q^{o}=-1$, $q^{c}=-N, p^{o}=0$, and $p^{c}=-N^{2}$. The formula (20) for an optimal $k^{*}$ becomes:\n\n$$\nk^{*}=\\frac{q^{o}\\left(h q^{o}-p^{o}\\right)}{q^{o}\\left(2 p^{c}-h q^{c}\\right)-p^{o} q^{c}} N=\\frac{-1(-N-0)}{-1\\left(-2 N^{2}+N^{2}\\right)-0} N=\\frac{1}{N} N=1,\n$$\n\nand thus by Theorem 2, 1-targeted disclosure is optimal. $\\square$\n\nProposition 3 shows that an optimal information structure takes a very simple form: the designer designates a single player and provides him with exclusive and full access to information. This player makes fully informed decisions. All other players invest the same amount regardless of the project quality. The same information structure is optimal for any number of investors, congestion and cost parameters, and project quality distribution.\n\nThe optimal designer's payoff equals\n\n$$\nv^{*}=\\mathbb{E}\\left[\\left(\\omega-\\frac{\\omega}{2}-\\frac{N-1}{2(N+1)} \\mathbb{E}[\\omega]\\right)\\left(\\frac{\\omega}{2}+\\frac{N-1}{2(N+1)} \\mathbb{E}[\\omega]\\right)\\right]=\\frac{N}{(N+1)^{2}} \\mathbb{E}^{2}[\\omega]+\\frac{1}{4} \\mathbb{V}[\\omega] .\n$$\n\nAs such, optimal information design avoids rent dissipation as the number of players grows to infinity. The project's total profit always stays above $\\mathbb{V}[\\omega] / 4$ and converges to this level as $N \\rightarrow \\infty$. The limiting payoff increases with the variance of project quality: under the optimal information structure, just as under full disclosure, riskier projects generate higher realized profits even when their expected quality is the same. ${ }^{14}$","text_sha256":"02461785e79c3f8937de382b2ec793767ad72da985bbb393babcb060567254ae"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0016","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"8.1 Persuading Investors","text":"Interestingly, this asymmetric treatment of investors mirrors common venture-capital practice: firms cultivate close relationships with a small set of venture capitalists who learn more about the firm than the broader pool of potential investors (e.g., Bernstein, Giroud, and Townsend (2016)). Although several explanations could account for this pattern, our results highlight its informational benefit: by creating a sharp information asymmetry among investors, it curbs congestion and raises both investment profits and investor welfare.","text_sha256":"00b1259b59e5d52b7ac7a54d95116c2a7bc5abde6c3e88d678dc39fc6eb5ba58"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0017","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"8.2 Polarizing Predictions","text":"### 8.2 Polarizing Predictions\n\nNumerous studies document societal polarization, with individuals reaching sharply divergent views of the same issues (e.g., Alesina, Miano, and Stantcheva (2020)). Conventional explanations stress behavioral biases or naïveté, yet recent evidence shows that most people can reliably distinguish fake news from accurate reporting (Angelucci and Prat (2024)). This tension prompts a sharper question: To what extent can polarization arise among fully rational agents solely because they receive different information, and which information structure generates the greatest polarization?\n\nWe address this question in a prediction game, in which multiple players try to predict a common underlying state. The state is one-dimensional $\\omega \\in \\mathbb{R}$ and distributed\n\n[^12]according to the prior $\\mu_{0}$. There are $N \\geq 2$ players, each choosing a prediction $a_{i} \\in$ $A_{i}=\\mathbb{R}$. The ex post payoff to player $i$ is\n$$\nu_{i}(a, \\omega)=-\\left(a_{i}-\\omega\\right)^{2} .\n$$\nBecause each player's payoff depends only on his own prediction and the state, for any given belief $\\mu \\in \\Delta\\left(A_{-i} \\times \\Omega\\right)$, player $i$ 's best prediction is simply the posterior expectation of the state:\n$$\na_{i}^{*}(\\mu)=\\mathbb{E}_{\\mu}[\\omega] .\n$$\n\nWe focus on the question of inducing maximal polarization of the players' predictions as measured by the pairwise squared sum:\n\n$$\nv(a, \\omega)=\\sum_{i, j}\\left(a_{i}-a_{j}\\right)^{2} .\n$$\n\nGiven the payoff, the designer benefits from sending private signals: any public information structure, including full disclosure or no disclosure, leads the players to make the same predictions and consequently minimizes the designer's objective. An optimal information structure, on the one hand, must provide some state information to move players' predictions and, on the other hand, should heterogeneously obfuscate the information to counteract truth drifting.\n\nProposition 4. (Polarizing by Segregation) Let the number of players $N$ be even. Then, an $N / 2$-targeted disclosure is optimal.\n\nProof. The setting is an instance of the general setting (11-12) with $h=0, q^{o}=-1$, $q^{c}=0, p^{o}=2 N^{2}$, and $p^{c}=-2 N^{2}$. The formula (20) for an optimal $k^{*}$ becomes:\n\n$$\nk^{*}=\\frac{q^{o}\\left(h q^{o}-p^{o}\\right)}{q^{o}\\left(2 p^{c}-h q^{c}\\right)-p^{o} q^{c}} N=\\frac{-1\\left(-2 N^{2}\\right)}{-1\\left(-4 N^{2}\\right)-0} N=\\frac{N}{2},\n$$\n\nand thus by Theorem 2, $N / 2$-targeted disclosure is optimal. $\\square$\n\nProposition 4 shows that, irrespective of the prior, expectation polarization can be achieved by a strikingly simple information structure that informs only half of the population.\n\nThis finding aligns with earlier work in information design. In a two-player, binarystate model, Arieli et al. (2021) characterized the feasible joint belief distributions and reached a parallel conclusion: to maximize polarization of posterior expectations, it is optimal to inform one of the two players. Proposition 4 extends this insight to any number of players and states. Arieli and Babichenko (2024) continue the study of polarization of posterior beliefs. They show that for any even number of players, informing half of them is optimal, because it achieves appropriate statistical bounds. Interestingly, even though belief polarization and expectation polarization differ once the state space is larger than binary, and the underlying arguments are not interchangeable, both problems share the same optimal information structure.\n\nThese results contribute to the debate over the sources of societal polarization. Recent studies show that political news consumption is sharply segregated, either because it flows through endogenous peer networks (Bowen et al. (2023)) or because it is curated by social-media algorithms (Braghieri et al. (2024)). As a result, different groups may learn about different topics. This kind of segregation-learning about different topics, rather than learning different facts about the same topic-can maximally polarize a society.","text_sha256":"7eff1fece4e6bdcf679f1bf218ebfff9e1258ecd450b6976097be0df2b66a30d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0018","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"8.3 Informing Competitive Pricing","text":"### 8.3 Informing Competitive Pricing\n\nThe information design machinery is useful to understand and guide the design of digital platforms and, more generally, of algorithmic information processing (e.g., Bergemann and Bonatti (2024), Ichihashi and Smolin (2025)). In this section, we view the designer as a platform that knows demand conditions better than firms do, thanks to a larger, more recent sales dataset and superior analytics. The platform can privately convey this information to each firm by granting it access to personalized data analysis or direct price recommendations, thereby persuading the firm regarding its pricing decisions. The\nplatform aims to maximize a weighted average of consumer and producer surplus. We characterize the information structure such a designer would optimally design and the resulting price behavior.\n\nFormally, we consider a differentiated duopoly game in the spirit of Vives (1984) and Gal-Or (1985). ${ }^{15}$ There are two firms that operate in the market. Each firm sells a single product and competes in price with its opponent, so action $a_{i}$ is the price set by firm $i$. Demand is ex ante symmetric across firms and is linear in prices and demand shocks:\n\n$$\nq_{i}(a, \\omega)=\\omega_{i}-a_{i}+\\eta a_{-i},\n$$\n\nwhere $\\omega_{i}$ captures an individual demand shock for firm $i$ 's product and $\\eta$ is a crossprice sensitivity that satisfies $|\\eta| \\in(0,1)$, so that $-1<\\eta<0$ corresponds to the case of complementary products, whereas $0<\\eta<1$ corresponds to the case of substitute products.\n\nThe state is two-dimensional $\\omega=\\left(\\omega_{1}, \\omega_{2}\\right) \\in \\mathbb{R}^{2}$ and comprises the individual demand shocks. The shocks are independently and identically distributed according to a normal distribution, $\\omega_{i} \\sim N\\left(\\omega_{0}, \\sigma^{2}\\right)$. ${ }^{16}$ The firms do not know the state but can be informed about it by the designer.\n\nThe firms have quadratic costs of production so that their profits are\n\n$$\nu_{i}(a, \\omega)=a_{i} q_{i}(a, \\omega)-c q_{i}(a, \\omega)^{2} .\n$$\n\nThe resulting ex post values of consumer surplus and producer surplus are\n\n$$\n\\begin{aligned}\n& C S(a, \\omega)=\\frac{a_{1}^{2}}{2}+\\frac{a_{2}^{2}}{2}-\\eta a_{1} a_{2}-a_{1} \\omega_{1}-a_{2} \\omega_{2}, \\\\\n& P S(a, \\omega)=u_{1}(a, \\omega)+u_{2}(a, \\omega) .\n\\end{aligned}\n$$\n\n[^13]The designer's payoff is a convex combination of consumer and producer surpluses, with $\\delta \\in[0,1]$ representing the weight assigned to the consumer surplus:\n\n$$\nv(a, \\omega)=\\delta C S(a, \\omega)+(1-\\delta) P S(a, \\omega) .\n$$\n\nThe optimal designer's choices in the extreme cases $\\delta=1$ and $\\delta=0$ correspond to consumer-optimal and producer-optimal information structures, respectively, whereas the choice in the case $\\delta=1 / 2$ corresponds to the socially efficient information structure. As the welfare weight $\\delta$ spans the interval [0,1], the corresponding solutions span the Pareto frontier in the space of consumer and producer surpluses.\n\nPrice Control We begin the analysis by studying a hypothetical scenario in which the designer can directly control the prices set by the firms. This scenario constitutes a first-best benchmark; it provides an upper bound on the designer's payoff and illustrates the designer's preferred pricing.\n\nThis problem admits a solution only if $\\delta$ is not excessively high: there is a threshold value $\\bar{\\delta} \\in(0,1)$, such that if $\\delta>\\bar{\\delta}$, then the designer can arbitrarily increase her payoff by setting arbitrarily large negative prices, because the monetary transfer to consumers outweighs any allocation inefficiency. In contrast, if $\\delta<\\bar{\\delta}$, then the designer's problem is well-behaved; it is concave and admits a unique solution that can be found by first-order conditions to (51):\n\n$$\na_{i}^{F B}(\\omega)=r_{o}^{F B} \\omega_{i}+r_{c}^{F B} \\omega_{-i},\n$$\n\nwhere $r_{o}^{F B}$ and $r_{c}^{F B}$ measure the optimal responsiveness of each firm's price to its own demand shock and to the shock of its competitor (explicit formulas appear in Section A.6).\n\nNo Disclosure and Full Disclosure The designer does not control prices directly but rather indirectly through demand information she supplies to firms. Before deriving the generally optimal policy, it is instructive to analyze two information benchmarks:\nno disclosure and full disclosure.\nUnder no disclosure, the firms' beliefs stay at the prior, and the equilibrium prices satisfy the first-order conditions derived from (48). In equilibrium, each firm sets a price\n\n$$\na_{i}^{N D}=\\frac{1+2 c}{2(1+c)-\\eta(1+2 c)} \\omega_{0} .\n$$\n\nLacking demand information, firms fix their prices at a level proportional to the expected demand.\n\nIn contrast, under full disclosure, the demand shocks are always commonly known. In equilibrium, each firm responds linearly to the shocks perfectly anticipating the price of its opponent:\n\n$$\na_{i}^{F D}(\\omega)=r_{o}^{F D} \\omega_{i}+r_{c}^{F D} \\omega_{-i}=\\frac{2(1+c)(1+2 c)}{4(1+c)^{2}-\\eta^{2}(1+2 c)^{2}} \\omega_{i}+\\frac{\\eta(1+2 c)^{2}}{4(1+c)^{2}-\\eta^{2}(1+2 c)^{2}} \\omega_{-i} .\n$$\n\nIn a sense, this behavior enriches price-setting under no disclosure. If $\\omega_{1}=\\omega_{2}=\\omega_{0}$, then prices are the same as those under no disclosure, $a_{i}^{F D}\\left(\\omega_{0}, \\omega_{0}\\right)=a_{i}^{N D}$. If $\\omega_{1} \\neq \\omega_{2}$, then demand is asymmetric across firms, and prices are adjusted to reflect competitive advantages. However, the prices average to the prices under no disclosure, $\\mathbb{E}\\left[a_{i}^{F D}(\\omega)\\right]=$ $a_{i}^{N D}$.\n\nOptimal Information The choice of any of the extreme information structures has drawbacks. Providing no disclosure misses the opportunity to strengthen the link between demand and allocation and thus potentially limits efficiency. Providing full disclosure may exacerbate competition and dissipate firm profits. Instead, and as a direct consequence of Theorem 4, the optimal information structure is partially informative and takes a form of a linear disclosure.\n\nProposition 5. (Managing Competition by Linear Statistics) There is a parameter $\\hat{\\delta}(\\eta, c)$ such that if $\\delta \\neq \\hat{\\delta}(\\eta, c)$, an optimal direct information structure exists, is unique,\nand recommends for some coefficients $a_{0}^{*}, r_{o}^{*}$ and $r_{c}^{*}$ prices\n\n$$\na_{i}(\\omega)=a_{0}^{*}+r_{o}^{*} \\omega_{i}+r_{c}^{*} \\omega_{-i} .\n$$","text_sha256":"93a89830a1cc299bb1e1d902b6a19ce24eee94b8dc631f61a6519781d05bbec6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0019","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"8.3 Informing Competitive Pricing","text":"The uniquely optimal direct information structure treats the firms symmetrically. Under it, each firm can only infer a linear combination of the individual demand shocks, and thus not able to infer whether the recommendation of a given price stems from its own demand conditions or the conditions of its competitor. Generically, $r_{c}^{*} \\neq r_{c}^{F D}$ and $r_{o}^{*} \\neq r_{o}^{F D}$, so providing full disclosure is suboptimal; instead, each firm receives personalized information that differs from its competitor's. This finding suggests that restricting disclosure to be public, as often done in the oligopoly literature on information sharing (Vives (1990, 1999)), though natural in some contexts, may entail losses.\n\nProposition 5 enables the calculation of optimal information structures. It can be shown that the optimal information structure may exhibit a discontinuous change with respect to the consumer weight $\\delta$ at $\\hat{\\delta}(\\eta, c)$, accompanied by a change in equilibrium price volatility and correlations. ${ }^{17}$ Intuitively, the pricing induced under the optimal information structure seeks to approximate its first-best counterpart under direct price control. However, lacking the ability to enforce prices directly, the designer chooses an information regime that better approximates it.\n\nThis observation yields two insights. First, patterns of price volatility and correlations can serve as effective diagnostics for discerning which side of the market, consumers or producers, the information structure is designed to favor. Second, even a slight shift in the design objective can trigger a dramatic change in the information provision and in resulting market behavior. This, in turn, can serve as a cautionary tale, underscoring the potential for market instability introduced by algorithms, which may pursue shifting objectives and adapt their policies more rapidly than human decision makers.\n\n[^14]","text_sha256":"c083e2c2b547911b2fb74d44b1f79110b474793fc29022326e65ec36ddf685dd"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0020","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"9 Conclusion","text":"## 9 Conclusion\n\nIn this paper, we introduced a certification approach to solving information-design problems in games and demonstrated its effectiveness and tractability in symmetric linearquadratic settings. In doing so, we provided theoretical justification for the use of targeted disclosure, Gaussian coupling, and linear disclosure. Our findings shed light on disclosure practices that guide investment, on the socially efficient control of information in markets, and on the limits of Bayesian polarization.\n\nOur analysis lays the groundwork and offers tools for studying information design in general smooth games. We see at least three promising avenues for further research. First, our theoretical framework could be applied to other important economic settings, such as contests, public-good provision, and labor or financial markets. Second, one could develop more general sufficient conditions for strong duality in non-compact settings. Third, the framework could be expanded to incorporate elements like information elicitation, information spillovers across players, and dynamic interaction. We expect all of these extensions to be feasible within the approach we have outlined.","text_sha256":"9d6907f6347c1122de048f5051ff1b06dafe060b22cff97ae74ddb5fff84ebfb"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0021","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAlesina, A., A. Miano, and S. Stantcheva (2020): \"The Polarization of Reality,\" in AEA Papers and Proceedings, vol. 110, 324-328.\n\nAmador, M. and K. Bagwell (2013): \"The Theory of Optimal Delegation with an Application to Tariff Caps,\" Econometrica, 81, 1541-1599.\n\nAmador, M., I. Werning, and G.-M. Angeletos (2006): \"Commitment vs. Flexibility,\" Econometrica, 74, 365-396.\n\nAnderson, E. and P. Nash (1987): Linear Programming in Infinite-dimensional Spaces: Theory and Applications, Wiley.\n\nAngeletos, G.-M. and A. Pavan (2007): \"Efficient Use of Information and Social Value of Information,\" Econometrica, 75, 1103-1142.\n\n- (2009): \"Policy with Dispersed Information,\" Journal of the European Economic Association, 7, 11-60.\n\nAngelucci, C. and A. Prat (2024): \"Is Journalistic Truth Dead? Measuring How Informed Voters Are about Political News,\" American Economic Review, 114, 887-925.\n\nArieli, I. and Y. Babichenko (2019): \"Private Bayesian Persuasion,\" Journal of Economic Theory, 182, 185-217.\n\n- (2024): \"A Population's Feasible Posterior Beliefs,\" Journal of Economic Theory, 215, 105764.\n\nArieli, I., Y. Babichenko, and F. Sandomirskiy (2023): \"Persuasion as Transportation,\" Working paper.\n\nArieli, I., Y. Babichenko, F. Sandomirskiy, and O. Tamuz (2021): \"Feasible Joint Posterior Beliefs,\" Journal of Political Economy, 129, 2546-2594.\n\nBergemann, D. and A. Bonatti (2024): \"Data, Competition, and Digital Platforms,\" American Economic Review, 114, 2553-2595.\n\nBergemann, D., T. Heumann, and S. Morris (2015): \"Information and Volatility,\" Journal of Economic Theory, 158, 427-465.\n\n- (2017): \"Information and Interaction,\" Working paper.\n- (2021): \"Information, Market Power, and Price Volatility,\" RAND Journal of Economics, 52, 125-150.\n\nBergemann, D. and S. Morris (2013): \"Robust Predictions in Games with Incomplete Information,\" Econometrica, 81, 1251-1308.\n(2016): \"Bayes Correlated Equilibrium and the Comparison of Information Structures in Games,\" Theoretical Economics, 11, 487-522.\n\nBernstein, S., X. Giroud, and R. R. Townsend (2016): \"The Impact of Venture Capital Monitoring,\" Journal of Finance, 71, 1591-1622.\n\nBowen, T. R., D. Dmitriev, and S. Galperti (2023): \"Learning from Shared News: When Abundant Information Leads to Belief Polarization,\" Quarterly Journal of Economics, 138, 955-1000.\n\nBraghieri, L., S. Eichmeyer, R. Levy, M. M. Mobius, J. Steinhardt, and R. Zhong (2024): \"Level Slant and Polarization of News Consumption on Social Media,\" Working paper.\n\nBrooks, B. and S. Du (2021): \"Optimal Auction Design with Common Values: An Informationally Robust Approach,\" Econometrica, 89, 1313-1360.\n\nCandogan, O. and P. Strack (2023): \"Optimal Disclosure of Information to Privately Informed Agents,\" Theoretical Economics, 18, 1225-1269.\n\nCarroll, G. (2017): \"Robustness and Separation in Multidimensional Screening,\" Econometrica, 85, 453-488.\n\nCarvajal, A., M. Rostek, and G. Sublet (2018): \"Information Design and Capital Formation,\" Journal of Economic Theory, 176, 255-292.\n\nChan, J., S. Gupta, F. Li, and Y. Wang (2019): \"Pivotal Persuasion,\" Journal of Economic theory, 180, 178-202.\n\nChiappori, P.-A., B. Salanié, and Y. Weiss (2017): \"Partner Choice, Investment in Children, and the Marital College Premium,\" American Economic Review, 107, 2109-2167.\n\nDizdar, D. and E. Kováč (2020): \"A Simple Proof of Strong Duality in the Linear Persuasion Problem,\" Games and Economic Behavior, 122, 407-412.\n\nDu, S. (2018): \"Robust Mechanisms under Common Valuation,\" Econometrica, 86, 1569-1588.\n\nDworczak, P. and A. Kolotilin (2024): \"The Persuasion Duality,\" Theoretical Economics, 19, 1701-1755.\n\nDworczak, P. and G. Martini (2019): \"The Simple Economics of Optimal Persuasion,\" Journal of Political Economy, 127, 1993-2048.\n\nElliott, M., A. Galeotti, A. Koh, and W. Li (2022): \"Market Segmentation through Information,\" Working paper.\n\nGal-Or, E. (1985): \"Information Sharing in Oligopoly,\" Econometrica, 329-343.\n\nGalichon, A. and B. Salanié (2022): \"Cupid's Invisible hand: Social Surplus and Identification in Matching Models,\" Review of Economic Studies, 89, 2600-2629.\n\nGalperti, S., A. Levkun, and J. Perego (2024): \"The Value of Data Records,\" Review of Economic Studies, 91, 1007-1038.\n\nGalperti, S. and J. Perego (2018): \"A Dual Perspective on Information Design,\" Working Paper.\n\nHolmström, B. (1979): \"Moral Hazard and Observability,\" Bell Journal of Economics, 74-91.\n\nIchihashi, S. and A. Smolin (2025): \"Buyer-Optimal Algorithmic Recommendations,\" Working paper.\n\nInostroza, N. and A. Pavan (2025): \"Adversarial Coordination and Public Information Design,\" Theoretical Economics, 20, 763-813.\n\nJehiel, P. (2015): \"On Transparency in Organizations,\" The Review of Economic Studies, 82, 736-761.\n\nKamenica, E. and M. Gentzkow (2011): \"Bayesian Persuasion,\" American Economic Review, 101, 2590-2615.\n\nKirby, A. J. (1988): \"Trade Associations as Information Exchange Mechanisms,\" The RAND Journal of Economics, 138-146.\n\nKolotilin, A. (2012): \"Optimal Information Disclosure: Quantity vs. Quality,\" Working paper.\n\n- (2018): \"Optimal Information Disclosure: A Linear Programming Approach,\" Theoretical Economics, 13, 607-635.\n\nKolotilin, A., R. Corrao, and A. Wolitzky (2025): \"Persuasion and Matching: Optimal Productive Transport,\" Journal of Political Economy, 133, 1334-1381.\n\nLin, X. and C. Liu (2024): \"Credible Persuasion,\" Journal of Political Economy, 132, 2228-2273.\n\nMathevet, L., J. Perego, and I. Taneva (2020): \"On Information Design in Games,\" Journal of Political Economy, 128, 1370-1404.\n\nMirrlees, J. A. (1999): \"The Theory of Moral Hazard and Unobservable Behaviour: Part I,\" Review of Economic Studies, 66, 3-21.\n\nMiyashita, M. and T. Ui (2023): \"LQG Information Design,\" Working paper.\n\nMorris, S., D. Oyama, and S. Takahashi (2024): \"Implementation via Information Design in Binary-Action Supermodular Games,\" Econometrica, 92, 775-813.\n\nMyerson, R. (1982): \"Optimal Coordination Mechanism in Generalized PrincipalAgent Problems,\" Journal of Mathematical Economics, 10, 67-81.\n\nOrtner, J., T. Sugaya, and A. Wolitzky (2024): \"Mediated Collusion,\" Journal of Political Economy, 132, 1247-1289.\n\nRayo, L. and I. Segal (2010): \"Optimal Information Disclosure,\" Journal of Political Economy, 118, 949-987.\n\nRockafellar, R. T. (1967): \"Duality and Stability in Extremum Problems Involving Convex Functions,\" Pacific Journal of Mathematics, 21, 167-187.","text_sha256":"28c108df0045b4ce84c0669c42adde1cd78b30dd789852d25bf66a15e797b2d6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0022","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"Romanyuk, G. and A. Smolin (2019): \"Cream Skimming and Information Design in Matching Markets,\" American Economic Journal: Microeconomics, 11, 250-276.\n\nRosen, J. (1965): \"Existence and Uniqueness of Equilibrium Points for Concave N-Person Games,\" Econometrica, 33, 520-534.\n\nRudin, W. (1976): Principles of Mathematical Analysis, New York: McGraw-Hill.\n\n- (1987): Real and Complex Analysis, McGraw-Hill, 3 ed.\n\nSalamanca, A. (2021): \"The Value of Mediated Communication,\" Journal of Economic Theory, 192, 105191.\n\nTamura, W. (2012): \"A Theory of Multidimensional Information Disclosure,\" Working paper.\n\n- (2018): \"Bayesian Persuasion with Quadratic Preferences,\" Working paper.\n\nTaneva, I. (2019): \"Information Design,\" American Economic Journal: Microeconomics, 11, 151-85.\n\nUi, T. (2020): \"LQG Information Design,\" Working paper.\nVives, X. (1984): \"Duopoly Information Equilibrium: Cournot and Bertrand,\" Journal of Economic Theory, 34, 71-94.\n\n- (1990): \"Trade Association Disclosure Rules, Incentives to Share Information, and Welfare,\" RAND Journal of Economics, 409-430.\n- (1999): Oligopoly Pricing: Old Ideas and New Tools, MIT press.\n\nVohra, R. (2011): Mechanism Design: a Linear Programming Approach, vol. 47, Cambridge University Press.","text_sha256":"df6d93f2d9acd734f9a5bad1ccd702ffe63f0cc09fc9f0c9fbada5d129b95026"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0023","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Appendix","text":"## A Appendix","text_sha256":"670602cc2839b357e79388e1de36a6f430e75ef165b68a510f6737e8ab36dbcf"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0024","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 1 Formalism Omitted in Section 2","text":"## A. 1 Formalism Omitted in Section 2\n\nDerivation of First-Order Condition (4) For $\\varepsilon>0$, define\n\n$$\n\\begin{aligned}\n& \\phi_{i \\varepsilon}^{+}\\left(a_{i}, a_{-i}, \\omega\\right) \\triangleq \\frac{u_{i}\\left(a_{i}+\\varepsilon, a_{-i}, \\omega\\right)-u_{i}\\left(a_{i}, a_{-i}, \\omega\\right)}{\\varepsilon}, \\\\\n& \\phi_{i \\varepsilon}^{-}\\left(a_{i}, a_{-i}, \\omega\\right) \\triangleq \\frac{u_{i}\\left(a_{i}, a_{-i}, \\omega\\right)-u_{i}\\left(a_{i}-\\varepsilon, a_{-i}, \\omega\\right)}{\\varepsilon} .\n\\end{aligned}\n$$\n\nDefinition 6. (Admissible Equilibria) An equilibrium is admissible, if for any player i and equilibrium belief $\\mu_{i} \\in \\Delta\\left(A_{-i} \\times \\Omega\\right)$, there exists a best response $a_{i}^{*} \\in A_{i}, \\Delta>0$, and $\\psi:$ $A_{-i} \\times \\Omega \\rightarrow \\mathbb{R}$ such that $\\int_{A_{-i} \\times \\Omega} \\psi\\left(a_{-i}, \\omega\\right) d \\mu_{i}<+\\infty$ and for all $\\varepsilon \\in(0, \\Delta),\\left(a_{-i}, \\omega\\right) \\in A_{-i} \\times \\Omega$, $\\left|\\phi_{i \\varepsilon}^{+}\\left(a_{i}^{*}, a_{-i}, \\omega\\right)\\right| \\leq \\psi\\left(a_{-i}, \\omega\\right)$ and $\\left|\\phi_{i \\varepsilon}^{-}\\left(a_{i}^{*}, a_{-i}, \\omega\\right)\\right| \\leq \\psi\\left(a_{-i}, \\omega\\right)$.\n\nThis admissibility effectively requires players' payoff differences to be locally well behaved at the best responses. It ensures the interchangeability of operators in (4) and is satisfied in all equilibria that we characterize. (If $A$ and $\\Omega$ were compact, then admissibility would be trivially satisfied whenever each $u_{i}(a, \\omega)$ was continuously differentiable in $a_{i}$. For general $A$ and $\\Omega$, any equilibrium is admissible if, for instance, each $u_{i}(a, \\omega)$ is polynomial.)\n\nLemma 3. (First-Order Approach) Under Assumption 1, for any player $i$ and admissible equilibrium belief $\\mu_{i} \\in \\Delta\\left(A_{-i} \\times \\Omega\\right), a_{i}$ is player $i$ 's best response if and only if condition (4) is satisfied.\n\nProof. For any $i$ and admissible equilibrium belief $\\mu_{i}, a_{i}^{*}$ satisfies the following condition:\n\n$$\n\\int_{A_{-i} \\times \\Omega} u_{i}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i} \\geq \\int_{A_{-i} \\times \\Omega} u_{i}\\left(a_{i}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i}, \\quad \\forall a_{i} \\in A_{i} .\n$$\n\nUnder Assumption 1, condition (53) is satisfied if and only if for any $\\varepsilon>0$, the following holds:\n\n$$\n\\begin{aligned}\n& \\int_{A_{-i} \\times \\Omega} \\phi_{i \\varepsilon}^{+}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i} \\leq 0 \\\\\n& \\int_{A_{-i} \\times \\Omega} \\phi_{i \\varepsilon}^{-}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i} \\geq 0\n\\end{aligned}\n$$\n\nMoreover, $\\phi_{i \\varepsilon}^{+}$is decreasing in $\\varepsilon$ and $\\phi_{i \\varepsilon}^{-}$is increasing in $\\varepsilon$; thus, conditions (54), (55) are equivalent to\n\n$$\n\\begin{aligned}\n0 \\geq \\lim _{\\varepsilon \\rightarrow 0} \\int_{A_{-i} \\times \\Omega} \\phi_{i \\varepsilon}^{+}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i} & =\\lim _{\\varepsilon \\rightarrow 0} \\frac{1}{\\varepsilon} \\int_{A_{-i} \\times \\Omega} u_{i}\\left(a_{i}^{*}+\\varepsilon, a_{-i}, \\omega\\right)-u_{i}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i} \\\\\n& =\\left.\\frac{\\partial^{+}}{\\partial a_{i}} \\int_{A_{-i} \\times \\Omega} u_{i}\\left(a_{i}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i}\\right|_{a_{i}=a_{i}^{*}}\n\\end{aligned}\n$$\n\n$$\n\\begin{aligned}\n0 \\leq \\lim _{\\varepsilon \\rightarrow 0} \\int_{A_{-i} \\times \\Omega} \\phi_{i \\varepsilon}^{-}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i} & =\\lim _{\\varepsilon \\rightarrow 0} \\frac{1}{\\varepsilon} \\int_{A_{-i} \\times \\Omega} u_{i}\\left(a_{i}^{*}, a_{-i}, \\omega\\right)-u_{i}\\left(a_{i}^{*}-\\varepsilon, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i} \\\\\n& =\\left.\\frac{\\partial^{-}}{\\partial a_{i}} \\int_{A_{-i} \\times \\Omega} u_{i}\\left(a_{i}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i}\\right|_{a_{i}=a_{i}^{*}}\n\\end{aligned}\n$$\n\nBy Lebesgue's dominated convergence theorem (Theorem 11.32, Rudin (1976)), because belief $\\mu_{i}$ arises in an admissible equilibrium, the following holds:\n\n$$\n\\begin{aligned}\n& \\lim _{\\varepsilon \\rightarrow 0} \\int_{A_{-i} \\times \\Omega} \\phi_{i \\varepsilon}^{+}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i}=\\int_{A_{-i} \\times \\Omega} \\lim _{\\varepsilon \\rightarrow 0} \\phi_{i \\varepsilon}^{+}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i}=\\int_{A_{-i} \\times \\Omega} \\dot{u}_{i}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i}, \\\\\n& \\lim _{\\varepsilon \\rightarrow 0} \\int_{A_{-i} \\times \\Omega} \\phi_{i \\varepsilon}^{-}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i}=\\int_{A_{-i} \\times \\Omega} \\lim _{\\varepsilon \\rightarrow 0} \\phi_{i \\varepsilon}^{-}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i}=\\int_{A_{-i} \\times \\Omega} \\dot{u}_{i}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i},\n\\end{aligned}\n$$\n\nand thus\n\n$$\n\\left.\\frac{\\partial}{\\partial a_{i}} \\int_{A_{-i} \\times \\Omega} u_{i}\\left(a_{i}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i}\\right|_{a_{i}=a_{i}^{*}}=\\int_{A_{-i} \\times \\Omega} \\dot{u}_{i}\\left(a_{i}^{*}, a_{-i}, \\omega\\right) \\mathrm{d} \\mu_{i}=0 .\n$$ $\\square$","text_sha256":"4ee1b222801c3d3d93660bad0d1eea8c70e2e830145d5d9a463c64454c671e3e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0025","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 2 Formalism Omitted in Section 3","text":"## A. 2 Formalism Omitted in Section 3\n\nProof of Theorem 1 The first step is\nLemma 4. (Weak Duality) $V^{P} \\leq V^{D}$.\n\nProof. Take any dual variables $(\\lambda, \\gamma)$ that satisfy the constraints of the dual problem (8). Take any measure $\\pi$ that satisfies the constraints of the primal problem (5). Integrating both sides\nof the dual constraints over $a \\in A$ and $\\omega \\in \\Omega$ against measure $\\pi$ yields:\n\n$$\n\\int_{A \\times \\Omega} v(a, \\omega) \\mathrm{d} \\pi \\leq \\int_{A \\times \\Omega} \\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega) \\mathrm{d} \\pi+\\int_{A \\times \\Omega} \\gamma(\\omega) \\mathrm{d} \\pi=\\int_{\\Omega} \\gamma(\\omega) \\mathrm{d} \\mu_{0},\n$$\n\nwhere the equality follows because $\\pi$ satisfies the primal constraints. The left-hand side of (56) is the value of the primal problem given measure $\\pi$. The right-hand side of (56) is the value of the dual problem given dual variables $(\\lambda, \\gamma)$. As the inequality (56) holds for any allowed values of primal measure and dual variables, it also holds at the respective maximization and minimization limits. $\\square$\n\nContinuing with the proof of Theorem 1, take any primal measure $\\pi$ implementable by information, i.e., that satisfies the constraints of the primal problem (5). If it is implementable by incentives, then there exist dual variables $\\lambda$ that implement this measure in the dual problem (10), and\n\n$$\n\\begin{aligned}\nV^{D} & =\\inf _{\\lambda^{\\prime} \\in \\times_{i} \\mathcal{M}\\left(A_{i}\\right)} \\mathbb{E}_{\\mu_{0}}\\left[\\sup _{a \\in A} w\\left(a, \\omega, \\lambda^{\\prime}\\right)\\right] \\\\\n& \\leq \\mathbb{E}_{\\pi}[w(a, \\omega, \\lambda)] \\\\\n& =\\int_{A \\times \\Omega} v(a, \\omega) \\mathrm{d} \\pi-\\int_{A \\times \\Omega} \\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega) \\mathrm{d} \\pi \\\\\n& =\\int_{A \\times \\Omega} v(a, \\omega) \\mathrm{d} \\pi \\leq V^{P},\n\\end{aligned}\n$$\n\nwhere the first inequality follows from the implementability of $\\pi$ in the dual problem and the last three steps follow from the feasibility of $\\pi$ in the primal problem.\n\nFurthermore, by Lemma $4, V^{D} \\geq V^{P}$. Combining the two inequalities, we obtain\n\n$$\nV^{D}=\\int_{A \\times \\Omega} v(a, \\omega) \\mathrm{d} \\pi=V^{P},\n$$\n\nwhich proves the optimality of measure $\\pi$.\n\nProof of Proposition 1 For any optimal measure $\\pi$, the following holds:\n\n$$\n\\begin{gathered}\n\\mathbb{E}_{\\pi}[v(a, \\omega)]=V^{P}, \\\\\n\\int_{A_{i}^{\\prime} \\times A_{-i} \\times \\Omega} \\dot{u}_{i}(a, \\omega) \\mathrm{d} \\pi=0 \\quad \\forall i=1, \\ldots, N, \\text { measurable } A_{i}^{\\prime} \\subseteq A_{i} .\n\\end{gathered}\n$$\n\nAs such, if the dual agent is offered contract $\\lambda$ and plays according to an optimal measure then his expected payoff is\n\n$$\n\\begin{aligned}\n\\mathbb{E}_{\\pi}[w(a, \\omega, \\lambda)] & =\\mathbb{E}_{\\pi}\\left[v(a, \\omega)-\\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega)\\right] \\\\\n& =\\mathbb{E}_{\\pi}[v(a, \\omega)]-\\mathbb{E}_{\\pi}\\left[\\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega)\\right]=V^{P} .\n\\end{aligned}\n$$\n\nThat is, the dual agent obtains the same expected payoff of $V^{P}$ by playing according to an optimal measure, whether $\\pi$ or $\\pi^{\\prime}$.\n\nMoreover, if $\\lambda$ certifies the optimality of measure $\\pi$, then playing according to $\\pi$ is a best response of the dual agent to contract $\\lambda$. However, since, as shown above, $\\pi^{\\prime}$ delivers the same expected dual payoff as $\\pi$, it follows that $\\pi^{\\prime}$ is also a best response of the dual agent to contract $\\lambda$. Indeed, for every $\\omega \\in \\Omega$ and $a \\in A, w(a, \\omega, \\lambda) \\leq \\sup _{\\tilde{a} \\in A} w(\\tilde{a}, \\omega, \\lambda)$. Since $\\lambda$ certifies $\\pi$, we have $V^{D}=V^{P}$, and since $\\pi^{\\prime}$ is optimal, $\\mathbb{E}_{\\pi^{\\prime}}[w(a, \\omega, \\lambda)]=V^{P}$. Hence\n\n$$\n0=V^{D}-\\mathbb{E}_{\\pi^{\\prime}}[w(a, \\omega, \\lambda)]=\\int_{\\Omega}\\left(\\sup _{a \\in A} w(a, \\omega, \\lambda)-\\int_{A} w(a, \\omega, \\lambda) \\pi^{\\prime}(d a \\mid \\omega)\\right) \\mu_{0}(d \\omega)\n$$\n\nThe integrand is nonnegative, so it must be equal to zero for $\\mu_{0}$-almost every $\\omega$. Therefore, for $\\mu_{0}$-almost every $\\omega$, the measure $\\pi^{\\prime}(\\cdot \\mid \\omega)$ is supported on $\\arg \\max _{a \\in A} w(a, \\omega, \\lambda)$, that is, $\\pi^{\\prime}$ is also implementable by incentives under contract $\\lambda$.\n\nFinally, as an optimal measure, $\\pi^{\\prime}$ satisfies the constraints of the primal problem. Hence, $\\pi^{\\prime}$ is implementable by information and by incentives; by definition, $\\lambda$ certifies the optimality of $\\pi^{\\prime}$.\n\nProof of Claim 1 Assume that $\\Omega$ and all $A_{i}$ are compact (metric) spaces, and that $v$ and each $\\dot{u}_{i}$ are continuous on $X \\triangleq \\Omega \\times A$. We first introduce an auxiliary dual problem in which\nthe multipliers are restricted to be continuous:\n\n$$\n\\begin{aligned}\nV^{D, c} \\triangleq & \\inf _{\\gamma \\in C(\\Omega), \\lambda_{i} \\in C\\left(A_{i}\\right)} \\int_{\\Omega} \\gamma(\\omega) d \\mu_{0} \\\\\n& \\text { s.t. } \\quad \\gamma(\\omega)+\\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega) \\geq v(a, \\omega) \\quad \\forall(a, \\omega) \\in A \\times \\Omega\n\\end{aligned}\n$$\n\nBecause every continuous function is measurable, the feasible set of $V^{D, c}$ is contained in the feasible set of $V^{D}$. Hence $V^{D} \\leq V^{D, c}$. By weak duality, $V^{P} \\leq V^{D}$. Therefore it is enough to prove that $V^{D, c}=V^{P}$.\n\nDefine $E \\triangleq C(X), Y \\triangleq C(\\Omega) \\times \\prod_{i=1}^{N} C\\left(A_{i}\\right)$, each endowed with the sup norm. Since $X$ is compact metric, both are Banach spaces, and by the Riesz-Markov theorem (Rudin, 1987, Theorem 6.19) $E^{*}=M(X), Y^{*}=M(\\Omega) \\times \\prod_{i=1}^{N} M\\left(A_{i}\\right)$, where $M(\\cdot)$ denotes the finite signed Radon measures.\n\nDefine the continuous linear operator $T: Y \\rightarrow E$ by\n\n$$\nT(\\gamma, \\lambda)(a, \\omega) \\triangleq \\gamma(\\omega)+\\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega) .\n$$\n\nAlso define the proper convex functions $\\phi: E \\rightarrow \\mathbb{R} \\cup\\{+\\infty\\}$ and $\\psi: Y \\rightarrow \\mathbb{R}$ by\n\n$$\n\\phi(f) \\triangleq\\left\\{\\begin{array}{ll}\n0, & \\text { if } f(x) \\geq v(x) \\forall x \\in X, \\\\\n+\\infty, & \\text { otherwise }\n\\end{array} \\quad \\psi(\\gamma, \\lambda) \\triangleq \\int_{\\Omega} \\gamma(\\omega) d \\mu_{0}\\right.\n$$\n\nThen,","text_sha256":"804122435d535dfcaa0aef67fb70258248278e8d39ee51b72375037dd63b9d5d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0026","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 2 Formalism Omitted in Section 3","text":"$$\nV^{D, c}=\\inf _{y \\in Y}\\{\\psi(y)+\\phi(T y)\\} .\n$$\n\nNow choose $M>\\|v\\|_{\\infty}$ and let $y_{0}=(M, 0, \\ldots, 0) \\in Y$. Then $T y_{0} \\equiv M>v$ on $X$, so $\\phi$ is continuous at $T y_{0}$. Hence the Banach-space Fenchel-Rockafellar theorem applies and yields (Rockafellar, 1967, Theorems 1 and 3)\n\n$$\nV^{D, c}=\\sup _{\\pi \\in E^{*}}\\left\\{-\\psi^{*}\\left(T^{*} \\pi\\right)-\\phi^{*}(-\\pi)\\right\\} .\n$$\n\nWe now compute the two conjugates. First,\n\n$$\n\\begin{aligned}\n-\\phi^{*}(-\\pi) & =-\\sup _{f \\in E}\\left\\{\\int_{X} f d(-\\pi)-\\phi(f)\\right\\} \\\\\n& =\\inf _{f \\in C(X), f \\geq v} \\int_{X} f d \\pi\n\\end{aligned}\n$$\n\nTherefore\n\n$$\n-\\phi^{*}(-\\pi)= \\begin{cases}\\int_{X} v d \\pi, & \\text { if } \\pi \\in M_{+}(X) \\\\ -\\infty, & \\text { otherwise }\\end{cases}\n$$\n\nIndeed, if $\\pi \\in M_{+}(X)$, the infimum is attained at $f=v$. If $\\pi \\notin M_{+}(X)$, then by the Riesz-Markov theorem there exists $h \\in C(X)$ with $h \\geq 0$ and $\\int_{X} h d \\pi<0$; hence for $f_{n} \\triangleq v+n h$ we have $f_{n} \\geq v$ and\n\n$$\n\\int_{X} f_{n} d \\pi=\\int_{X} v d \\pi+n \\int_{X} h d \\pi \\rightarrow-\\infty .\n$$\n\nSecond, identify $T^{*} \\pi \\in Y^{*}$ explicitly. For $\\pi \\in M(X)$, let $\\pi_{0} \\in M(\\Omega)$ be the $\\Omega$-marginal of $\\pi$, and for each $i$ define a finite signed measure $\\nu_{i} \\in M\\left(A_{i}\\right)$ by\n\n$$\n\\nu_{i}(B) \\triangleq \\int_{B \\times A_{-i} \\times \\Omega} \\dot{u}_{i}(a, \\omega) d \\pi \\quad \\forall \\text { Borel } B \\subseteq A_{i}\n$$\n\nSince $\\dot{u}_{i}$ is continuous on the compact set $X$, it is bounded, so $\\nu_{i}$ is well defined. Moreover, for every $(\\gamma, \\lambda) \\in Y$,\n\n$$\n\\begin{aligned}\n\\left\\langle T^{*} \\pi,(\\gamma, \\lambda)\\right\\rangle & =\\int_{X} \\gamma(\\omega) d \\pi+\\sum_{i=1}^{N} \\int_{X} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega) d \\pi \\\\\n& =\\int_{\\Omega} \\gamma d \\pi_{0}+\\sum_{i=1}^{N} \\int_{A_{i}} \\lambda_{i} d \\nu_{i}\n\\end{aligned}\n$$\n\nHence, under the identification $Y^{*}=M(\\Omega) \\times \\prod_{i=1}^{N} M\\left(A_{i}\\right)$,\n\n$$\nT^{*} \\pi=\\left(\\pi_{0}, \\nu_{1}, \\ldots, \\nu_{N}\\right) .\n$$\n\nNow $\\psi(\\gamma, \\lambda)=\\int_{\\Omega} \\gamma d \\mu_{0}$ depends only on $\\gamma$, so its conjugate is\n\n$$\n\\psi^{*}\\left(\\eta_{0}, \\eta_{1}, \\ldots, \\eta_{N}\\right)= \\begin{cases}0, & \\text { if } \\eta_{0}=\\mu_{0} \\text { and } \\eta_{i}=0 \\forall i, \\\\ +\\infty, & \\text { otherwise }\\end{cases}\n$$\n\nTherefore\n\n$$\n-\\psi^{*}\\left(T^{*} \\pi\\right)= \\begin{cases}0, & \\text { if } \\pi_{0}=\\mu_{0} \\text { and } \\nu_{i}=0 \\forall i \\\\ -\\infty, & \\text { otherwise }\\end{cases}\n$$\n\nCombining the two conjugate calculations gives\n\n$$\nV^{D, c}=\\sup _{\\pi \\in M_{+}(X)}\\left\\{\\int_{X} v d \\pi \\mid \\pi_{0}=\\mu_{0}, \\nu_{i}=0 \\forall i\\right\\} .\n$$\n\nSince $\\pi_{0}=\\mu_{0}$ and $\\mu_{0}$ is a probability measure, every such $\\pi$ has total mass one, so $\\pi \\in \\Delta(X)$.\nFinally, the condition $\\pi_{0}=\\mu_{0}$ is exactly the Bayes-plausibility constraint. Also, $\\nu_{i}=0$ is equivalent, by the uniqueness part of the Riesz-Markov theorem, to\n\n$$\n\\int_{A_{i}^{\\prime} \\times A_{-i} \\times \\Omega} \\dot{u}_{i}(a, \\omega) d \\pi=0 \\quad \\forall \\text { Borel } A_{i}^{\\prime} \\subseteq A_{i}\n$$\n\nwhich is exactly the obedience constraint in the primal problem. Hence the feasible set above is precisely the primal feasible set, and therefore $V^{D, c}=V^{P}$. We conclude that\n\n$$\nV^{P} \\leq V^{D} \\leq V^{D, c}=V^{P},\n$$\n\nso $V^{D}=V^{P}$.","text_sha256":"03d2491fa1b3bc94cf3d03a532f38d7ed0b52f77645fd7c10302a10a275a44ac"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0027","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 3 Formalism Omitted in Section 5","text":"## A. 3 Formalism Omitted in Section 5\n\nGame Normalization The game normalization is achieved by defining $\\omega^{\\prime}=\\omega-\\mathbb{E}[\\omega], a_{i}^{\\prime}=$ $a_{i}+\\frac{\\mathbb{E}[\\omega]+b_{1}}{q^{o}+q^{c}}, b=h \\mathbb{E}[\\omega]+b_{2}-2\\left(p^{o}+p^{c}\\right) \\frac{\\mathbb{E}[\\omega]+b_{1}}{q^{o}+q^{c}}$, and ignoring the strategically irrelevant additive terms, i.e., those in $u_{i}$ that do not depend on $a_{i}$ and those in $v$ that do not depend on $a$.\n\nProof of Theorem 2, continued. If $k^{*} \\notin[0, N]$, then, since $q^{o} p^{c}>p^{o} q^{c}$, either (i) $p^{o}<h q^{o}$, or (ii) $\\left(h q^{o}-p^{o}\\right)\\left(q^{o}+q^{c}\\right)>2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)$.\n\nIndeed, if neither (i) or (ii) holds then it must be the case that $p^{o} \\geq h q^{o}$, and $\\left(h q^{o}-p^{o}\\right)\\left(q^{o}+\\right.$ $\\left.q^{c}\\right) \\leq 2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)$. But in this case, because of our assumptions $q^{o}<0, q^{o}\\left(h q^{o}-p^{o}\\right) \\geq 0$ and therefore,\n\n$$\n\\frac{k^{*}}{N}=\\frac{q^{o}\\left(h q^{o}-p^{o}\\right)}{\\left(2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)-\\left(h q^{o}-p^{o}\\right)\\left(q^{o}+q^{c}\\right)\\right)+q^{o}\\left(h q^{o}-p^{o}\\right)} \\in[0,1] .\n$$\n\nConsequently, if $k^{*} \\notin[0, N]$, then either (i) or (ii) holds.\nCase (i). If (i) holds, then no disclosure is optimal, which can be certified by an affine contract with $\\left(x, x_{0}\\right)=\\left(-h,-\\frac{b}{q^{o}+q^{c}}\\right)$. Under this contract, the dual payoff function is\n\n$$\nw(a, \\omega)=x_{0} \\omega+\\left(p^{o}-h q^{o}\\right) \\check{a}+\\left(p^{c}-h q^{c}\\right) \\bar{a}^{2}=x_{0} \\omega+\\frac{1}{N^{2}} a^{T} M\\left(\\left(p^{o}-h q^{o}\\right) N+p^{c}-h q^{c}, p^{c}-h q^{c}\\right) a,\n$$\n\nwhere $M(x, y)$ denotes the $N \\times N$ matrix whose diagonal entries equal $x$ and off-diagonal entries equal $y$. By Theorem 1, no disclosure is optimal if $a=(0, \\ldots, 0)$ maximizes $w(a, \\omega)$ for all $\\omega$. This happens if $M\\left(\\left(p^{o}-h q^{o}\\right) N+p^{c}-h q^{c}, p^{c}-h q^{c}\\right)$ is negative semidefinite (NSD). Note that matrix $M(x, y)$ has two distinct eigenvalues: $x+(N-1) y$ and $x-y$. Therefore, $M(x, y)$ is NSD if $x+(N-1) y \\leq 0$ and $x-y \\leq 0$ which for $M\\left(\\left(p^{o}-h q^{o}\\right) N+p^{c}-h q^{c}, p^{c}-h q^{c}\\right)$ corresponds to:\n\n$$\n\\begin{aligned}\np^{o} & \\leq h q^{o} \\\\\np^{o}+p^{c} & \\leq h\\left(q^{o}+q^{c}\\right)\n\\end{aligned}\n$$\n\nSubcase 1: $h q^{c} \\geq p^{c}$. In this case, NSD is equivalent to $p^{o}-h q^{o} \\leq 0$. In case $q^{o} p^{c}>p^{o} q^{c}$ and $p^{o}<h q^{o}$, this is satisfied.\n\nSubcase 2: $h q^{c}<p^{c}$. In this case, NSD is equivalent to $p^{o}-h q^{o}+p^{c}-h q^{c} \\leq 0$. In case $q^{o} p^{c}>p^{o} q^{c}$ and $p^{o}<h q^{o}$, note that $p^{c}=-p^{o}-\\eta$ and $q^{c}=-q^{o}-\\varepsilon$ for some $\\eta$ and $\\varepsilon>0$. Because $q^{o} p^{c}>p^{o} q^{c}$, we have $-q^{o} \\eta>-p^{o} \\varepsilon$, which implies $\\eta / \\varepsilon>p^{o} / q^{o}>h$ because $p^{o}<h q^{o}<0$. Thus, $\\eta>h \\varepsilon$; equivalently, $p^{o}+p^{c}<h\\left(q^{o}+q^{c}\\right)$.\n\nCase (ii). If (ii) holds, then full disclosure is optimal, which can be certified by an affine\ncontract with $\\left(x, x_{0}\\right)=\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}},-\\frac{b}{q^{o}+q^{c}}\\right)$. Under this contract, the dual payoff function is\n\n$$\n\\begin{aligned}\nw(a, \\omega) & =x_{0} \\omega+\\left(2 h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right) \\omega \\bar{a}+\\left(p^{o}+q^{o}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right)\\right) \\check{a}+\\left(p^{c}+q^{c}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right)\\right) \\bar{a}^{2} \\\\\n& =x_{0} \\omega+\\frac{1}{N^{2}}\\left(\\left(2 h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right) N \\omega 1_{N}^{T} a+a^{T} M(x, y) a\\right),\n\\end{aligned}\n$$\n\nwhere $1_{N}=(1, \\ldots, 1)^{T}$ is an $N$-dimensional vector of ones, $x=N\\left(p^{o}+q^{o}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right)\\right)+p^{c}+$ $q^{c}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right)$ and $y=p^{c}+q^{c}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right)$.\n\nBy Theorem 1, full disclosure is optimal if $a=\\left(-\\frac{\\omega}{q^{o}+q^{c}}, \\ldots,-\\frac{\\omega}{q^{o}+q^{c}}\\right)$ maximizes $w(a, \\omega)$ for all $\\omega$. If $M(x, y)$ is NSD, then the maximizing $a$ can be found via F.O.C.:\n\n$$\n\\left(2 h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right) N \\omega 1_{N}+2 M(x, y) a=0,\n$$\n\nwith a solution $a=\\left(-\\frac{\\omega}{q^{o}+q^{c}}, \\ldots,-\\frac{\\omega}{q^{o}+q^{c}}\\right)$. Therefore, it remains to establish that $M(x, y)$ is NSD, $x+(N-1) y \\leq 0$ and $x-y \\leq 0$ which corresponds to\n\n$$\n\\begin{aligned}\np^{o}+q^{o}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right) & \\leq 0, \\\\\np^{o}+q^{o}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right)+p^{c}+q^{c}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right) & \\leq 0 .\n\\end{aligned}\n$$\n\nSubcase 1: $p^{c}+q^{c}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right) \\leq 0$ and $p^{o}+p^{c} \\neq h\\left(q^{o}+q^{c}\\right)$. We show that the inequalities hold strictly. It suffices to show $p^{o}+q^{o}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right)<0$; equivalently, $\\left(h q^{o}-p^{o}\\right)\\left(q^{o}+q^{c}\\right)>$ $2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)$. In case $p^{c} q^{o}>p^{o} q^{c}$ and $\\left(h q^{o}-p^{o}\\right)\\left(q^{o}+q^{c}\\right)>2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)$, this is clearly satisfied.\n\nSubcase 2: $p^{c}+q^{c}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right)>0$ and $p^{o}+p^{c} \\neq h\\left(q^{o}+q^{c}\\right)$. We show that the inequalities hold strictly. It suffices to show $p^{o}+q^{o}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right)+p^{c}+q^{c}\\left(h-\\frac{2\\left(p^{o}+p^{c}\\right)}{q^{o}+q^{c}}\\right)<0$; equivalently, $h\\left(q^{o}+q^{c}\\right)-p^{o}-p^{c}<0$. In case $q^{o} p^{c}-p^{o} q^{c}>0$ and $\\left(h q^{o}-p^{o}\\right)\\left(q^{o}+q^{c}\\right)>2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)$, noting that $p^{c}=-p^{o}-\\eta$ and $q^{c}=-q^{o}-\\varepsilon$ for some $\\eta \\in \\mathbb{R}$ and $\\varepsilon>0$, we have $-q^{o} \\eta>-p^{o} \\varepsilon$ and $q^{o}(2 \\eta-h \\varepsilon)>p^{o} \\varepsilon$, implying (by summing them side-by-side) $q^{o}(\\eta-h \\varepsilon)>0$; equivalently,\n$\\eta-h \\varepsilon<0$. Thus,\n\n$$\nh\\left(q^{o}+q^{c}\\right)-p^{o}-p^{c}=\\eta-h \\varepsilon<0 .\n$$","text_sha256":"26cf54d35bc1312479572d33633a428254b8e238bc90f3b304c7eb43bff371fa"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0028","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 3 Formalism Omitted in Section 5","text":"Proof of Theorem 3, continued. By concavity conditions $-q^{o}>0$, and thus $\\sigma^{2}>0$ if and only if\n\n$$\n\\begin{aligned}\n\\beta\\left(1+q^{c} \\beta+q^{o} \\beta\\right) & >0, \\\\\n\\frac{\\left(p^{o}-h q^{o}\\right)}{2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)}\\left(1+\\frac{\\left(q^{c}+q^{o}\\right)\\left(p^{o}-h q^{o}\\right)}{2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)}\\right) & >0, \\\\\n\\frac{\\left(p^{o}-h q^{o}\\right)\\left(2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)+\\left(q^{c}+q^{o}\\right)\\left(p^{o}-h q^{o}\\right)\\right)}{4\\left(p^{c} q^{o}-p^{o} q^{c}\\right)^{2}} & >0 .\n\\end{aligned}\n$$\n\nBecause $q^{o}<0$, setting $c_{1} \\triangleq q^{o}\\left(h q^{o}-p^{o}\\right)$ and $c_{2} \\triangleq q^{o}\\left(2 p^{c}-h q^{c}\\right)-p^{o} q^{c}$, the inequality holds if and only if\n\n$$\n\\begin{aligned}\nq^{o}\\left(h q^{o}-p^{o}\\right)\\left(2\\left(p^{c} q^{o}-p^{o} q^{c}\\right)+\\left(q^{c}+q^{o}\\right)\\left(p^{o}-h q^{o}\\right)\\right) & >0 \\\\\nc_{1}\\left(c_{2}-c_{1}\\right) & >0\n\\end{aligned}\n$$\n\nMoreover,\n\n$$\n\\frac{k^{*}}{N}=\\frac{q^{o}\\left(h q^{o}-p^{o}\\right)}{q^{o}\\left(2 p^{c}-h q^{c}\\right)-p^{o} q^{c}}=\\frac{c_{1}}{c_{2}} .\n$$\n\nThe result follows, because for any $c_{1}, c_{2} \\in \\mathbb{R}, c_{1} / c_{2} \\in(0,1)$ if and only if $c_{1}\\left(c_{2}-c_{1}\\right)>0$.","text_sha256":"e86cadfc7ebd6c0a06fd771892b2ce39a80d1c00fa213de519a83c7ecf599162"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0029","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 4 Formalism Omitted in Section 8.1","text":"## A. 4 Formalism Omitted in Section 8.1\n\nNo Disclosure Under no disclosure, each player's action cannot depend on the state and is thus uniquely determined by condition\n\n$$\n\\mathbb{E}\\left[a_{i}\\right]=\\frac{1}{N+1} \\mathbb{E}[\\omega] .\n$$\n\nFull Disclosure For any $\\omega$, the ensuing game admits a strictly concave potential $\\Psi(a, \\omega)=$ $\\left(\\omega-\\frac{\\breve{a}}{2}\\right) \\breve{a}-\\frac{1}{2} \\sum_{i=1}^{N} a_{i}^{2}$ and thus has a unique equilibrium. Parameterize a symmetric linear strategy profile:\n\n$$\na_{i}(\\omega)=k_{0}+k_{1} \\omega .\n$$\n\nThe best-response condition (43) can be rewritten as\n\n$$\na_{i}(\\omega)=-\\frac{N-1}{2} k_{0}+\\frac{1-k_{1}(N-1)}{2} \\omega,\n$$\n\nwhich determines the equilibrium parameters at $k_{0}=0$ and $k_{1}=\\frac{1}{N+1}$.","text_sha256":"a5646f283af44daf695532369d5a6fde1eac64452926762b02c465663309cbc9"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0030","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 5 Formalism Omitted in Section 6","text":"## A. 5 Formalism Omitted in Section 6\n\nGame Normalization Under any information structure, the expected marginal payoffs must equal zero. Thus,\n\n$$\n\\mathbb{E}\\left[\\omega_{i} p^{o}+\\breve{\\omega}_{-i} p^{c}+b_{1}+q^{o} a_{i}+q^{c} \\breve{a}_{-i}\\right]=\\mu\\left(p^{o}+(N-1) p^{c}\\right)+b_{1}+\\mathbb{E}\\left[q^{o} a_{i}+q^{c} \\breve{a}_{-i}\\right]=0 .\n$$\n\nBy symmetry, it follows that for all $i$,\n\n$$\n\\mathbb{E}\\left[a_{i}\\right]=a_{0} \\triangleq-\\frac{\\mu\\left(p^{o}+(N-1) p^{c}\\right)+b_{1}}{q^{o}+(N-1) q^{c}} .\n$$\n\nIn the normalized game, the expected actions equal zero. The normalization is achieved by defining $\\omega^{\\prime}=(\\omega-\\mu) / \\sigma, a_{i}^{\\prime}=\\left(a_{i}-a_{0}\\right) / \\sigma, b=\\left(b_{2}+\\left(\\tilde{p}^{o}+(N-1) \\tilde{p}^{c}\\right) \\mu+2\\left(\\tilde{q}^{o}+(N-1) \\tilde{q}^{c}\\right) a_{0}\\right) / \\sigma$, rescaling the payoffs by $\\sigma^{2}$, and ignoring the strategically irrelevant additive terms, i.e., those in $u_{i}$ that do not depend on $a_{i}$ and those in $v$ that do not depend on $a$.\n\nProof of Lemma 1 Matrix $M(x, y)$ has two distinct eigenvalues: $x+(N-1) y$ and $x-y$. Therefore, it is negative semidefinite if and only if $x+(N-1) y \\leq 0$ and $x-y \\leq 0$; it is negative definite if and only if $x+(N-1) y<0$ and $x-y<0$. Therefore, $M\\left(\\bar{q}^{o}, \\bar{q}^{c}\\right)=M\\left(\\tilde{q}^{o}+x q^{o}, \\tilde{q}^{c}+x q^{c}\\right)$\nis negative semidefinite if and only if\n\n$$\n\\begin{aligned}\nx\\left(q^{o}+(N-1) q^{c}\\right) & \\leq-\\tilde{q}^{o}-(N-1) \\tilde{q}^{c} \\\\\nx\\left(q^{o}-q^{c}\\right) & \\leq-\\left(\\tilde{q}^{o}-\\tilde{q}^{c}\\right)\n\\end{aligned}\n$$\n\nBy concavity assumptions, $q^{o}+(N-1) q^{c}<0$ and $q^{o}-q^{c}<0$. The result follows.\n\nProof of Lemma 2 Note that $M(x, y) M(z, w)=M(x z+(N-1) y w, x w+y z+(N-2) y w)$ and $M(x, y)^{-1}=M\\left(\\frac{x+(N-2) y}{(x-y)(x+(N-1) y)},-\\frac{y}{(x-y)(x+(N-1) y)}\\right)$, if invertible. Thus,\n\n$$\n\\begin{aligned}\na^{*}(\\omega, x) & =-\\frac{1}{2} M^{-1}\\left(\\bar{q}^{o}(x), \\bar{q}^{c}(x)\\right) M\\left(\\bar{p}^{o}(x), \\bar{p}^{c}(x)\\right) \\omega \\\\\n& =-\\frac{1}{2} \\frac{M\\left(\\bar{q}^{o}(x) \\bar{p}^{o}(x)+(N-2) \\bar{q}^{c}(x) \\bar{p}^{o}(x)-(N-1) \\bar{q}^{c}(x) \\bar{p}^{c}(x), \\bar{q}^{o}(x) \\bar{p}^{c}(x)-\\bar{q}^{c}(x) \\bar{p}^{o}(x)\\right)}{\\left(\\bar{q}^{o}(x)-\\bar{q}^{c}(x)\\right)\\left(\\bar{q}^{o}(x)+(N-1) \\bar{q}^{c}(x)\\right)} \\omega,\n\\end{aligned}\n$$\n\nand\n\n$$\n\\begin{aligned}\nW(x) & =-\\frac{1}{4} \\mathbb{E}\\left[\\omega^{T} M\\left(\\bar{p}^{o}(x), \\bar{p}^{c}(x)\\right) M^{-1}\\left(\\bar{q}^{o}(x), \\bar{q}^{c}(x)\\right) M\\left(\\bar{p}^{o}(x), \\bar{p}^{c}(x)\\right) \\omega\\right] \\\\\n& =-\\frac{N}{4} \\frac{Z(x)}{\\left(\\bar{q}^{o}(x)-\\bar{q}^{c}(x)\\right)\\left(\\bar{q}^{o}(x)+(N-1) \\bar{q}^{c}(x)\\right)},\n\\end{aligned}\n$$\n\nwhere\n\n$$\n\\begin{aligned}\nZ(x) \\triangleq & \\bar{q}^{o}(x)\\left(\\bar{p}^{o}(x)^{2}+\\bar{p}^{c}(x)^{2}(N-1)(1+(N-2) \\rho)+2(N-1) \\rho \\bar{p}^{o}(x) \\bar{p}^{c}(x)\\right)+ \\\\\n& +\\bar{q}^{c}(x)\\left(\\bar{p}^{o}(x)^{2}(N-2-\\rho(N-1))-\\rho(N-1)^{2} \\bar{p}^{c}(x)^{2}-2(N-1) \\bar{p}^{o}(x) \\bar{p}^{c}(x)\\right) .\n\\end{aligned}\n$$\n\nFirst, consider the case of $x \\downarrow \\underline{x}$. In this case, the term dividing $Z(x)$ is going to zero (from above). Thus, if $Z(\\underline{x}) \\neq 0$, then as $x \\downarrow \\underline{x}, W(x)$ diverges to $+\\infty$.\n(i) $\\underline{x}=-\\frac{\\tilde{q}^{o}-\\tilde{q}^{c}}{q^{o}-q^{c}}$. As $-\\frac{\\tilde{q}^{o}-\\tilde{q}^{c}}{q^{o}-q^{c}} \\geq-\\frac{\\tilde{q}^{o}+(N-1) \\tilde{q}^{c}}{q^{o}+(N-1) q^{c}}$, we have $\\tilde{q}^{o} q^{c} \\leq q^{o} \\tilde{q}^{c}$, and by the genericity assumption, $\\tilde{q}^{o} q^{c}<q^{o} \\tilde{q}^{c}$. This implies that $\\bar{q}^{o}=\\bar{q}^{c}<0$. Thus,\n\n$$\nZ(\\underline{x})=\\bar{q}^{c}(N-1)(1-\\rho)\\left(\\bar{p}^{o}-\\bar{p}^{c}\\right)^{2}<0 .\n$$\n\n(ii) $\\underline{x}=-\\frac{\\tilde{q}^{o}+(N-1) \\tilde{q}^{c}}{q^{o}+(N-1) q^{c}}$. As $-\\frac{\\tilde{q}^{o}-\\tilde{q}^{c}}{q^{o}-q^{c}} \\leq-\\frac{\\tilde{q}^{o}+(N-1) \\tilde{q}^{c}}{q^{o}+(N-1) q^{c}}$, we have $\\tilde{q}^{o} q^{c} \\geq q^{o} \\tilde{q}^{c}$, and by the genericity assumption, $\\tilde{q}^{o} q^{c}>q^{o} \\tilde{q}^{c}$. This implies that $\\bar{q}^{o}+(N-1) \\bar{q}^{c}=0$, and $\\bar{q}^{c}>0$. Thus,\n\n$$\nZ(\\underline{x})=-\\bar{q}^{c}(1+\\rho(N-1))\\left(\\bar{p}^{o}+(N-1) \\bar{p}^{c}\\right)^{2}<0 .\n$$\n\nNow, consider the case of $x \\uparrow+\\infty$. In this case, for the denominator of the dual payoff, we have\n\n$$\n\\frac{\\left(\\bar{q}^{o}-\\bar{q}^{c}\\right)\\left(\\bar{q}^{o}+(N-1) \\bar{q}^{c}\\right)}{x^{2}} \\rightarrow\\left(q^{o}-q^{c}\\right)\\left(q^{o}+(N-1) q^{c}\\right)>0,\n$$\n\nwhile for the numerator we have\n\n$$\n\\begin{aligned}\n\\frac{Z(x)}{x^{3}} \\rightarrow & q^{o}\\left(\\left(p^{o}\\right)^{2}+\\left(p^{c}\\right)^{2}(N-1)(1+(N-2) \\rho)+2(N-1) \\rho p^{o} p^{c}\\right) \\\\\n& +q^{c}\\left(\\left(p^{o}\\right)^{2}(N-2-\\rho(N-1))-\\rho(N-1)^{2}\\left(p^{c}\\right)^{2}-2(N-1) p^{o} p^{c}\\right) .\n\\end{aligned}\n$$\n\nIt suffices to show that the right-hand side of (58) is strictly negative, because then the dual payoff is $x$ times a strictly positive constant, which diverges to infinity as $x$ grows.\n\nFirst, note that\n\n$$\n\\begin{aligned}\n& \\left(p^{o}\\right)^{2}+\\left(p^{c}\\right)^{2}(N-1)(1+(N-2) \\rho)+2(N-1) \\rho p^{o} p^{c} \\\\\n& =\\left(p^{o}+\\rho(N-1) p^{c}\\right)^{2}+\\left(p^{c}\\right)^{2}\\left((N-1)(1+(N-2) \\rho)-\\rho^{2}(N-1)^{2}\\right) \\\\\n& >\\left(p^{c}\\right)^{2}(N-1)\\left(1+(N-2) \\rho-\\rho^{2}(N-1)\\right) \\\\\n& =\\left(p^{c}\\right)^{2}(N-1)(1+(N-1) \\rho)(1-\\rho) \\geq 0 .\n\\end{aligned}\n$$\n\n(i) $q^{c}<0$. Because $q^{o}<q^{c}$, the right-hand side of (58) is lower than\n\n$$\n\\begin{aligned}\n& q^{c}\\left(\\left(p^{o}\\right)^{2}+\\left(p^{c}\\right)^{2}(N-1)(1+(N-2) \\rho)+2(N-1) \\rho p^{o} p^{c}\\right) \\\\\n& +q^{c}\\left(\\left(p^{o}\\right)^{2}(N-2-\\rho(N-1))-\\rho(N-1)^{2}\\left(p^{c}\\right)^{2}-2(N-1) p^{o} p^{c}\\right) \\\\\n& =q^{c}\\left(p^{o}-p^{c}\\right)^{2}(N-1)(1-\\rho)<0 .\n\\end{aligned}\n$$\n\n(ii) $q^{c}>0$. Because $q^{o}+(N-1) q^{c}<0$, the right-hand side of (58) is lower than\n$$\n\\begin{aligned}\n& -(N-1) q^{c}\\left(\\left(p^{o}\\right)^{2}+\\left(p^{c}\\right)^{2}(N-1)(1+(N-2) \\rho)+2(N-1) \\rho p^{o} p^{c}\\right) \\\\\n& +q^{c}\\left(\\left(p^{o}\\right)^{2}(N-2-\\rho(N-1))-\\rho(N-1)^{2}\\left(p^{c}\\right)^{2}-2(N-1) p^{o} p^{c}\\right) \\\\\n& =-q^{c}\\left(p^{o}+(N-1) p^{c}\\right)^{2}(1+\\rho(N-1))<0\n\\end{aligned}\n$$\n\nThe result follows.","text_sha256":"77b5874cba87668889c075595473be924de3750038aaaee53d2fc9f678ef3004"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0031","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 6 Formalism Omitted in Section 8.3","text":"## A. 6 Formalism Omitted in Section 8.3\n\nWe can microfound the linear demand as being generated by a continuum of consumers that differ in their tastes. Each consumer has a type $\\omega=\\left(\\omega_{1}, \\omega_{2}\\right) \\in \\mathbb{R}^{2}$ and decides how much of the firms' products to consume, $q=\\left(q_{1}, q_{2}\\right)$. The payoff of a type- $\\omega$ consumer who consumes quantities $q$ at prices $a$ is\n\n$$\n-a^{T} q+\\frac{1}{2}(\\omega-q)^{T} H^{-1}(\\omega-q),\n$$\n\nwhere $H$ is an $N \\times N$ negative semidefinite matrix:\n\n$$\nH \\triangleq\\left(\\begin{array}{cc}\n-1 & \\eta \\\\\n\\eta & -1\n\\end{array}\\right) .\n$$\n\nFor any price vector $a \\in A$, the quantities demanded by a type- $\\omega$ consumer are $q(a, \\omega)=\\omega+H a$. Consumer and producer surpluses can be written as\n\n$$\n\\begin{aligned}\n& C S(a, \\omega)=a^{T} B_{C S} \\omega-\\frac{1}{2} a^{T} C_{C S} a \\\\\n& P S(a, \\omega)=a^{T}(\\omega+H a)-c(\\omega+H a)^{T}(\\omega+H a)=-c \\omega^{T} \\omega+a^{T} B_{P S} \\omega-\\frac{1}{2} a^{T} C_{P S} a,\n\\end{aligned}\n$$\n\nwhere the payoff coefficient matrices are\n\n$$\n\\begin{aligned}\n& B_{C S} \\triangleq-I, C_{C S} \\triangleq H \\\\\n& B_{P S} \\triangleq I-2 c H, C_{P S} \\triangleq-2 H+2 c H^{2} \\\\\n& B \\triangleq \\delta B_{C S}+(1-\\delta) B_{P S}, C \\triangleq \\delta C_{C S}+(1-\\delta) C_{P S}\n\\end{aligned}\n$$\n\nPrice Control The designer's first-order condition is\n\n$$\nB \\omega-C a=0,\n$$\n\nwhich results in the first-best responsiveness matrix $R^{F B}=C^{-1} B$, and thus,\n\n$$\n\\begin{aligned}\nr_{o}^{F B} & =\\frac{2+\\delta(6 \\delta-7)+4 c^{2}(1-\\delta)^{2}\\left(1-\\eta^{2}\\right)+2 c(1-\\delta)\\left(3-5 \\delta-(1-\\delta) \\eta^{2}\\right)}{\\left(1-\\eta^{2}\\right)(2-3 \\delta+2 c(1-\\delta)(1-\\eta))(2-3 \\delta+2 c(1-\\delta)(1+\\eta))}, \\\\\nr_{c}^{F B} & =\\frac{\\eta\\left(2+\\delta(6 \\delta-7)+4 c^{2}(1-\\delta)^{2}\\left(1-\\eta^{2}\\right)+4 c(1-\\delta)(1-2 \\delta)\\right)}{\\left(1-\\eta^{2}\\right)(2-3 \\delta+2 c(1-\\delta)(1-\\eta))(2-3 \\delta+2 c(1-\\delta)(1+\\eta))} .\n\\end{aligned}\n$$\n\nThe threshold value $\\bar{\\delta}$ is the one that equalizes the determinant of $C$ to zero:\n\n$$\n\\bar{\\delta}=\\frac{2+2 c(1-|\\eta|)}{3+2 c(1-|\\eta|)} .\n$$\n\nNo Disclosure and Full Disclosure Equilibrium pricing behavior is derived from the system of first-order conditions:\n\n$$\n\\mathbb{E}_{\\mu}\\left[q_{i}\\left(a_{i}, a_{-i}, \\omega\\right)+\\frac{\\partial q_{i}\\left(a_{i}, a_{-i}, \\omega\\right)}{\\partial a_{i}}\\left(a_{i}-2 c q_{i}\\left(a_{i}, a_{-i}, \\omega\\right)\\right)\\right]=0, \\quad i=1,2 .\n$$\n\nProof of Proposition 5 The setting is an instance of the general framework (23-24) with $N=2, \\rho=0, b_{1}=b_{2}=0$ and $p_{o}=1+2 c, p_{c}=0, q_{o}=-2(1+c), q_{c}=\\eta(1+2 c)$, $\\tilde{p}_{o}=\\delta(-1)+(1-\\delta)(1+2 c), \\tilde{p}_{c}=(1-\\delta)(-2 c \\eta), \\tilde{q}_{o}=\\delta / 2+(1-\\delta)\\left(-1-c\\left(1+\\eta^{2}\\right)\\right)$, $\\tilde{q}_{c}=-\\delta \\eta / 2+(1-\\delta) \\eta(1+2 c)$.\n\nThe non-genericity conditions (26-28) reduce to\n\n$$\n\\delta \\neq \\hat{\\delta}(\\eta, c) \\triangleq \\frac{2+6 c+4 c^{2}-8 c|\\eta|-8 c^{2}|\\eta|+2 c \\eta^{2}+4 c^{2} \\eta^{2}}{5+8 c+4 c^{2}-|\\eta|-10 c|\\eta|-8 c^{2}|\\eta|+2 c \\eta^{2}+4 c^{2} \\eta^{2}} .\n$$\n\nIf $\\eta<0$, at $\\delta=\\hat{\\delta}$ condition (27) is violated. If $\\eta>0$, at $\\delta=\\hat{\\delta}$ condition (28) is violated. Otherwise, the conditions hold.\n\nThe resulting lower bound on $x$ is\n\n$$\n\\underline{x}=\\max \\left\\{-\\frac{\\tilde{q}_{o}-\\tilde{q}_{c}}{q_{o}-q_{c}},-\\frac{\\tilde{q}_{o}+\\tilde{q}_{c}}{q_{o}+q_{c}}\\right\\} .\n$$\n\nThe expected dual payoff reduces to\n\n$$\nW(x)=-\\frac{\\sigma^{2}\\left(\\bar{q}_{o}(x) \\bar{p}_{o}(x)^{2}+\\bar{q}_{o}(x) \\bar{p}_{c}(x)^{2}-2 \\bar{q}_{c}(x) \\bar{p}_{o}(x) \\bar{p}_{c}(x)\\right)}{2\\left(\\bar{q}_{o}(x)^{2}-\\bar{q}_{c}(x)^{2}\\right)} .\n$$\n\nBy Theorem 4 and normalization (A.5), whenever $\\delta \\neq \\hat{\\delta}$, the optimal direct information structure is unique and is given by (52) with\n\n$$\n\\begin{aligned}\n& r_{o}^{*}=-\\frac{\\bar{q}_{o}\\left(x^{*}\\right) \\bar{p}_{o}\\left(x^{*}\\right)-\\bar{q}_{c}\\left(x^{*}\\right) \\bar{p}_{c}\\left(x^{*}\\right)}{2\\left(\\bar{q}_{o}\\left(x^{*}\\right)^{2}-\\bar{q}_{c}\\left(x^{*}\\right)^{2}\\right)}, \\\\\n& r_{c}^{*}=-\\frac{\\bar{q}_{o}\\left(x^{*}\\right) \\bar{p}_{c}\\left(x^{*}\\right)-\\bar{q}_{c}\\left(x^{*}\\right) \\bar{p}_{o}\\left(x^{*}\\right)}{2\\left(\\bar{q}_{o}\\left(x^{*}\\right)^{2}-\\bar{q}_{c}\\left(x^{*}\\right)^{2}\\right)},\n\\end{aligned}\n$$\n\nand $x^{*}$ being any minimizer of (61) over $x^{*}>\\underline{x}$. The constant term $a_{0}^{*}$ can be found via mean action invariance: $a_{0}^{*}=a_{i}^{N D}-\\left(r_{o}^{*}+r_{c}^{*}\\right) \\omega_{0}$.","text_sha256":"875c953d19151f0f3a07801d041a8217509ae0b25ae3ab59611c10b75b30d4d0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0032","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B Bounded Action Spaces","text":"## B Bounded Action Spaces\n\nConsider the concave information-design problem as in the main text, but let the action space of each player be $A_{i}=\\left[\\underline{a}_{i}, \\bar{a}_{i}\\right],-\\infty<\\underline{a}_{i}<\\bar{a}_{i}<+\\infty .{ }^{18}$ For any given $\\mu \\in \\Delta\\left(A_{-i} \\times \\Omega\\right)$, the player's best-response action $a_{i}^{*}(\\mu)$, if interior, must be unimprovable by local deviations to\n\n[^15]lower and higher actions and hence satisfies the first-order condition\n$$\n\\mathbb{E}_{\\mu}\\left[\\dot{u}_{i}\\left(a_{i}^{*}, a_{-i}, \\omega\\right)\\right]=0 .\n$$\nIn contrast, the optimal boundary actions must only be unimprovable by one-sided local deviations. As such, the player's best response if located on the boundary must satisfy the following:\n$$\n\\begin{aligned}\n& \\mathbb{E}_{\\mu}\\left[\\dot{u}_{i}\\left(\\underline{a}_{i}^{*}, a_{-i}, \\omega\\right)\\right] \\leq 0, \\\\\n& \\mathbb{E}_{\\mu}\\left[\\dot{u}_{i}\\left(\\bar{a}_{i}^{*}, a_{-i}, \\omega\\right)\\right] \\geq 0 .\n\\end{aligned}\n$$\nWe can write the resulting primal information-design problem as follows:\n$$\n\\begin{aligned}\nV_{B}^{P} \\triangleq & \\sup _{\\pi \\in \\Delta(A \\times \\Omega)} \\int_{A \\times \\Omega} v(a, \\omega) \\mathrm{d} \\pi \\\\\n& \\text { s.t. } \\int_{A_{i}^{\\prime} \\times A_{-i} \\times \\Omega} \\dot{u}_{i}(a, \\omega) \\mathrm{d} \\pi=0 \\quad \\forall i=1, \\ldots, N, \\text { measurable } A_{i}^{\\prime} \\subseteq\\left(\\underline{a}_{i}, \\bar{a}_{i}\\right), \\\\\n& \\int_{A_{i}^{\\prime} \\times A_{-i} \\times \\Omega} \\dot{u}_{i}(a, \\omega) \\mathrm{d} \\pi \\leq 0 \\quad \\forall i=1, \\ldots, N, \\text { measurable } A_{i}^{\\prime} \\subseteq\\left[\\underline{a}_{i}, \\bar{a}_{i}\\right), \\\\\n& \\int_{A_{i}^{\\prime} \\times A_{-i} \\times \\Omega} \\dot{u}_{i}(a, \\omega) \\mathrm{d} \\pi \\geq 0 \\quad \\forall i=1, \\ldots, N, \\text { measurable } A_{i}^{\\prime} \\subseteq\\left(\\underline{a}_{i}, \\bar{a}_{i}\\right], \\\\\n& \\int_{A \\times \\Omega^{\\prime}} \\mathrm{d} \\pi=\\int_{\\Omega^{\\prime}} \\mathrm{d} \\mu_{0} \\quad \\forall \\text { measurable } \\Omega^{\\prime} \\subseteq \\Omega .\n\\end{aligned}\n$$\nThis primal problem entails the dual problem\n$$\n\\begin{aligned}\nV_{B}^{D} \\triangleq & \\inf _{\\lambda \\in x_{i} \\mathcal{M}\\left(A_{i}\\right), \\gamma \\in \\mathcal{M}(\\Omega)} \\int_{\\Omega} \\gamma(\\omega) \\mathrm{d} \\mu_{0} \\\\\n& \\text { s.t. } \\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega)+\\gamma(\\omega) \\geq v(a, \\omega) \\forall a \\in A, \\omega \\in \\Omega, \\\\\n& \\quad \\lambda_{i}\\left(\\underline{a}_{i}\\right) \\geq 0, \\lambda_{i}\\left(\\bar{a}_{i}\\right) \\leq 0 \\quad \\forall i=1, \\ldots, N .\n\\end{aligned}\n$$\nThe presence of additional obedience constraints (66), (67) in the primal problem translates into the sign constraints on the Lagrange multipliers (71) in the dual problem. As in the case\nof unbounded actions, the dual problem (69) can be simplified and rewritten as\n$$\n\\begin{aligned}\nV_{B}^{D} & =\\inf _{\\lambda \\in \\times_{i} \\mathcal{M}\\left(A_{i}\\right)} \\mathbb{E}_{\\mu_{0}}\\left[\\sup _{a \\in A} w(a, \\omega, \\lambda)\\right] \\\\\n\\text { s.t. } & \\lambda_{i}\\left(\\underline{a}_{i}\\right) \\geq 0, \\lambda_{i}\\left(\\bar{a}_{i}\\right) \\leq 0 \\quad \\forall i=1, \\ldots, N\n\\end{aligned}\n$$\nThe adversarial-contracting interpretation remains intact, but the space of allowed contracts is limited at the boundary actions by the presence of sign constraints.\n\nLemma 5. (Weak Duality with Bounded Action Spaces) $V_{B}^{P} \\leq V_{B}^{D}$.\nProof. Take any dual variables $(\\lambda, \\gamma)$ that satisfy the constraints of the dual problem (69). Take any measure $\\pi$ that satisfies the constraints of primal problem (64). Integrating both sides of the dual constraints over $a \\in A$ and $\\omega \\in \\Omega$ against measure $\\pi$ yields\n\n$$\n\\int_{A \\times \\Omega} v(a, \\omega) \\mathrm{d} \\pi \\leq \\int_{A \\times \\Omega} \\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega) \\mathrm{d} \\pi+\\int_{A \\times \\Omega} \\gamma(\\omega) \\mathrm{d} \\pi \\leq \\int_{\\Omega} \\gamma(\\omega) \\mathrm{d} \\mu_{0},\n$$\n\nwhere the second inequality follows because $\\pi$ satisfies the primal constraints and the Lagrange multipliers satisfy the dual constraints. (This inequality holds as equality in the case of unbounded actions.) The left-hand side of (74) is the value of the primal problem given measure $\\pi$, whereas the right-hand side of (74) is the value of the dual problem given dual variables $(\\lambda, \\gamma)$. As inequality (74) holds for any allowed values of primal measure and dual variables, it also holds at the respective maximization and minimization limits. $\\square$\n\nIn the case of bounded actions, we call a measure $\\pi \\in \\Delta(A \\times \\Omega)$ implementable by information if it satisfies the constraints of the primal problem (64) and implementable by incentives if there exists a feasible contract in the dual problem (72) that induces this measure as a best response and satisfies the complementarity slackness condition: for all $i$ and measurable $A_{i}^{\\prime} \\subseteq\\left[\\underline{a}_{i}, \\bar{a}_{i}\\right], \\int_{A_{i}^{\\prime} \\times A_{-i} \\times \\Omega} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega) \\mathrm{d} \\pi=0$.\n\nTheorem 5. (Optimality Certification with Bounded Action Spaces) In the case of bounded actions, if $\\pi \\in \\Delta(A \\times \\Omega)$ is implementable by information and is implementable by incentives by contract $\\lambda$, then (i) $\\pi$ solves the information-design problem, (ii) $\\lambda$ solves the adversarialcontracting problem, and (iii) $V_{B}^{P}=V_{B}^{D}$.\n\nProof. Take any primal measure $\\pi$ implementable by information, i.e., one that satisfies the constraints of the primal problem (64). If it is implementable by incentives, then there exist dual variables $\\lambda$ that implement this measure in the dual problem (72) and satisfy complementarity slackness. Therefore,\n\n$$\n\\begin{aligned}\nV_{B}^{D} & =\\inf _{\\lambda^{\\prime} \\in \\times_{i} \\mathcal{M}\\left(A_{i}\\right)+(73)} \\mathbb{E}_{\\mu_{0}}\\left[\\sup _{a \\in A} w\\left(a, \\omega, \\lambda^{\\prime}\\right)\\right] \\\\\n& \\leq \\mathbb{E}_{\\pi}[w(a, \\omega, \\lambda)] \\\\\n& =\\int_{A \\times \\Omega} v(a, \\omega) \\mathrm{d} \\pi-\\int_{A \\times \\Omega} \\sum_{i=1}^{N} \\lambda_{i}\\left(a_{i}\\right) \\dot{u}_{i}(a, \\omega) \\mathrm{d} \\pi \\\\\n& =\\int_{A \\times \\Omega} v(a, \\omega) \\mathrm{d} \\pi \\leq V_{B}^{P}\n\\end{aligned}\n$$","text_sha256":"9655ade3b27bb592e424390e14d91f93808b35b7c4447bf3a09647ac55437f63"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0033","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B Bounded Action Spaces","text":"where the first inequality follows from the implementability of $\\pi$ in the dual problem, whereas the last three steps follow from the feasibility of $\\pi$ in the primal problem and complementarity slackness.\n\nFurthermore, by Lemma 5, $V_{B}^{D} \\geq V_{B}^{P}$. Hence,\n\n$$\nV_{B}^{D}=\\int_{A \\times \\Omega} v(a, \\omega) \\mathrm{d} \\pi=V_{B}^{P}\n$$\n\nwhich proves the optimality of measure $\\pi$. $\\square$\n\nClaim 2. (Strong Duality with Bounded Action Spaces) If $\\Omega$ is compact, and $v$ and each $\\dot{u}_{i}$ are continuous, then $V_{B}^{D}=V_{B}^{P}$.\n\nProof. The proof is analogous to that of Claim 1, except that one carries the boundary sign restrictions through the auxiliary continuous dual. In particular, one defines $V_{B}^{D, c}$ exactly as in the proof of Claim 1, but with continuous multipliers $\\lambda_{i} \\in C\\left(A_{i}\\right)$ constrained to satisfy $\\lambda_{i}\\left(\\underline{a}_{i}\\right) \\geq 0$ and $\\lambda_{i}\\left(\\bar{a}_{i}\\right) \\leq 0$ for each $i$. Then $V_{B}^{D} \\leq V_{B}^{D, c}$, while weak duality gives $V_{B}^{P} \\leq V_{B}^{D}$, so it suffices to show $V_{B}^{D, c}=V_{B}^{P}$. Applying the Fenchel-Rockafellar duality with the same spaces and linear operator as in the proof of Claim 1, the only change is that the functional on the multiplier space now includes the indicators of the cones $K_{i}=\\left\\{\\lambda_{i} \\in C\\left(A_{i}\\right): \\lambda_{i}\\left(\\underline{a}_{i}\\right) \\geq 0, \\lambda_{i}\\left(\\bar{a}_{i}\\right) \\leq 0\\right\\}$. Accordingly, the conjugate condition in the dual becomes $\\pi_{0}=\\mu_{0}$ together with $\\nu_{i} \\in K_{i}^{\\circ}$ for\nevery $i$, where $\\nu_{i}$ is the weighted marginal measure induced by $\\dot{u}_{i}$ and $K_{i}^{\\circ}$ is the polar cone of $K_{i}$. A direct characterization of $K_{i}^{\\circ}$ shows that this is equivalent to the bounded-action obedience constraints (65)-(67): zero weighted mass on interior subsets and the appropriate weak inequalities on sets touching the lower or upper boundary. Thus the feasible set in the Fenchel-Rockafellar dual coincides exactly with the primal feasible set, so $V_{B}^{D, c}=V_{B}^{P}$, and therefore $V_{B}^{P} \\leq V_{B}^{D} \\leq V_{B}^{D, c}=V_{B}^{P}$, implying $V_{B}^{D}=V_{B}^{P}$. $\\square$\n\n[^0]:    Smolin: Toulouse School of Economics, University of Toulouse Capitole and CEPR, alexey.v.smolin@gmail.com. Yamashita: Osaka University, tytakuroy@gmail.com. For valuable suggestions and comments, we would like to thank the Coeditor Rakesh Vohra, two anonymous referees, Refine.ink, Dirk Bergemann, Deniz Dizdar, Laura Doval, Philippe Jehiel, Jiangtao Li, Xiao Lin, Elliot Lipnowski, Stephen Morris, Alessandro Pavan, Antonio Penta, Jacopo Perego, Fedor Sandomirskiy, Ludvig Sinander, Takashi Ui, Xavier Vives, and Alexander Wolitzky, as well as seminar participants at Toulouse School of Economics, Western University, University of Toronto, University of Surrey, University of Oxford, Hitotsubashi University, Singapore Management University, CMU/Pittsburgh, Pennsylvania State University, MIT/Harvard, Bonn Winter Theory Workshop 2022, SWET 2022, CMid2022, EC 2022, EEA-ESEM 2022, EARIE 2022, and Venice Winter Theory Workshop 2023. An extended abstract of a previous version of this paper appeared as \"Information Design in Concave Games\" in the proceedings to EC'22. Smolin gratefully acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (grant ANR-17-EURE-0010) and through the AI Interdisciplinary Institute ANITI (grant ANR-23-IACL-0002), as well as the hospitality of Northwestern University and Columbia Business School, where parts of this paper were completed. Yamashita gratefully acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (grant ANR-17-EURE-0010), funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant 714693), and JSPS KAKENHI (grant 23K01311).\n\n[^1]:    ${ }^{1}$ See, for example, Dworczak and Martini (2019), Dizdar and Kováč (2020), and Dworczak and Kolotilin (2024). The single-player case also covers scenarios in which there are many players but the information is required to be public.\n\n[^2]:    ${ }^{2}$ The duality methodology is routinely used in many disciplines to solve optimization problems. In mechanism design, duality methods have been recently used to study optimal delegation (Amador, Werning, and Angeletos (2006); Amador and Bagwell (2013)), matching (Chiappori, Salanié, and Weiss (2017); Galichon and Salanié (2022)), robust selling mechanisms (Carroll (2017); Du (2018); Brooks and Du (2021)), mediation (Salamanca (2021), Ortner, Sugaya, and Wolitzky (2024)), and limited commitment (Lin and Liu (2024)) among others. For a unified treatment, see Vohra (2011).\n    ${ }^{3}$ Alternatively, one can develop original arguments tailored to the studied problem; see, for example, Arieli and Babichenko (2019), Chan, Gupta, Li, and Wang (2019), Elliott, Galeotti, Koh, and Li (2022), Arieli, Babichenko, and Sandomirskiy (2023), and Candogan and Strack (2023).\n\n[^3]:    ${ }^{4}$ In their Section 4.4, Bergemann et al. (2017) argue that Gaussian structures, possibly asymmetric and with extraneous noise, span all implementable covariance matrices of equilibrium actions.\n\n[^4]:    ${ }^{5}$ We write $\\mathbb{E}_{\\mathcal{I}, \\sigma_{i}, \\sigma_{-i}}[\\cdot]$ for the expectation under information structure $\\mathcal{I}$ and strategy profile $\\left(\\sigma_{i}, \\sigma_{-i}\\right)$. Throughout the paper an integral is left undefined whenever the integrand is not integrable with respect to the relevant measure.\n\n[^5]:    ${ }^{6}$ This notion of a concave game is related to but distinct from the notion of a concave game of Rosen (1965). In particular, it requires neither differentiability nor continuity of a player's payoff in the other players' actions or the state.\n\n[^6]:    ${ }^{7}$ This observation resonates with the research on robust information design in regime change games, such as Inostroza and Pavan (2025) and Morris, Oyama, and Takahashi (2024), which is fundamentally concerned with multiple informational implementations of the same aggregate behavior.","text_sha256":"b4d9b4bca112dc04e2ef8db3434787f488a13fc72946916ae908b60130139464"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0034","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B Bounded Action Spaces","text":"[^7]:    ${ }^{8}$ This includes the case when the denominator in (20) is zero.\n    ${ }^{9}$ Note that conditions (i) and (ii) are not equivalent to $k^{*}<0$ and $k^{*}>N$. Examples exist in which full disclosure is optimal even when $k^{*}<0$, and others in which no disclosure is optimal even when $k^{*}>N$.\n\n[^8]:    ${ }^{10}$ By Theorem 2, if $k^{*} \\notin(0, N)$, then either full disclosure or no disclosure is optimal.\n\n[^9]:    ${ }^{11}$ For a more detailed analysis, see the working version of the paper.\n\n[^10]:    ${ }^{12}$ This phenomenon parallels cream-skimming among competing contractors on online platforms, which can likewise be mitigated through information design (Romanyuk and Smolin (2019)).\n\n[^11]:    ${ }^{13}$ Because the expected investment and the expected investment costs are invariant to information, this payoff also captures the objective of maximizing the total investor welfare.\n\n[^12]:    ${ }^{14}$ The setting in this section can also be viewed as Cournot competition with linear production costs and uncertain linear demand. In that interpretation, the rent dissipation under both the no-disclosure and full-disclosure benchmarks parallels the zero-profit outcome of large competitive markets. Exclusive disclosure then amounts to informing only a single firm about the demand state, and this policy maximizes total producer surplus. The structure bears some resemblance to collusion that designates one firm as a monopolist, but information control is weaker than direct production control, because every firm still chooses a positive output even when uninformed. This perspective helps illuminate the role of trade associations, which cannot engage in illegal collusion yet can manage information flows (see, e.g., Kirby (1988); Vives (1990)).\n\n[^13]:    ${ }^{15}$ This formalization complements that adopted by Elliott et al. (2022), who study information design in unit-demand competition. Naturally, the details of their optimal information structures differ from ours; however, their interpretation of information provision as market segmentation and their broader discussion of how information shapes competitive outcomes can be applied in our setting.\n    ${ }^{16}$ As standard, this specification allows the prices and quantities to be negative.\n\n[^14]:    ${ }^{17}$ Formally, while the optimal certificate changes continuously for all $\\delta \\in(0,1)$, at $\\delta=\\hat{\\delta}$ the certificate allows a range of the dual agent's best responses connecting one branch of best responses to another, so that the implemented allocation rule displays a jump.\n\n[^15]:    ${ }^{18}$ The extension to half-bounded spaces is straightforward.","text_sha256":"1c0fb0518c6699fb63f828d7d434a091db648627f4b21c00402e19fda51b0f3a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:information-design-in-smooth-games:2026-03-11:0035","work_id":"alex-smolin:information-design-in-smooth-games","paper_id":"alex-smolin:information-design-in-smooth-games:2026-03-11","title":"Information Design in Smooth Games","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Takuro Yamashita","url":"https://sites.google.com/view/takuroyamashita/"}],"manuscript_date":"2026-03-11","language":"en","version_type":"accepted-author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md","source_record":"https://arxiv.org/abs/2202.10883","citation":"Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Alex Smolin; Takuro Yamashita\n\n**Canonical citation:** Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/information-design-in-smooth-games.md\n\n**Source record:** https://arxiv.org/abs/2202.10883\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"31499df942e3e920e7dbad5174e4e6a7afb5cab38ab058551de485a4996038a4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0001","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Shota Ichihashi; Alex Smolin.\n> Canonical citation: Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"8a1e385ed7c2e4bc0f80045178e4c3fa33b1d052111d8371b3df5a3af96d48db"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0002","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Data Provision to an Informed Seller","text":"# Data Provision to an Informed Seller\n\n**Authors:** Shota Ichihashi; Alex Smolin\n\n**Manuscript date:** 2023-03-03\n\n#### Abstract\n\nA monopoly seller is privately and imperfectly informed about the buyer's value of the product. The seller uses information to price discriminate the buyer. A designer offers a mechanism that provides the seller with additional information based on the seller's report about her type. We establish the impossibility of screening for welfare purposes-i.e., the designer can attain any implementable combination of buyer surplus and seller profit by providing the same signal to all seller types. We use this result to characterize the set of implementable welfare outcomes, study the seller's incentive to acquire third-party data, and demonstrate the trade-off between buyer surplus and efficiency.\n\n[^0]","text_sha256":"04339b06bf81b4240351058d27a93205479b7a331578107c17705b75db7e1e76"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0003","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nThe use of consumer data has become an important and ubiquitous aspect of the interactions between companies and consumers in the digital economy. Companies collect personal information and track consumers as they browse entertainment portals, post content on social media, and buy products on e-commerce websites. This data is used to adapt online services directly at the point of collection or later if transferred elsewhere. One prominent case is the use of consumer data for price discrimination: A seller may use data on its customers to learn about their preferences and tailor the prices of services and products via outright personalized pricing, providing discount coupons, or steering customers to a more expensive version of similar products.\n\nAs the importance of consumer data grows, various parties-such as policymakers, platforms, and consumers themselves-are attempting to control the flow of consumer data to sellers. However, the ability of these parties to control the allocation of data across sellers is likely to be limited because of information asymmetry-i.e., sellers may be privately informed about their market demands and customers. For example, a regulator might prefer that sellers within a certain group not use certain data, but the regulator may not know which sellers belong to that group. This raises questions about the scope of data provision to privately informed sellers and its potential impact on consumer surplus, seller profits, and overall efficiency.\n\nMotivated by the above discussion, we study a model of data provision to a privately informed seller. The seller is a monopolist and has a product for sale. The buyer has a binary uncertain value for the product, which is either high or low. The seller's interim belief that the value is high, which we call the seller's type, is her private information. The seller may improve her pricing by obtaining additional information, which we model as a statistical signal that is informative about the value. We take a mechanism-design approach and consider a designer who offers a menu of signals to the seller. In practice, a menu corresponds to a restriction imposed by a regulator or a platform regarding what data sellers can use. Given a menu, the seller selects a signal and learns about the buyer's value. Finally, the seller sets a price and the buyer decides whether to purchase the product.\n\nOur focus is the set of all possible welfare outcomes the designer can implement by offering an arbitrary menu of signals. If the designer observes the seller's type, the set of possible outcomes becomes the \"surplus triangle,\" as described by Bergemann, Brooks, and Morris (2015)-i.e., any division of the total surplus between the buyer and the seller can be achieved provided that the total surplus is not higher than the efficient surplus, the buyer surplus is nonnegative, and the seller's profit is no lower than the profit she can achieve without additional information.\n\nWhen the seller's type is private, the designer can no longer attain the surplus triangle. For example, the first-best buyer surplus, which attains efficiency and gives no extra rents to the seller, requires that different types of sellers obtain different signals. However, if the designer were to implement such an outcome by offering a menu of those signals, some types would choose signals intended for other types in order to extract surplus from the buyer.\n\nOur first main result shows that the designer cannot effectively screen the seller at all: An outcome is implementable through some menu of signals if and only if it is implementable by providing the same public signal to all seller types. This result illustrates the difficulty of eliciting the seller's private information and providing personalized data. To prove the result, we develop novel structural properties of how personalized data provision can impact prices across different seller types and buyer values. We then use these properties to explicitly construct a public signal that replicates the equilibrium pricing behavior of any menu of signals.\n\nWe build on this result and geometrically characterize the set of implementable outcomes as the convex hull of an aggregate surplus function that averages across seller types. We show that not only can the designer focus on public signals, but these signals do not need to be complex: Any implementable outcome can be achieved using a signal with at most three signal realizations, and any extreme implementable outcome requires at most two signal realizations.\n\nWe use the above results to derive the surplus set in a closed form when the seller's type is uniformly distributed. Figure 1 depicts the set of implementable outcomes for observable types (light blue triangle) and unobservable types (dark blue area). The seller's private information shrinks the implementable set. In this example, providing any signal reduces\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: The sets of implementable outcomes for observable and unobservable seller types when low value equals 1, high value equals 2, and types are uniformly distributed on [0, 1]. The horizontal axis and vertical axis represent the ex ante expected payoffs of the buyer and the seller, respectively.\n\nthe buyer's ex ante expected payoff because a large mass of seller types will use it to extract surplus from the buyer. Moreover, the only efficient outcome leads to zero buyer surplus. The right boundary of the surplus set is spanned by signals that reveal that the buyer has the high value with some probability, and symmetrically, the left boundary is spanned by signals that reveal the low value.","text_sha256":"0182ca877c1a4526dc71c1061b4ef42c4e2606ff72f3945a43f56c1942a5341b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0004","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"Our result on the impossibility of screening has three implications. First, the result highlights a tension between consumer protection and efficiency regarding the provision of consumer data to sellers. The provision of data may enable sellers to tailor pricing, benefit consumers, and enhance efficiency. At the same time, how much and what kind of data sellers should be allowed to use for such beneficial pricing will depend on the specific market conditions and the prior information each seller faces. However, in reality, a regulator would not be able to apply different rules to different firms, partly because of the informational friction we highlight. In such a situation, allowing some sellers to use a certain piece of information could increase total surplus, but other sellers may use the same information to merely extract consumer surplus. In some settings, this trade-off can be so stark that it may be impossible to attain efficiency without giving the entire total surplus to the seller.\n\nSecond, we study the seller's incentive to acquire third-party data. Here, third-party\ndata-such as the data a seller may purchase from a data broker-refers to the source of information the seller can acquire but is out of the designer's control. We can model this as the seller's choice to be privately informed about the buyer. From the designer's perspective, the seller's private information shrinks the set of implementable outcomes, and in particular, shifts the Pareto frontier downward. Thus, third-party data typically hurts the designer who cares about social welfare. In contrast, the seller's incentives to acquire third-party data depend on which implementable outcome is selected. In particular, if the seller could propose to acquire additional \"first-party\" data directly from the buyer but the buyer has the authority to veto the proposal, the seller may prefer not to acquire third-party data. Indeed, because of adverse selection, the buyer may optimally veto acquisition of any additional data by a privately informed seller. As a result, the third-party data could crowd out the first-party data and reduce the overall information available to the seller.\n\nThird, we show that the surplus set we characterize subsumes the set of equilibrium outcomes of various games in which the buyer and the seller exchange information with each other, including cheap talk communication, voluntary disclosure of the buyer's value, and a request-consent protocol in which a privately informed seller chooses a signal subject to the buyer's consent. This observation underlies the fact that our mechanism-design approach is useful for evaluating various communication protocols and data collection policies.\n\nOur baseline model assumes that the buyer's value is binary. On the one hand, this assumption is crucial for our result whereby the designer cannot effectively screen the seller. On the other hand, many of our key insights extend to the case of general multiple values. In particular, we show that the seller's private information generally shrinks the set of implementable outcomes and creates a tension between consumer protection and efficiency.\n\nRelated Literature First and foremost, our paper relates to the recent theoretical literature on the impact of information under third-degree price discrimination or, equivalently, of market segmentation on market outcomes. In their seminal paper, Bergemann, Brooks, and Morris (2015) show that all individually rational outcomes can arise in a single-product monopoly setting. Their analysis was later extended to multiproduct markets (Haghpanah and Siegel (2022, forthcoming)); competitive markets (Shi and Zhang (2020); Rhodes and\n\nZhou (2022); Elliott, Galeotti, Koh, and Li (2022)); and two-sided markets (Condorelli and Szentes (2022)). ${ }^{1}$ We contribute to this literature by highlighting the importance of the seller's private information, which is absent in those papers. The presence of such private information is relevant in practice for regulators and platforms, which aim to control the flow of consumer data but are likely to face information asymmetry vis-a-vis sellers. We show that the seller's private information limits possible welfare outcomes and introduces the trade-off between consumer welfare and efficiency.\n\nSecond, our paper contributes to the literature on consumer privacy and privacy regulation (Acquisti, Taylor, and Wagman, 2016; Choi, Jeon, and Kim, 2019; Fainmesser, Galeotti, and Momot, 2022; Argenziano and Bonatti, 2021; Bergemann, Bonatti, and Gan, 2022). We highlight the difficulty of tailoring a privacy regulation to unknown market conditions, assess the impact of access to third-party data on first-party data collection, and compare the performance of several communication protocols. To focus on the role of the seller's private information, we abstract away from other economic forces studied in the above papers, such as the use of data for product selection and service improvement as well as information externalities between consumers.\n\nOur mechanism-design framework to study the regulation of a privately informed monopolist follows that of Baron and Myerson (1982). We employ this framework in the context of restricting the use of consumer data by a monopolist. Similar mechanism-design machinery in the context of information provision has recently been used by Kolotilin, Mylovanov, Zapechelnyuk, and Li (2017), Bergemann, Bonatti, and Smolin (2018); Smolin (forthcoming); and Yang (2022). All of these allow for a fully flexible way of designing information, following the Bayesian persuasion literature (Rayo and Segal (2010); Kamenica and Gentzkow (2011)).\n\n[^1]","text_sha256":"dca3c69a3af9d1d8782ae04e0809f68fba783f22ec95994037602c925a557844"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0005","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"## 2 Model\n\nThere is a seller and a buyer. The seller has a unit good for sale. The buyer's value $v$ for the good is uncertain and either high or low: $v \\in V \\triangleq\\{L, H\\}$ with $H>L>0$. The seller is privately informed about the value. The seller's private information is captured by $\\theta \\in \\Theta \\subseteq[0,1]$ and represents the seller's belief that $v=H$. The type is distributed according to measure $F \\in \\Delta([0,1]) .^{2}$\n\nTo improve her pricing, the seller seeks to acquire additional data. We model data as a statistical signal that can be arbitrarily informative about the value. Formally, a signal $\\mathcal{I}=(S, \\pi)$ consists of a set $S$ of signal realizations $s$ and a family of distributions $\\{\\pi(\\cdot \\mid v)\\}_{v \\in V}$ over $S$. Where it does not cause confusion, we write conditional distribution $\\pi(\\cdot \\mid v)$ as $\\pi(v)$.\n\nWe adopt a mechanism-design approach and assume that signals are provided by a designer. At the outset, the designer posts a menu $\\mathcal{M}$ of signals to the seller. Then the game between the seller and the buyer proceeds as follows. First, the nature draws the seller's type $\\theta$ according to prior distribution $F$ and the buyer's value $v$ according to $\\theta .{ }^{3}$ Second, the seller privately observes her type $\\theta$ and chooses a signal $\\mathcal{I}=(S, \\pi) \\in \\mathcal{M}$. Third, the seller observes signal realization $s$ drawn according to $\\pi(v)$ and posts a price $p \\in \\mathbb{R}$ for the product. Finally, the buyer observes value $v$ and price $p$ and decides whether to buy the product. If the trade occurs, the buyer obtains payoff $v-p$ and the seller obtains payoff $p$. Otherwise, both players obtain zero payoffs.\n\nFor any given menu $\\mathcal{M}$ of signals, the solution concept is a perfect Bayesian equilibrium. Any equilibrium induces an allocation rule $a: V \\rightarrow[0,1] \\times \\mathbb{R}$, which specifies for each value the probability of a trade and the expected payment from the buyer to the seller. We call the corresponding ex ante expected payoffs of the buyer and the seller as the buyer surplus and seller profit, respectively. A welfare outcome, or simply outcome, refers to a pair of buyer surplus and seller profit. Each allocation rule leads to a unique outcome, but a given outcome may come from multiple allocation rules. An allocation rule and an outcome are implementable if they can arise in an equilibrium of some menu.\n\n[^2]Our goal is twofold. First, we study how the seller's private information limits the designer's ability to implement certain outcomes. To do so, in Section 3, we characterize the set of implementable outcomes and compare it with the case in which the seller's type is observed. Second, we aim to derive implications on data policies for regulators and sellers when the sellers may have private information or face heterogeneous demand conditions. To this end, in Section 4, we use the characterization result to highlight a general tension between consumer protection and efficiency, and we examine the seller's incentives to acquire external data that is outside the control of the designer. Finally, we show that the set of implementable outcomes we characterize subsumes the equilibrium outcomes of various communication protocols motivated by applications.","text_sha256":"e379665dd95577ac491f4149a2061b3faecd8c9cabe15eecfd4c002811ccc1b4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0006","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2.1 Discussion of Modeling Assumptions","text":"### 2.1 Discussion of Modeling Assumptions\n\nBefore proceeding with the analysis, we briefly discuss several modeling choices that are characteristic of our model.\n\nData as a Signal. As is common in the literature on information design, we model data as a statistical signal that is informative about the buyer's value. This approach bypasses the technical aspects of data analysis and algorithmic implementation and focuses directly on the seller's economic assessments of values. Different kinds of data, such as web cookies, geolocation data, and consumer behavior on the seller's website, are indicative of consumer preferences and thus can be considered to be signals. In turn, a signal realization refers to the specific instance of data, such as the contents of a consumer's cookies, his geolocation, or his web-browsing history. If we view the model as consisting of a continuum of buyers with heterogeneous values and sellers with heterogeneous types, any given signal induces a market segmentation with buyers belonging to the same segment if they have the same realized signal. However, since sellers have different types, the same signal can lead to different market segmentations depending on their types.\n\nRole of the Designer. We abstract away from the designer's objective and focus on all implementable outcomes. This enables us to make predictions that do not depend on the\ndesigner's preferences. This approach is also useful when considering a player who controls consumer data with a particular objective. For example, the designer may be a regulator trying to maximize consumer welfare by restricting the data that sellers can use about consumers. Alternatively, the designer could be a platform or an information intermediary that provides a seller with information about buyers. Specifying the designer's objective or a specific data collection protocol will select certain implementable outcomes.\n\nInformed Seller. The seller's private information about the value of the product to the buyer can come from two sources. First, the seller may have some information about the buyer that is beyond the designer's control. Information could come from technological constraints like knowledge of the consumer's IP address or from uncontrolled consequences of online interactions such as purchasing decisions. Second, the seller may be more informed than the designer about the general quality of their product. If the seller has a high-quality product, they believe that the consumer's value is more likely to be high.\n\nPrice Discrimination. We assume that the seller uses data for third-degree price discrimination. In practice, while sellers may be reluctant to display different base prices to different consumers, there are at least two common and indirect ways to price discriminate in the digital economy. First, sellers may offer personalized discounts to consumers by setting the base price high and changing the size or frequency of discounts to implement discriminatory pricing. Second, sellers may offer personalized recommendations that point consumers toward similar products that vary in price. By maintaining a large inventory of such products, sellers can effectively engage in price discrimination.","text_sha256":"564be057b73d871dded850352b9af140725ffae4466cbbc7335b2e87b47590a7"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0007","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Surplus Set Characterization","text":"## 3 Surplus Set Characterization\n\nTo study how the seller's private information affects implementable outcomes, we begin our analysis with the benchmark case in which the seller's type is observable to the designer. We then proceed to the primary scenario of unobservable types and present our main results.","text_sha256":"c0f53ac7c262fa661c656aa56f32be3499b82a7c119bf2253996f0b5bff324e8"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0008","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.1 Observable Seller Type","text":"### 3.1 Observable Seller Type\n\nFirst, suppose that the seller's type is deterministic, i.e., $\\Theta=\\left\\{\\theta_{0}\\right\\}$. Denote the maximum feasible total surplus by $\\overline{\\mathrm{W}}\\left(\\theta_{0}\\right) \\triangleq \\theta_{0} H+\\left(1-\\theta_{0}\\right) L$ and the seller's profit from optimal uniform pricing by $\\underline{\\Pi}\\left(\\theta_{0}\\right) \\triangleq \\max \\left\\{\\theta_{0} H, L\\right\\}$. Clearly, any feasible welfare outcome (U, П) must belong to the \"surplus triangle\" characterized by constraints $\\mathrm{U} \\geq 0, \\Pi \\geq \\underline{\\Pi}\\left(\\theta_{0}\\right)$, and $\\mathrm{U}+\\Pi \\leq \\overline{\\mathrm{W}}\\left(\\theta_{0}\\right)$. Bergemann et al. (2015) demonstrate that any such welfare outcome can be achieved by some signal.\n\nClaim 1. (Bergemann, Brooks, Morris (2015)) If the seller type is commonly known to be $\\theta_{0}$, then outcome ( $\\mathrm{U}, \\Pi$ ) is implementable if and only if $\\mathrm{U} \\geq 0, \\Pi \\geq \\underline{\\Pi}\\left(\\theta_{0}\\right)$, and $\\mathrm{U}+\\Pi \\leq \\overline{\\mathrm{W}}\\left(\\theta_{0}\\right)$.\n\nThis result immediately generalizes to the case in which the seller's type is drawn according to distribution $F$ but the designer observes the realized type. The aggregate set of implementable outcomes is a Minkowski average of the surplus sets for each realized type. For each type $\\theta$, by Claim 1, the implementable outcomes satisfy three linear constraints, two of which feature type-dependent terms $\\underline{\\Pi}(\\theta)$ and $\\overline{\\mathrm{W}}(\\theta)$. Denote their aggregate values averaged across types by\n\n$$\n\\begin{aligned}\n& \\underline{\\Pi} \\triangleq \\int_{0}^{1} \\underline{\\Pi}(\\theta) \\mathrm{d} F(\\theta) \\\\\n& \\overline{\\mathrm{W}} \\triangleq \\int_{0}^{1} \\overline{\\mathrm{~W}}(\\theta) \\mathrm{d} F(\\theta)\n\\end{aligned}\n$$\n\nFrom the ex ante perspective, $\\underline{\\Pi}$ is the profit the seller can guarantee, $\\overline{\\mathrm{W}}$ is the maximum feasible total surplus, and 0 is the welfare level the buyer can guarantee. Hence, the aggregate welfare outcome must belong to a triangle outlined by these constraints. At the same time, as in the case of a known type, the converse is also true (the omitted proofs are in Appendix).\n\nClaim 2. (Observable type) If the seller type is commonly known and distributed according to $F$, then outcome ( $\\mathrm{U}, \\Pi$ ) is implementable if and only if $\\mathrm{U} \\geq 0, \\Pi \\geq \\underline{\\Pi}$, and $\\mathrm{U}+\\Pi \\leq \\overline{\\mathrm{W}}$.\n\nThe result implies that any efficient outcome is implementable as long as the seller's profit exceeds the profit under no additional data. One notable implementable outcome is the buyer-optimal outcome $(\\overline{\\mathrm{W}}-\\underline{\\Pi}, \\underline{\\Pi})$, under which the allocation rule is efficient but the\nseller's profit stays at the level of no additional information. At this outcome the buyer obtains all surplus created by the data. As a result, if the seller's type is observable, there is no inherent trade-off between consumer protection and efficiency.\n\nHowever, the buyer-optimal outcome generally requires that different seller types obtain different signals. When the seller type is her private information this outcome may not be implementable, and the trade-off between consumer surplus and efficiency reappears. To see this, suppose that the seller's type is either $\\theta_{1} \\in(0, L / H)$ or $\\theta_{2} \\in(L / H, 1)$-i.e., the optimal uniform price is $L$ for type $\\theta_{1}$ and $H$ for type $\\theta_{2}$. At the buyer-optimal outcome, type $\\theta_{1}$ must not benefit from the data, because it is already willing to set price $L$. In contrast, for the outcome to be efficient, type $\\theta_{2}$ must be provided with an informative signal that reveals that the buyer's value is $H$ with a positive probability. ${ }^{4}$ However, in that case type $\\theta_{1}$ could strictly benefit from the signal of $\\theta_{2}$, which leads to a contradiction.\n\nThis example shows that the seller's private information restricts the set of implementable outcomes. In the next section, we analyze the structure of the seller's incentive compatibility constraints and characterize the implementable allocation rules and welfare outcomes.","text_sha256":"b62474d257c1c5b351c34767c4f3c1b31a306bf6986962538df67a3cec4669c3"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0009","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.2 Unobservable Seller Type","text":"### 3.2 Unobservable Seller Type\n\nFrom now on we focus on the case in which the seller's type is unobservable and characterize the set of implementable outcomes. We show that the seller's private information leads to an extreme adverse selection: Any outcome with heterogeneous data allocation, in which different types self-select into different signals, can be implemented by providing all types with the same public signal.\n\nAn important class of menus is a class of direct mechanisms. A direct mechanism is a menu of direct signals indexed by $\\theta$ so that $\\mathcal{I}(\\theta)$ sends two signal realizations, $S=\\left\\{s_{L}, s_{H}\\right\\}$, and the likelihood functions $\\pi(\\theta)$ are such that each type $\\theta$ is willing to choose signal $\\mathcal{I}(\\theta)$ and to set prices $p=L$ and $p=H$ after observing signal realizations $s_{L}$ and $s_{H}$ of $\\mathcal{I}(\\theta)$, respectively. Interpreting signal realization $s_{v}$ as a recommendation to set price $v \\in\\{H, L\\}$, we can view a direct mechanism as sending a value-dependent price recommendation based\n\n[^3]on the seller's reported type. The following result shows that the designer can without loss of generality focus on direct mechanisms (the proof is standard and omitted; it follows the revelation principle argument of Myerson (1982) and Bergemann et al. (2018)).\n\nClaim 3. (Direct Mechanisms) An allocation rule is implementable if and only if it is implementable by a direct mechanism.\n\nGiven a direct mechanism, we can parameterize the direct signal $\\mathcal{I}(\\theta)$ for each type $\\theta$ by probabilities $\\alpha(\\theta)$ and $\\beta(\\theta)$ with which the signal sends realization $s_{H}$ conditional on values $H$ and $L$, respectively. Without loss of generality, we assume $\\beta(\\theta) \\geq \\alpha(\\theta)$. We can then express signal $\\mathcal{I}(\\theta)$ in matrix form as\n\n| $\\mathcal{I}(\\theta)$ | $s_{L}$ | $s_{H}$ |\n| :--- | :--- | :--- |\n| $v=L$ | $1-\\alpha(\\theta)$ | $\\alpha(\\theta)$ |\n| $v=H$ | $1-\\beta(\\theta)$ | $\\beta(\\theta)$ |\n\nFor truth-telling to be optimal, each type should prefer her own signal $\\mathcal{I}(\\theta)$ to all other alternatives. The value of a signal depends on the type's response to recommendations, which in turn depends on the posterior beliefs the recommendations induce. Because $\\beta(\\theta) \\geq \\alpha(\\theta)$, the posterior belief rank is the same for all types: The posterior probability of $v=H$ is higher after observing $s_{H}$ than after observing $s_{L}$. As such, no type would be willing to swap the pricing decisions-i.e., set $p=H$ after $s_{L}$ and set $p=L$ after $s_{H}$. Hence, the relevant incentive constraints are those under which the seller misreports the type and follows the recommendation. Type $\\theta$ 's profit after such a deviation to type $\\theta^{\\prime}$ is\n\n$$\n\\begin{aligned}\n\\Pi\\left(\\theta, \\theta^{\\prime}\\right) & \\triangleq(1-\\theta)\\left(1-\\alpha\\left(\\theta^{\\prime}\\right)\\right) L+\\theta\\left(\\left(1-\\beta\\left(\\theta^{\\prime}\\right)\\right) L+\\beta\\left(\\theta^{\\prime}\\right) H\\right) \\\\\n& =\\left(1-\\alpha\\left(\\theta^{\\prime}\\right)\\right) L+\\theta\\left(\\alpha\\left(\\theta^{\\prime}\\right) L+\\beta\\left(\\theta^{\\prime}\\right)(H-L)\\right)\n\\end{aligned}\n$$\n\nIncentive compatibility requires that $\\Pi(\\theta, \\theta) \\geq \\Pi\\left(\\theta, \\theta^{\\prime}\\right)$ for all $\\theta, \\theta^{\\prime} \\in \\Theta$. This property must hold in any direct mechanism and allows us to pin down the structural properties of any implementable allocation rule.\n\nProposition 1. (Allocation Properties) In any direct mechanism, for any $\\theta_{1}, \\theta_{2}, \\theta_{3} \\in \\Theta$ such that $\\theta_{1}<\\theta_{2}<\\theta_{3}$, the following hold:\n\n1. (Monotonicity) $\\alpha\\left(\\theta_{1}\\right) \\leq \\alpha\\left(\\theta_{2}\\right)$ and $\\beta\\left(\\theta_{1}\\right) \\leq \\beta\\left(\\theta_{2}\\right)$; and\n2. (Relative Impact) $\\left(\\beta\\left(\\theta_{3}\\right)-\\beta\\left(\\theta_{2}\\right)\\right)\\left(\\alpha\\left(\\theta_{2}\\right)-\\alpha\\left(\\theta_{1}\\right)\\right) \\leq\\left(\\beta\\left(\\theta_{2}\\right)-\\beta\\left(\\theta_{1}\\right)\\right)\\left(\\alpha\\left(\\theta_{3}\\right)-\\alpha\\left(\\theta_{2}\\right)\\right)$.\n\nThe equilibrium properties in Proposition 1 are direct consequences of the seller's local incentive constraints and do not necessarily hold if the seller's type is observable. Both properties can be interpreted in terms of comparative statics of the seller's behavior with respect to her type. The first property means that the higher the type-i.e., the more likely high-value buyers are in the seller's market-the greater the probability of a higher price for all buyers, irrespective of the seller's additional information. In other words, regardless of the data-provision mechanism, high-value buyers impose negative externalities on low-value buyers because the presence of high-value buyers increases the likelihood that low-value buyers will face higher prices.\n\nThe second property evaluates the relative price impact of an increase in the seller's type on low-value and high-value buyers, and is more easily interpreted when viewed as ratio monotonicity-i.e., $\\frac{\\alpha\\left(\\theta_{3}\\right)-\\alpha\\left(\\theta_{2}\\right)}{\\beta\\left(\\theta_{3}\\right)-\\beta\\left(\\theta_{2}\\right)} \\geq \\frac{\\alpha\\left(\\theta_{2}\\right)-\\alpha\\left(\\theta_{1}\\right)}{\\beta\\left(\\theta_{2}\\right)-\\beta\\left(\\theta_{1}\\right)}$. Essentially, it states that the seller type's increasing disproportionately affects low-value buyers at higher types compared with lower types. This property follows from the optimal change in the seller's data strategy. Higher seller types value and primarily choose signals in the menu that help identify low-value buyers, because such signals are more likely to change the types' prior behavior. These signals pool low-value buyers with high-value buyers at high-price recommendations, thus exacerbating the price impact of the type's increase on those buyers. In contrast, lower seller types value and choose signals that help identify high-value buyers. These signals pool low-value buyers with high-value buyers at low-price recommendations, thus mitigating the price impact of the type's increase on those buyers.","text_sha256":"ef2f7f665a56dcfc1af5463111415384929b54fc2444b5fc9b20d9f69ac15534"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0010","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.2 Unobservable Seller Type","text":"The allocation properties of Proposition 1 outline the constraints imposed by the seller's private information in direct mechanisms. By Claim 3, this result applies to any mechanism, with $\\alpha(\\theta)$ and $\\beta(\\theta)$ being interpreted as the equilibrium pricing probabilities of different types. In particular, it implies that if the designer provides a single public signal to all\ntypes, they optimally respond in such a way that the induced pricing behavior conforms with Proposition 1. It turns out that the opposite is also true: The allocation rule of any mechanism can be replicated by providing a single public signal. ${ }^{5}$\n\nProposition 2. (Public Signals) Any implementable allocation rule can be implemented by a menu with a single signal in it.\n\nProof. The proof is constructive. Consider any direct mechanism $(\\alpha(\\theta), \\beta(\\theta))_{\\theta \\in \\Theta}$. We construct a public signal $\\hat{\\mathcal{I}}$ that implements the same outcome. The signal realization space of $\\hat{\\mathcal{I}}$ is $S=[0,1]$. The likelihood function $\\pi$ is such that for all $x \\in \\Theta, \\operatorname{Pr}(s \\leq x \\mid v=L)=\\alpha(x)$ and $\\operatorname{Pr}(s \\leq x \\mid v=H)=\\beta(x)$. Doing so is possible, because Proposition 1 ensures that functions $\\alpha$ and $\\beta$ are increasing and take values in [0,1]. ${ }^{6}$\n\nThe defining feature of signal $\\hat{\\mathcal{I}}$ is that pooling signal realizations $s$ below and above $\\theta$ results in signal $\\mathcal{I}(\\theta)=(\\alpha(\\theta), \\beta(\\theta))$. That is, $\\hat{\\mathcal{I}}$ is Blackwell more informative than either of the signals in the original mechanism. However, no type can benefit from the extra informativeness of $\\hat{\\mathcal{I}}$. Indeed, the second property of Proposition 1 implies that signal $\\hat{\\mathcal{I}}$ satisfies a monotone likelihood ratio property. ${ }^{7}$ As such, lower signal realizations induce higher posterior beliefs across all types, and for each type, a best response to $\\hat{\\mathcal{I}}$ is characterized by a threshold $\\tilde{s}$ such that the type sets price $p=H$ for all $s<\\tilde{s}$ and price $p=L$ for all $s>\\tilde{s}$. By construction, the choice between different thresholds is equivalent to the choice between different signals in the original mechanism. Hence, by the incentive compatibility of the original mechanism, type $\\theta$ optimally chooses the threshold $\\tilde{s}=\\theta$. The resulting allocation rule mimics the allocation rule in the original direct mechanism type by type. $\\square$\n\n[^4]","text_sha256":"baf601d35cc8c201e1431fdd1480ccfa3c7cdae3f4de5ffe088e1e8aa98d770d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0011","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.3 Implementable Outcome Characterization","text":"### 3.3 Implementable Outcome Characterization\n\nProposition 2 implies that the set of all implementable outcomes coincides with the set of all outcomes that are implementable by public signals. We derive the implementable outcomes under public signals in two steps: First, for each signal, we derive the payoffs of the buyer and the seller conditional on each signal realization but unconditional on the seller's type. Second, we then aggregate these payoffs across all signal realizations.\n\nSpecifically, for any given signal $\\mathcal{I}$ and realization $s$, define by $\\mu \\in[0,1]$ the posterior belief that $s$ induces in a hypothetical uninformed seller with a prior belief $\\mu_{0} \\triangleq \\mathbb{E}[\\theta]$, so that $\\mu \\triangleq \\operatorname{Pr}\\left(v=H \\mid s, \\mu_{0}\\right)$. We call this belief $\\mu$ a basic posterior belief. By Bayes' rule, the same signal realization observed by the seller of type $\\theta$ results in the posterior belief $t \\triangleq \\operatorname{Pr}(v=H \\mid s, \\theta)$ equal to ${ }^{8}$\n\n$$\nt(\\mu, \\theta)=\\frac{\\theta \\mu\\left(1-\\mu_{0}\\right)}{\\theta \\mu\\left(1-\\mu_{0}\\right)+(1-\\theta)(1-\\mu) \\mu_{0}} .\n$$\n\nThe posterior belief increases both in $\\mu$ and $\\theta$, equals zero if either of the arguments equals zero, and equals one if either of the arguments equals one. Formula (5) applies to all types. Therefore, the basic posterior belief $\\mu$ fully determines the distribution of the seller's posterior beliefs. The higher the $\\mu$, the higher the corresponding distribution of the seller's posterior beliefs in the sense of first-order stochastic dominance-i.e., the beliefs of different seller types move in concordance upon observing the same signal realization.\n\nIn turn, the seller's posterior belief determines her pricing decision: The seller sets price $p=H$ if $t>L / H$ and price $p=L$ if $t<L / H$. Equivalently, when observing a signal that induces a basic posterior belief $\\mu$, the informed seller with type $\\theta$ sets a high price if $\\theta>\\tilde{\\theta}(\\mu)$ and a low price if $\\theta<\\tilde{\\theta}(\\mu)$, where the threshold type $\\tilde{\\theta}$ is uniquely defined by the condition $t(\\mu, \\tilde{\\theta}(\\mu)))=L / H$.\n\nBy Bayes' consistency, if the seller's posterior belief is $t$, then the probability that the consumer's value is $H$ is indeed $t$. Consequently, the players' expected payoffs after signal\n\n[^5]realization $s$ that induces a basic posterior belief $\\mu$ can be written as\n$$\n\\begin{aligned}\n& \\mathrm{U}(\\mu)=\\int_{0}^{\\tilde{\\theta}(\\mu)} t(\\mu, \\theta)(H-L) \\mathrm{d} F(\\theta), \\\\\n& \\Pi(\\mu)=\\int_{0}^{\\tilde{\\theta}(\\mu)} L \\mathrm{~d} F(\\theta)+\\int_{\\tilde{\\theta}(\\mu)}^{1} t(\\mu, \\theta) H \\mathrm{~d} F(\\theta) .\n\\end{aligned}\n$$\nThe welfare functions $\\mathrm{U}(\\mu)$ and $\\Pi(\\mu)$ aggregate consumer surplus and seller profit respectively across seller types for any basic posterior belief. We can use these functions to characterize the set of implementable outcomes. Define by $\\operatorname{graph}(U, \\Pi)$ a graph of the vector function that keeps track of players' payoffs $(\\mathrm{U}(\\mu), \\Pi(\\mu))$ at different basic beliefs $\\mu$. Any public signal is characterized by a distribution of the basic beliefs, which by Bayes' rule must average to the prior belief $\\mu_{0}$. Therefore, any public signal implements an outcome in a convex hull of graph $(\\mathrm{U}, \\Pi)$ with the first component equal to $\\mu_{0}$. Vice versa, Aumann and Maschler (1995) and Kamenica and Gentzkow (2011) show that any belief distribution that averages to the prior can be induced by some signal. Therefore, any point in the convex hull of $\\operatorname{graph}(\\mathrm{U}, \\Pi)$ such that the first component is $\\mu_{0}$ can be implemented by some signal (cf. Doval and Smolin (2023)).\n\nProposition 3. (Implementable Outcomes) The set of all implementable outcomes is\n\n$$\n\\mathcal{E}=\\left\\{(x, y):\\left(\\mu_{0}, x, y\\right) \\in \\operatorname{co}(\\operatorname{graph}(\\mathrm{U}, \\Pi))\\right\\},\n$$\n\nwhere functions U and $\\Pi$ are given by (6) and (7).\nFor any given type distribution, Proposition 3 enables a geometric characterization of the set of equilibrium outcomes. The result highlights the fact that the type distribution $F$ affects the set of implementable outcomes via its impact on the indirect payoffs (6) and (7).\n\nTo simplify exposition, for the rest of this section we assume that $\\theta$ is continuously distributed over [0, 1]. The indirect payoffs are then continuous functions of basic belief $\\mu$, because $\\tilde{\\theta}(\\mu)$ is continuous and the types are continuously distributed over [0, 1]. Hence, by Proposition 3, the set $\\mathcal{E}$ is compact and convex, and we can characterize its extreme points via supporting Bayesian persuasion problems. Indeed, any extreme point of $\\mathcal{E}$ maximizes\nsome linear combination of buyer surplus and seller profit. This maximization problem corresponds to a Bayesian persuasion problem over a public split of basic posterior beliefs to maximize\n\n$$\nW^{\\lambda}(\\mu) \\triangleq \\lambda_{U} \\mathrm{U}(\\mu)+\\lambda_{\\Pi} \\Pi(\\mu)\n$$\n\nfor some $\\left(\\lambda_{U}, \\lambda_{\\Pi}\\right) \\in \\mathbb{R}^{2}$. We then obtain the following result:\nProposition 4. (Extreme Outcomes) If the seller type is continuously distributed over $[0,1]$, then the set of extreme implementable outcomes is spanned by solutions to Bayesian persuasion problems parameterized by $\\lambda=\\left(\\lambda_{U}, \\lambda_{\\Pi}\\right) \\in \\mathbb{R}^{2}$ :\n\n$$\n\\max _{\\tau \\in \\Delta(\\Delta(V))} \\mathbb{E}_{\\mu \\sim \\tau}\\left[W^{\\lambda}(\\mu)\\right] \\quad \\text { subject to } \\int_{\\Delta(V)} \\mu \\tau(d \\mu)=\\mu_{0}\n$$\n\nPropositions 3 and 4 imply bounds on the necessary complexity of the signals used. When the type is continuously distributed, the indirect payoffs are continuous functions of basic belief $\\mu$ and graph( $\\mathrm{U}, \\Pi$ ) features a single connected component. Thus, by Fenchel-Bunt's theorem, every point in the convex hull of graph(U, П) can be generated by a randomization over at most $|V|-1+2=3$ of its points. Such randomization corresponds to a signal with at most 3 signal realizations. Moreover, any extreme point of $\\mathcal{E}$ is implemented by a solution of a Bayesian persuasion problem that can be set to contain at most $|V|=2$ signal realizations.","text_sha256":"7fea33deb65d9fef8f5652a2b04bb94a721fafc518c6f1cbbbf8ece2cee743d7"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0012","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.3 Implementable Outcome Characterization","text":"Corollary 1. (Signal Complexity) If the seller type is continuously distributed over [0, 1], then any implementable outcome is implementable by a public signal with at most 3 signal realizations. Any extreme point of the set of implementable outcomes is implementable by a public signal with at most 2 signal realizations.\n\nCorollary 1 shows that even when there are many seller types, the ex ante consumer and seller payoffs can be obtained if the seller obtains coarse data that features at most 3 labels. Furthermore, extreme outcomes, such as a buyer-optimal outcome, can be obtained with even coarser data that features at most 2 labels.","text_sha256":"294e92a6b72021aea0ba7800da4384365b54582c5168c83a996dcae3eb148d3a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0013","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.4 Uniform Type Distribution","text":"### 3.4 Uniform Type Distribution\n\nSo far we have established the general properties of implementable allocations, characterized the set of implementable outcomes, and placed upper bounds on the complexity of necessary signals. However, we have been silent with respect to a few important questions, such as when providing some data can benefit the buyer or how to achieve a buyer-optimal outcome. To tackle these questions and to further illustrate our results, we now consider the special case of the seller's type being uniformly distributed over the unit interval. This case captures a natural benchmark in which all seller types are possible and equally likely.\n\nTo state the next result, we introduce two classes of signals. Consider a signal whose likelihood function in tabular form is\n\n| $\\mathcal{I}$ | $s_{L}$ | $s_{H}$ |\n| :--- | :--- | :--- |\n| $v=L$ | $1-\\alpha$ | $\\alpha$ |\n| $v=H$ | $1-\\beta$ | $\\beta$ |\n\nfor some $\\alpha, \\beta \\in[0,1]$. A signal is high-value flagging if $\\alpha=0$, so that realization $s_{H}$ can arise only if $v=H$. A signal is low-value flagging if $\\beta=1$, so that realization $s_{L}$ can arise only if $v=L$. Note that the fully informative signal ( $\\alpha=0$ and $\\beta=1$ ) and the uninformative signal $(\\alpha=\\beta=0)$ belong to these classes.\n\nProposition 5. (Uniform Types) Let the seller type be uniformly distributed on [0, 1]. The full-information outcome and the no-information outcome are extreme points of $\\mathcal{E}$. The left and right boundaries of $\\mathcal{E}$ connect these two outcomes and are spanned by low-value and high-value flagging signals, respectively. Moreover, the following hold:\n\n1. If $\\frac{L}{H} \\geq \\frac{1}{2}$, then the buyer-optimal outcome in $\\mathcal{E}$ is generated by the uninformative signal.\n2. If $\\frac{L}{H}<\\frac{1}{2}$, then the buyer-optimal outcome in $\\mathcal{E}$ is generated by the high-value flagging signal with flagging rate $\\beta=\\frac{H-2 L}{H-L}$.\n\nThe proof of Proposition 5 builds on Proposition 4. Any extreme point of $\\mathcal{E}$ solves a Bayesian persuasion problem (10) and is generated by a public signal with two signal\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 2: Implementable outcomes when seller types are uniformly distributed on [0, 1]. Light blue denotes the case of observable type. Dark blue denotes the case of unobservable type. Black boundaries are spanned by flagging signals. Points $C$ indicate buyer-optimal outcomes in $\\mathcal{E}$.\n\nrealizations. We can thus present the persuasion problem as a maximization problem with respect to $\\alpha$ and $\\beta$ and solve it in closed form.\n\nIntuitively, data provision affects the buyer surplus in two ways. On the one hand, it can benefit the buyer by persuading some seller types $\\theta>L / H$ to set price $p=L$ even when the buyer has value $H$. On the other hand, data provision may harm the buyer if $\\theta<L / H$, because such seller types would set price $p=L$ in the absence of data. Because all implementable outcomes are spanned by public signals, any data provision increases the buyer's surplus for some seller types and decreases it for other types. If $L / H$ is high-i.e., a large fraction of seller types set price $L$ in the absence of data-then the negative effect dominates and any data provision is detrimental for the buyer (Part 1). If $L / H$ is low, then some data provision is Pareto improving (Part 2).\n\nFigure 2 depicts the sets of implementable outcomes for two concrete cases of value distribution. When the seller type is observable, any individually rational outcome-i.e., the whole surplus triangle-can be implemented by data provision. In contrast, when the seller's type is unobserved, the sets of implementable outcomes shrink. The uninformative signal implements the \"south\" end of $\\mathcal{E}$ as an extreme point, which is also the seller-worst\noutcome. The right boundary is spanned by high-value flagging and the left boundary by low-value flagging. As we move along each boundary from south to north, the corresponding signals have higher flagging rates and become more informative. ${ }^{9}$ The two boundaries meet at the \"north\" end of $\\mathcal{E}$, which is the seller-optimal outcome and implemented by providing full information. In fact, this is the only efficient outcome and gives zero surplus to the buyer. The buyer-optimal outcome differs in the two cases. Figure 2(a) depicts part 2 of the proposition: As $L / H=1 / 3<1 / 2$, we have $\\beta=1 / 2$, so the buyer surplus is maximized by flagging one-half of high-value buyers. Figure 2(b) depicts part 1: As $L / H=2 / 3>1 / 2$, the buyer surplus is maximized by providing no data. In this case, adverse selection is so severe that any additional information would on average hurt the buyer.","text_sha256":"fb8b29828e16fec185074a6d50d7e9950123ac3dd5108a8555adee3662572ff0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0014","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Implications","text":"## 4 Implications\n\nWe derive three implications of the above results. First, we show that the seller's private information creates a trade-off between buyer protection and efficiency. Second, we study the impact of the seller's use of third-party data, which we capture as the seller's choice to acquire private information. Finally, we demonstrate that the implementable set subsumes the equilibrium outcomes of various games in which the buyer and the seller exchange information with each other.","text_sha256":"a244d8c5e486c6ea3abcdef033a526019360ffc8c1b854c5e9ef93b704ef3137"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0015","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.1 The Trade-off Between Buyer Protection and Efficiency","text":"### 4.1 The Trade-off Between Buyer Protection and Efficiency\n\nIf the seller's type is observable, the designer can distribute the efficient total surplus flexibly between the buyer and the seller (Claim 2). However, implementing an efficient outcome typically requires providing different signals to different types, which is infeasible when the seller's type is her private information (Proposition 2) . The impossibility of heterogeneous data provision introduces a tension between enhancing consumer surplus and total surplus-and in many cases, achieving efficiency implies that the buyer receives no surplus at all. Below, the seller's \"rent\" means the seller's expected profit under a given outcome minus\n\n[^6]her profit in the absence of additional data.\nProposition 6. (Efficiency and Buyer Surplus) Assume that the type distribution is nondegenerate and places probability 1 on (0, 1). Then the following hold:\n\n1. If there is a positive measure of types strictly above $L / H$, then any efficient outcome gives a strictly positive rent to the seller.\n2. If there is a strictly positive measure of types in any neighborhood of 1, then the only efficient outcome is the outcome in which all seller types obtain full information and perfectly price discriminate the buyer.\n\nThe intuition is as follows. To incentivize type $\\theta>L / H$ to price efficiently, a signal has to reveal the buyer's value with some probability $\\beta>0$ when $v=H$, so that in the remaining event, the seller sets price $L$, believing that the value is likely to be $L$. The higher the $\\theta$, the higher the probability $\\beta$ must be and the more informative the signal becomes. While it is possible for the designer to choose $\\beta$ so that the seller with a known type achieves the same profit as with no additional information, this is not the case for a privately informed seller. With the privately informed seller, a signal provided to one type for efficient pricing will also be used by lower types to earn positive rents. Moreover, a type that is arbitrarily close to 1 must receive the (almost) fully informative signal to price efficiently. The impossibility of screening implies that all seller types receive full information and the buyer receives no surplus.\n\nThe result has an implication for policies regarding the use of consumer data by firms. Consumer data may enable sellers to tailor pricing, which can benefit consumers and enhance efficiency. At the same time, how much and what kind of data sellers should be allowed to use for such beneficial pricing will depend on the specific market conditions and the initial endowment of data and technology for each seller. However, in practice, a regulator would not be able to tailor a regulation to each firm, partly because of the informational friction highlighted in this paper. In such a case, the same regulation is applied to heterogeneous sellers. Our result yields two relevant distortions in such a situation: First, some sellers obtain too little information, which leads to inefficient pricing. Second, some sellers obtain too much information, which not only allows them to create surplus but also to extract\nsurplus from consumers. Our result clarifies this economic force and further shows that the distortion can lead to an extreme outcome in which ensuring efficient pricing erodes the entire buyer surplus.","text_sha256":"b87d39187d71c348a64c2b12c94195b65c4faa387d144df32c9f7a9d831abbf1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0016","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.2 Third-party Data","text":"### 4.2 Third-party Data\n\nIn some applications, the seller can obtain third-party data at her will from outside sources. For example, a seller may learn about buyers from tracking tools or data brokers. We can analyze the acquisition of such data by interpreting it as the seller's choice to be privately informed.\n\nSpecifically, we can study the welfare impact of third-party data by comparing the informed seller-who has private type $\\theta \\sim F$ with a strictly positive probability of being below and being above $L / H$-with the uninformed seller, whose type is deterministic and known to be $\\theta_{0} \\triangleq \\mathbb{E}_{\\tilde{\\theta} \\sim F}[\\tilde{\\theta}]$. In general, the seller's private information shrinks the set of implementable outcomes (e.g., Figure 2). For example, if the type distribution has full support over [0, 1], then no efficient outcome, except for the outcome in which the seller extracts full surplus, is implementable when the seller has private information (Proposition 6).\n\nA natural question is how the seller's acquisition of third-party data affects the seller and the buyer. The answer depends on which implementable outcome is selected. As an illustration, suppose that the selected outcome is the constrained seller-optimal outcome- i.e., the implementable outcome that maximizes seller surplus subject to the constraint whereby the buyer must weakly benefit from the data provision in terms of his ex ante expected surplus. This outcome reflects a commonly observed feature of the digital economy, in which firms specify the terms of data collection and usage, but harmful data collection may be prevented by a user's opting out or by a regulator's intervention.\n\nProposition 7. (Third-party Data) Let $(\\mathrm{U}, \\Pi)$ and $\\left(\\mathrm{U}^{\\prime}, \\Pi^{\\prime}\\right)$ be the constrained seller-optimal outcomes given the uninformed seller and the informed seller. Then the following hold:\n\n1. If $\\theta_{0}>\\frac{L}{H}$, then $\\Pi>\\Pi^{\\prime}$ and $\\mathrm{U}<\\mathrm{U}^{\\prime}$-i.e., the seller is worse off and the buyer is better off when the seller has private information.\n2. If $\\theta_{0}<\\frac{L}{H}$, then $\\Pi<\\Pi^{\\prime}$ and $\\mathrm{U}>\\mathrm{U}^{\\prime}$-i.e., the seller is better off and the buyer is worse off when the seller has private information.\n\nThe intuition for Part 1 is as follows. The condition $\\theta_{0}>L / H$ means that the uninformed seller believes that the buyer's value is likely to be high, so she optimally sets price $H$, which leads to zero buyer surplus. In such a case, the constrained seller-optimal outcome is that the seller obtains full information and extracts the efficient total surplus. In contrast, the constrained seller-optimal outcome for the privately informed seller dictates that the seller never obtains full information, because the buyer surplus at no additional data provision is still positive. Consequently, the seller's private information decreases the seller's profit and increases buyer surplus. Part 1 of the result suggests that the seller may choose not to be privately informed about the buyer in order to eliminate adverse selection and obtain superior first-party data. This choice, while seemingly privacy friendly, could in fact harm the buyer. ${ }^{10}$\n\nThe argument for Part 2 is symmetric: Given that the uninformed seller sets price $L$, the constrained seller-optimal outcome is to provide no data. Compared with this outcome, the seller's private information increases her profit at the expense of buyer surplus and efficiency.","text_sha256":"3a40ff3f036aae6523453dc4dd70e6fd6a085a580b1d68d1b24ddcddda2759d1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0017","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.3 Communication Protocols","text":"### 4.3 Communication Protocols\n\nSo far, we have adopted a mechanism-design approach and studied the set of implementable outcomes. We now show that this set contains the equilibrium outcomes in a large class of communication protocols with an informed buyer and seller, such as cheap-talk communication, voluntary disclosure, and the collection of verifiable data.\n\n[^7]Formally, assume that the buyer and seller are privately informed about value $v$, having types $t$ and $\\theta$, respectively, which are independent conditional on $v$. Before trade, the players can communicate via a predefined two-stage protocol. ${ }^{11}$ A protocol $\\mathcal{P}$ is a triple $\\left(A_{S},\\left\\{A_{B}(t)\\right\\}_{t \\in T}, \\Psi\\right)$. The sets $A_{S}$ and $\\left\\{A_{B}(t)\\right\\}_{t \\in T}$ denote the seller's action space and the buyer's action space, respectively. The buyer's action space $A_{B}(t)$ can depend on his type $t$, which enables us to capture, e.g., voluntary disclosure settings. The signal scheme is denoted by $\\Psi=(S, \\pi)$. Here, $S$ is the set of signal realizations the seller can possibly observe in the protocol and $\\pi: V \\times A_{S} \\times A_{B} \\rightarrow \\Delta(S)$ is the likelihood function of different signal realizations, which can depend on the value and, importantly, on the players' actions. In the protocol, first the seller takes an action. Second, the buyer observes the seller's action and chooses his own action. A seller strategy is $\\sigma_{S}: \\Theta \\rightarrow \\Delta\\left(A_{S}\\right)$ and a buyer strategy is $\\sigma_{B}: T \\times A_{S} \\rightarrow \\Delta\\left(A_{B}(t)\\right)$. At the end of the communication, the seller observes signal realization $s$ according to $\\Psi$.\n\nAfter communicating within the protocol, the seller sets the price $p$. The buyer learns the value $v$, observes the price, and decides whether to purchase the good. If trade occurs, the buyer's ex post payoff is $v-p$ and the seller's ex post payoff is $p$. If trade does not occur, both players obtain a zero payoff. The solution concept is a perfect Bayesian equilibrium.\n\nAny equilibrium in a protocol results in an allocation rule $a: V \\rightarrow[0,1] \\times \\mathbb{R}$, which as before specifies the probability of a trade and the expected payment from the buyer to the seller for each value. We continue to use (welfare) outcome, buyer surplus, and seller profit to mean relevant ex ante expected payoffs.\n\nProposition 8. (Menu Mechanisms) An allocation rule can arise in an equilibrium with some communication protocol if and only if it can arise in an equilibrium of a menu mechanism.\n\nProof. The proof follows the revelation principle argument of Myerson (1982, 1983), but is adapted to fit the specific details of our environment. For the \"if\" direction, note that any menu mechanism is itself an example of a communication protocol with $A_{B}(t) \\equiv\\left\\{a_{0}\\right\\}$. For the \"only if\" direction, consider any protocol and an equilibrium in it. In this equilibrium,\n\n[^8]each action of the seller induces a signal $\\mathcal{I}(a)=(S, \\pi(a))$ with $\\pi(a): V \\rightarrow \\Delta(S)$, where the likelihood function averages over the buyer's equilibrium strategy and the signal scheme. We can replace this protocol with a menu mechanism $\\mathcal{M}=\\{\\mathcal{I}(a)\\}_{a \\in A_{S}}$. This change does not alter the seller's equilibrium strategy and results in the same allocation rule as the original protocol. Indeed, the only thing that matters at the trading stage is the seller's estimate of the buyer's value. It does not matter whether this estimate is obtained through direct data provision or equilibrium inference. Moreover, the buyer type is not useful for screening the seller type because they are conditionally independent. As a result, menu mechanisms with direct data provision implement all equilibrium allocation rules. $\\square$\n\nBelow, we describe protocols that select different subsets of the implementable outcomes, as depicted in Figure 3. For simplicity, assume that the seller's type distribution $F$ has a full support on [0, 1].\n\nCheap Talk. As a benchmark, suppose that the buyer knows the value and can inform the seller about it via an unverifiable, cheap-talk message-i.e., $t \\equiv v, A_{S}=\\left\\{a_{0}\\right\\}, A_{B}(H)=$ $A_{B}(L)=A_{B}$, and $s \\equiv a_{B}$. This protocol leads to a no-information outcome. Indeed, if an equilibrium entailed some nontrivial communication-i.e., if $v=L$ were strictly more likely after message $m \\in A_{B}$ than after message $m^{\\prime} \\in A_{B}$-then sending $m$ would lead to lower prices for all seller types and strictly so, given the full-support assumption. An $H$-buyer would then never send message $m^{\\prime}$, which would contradict the presumption that message $m$ indicates a higher likelihood of an $L$-buyer. The result implies that for informative communication to occur, information provision must be verifiable to some extent.\n\nVoluntary Disclosure. We turn to the buyer's voluntary disclosure of verifiable information. ${ }^{12}$ In this protocol, given the full-support assumption, the buyer is never willing to verify that he has value $H$, because the buyer would then face a high price and lose a positive surplus. As a result, any equilibrium strategy of the buyer generates a low-value flagging signal.\n\n[^9]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 3: Equilibrium outcomes selected by different communication protocols. Seller types are uniformly distributed on $[0,1]$, and $(L, H)=(1,3)$.","text_sha256":"44e5e5dbc6ab321140049daf68295e58338219783dad4504d45510e34004d744"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0018","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.3 Communication Protocols","text":"Formally, we consider the following protocol: $t \\equiv v, A_{S}=\\left\\{a_{0}\\right\\}, A_{B}(v)=\\{\\{v\\}, \\emptyset\\}$ for each $v \\in\\{H, L\\}$, and $s \\equiv a_{B}$, so that the buyer can choose whether to disclose his value. We argue that the set of equilibrium outcomes coincides with all implementable outcomes spanned by the low-value flagging signals. Indeed, if an $L$-buyer sends message $\\emptyset$ with a positive probability, then the seller with a sufficiently low type $\\theta$ sets price $L$ after observing message $\\emptyset$. But then, an $H$-buyer strictly prefers sending $\\emptyset$ to $\\{H\\}$. Thus, there could be only two kinds of equilibrium: (i) an $L$-buyer mixes between messages $\\{L\\}$ and $\\emptyset$ and an $H$-buyer sends $\\emptyset$ with probability 1, and (ii) an $L$-buyer sends message $\\{L\\}$ for sure, and an $H$-buyer sends $\\emptyset$ or $\\{H\\}$ revealing his value $H$. In fact, any such buyer strategy can arise in equilibrium, because an $L$-buyer never obtains a positive surplus and thus is indifferent between any messages, and any possible deviation by an $H$-buyer can be deterred by the seller's skeptical belief that places probability 1 on $H$.\n\nThe low-value flagging characterization, combined with Proposition 5, implies that if the seller's type is uniformly distributed, the equilibrium outcomes of the voluntary disclosure game span the left boundary of the surplus set (see Figure 2), which highlights the potential inefficiency of this protocol. This inefficiency remains even if the seller can communicate before the buyer's disclosure-i.e., if $A_{S}$ is general. In that case, in equilibrium, each message sent by the seller induces some low-value flagging signal, but any two low-value flagging\nsignals are ranked in Blackwell informativeness. Hence, all seller types send a message that induces the same low-value flagging signal.\n\nRequest-Consent Protocol. The seller's action space $A_{S}$ is given by the set of all signals with $S=[0,1]$, and the buyer's action space is given by $A_{B}(H)=A_{B}(L)=\\{$ accept , reject $\\}$. The seller first chooses a signal, and then the buyer observes the requested signal (but not its realization) and decides whether to accept it. The seller obtains the requested signal if the buyer chooses accept and does not observe any additional information otherwise.\n\nIn this protocol, if the buyer is perfectly informed (i.e., $t \\equiv v$ ), then any implementable outcome can arise in some equilibrium: Take any implementable outcome and the public signal $\\mathcal{I}$ that implements it. The following equilibrium has all seller types obtain signal $\\mathcal{I}$. On the equilibrium path, all seller types request signal $\\mathcal{I}$ and the buyer accepts it regardless of his value. The seller's deviation to an off-path signal can be deterred if the buyer, following the deviation, believes that the seller's type is $\\theta=0$. The buyer with this belief thinks that the seller will set price $L$, so the buyer never strictly benefits from providing additional information. In turn, if the seller believes that any buyer who deviates and rejects the signal request has value $H$, then the buyer finds it optimal to accept signal $\\mathcal{I}$.\n\nIn contrast, if the buyer is not informed when deciding on the request acceptance (i.e., $t \\equiv t_{0}$ ), then the set of equilibrium outcomes is smaller and coincides with the set of implementable outcomes such that the buyer is at least as well off as under no data provision. Indeed, if all seller types request the same signal $\\mathcal{I}$, the uninformed buyer will accept it if and only if signal $\\mathcal{I}$ weakly increases his ex ante payoff; the seller cannot use the off-path belief punishment because the buyer is uninformed. The result then follows from Proposition 2. Importantly, in some cases there may be no public signal that strictly increases buyer surplus (see Figure 2 (b)). In those cases, the unique equilibrium outcome of the request-consent protocol with an uninformed buyer would be no data collection.","text_sha256":"825d6d0c846fa3c99da2b5d52fa9876f6eabde7e9f124c23bf83214a5be292a3"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0019","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.4 Many Values","text":"### 4.4 Many Values\n\nThere are multiple ways in which our setting can be extended. One is to allow many possible buyer values, $V=\\left\\{v_{1}, \\ldots, v_{n}\\right\\}$ with $v_{1}<\\cdots<v_{n}, n>2$. The seller's type $\\theta \\in \\Delta(V)$ is now multidimensional, and the complete characterization of implementable outcomes is not attainable because it involves multidimensional screening. Nevertheless, we show that our main qualitative findings extend to this setting and provide necessary clarifications.\n\nFirst of all, private information held by the seller generally limits the set of implementable outcomes, as in the case of binary values. However, in the case of many values, the limits are less stark, since public signals do not necessarily span all implementable outcomes. Differentiated data provision can be welfare enhancing in this setting, and it can even attain the buyer's first-best outcome, which attains efficiency without increasing the seller's profits. The following example illustrates such a possibility.\n\nExample 1. (Benefits of Heterogeneous Data Provision) Let the possible values be $V=$ $\\{L, M, H\\}=\\{1,3,4\\}$. Let there be three seller types, $\\theta_{1}=(1 / 2,1 / 4,1 / 4), \\theta_{2}=(1 / 2,1 / 2,0)$, and $\\theta_{3}=(1 / 2,0,1 / 2)$, where the three types are equally likely. In the absence of additional data, the types set prices $p_{1}=M, p_{2}=M$, and $p_{3}=H$, and buyer surplus is $\\mathrm{U}_{0}=1 / 12$.\n\nWe derive the buyer-optimal outcome and show that it is efficient and cannot be attained by a public signal. Namely, consider a menu in which the types are provided with the following signals:\n\n| $\\mathcal{I}\\left(\\theta_{1}\\right)$ | $s_{1}$ | $s_{2}$ | $\\mathcal{I}\\left(\\theta_{2}\\right)$ | $s_{1}$ | $s_{2}$ | $\\mathcal{I}\\left(\\theta_{3}\\right)$ | $s_{1}$ | $s_{2}$ |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| $L$ | 1 | 0 | $L$ | 1 | 0 | $L$ | 1 | 0 |\n| $M$ | 1/2 | 1/2 | $M$ | 1/2 | 1/2 | $M$ | 2/3 | 1/3 |\n| $H$ | 1/2 | 1/2 | $H$ | 1/2 | 1/2 | $H$ | 1/3 | 2/3 |\n\nso that each signal has two possible realizations with the likelihood function $\\pi$ as presented in the tabular form. It is straightforward to verify that no type wants to deviate. Moreover, the outcome is efficient and each type earns the same profits as in the absence of additional information: Given these signals, types $\\theta_{1}$ and $\\theta_{2}$ set prices $p=L$ and $p=M$ after signal realizations $s_{1}$ and $s_{2}$, respectively, and type $\\theta_{3}$ sets prices $p=L$ and $p=H$, respectively. The\nbuyer surplus is then maximal among all feasible allocations, given the individual rationality of the seller.\n\nHowever, we cannot implement this outcome by providing the same signal to all seller types: Any efficient public signal must give strictly positive rents to type $\\theta_{1}$. Indeed, for signal $\\mathcal{I}=(S, \\pi)$ to attain efficiency, it must be that whenever $\\pi\\left(s_{0} \\mid L\\right)>0$ for some $s_{0} \\in S$ all types charge $p=L$ following that signal, which means that $\\pi\\left(s_{0} \\mid L\\right) \\geq 2 \\pi\\left(s_{0} \\mid M\\right)$ to persuade type $\\theta_{2}$ and $\\pi\\left(s_{0} \\mid L\\right) \\geq 3 \\pi\\left(s_{0} \\mid H\\right)$ to persuade type $\\theta_{3}$; in turn, these incentive constraints together guarantee that $\\theta_{1}$ also charges a low price. Since type $\\theta_{1}$ never charges a price below $p=L$, her profit decreases as we increase the frequency of signal $s_{0}$-i.e., as we increase $\\pi\\left(s_{0} \\mid L\\right), \\pi\\left(s_{0} \\mid M\\right)$, and $\\pi\\left(s_{0} \\mid H\\right)$ subject to the incentive constraints. Therefore, of all efficient public signals, the profit of type $\\theta_{1}$ is minimized at the signal\n\n| $\\hat{\\mathcal{I}}$ | $s_{1}$ | $s_{2}$ |\n| :--- | :--- | :--- |\n| $L$ | 1 | 0 |\n| $M$ | 1/2 | 1/2 |\n| $H$ | 1/3 | 2/3 |\n\nwhich is efficient and maximizes the probability of type $\\theta_{1}$ setting price $p=L$ while ensuring that type $\\theta_{1}$ offers the remaining buyers the second-lowest price $p=M$. The resulting $\\theta_{1}$ 's minimal profit equals 19/12, which is strictly greater than her outside option of 3/2 and, as such, necessarily results in lower buyer surplus than under the multi-item mechanism presented above. $\\square$\n\nIntuitively, as in most mechanism-design problems, the multidimensionality of the seller's type gives rise to a richer structure of incentive constraints and implementable outcomes. In particular, the buyer-optimal efficient signals are not necessarily Blackwell-comparable and can appeal to the \"right\" buyer type. As a result, complete characterization of the set of implementable outcomes becomes more complicated. Nevertheless, we show that the welfare implications of Section 4.1 generalize: Seller heterogeneity introduces a trade-off between efficiency and consumer welfare. ${ }^{13}$\n\n[^10]Proposition 9. (Efficiency and Buyer Surplus) Assume that the type distribution $F$ places probability 1 on the interior of $\\Delta(V)$. Then the following hold:\n\n1. If $F$ admits a positive density over an open set of types who, in the absence of data, charge prices $p>v_{1}$, then any efficient outcome gives a strictly positive rent to the seller.\n2. If there is a strictly positive measure of types in any neighborhood of the type that places probability 1 on $v_{n}$, then the only efficient outcome is that under full information.\n\nProposition 9 is a counterpart of Proposition 6 for the case of many values. The first part of Proposition 9 is based on the fact that signals that yield no rents to the seller must be carefully tailored to the seller's type. Specifically, consider the types over which $F$ has a density. Efficiency requires that these types occasionally set price $p=v_{1}$ to serve the lowestvalue buyer, and therefore must be given additional information. However, if some type weakly prefers to set the lowest price after receiving some signal realization, nearby types would strictly prefer to set the lowest price following that realization and would therefore strictly prefer to change their prior action. This means that they can earn strictly positive rents from that signal and must earn strictly positive rents when faced with any efficient menu.","text_sha256":"4efe5893359b831544108035ad7217b4835ee775ad71a02c637c5684b48f179d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0020","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.4 Many Values","text":"For the second part of Proposition 9, we can show that under the stated condition, there exists a sequence of types with the following properites. First, types converge to the extreme type that places probability 1 on the highest possible value. Second, along the sequence, the corresponding signals, which induce efficient outcomes, converge to the fully informative signal. In particular, we show that the types along this sequence need to be persuaded to charge all possible prices and thus must be provided with progressively more detailed information. The limit argument, coupled with incentive compatibility, then implies that all types must be offered a fully informative signal.","text_sha256":"e5de0ea85ee4698ea5ddd705472a33040a1f710f3967a49bcb62eb57aafb482f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0021","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Conclusion","text":"## 5 Conclusion\n\nWe studied the provision of consumer data to a monopoly seller who has imperfect private information about the value of its product. Without private information, the designer can\nflexibly provide the seller with data to distribute efficient total surplus between the buyer and the seller. We show that the seller's private information prevents the designer from providing different signals to different seller types, which leads to the impossibility of effective screening. Our results provide insights into data policies for policymakers and businesses. For policymakers or platforms that aim to control the follow of consumer data to sellers, the results highlight the trade-off between consumer protection and efficiency driven by adverse selection regarding data use. From the seller's perspective, our results clarify why the collection of third-party data without consumer consent might harm profits. To clarify the economic intuition, we have focused on a simple setting in which the buyer has binary values and the seller uses data for price discrimination. However, the issues that arise when a firm's private information hinders the effective allocation of consumer data-and the resulting challenges faced by regulators and firms with regard to data policies-would be relevant in broader contexts.","text_sha256":"1f2866dfeac64dd3bc78b14e577815bf5b3d0c2b291d81c6afae22d57f9f622c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0022","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAcquisti, A., C. Taylor, and L. Wagman (2016): \"The Economics of Privacy,\" Journal of Economic Literature, 54, 442-492.\n\nAli, S. N., G. Lewis, and S. Vasserman (forthcoming): \"Voluntary Disclosure and Personalized Pricing,\" Review of Economic Studies.\n\nAlonso, R. and O. Camara (2016): \"Bayesian Persuasion with Heterogeneous Priors,\" Journal of Economic Theory, 165, 672-706.\n\nAlonso, R. and K. Zachariadis (2021): \"Persuading Large Investors,\" Working paper.\n\nArgenziano, R. and A. Bonatti (2021): \"Information Revelation and Privacy Protection,\" Working paper.\n\nAumann, R. and M. Maschler (1995): Repeated Games with Incomplete Information, MIT Press.\n\nBaron, D. P. and R. B. Myerson (1982): \"Regulating a Monopolist with Unknown Costs,\" Econometrica, 911-930.\n\nBergemann, D., A. Bonatti, and T. Gan (2022): \"The Economics of Social Data,\" The RAND Journal of Economics, 53, 263-296.\n\nBergemann, D., A. Bonatti, and A. Smolin (2018): \"The Design and Price of Information,\" American Economic Review, 108, 1-48.\n\nBergemann, D., B. Brooks, and S. Morris (2015): \"The Limits of Price Discrimination,\" American Economic Review, 105, 921-957.\n\nChoi, J. P., D.-S. Jeon, and B.-C. Kim (2019): \"Privacy and Personal Data Collection with Information Externalities,\" Journal of Public Economics, 173, 113-124.\n\nCondorelli, D. and B. Szentes (2022): \"Buyer-Optimal Platform Design,\" Working paper.\n\nDeb, R. and A.-K. Roesler (2022): \"Multi-dimensional Screening: Buyer-Optimal Learning and Informational Robustness,\" Working paper.\n\nDoval, L. and A. Smolin (2023): \"Persuasion and Welfare,\" Working paper.\n\nElliott, M., A. Galeotti, A. Koh, and W. Li (2022): \"Market Segmentation through Information,\" Working paper.\n\nFainmesser, I. P., A. Galeotti, and R. Momot (2022): \"Digital privacy,\" Management Science.\n\nHaghpanah, N. and R. Siegel (2022): \"The Limits of Multiproduct Price Discrimination,\" American Economic Review: Insights, 4, 443-58.\n\n- (forthcoming): \"Pareto Improving Segmentation of Multi-product Markets,\" Journal of Political Economy.\n\nKamenica, E. and M. Gentzkow (2011): \"Bayesian Persuasion,\" American Economic Review, 101, 2590-2615.\n\nKolotilin, A., T. Mylovanov, A. Zapechelnyuk, and M. Li (2017): \"Persuasion of a Privately Informed Receiver,\" Econometrica, 85, 1949-1964.\n\nMyerson, R. B. (1982): \"Optimal Coordination Mechanisms in Generalized Principal-Agent Problems,\" Journal of Mathematical Economics, 10, 67-81.\n\n- (1983): \"Mechanism Design by an Informed Principal,\" Econometrica, 1767-1797.\n\nRayo, L. and I. Segal (2010): \"Optimal Information Disclosure,\" Journal of Political Economy, 118, 949-987.\n\nRhodes, A. and J. Zhou (2022): \"Personalized Pricing and Competition,\" Working paper.\n\nRoesler, A.-K. and B. Szentes (2017): \"Buyer-Optimal Learning and Monopoly Pricing,\" American Economic Review, 107, 2072-80.\n\nShaked, M. and J. G. Shanthikumar (2007): Stochastic Orders, Springer.\n\nShi, X. and J. Zhang (2020): \"Welfare of Price Discrimination and Market Segmentation in Duopoly,\" Working paper.\n\nSmolin, A. (forthcoming): \"Disclosure and Pricing of Attributes,\" The RAND Journal of Economics.\n\nYang, K. H. (2022): \"Selling Consumer Data for Profit: Optimal Market-Segmentation Design and Its Consequences,\" American Economic Review, 112, 1364-93.","text_sha256":"1324646d96eb5773e665d722b84cd4663d117d293f8c2524e31df93b0469ed63"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0023","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Appendix: Proofs Omitted from the Main Text","text":"## Appendix: Proofs Omitted from the Main Text\n\nProof of Claim 2. To show the \"only if\" direction, take any feasible welfare outcome\n\n$$\n(\\mathrm{U}, \\Pi)=\\left(\\int_{0}^{1} \\mathrm{U}(\\theta) \\mathrm{d} F(\\theta), \\int_{0}^{1} \\Pi(\\theta) \\mathrm{d} F(\\theta)\\right) .\n$$\n\nClaim 1 implies that for each $\\theta$, we have $\\mathrm{U}(\\theta) \\geq 0, \\Pi(\\theta) \\geq \\underline{\\Pi}(\\theta)$, and $\\mathrm{U}(\\theta)+\\Pi(\\theta) \\leq \\overline{\\mathrm{W}}(\\theta)$. Integrating both sides of each inequality with $F$, we obtain $\\mathrm{U} \\geq 0, \\Pi \\geq \\underline{\\Pi}$, and $\\mathrm{U}+\\Pi \\leq \\overline{\\mathrm{W}}$.\n\nTo show the \"if\" direction, take any $(\\mathrm{U}, \\Pi) \\in \\mathbb{R}^{2}$ such that $\\mathrm{U} \\geq 0, \\Pi \\geq \\underline{\\Pi}$, and $\\mathrm{U}+\\Pi \\leq \\overline{\\mathrm{W}}$. The point (U, П) belongs to the triangle whose vertices are $(0, \\overline{\\mathrm{~W}}),(0, \\underline{\\Pi})$, and $(\\overline{\\mathrm{W}}-\\underline{\\Pi}, \\underline{\\Pi})$. Let $\\alpha, \\beta \\in[0,1]$ with $\\alpha+\\beta \\leq 1$ satisfy\n\n$$\n\\begin{aligned}\n(\\mathrm{U}, \\Pi) & =\\alpha(0, \\overline{\\mathrm{~W}})+\\beta(0, \\underline{\\Pi})+(1-\\alpha-\\beta)(\\overline{\\mathrm{W}}-\\underline{\\Pi}, \\underline{\\Pi}) \\\\\n& =\\left(\\int_{0}^{1}(1-\\alpha-\\beta)[\\overline{\\mathrm{W}}(\\theta)-\\underline{\\Pi}(\\theta)] \\mathrm{d} F(\\theta), \\int_{0}^{1} \\alpha \\overline{\\mathrm{~W}}(\\theta)+\\beta \\underline{\\Pi}(\\theta)+(1-\\alpha-\\beta) \\underline{\\Pi}(\\theta) \\mathrm{d} F(\\theta)\\right) .\n\\end{aligned}\n$$\n\nFor each $\\theta$, the point\n\n$$\n(\\mathrm{U}(\\theta), \\Pi(\\theta)) \\triangleq((1-\\alpha-\\beta)[\\overline{\\mathrm{W}}(\\theta)-\\underline{\\Pi}(\\theta)], \\alpha \\overline{\\mathrm{W}}(\\theta)+\\beta \\underline{\\Pi}(\\theta)+(1-\\alpha-\\beta) \\underline{\\Pi}(\\theta))\n$$\n\nis in the surplus triangle and thus feasible. Aggregating the welfare outcome $(\\mathrm{U}(\\theta), \\Pi(\\theta))$ across possible $\\theta$, we conclude that $(\\mathrm{U}, \\Pi)=\\left(\\int_{0}^{1} \\mathrm{U}(\\theta) \\mathrm{d} F(\\theta), \\Pi(\\theta) \\mathrm{d} F(\\theta)\\right)$ is also feasible. □\n\nProof of Proposition 1. Part 1. Fix any $\\theta_{2}>\\theta_{1}$. The system of mutual incentive constraints is\n\n$$\n\\begin{aligned}\n& \\left(1-\\alpha\\left(\\theta_{1}\\right)\\right) L+\\theta_{1}\\left(\\alpha\\left(\\theta_{1}\\right) L+\\beta\\left(\\theta_{1}\\right)(H-L)\\right) \\geq\\left(1-\\alpha\\left(\\theta_{2}\\right)\\right) L+\\theta_{1}\\left(\\alpha\\left(\\theta_{2}\\right) L+\\beta\\left(\\theta_{2}\\right)(H-L)\\right), \\\\\n& \\left(1-\\alpha\\left(\\theta_{2}\\right)\\right) L+\\theta_{2}\\left(\\alpha\\left(\\theta_{2}\\right) L+\\beta\\left(\\theta_{2}\\right)(H-L)\\right) \\geq\\left(1-\\alpha\\left(\\theta_{1}\\right)\\right) L+\\theta_{2}\\left(\\alpha\\left(\\theta_{1}\\right) L+\\beta\\left(\\theta_{1}\\right)(H-L)\\right) .\n\\end{aligned}\n$$\n\nSumming over the inequalities (14) and (15) and using the fact that $\\theta_{2}>\\theta_{1}$ we obtain\n\n$$\n\\alpha\\left(\\theta_{2}\\right) L+\\beta\\left(\\theta_{2}\\right)(H-L) \\geq \\alpha\\left(\\theta_{1}\\right) L+\\beta\\left(\\theta_{1}\\right)(H-L) .\n$$\n\nIn turn, (14) and (16) together imply $\\alpha\\left(\\theta_{2}\\right) \\geq \\alpha\\left(\\theta_{1}\\right)$ because\n\n$$\n\\left(\\alpha\\left(\\theta_{2}\\right)-\\alpha\\left(\\theta_{1}\\right)\\right) L \\geq \\theta_{1}\\left(\\alpha\\left(\\theta_{2}\\right) L+\\beta\\left(\\theta_{2}\\right)(H-L)-\\alpha\\left(\\theta_{1}\\right) L-\\beta\\left(\\theta_{1}\\right)(H-L)\\right) \\geq 0 .\n$$\n\nFinally, (15) and (17) imply $\\beta\\left(\\theta_{2}\\right) \\geq \\beta\\left(\\theta_{1}\\right)$ because\n\n$$\n\\left(\\beta\\left(\\theta_{2}\\right)-\\beta\\left(\\theta_{1}\\right)\\right) \\theta_{2}(H-L) \\geq\\left(\\alpha\\left(\\theta_{2}\\right)-\\alpha\\left(\\theta_{1}\\right)\\right)\\left(1-\\theta_{2}\\right) L \\geq 0 .\n$$\n\nPart 2. Fix any $\\theta_{1}<\\theta_{2}<\\theta_{3}$. The system of incentive constraints of type $\\theta_{2}$ toward types $\\theta_{1}$ and $\\theta_{3}$ can be written as\n\n$$\n\\begin{aligned}\n& \\left(\\beta\\left(\\theta_{2}\\right)-\\beta\\left(\\theta_{1}\\right)\\right) \\theta_{2}(H-L) \\geq\\left(\\alpha\\left(\\theta_{2}\\right)-\\alpha\\left(\\theta_{1}\\right)\\right)\\left(1-\\theta_{2}\\right) L, \\\\\n& \\left(\\beta\\left(\\theta_{3}\\right)-\\beta\\left(\\theta_{2}\\right)\\right) \\theta_{2}(H-L) \\leq\\left(\\alpha\\left(\\theta_{3}\\right)-\\alpha\\left(\\theta_{2}\\right)\\right)\\left(1-\\theta_{2}\\right) L .\n\\end{aligned}\n$$\n\nBy property in Part 1, all sides of (19) and (20) are positive. Multiplying the respective smaller and larger parts and dividing the resulting inequality by $\\theta_{2}\\left(1-\\theta_{2}\\right)(H-L) L$, we obtain the desired inequality of Part 2. $\\square$\n\nProof of Proposition 5. Step 1. We use the \"boundary\" to mean the boundary of $\\mathcal{E}$. By Proposition 4 and Corollary 1, any extreme point on the boundary can arise with a signal that has two signal realizations. We parameterize such signals by $(\\alpha, \\beta), \\beta \\geq \\alpha$ as\n\n| $\\mathcal{I}(\\theta)$ | $s_{L}$ | $s_{H}$ |\n| :--- | :--- | :--- |\n| $v=L$ | $1-\\alpha$ | $\\alpha$. |\n| $v=H$ | $1-\\beta$ | $\\beta$ |\n\nGiven any such signal, we write the objective $\\lambda_{U} \\mathrm{U}+\\lambda_{\\Pi} \\Pi$ in terms of the highest and lowest types that respond to the signal realizations. In particular, for any given $(\\alpha, \\beta)$ with $\\beta \\geq \\alpha$, we can find two cutoffs $x, y$ with $x \\leq y$ such that types below $x$ set price $L$ after both realizations; types above $y$ set price $H$ after both realizations; and types between $x$ and $y$ will set prices $H$ and $L$ after $s_{H}$ and $s_{L}$, respectively. Cutoffs $x$ and $y$ solve\n\n$$\n\\begin{aligned}\nL & =(1-\\alpha) L+x(\\alpha L+\\beta(H-L)), \\\\\ny H & =(1-\\alpha) L+y(\\alpha L+\\beta(H-L)),\n\\end{aligned}\n$$\n\nwhich implies\n\n$$\nx=\\frac{\\alpha L}{\\alpha L+\\beta(H-L)}, \\quad y=\\frac{(1-\\alpha) L}{H-\\alpha L-\\beta(H-L)} .\n$$\n\nAlternatively, we can write $(\\alpha, \\beta)$ as functions of $(x, y)$ :\n\n$$\n\\alpha=\\frac{x(H y-L)}{L(y-x)}, \\quad \\beta=\\frac{(H y-L)(1-x)}{(y-x)(H-L)} .\n$$\n\nCutoffs $(x, y)$ can arise under some signal if and only if $0 \\leq x \\leq \\frac{L}{H} \\leq y \\leq 1$. The interim profit of type $\\theta \\in[x, y]$ is\n\n$$\n\\Pi(x, y \\mid \\theta)=(1-\\theta)(1-\\alpha) L+\\theta((1-\\beta) L+\\beta H))=L+\\frac{(H y-L)(\\theta-x)}{y-x} .\n$$\n\nThe interim profits of types $\\theta \\leq x$ and $\\theta \\geq y$ are $L$ and $\\theta H$, respectively. The ex ante seller profit is","text_sha256":"42b8273687ed4048b38ac504e9bc5eea1337ad06a75e932282374287c16ed987"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0024","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Appendix: Proofs Omitted from the Main Text","text":"$$\n\\begin{aligned}\n\\Pi(x, y) & =\\int_{0}^{x} L \\mathrm{~d} \\theta+\\int_{x}^{y} L+\\frac{H y-L}{y-x}(\\theta-x) \\mathrm{d} \\theta+\\int_{y}^{1} \\theta H \\mathrm{~d} \\theta \\\\\n& =L y+\\frac{1}{2}(H y-L)(y+x)-(H y-L) x+\\frac{H}{2}\\left(1-y^{2}\\right)\n\\end{aligned}\n$$\n\nThe buyer surplus is\n\n$$\n\\begin{aligned}\n\\mathrm{U}(x, y) & \\triangleq(H-L) \\int_{0}^{x} \\theta \\mathrm{~d} \\theta+(1-\\beta)(H-L) \\int_{x}^{y} \\theta \\mathrm{~d} \\theta \\\\\n& =(H-L) \\int_{0}^{x} \\theta \\mathrm{~d} \\theta+\\left(H-L-\\frac{(H y-L)(1-x)}{y-x}\\right) \\int_{x}^{y} \\theta \\mathrm{~d} \\theta \\\\\n& =(H-L) \\int_{0}^{y} \\theta \\mathrm{~d} \\theta-\\frac{(H y-L)(1-x)}{y-x} \\int_{x}^{y} \\theta \\mathrm{~d} \\theta \\\\\n& =\\frac{1}{2}(H-L) y^{2}-\\frac{1}{2}(H y-L)(1-x)(y+x)\n\\end{aligned}\n$$\n\nStep 2. We characterize signals that span the \"right\" boundary that corresponds to $\\lambda_{U} \\geq 0$. Take any $\\left(\\lambda_{U}, \\lambda_{\\Pi}\\right) \\in \\mathbb{R}^{2}$ with $\\lambda_{U} \\geq 0$. Because $\\Pi(x, y)$ is linear in $x$ and $\\mathrm{U}(x, y)$ is convex in $x, W(x, y)$ is convex in $x$ and maximized at $x=0$ or $x=L / H$. Note that $(x, y)=$ $(L / H, y)$ implies $(\\alpha, \\beta)=(1,1)$ and $(x, y)=(0, L / H)$ leads to $(0,0)$, but both are the same\nuninformative signal. Thus we can without loss of generality assume $x=0$ and focus on the problem $\\max _{\\frac{L}{H} \\leq y \\leq 1} W(0, y)$, where\n\n$$\n\\begin{aligned}\nW(0, y) & =\\lambda_{U}\\left[\\frac{1}{2}(H-L) y^{2}-\\frac{1}{2}(H y-L) y\\right]+\\lambda_{\\Pi}\\left[L y+\\frac{1}{2}(H y-L) y+\\frac{H}{2}\\left(1-y^{2}\\right)\\right] \\\\\n& =\\frac{\\lambda_{U}}{2} L y(1-y)+\\frac{\\lambda_{\\Pi}}{2}(L y+H) .\n\\end{aligned}\n$$\n\nIf $\\lambda_{U}=0$, then the function is maximized at $y=1$ for $\\lambda_{\\Pi}>0$ and at $y=0$ for $\\lambda_{\\Pi}<0$. Thus, the point that maximizes the seller profit is attained by the fully informative signal, and the point that minimizes the seller profit is attained by the uninformative signal.\n\nIf $\\lambda_{U}>0$, the function $W(0, y)$ is strictly concave in $y$ so we can use the first-order condition to determine an interior solution:\n\n$$\ny=\\frac{\\lambda_{U}+\\lambda_{\\Pi}}{2 \\lambda_{U}}=\\frac{1}{2}\\left(1+\\frac{\\lambda_{\\Pi}}{\\lambda_{U}}\\right) .\n$$\n\nIf $\\frac{L}{H} \\leq \\frac{1}{2}\\left(1+\\frac{\\lambda_{\\Pi}}{\\lambda_{U}}\\right) \\leq 1$, then the optimal $y$ is $\\frac{1}{2}\\left(1+\\frac{\\lambda_{\\Pi}}{\\lambda_{U}}\\right)$. Otherwise, there is a corner solution $y=\\frac{L}{H}$ or $y=1$ for low or large values of $\\lambda_{\\Pi}$, respectively.\n\nPlugging $x=0$ into $(\\alpha, \\beta)$ above, we obtain $\\alpha=0$-i.e., the signal sends realization $s_{H}$ only if the value is $H$, so it is a high-value flagging signal. Thus each point on the right boundary arises under some high-value flagging signal. As we move the right boundary from the seller-worst point to the seller-optimal point, cutoff $y$ increases (or equivalently, $\\beta$ increases) and the corresponding signal changes from the uninformative signal to the fully informative signal. If $\\frac{L}{H} \\geq \\frac{1}{2}$, the buyer optimal point is $\\lambda_{\\Pi}=0$, so we have $y^{*}=L / H$-i.e., the uninformative signal maximizes the buyer surplus. If $\\frac{L}{H}<\\frac{1}{2}$, the buyer optimal point is $y^{*}=1 / 2$-i.e., a partially informative high-value flagging signal maximizes buyer surplus. In this case, plugging the optimal $(x, y)$ into $(\\alpha, \\beta)$, we obtain $\\alpha=1$ and $\\beta=\\frac{H-2 L}{H-L}$.\n\nStep 3. We characterize the \"left\" boundary that corresponds to $\\lambda_{U}<0$. The seller profit $\\Pi(x, y)$ is linear in $y$ and the buyer surplus is strictly concave in $y$. Because $\\lambda_{U}<0$, the function $W(x, y)$ is strictly convex in $y$, which means that the optimal $y$ will be $\\frac{L}{H}$ or 1 . Because $y=\\frac{L}{H}$ is equivalent to $(x, y)=\\left(\\frac{L}{H}, 1\\right)$, we can without loss of generality assume\nthat $y=1$. We have\n\n$$\n\\begin{aligned}\nW(x, 1) & =\\lambda_{U}\\left[\\frac{1}{2}(H-L)-\\frac{1}{2}(H-L)(1-x)(1+x)\\right]+\\lambda_{\\Pi}\\left[L+\\frac{1}{2}(H-L)(1+x)-(H-L) x\\right] \\\\\n& =\\frac{\\lambda_{U}}{2}(H-L) x^{2}+\\frac{\\lambda_{\\Pi}}{2}[2 L+(H-L)(1-x)] .\n\\end{aligned}\n$$\n\nThe first-order condition with respect to $x$ yields\n\n$$\n\\lambda_{U} x-\\frac{\\lambda_{\\Pi}}{2}=0 \\Longleftrightarrow x=\\frac{\\lambda_{\\Pi}}{2 \\lambda_{U}} .\n$$\n\nThus any point on the left boundary can arise under a low-value flagging signal. As we move the left boundary from the seller-worst point to the seller-optimal point by raising $\\lambda_{\\Pi}$, the corresponding signal changes from no disclosure $(x, y)=(L / H, 1)$ to full disclosure $(x, y)=(0,1)$. $\\square$\n\nProof of Proposition 6. To prove part 1, take any efficient outcome. By Proposition 2, all types can be assumed to obtain the same signal. If the signal is fully revealing, then the seller obtains a strictly positive rent because almost all seller types belong to the interior $(0,1)$ of the type space. If the signal is not fully revealing, then there must exist signal realization $s$ such that the posterior is non-degenerate and induces all types to set price $L$. By the martingale property of belief, there must exist another signal realization $s^{\\prime}$ under which types in $(L / H, 1)$ optimally set price $H$. Because the outcome is efficient, $s^{\\prime}$ perfectly reveals that the value is $H$. Then, there are two possibilities. First, if all types in $(L / H, 1)$ are concentrated on a single point, then distribution $F$, which is non-degenerate, must place a positive probability on types in $[0, L / H]$. Those types would earn a strictly higher profit than without additional information, because they would strictly benefit from observing $s^{\\prime}$. Second, if types strictly above $L / H$ are not concentrated on a single point, then below the highest type in the support of $F$ there is a positive measure of types who strictly prefer to set price $L$ after observing signal realization $s$. These types earn a strictly higher profit than without additional information.","text_sha256":"f595aa5cbac8874bf79995b44e1cf5f92f459fae823273fee57d4fcdb910aabf"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0025","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Appendix: Proofs Omitted from the Main Text","text":"To prove part 2, consider an implementable outcome in which a positive mass of types do not obtain the fully informative signal. Profit $\\Pi(\\theta)$ of any such type satisfies $\\Pi(\\theta)<$\n$\\theta H+(1-\\theta) L$. Proposition 2 implies that there exists a public signal $\\mathcal{I}$ that implements the same outcome. Signal $\\mathcal{I}$ is not fully informative, because otherwise we would have $\\Pi(\\theta)=\\theta H+(1-\\theta) L$ for all types. Thus there is a set of signal realizations that can arise under both $v=H$ and $v=L$ with positive probabilities. After observing such realizations, types that are sufficiently close to 1 set price $H$, which leads to no trade when $v=L$. Because the set of any such types has a strictly positive measure, the efficiency loss is strictly positive. $\\square$\n\nProof of Proposition 7. Part 1. If the seller's type is known to be $\\theta_{0}>L / H$, then in the absence of additional data the seller would set price $H$, leading to zero buyer surplus. As a result, at the constrained seller-optimal outcome the seller extracts full surplus. In contrast, if the seller is privately informed, then the buyer surplus under no data provision is positive, because the seller sets price $L$ whenever her type is below $L / H$. As a result, the buyer surplus under the constrained seller-optimal outcome is strictly positive, and $0=\\mathrm{U}<\\mathrm{U}^{\\prime}$, $\\overline{\\mathrm{W}}\\left(\\theta_{0}\\right)=\\Pi>\\Pi^{\\prime}$.\n\nPart 2. If the seller's type is known to be $\\theta_{0}<L / H$, then in the absence of additional data the seller sets price $L$. Because this is the uniquely best possible outcome for the buyer, the constrained seller-optimal outcome is the same as under no data provision. In contrast, when the seller is privately informed, then she has a strictly positive measure of types above $L / H$ at which she sets price $H$ and earns strictly higher profits than under no data provision. Thus $\\underline{\\Pi}\\left(\\theta_{0}\\right)=\\Pi<\\Pi^{\\prime}, \\overline{\\mathrm{W}}\\left(\\theta_{0}\\right)-\\underline{\\Pi}\\left(\\theta_{0}\\right)=\\mathrm{U}>\\mathrm{U}^{\\prime}$. $\\square$\n\nProof of Proposition 9. Part 1. Assume the stated condition holds. Consider a direct menu that leads to an efficient outcome. Let $\\tilde{\\Theta} \\subseteq \\Delta(V)$ be the open set over which $F$ has a positive density. By the condition described in Part 1, we can take $\\tilde{\\Theta}$ so that any type in $\\tilde{\\Theta}$ chooses a price strictly above the lowest possible value $v_{1}$ in the absence of additional information. In this set, a positive measure of types assign positive probability to $v=v_{1}$, so there exists a type $\\tilde{\\theta} \\in \\tilde{\\Theta}$ such that the direct signal $\\mathcal{I}(\\tilde{\\theta})$ recommends $p=v_{1}$ with a strictly positive probability. Let $\\mu(\\tilde{\\theta})$ be the posterior belief of $\\tilde{\\theta}$ after that recommendation. If types in an open neighborhood of $\\tilde{\\theta}$ observe recommendation $p=v_{1}$, they would have posterior beliefs over an open neighborhood of $\\mu(\\tilde{\\theta})$, in accordance with equation (5). Because pricing\nindifference curves in $\\Delta(V)$ have measure zero, a strictly positive measure of types in $\\tilde{\\Theta}$ would strictly prefer to follow that recommendation, and would thus strictly benefit from $\\mathcal{I}(\\tilde{\\theta})$. It follows by incentive compatibility that the seller's rents are strictly positive.\n\nPart 2. Assume the stated condition holds. Consider any direct menu that leads to an efficient outcome. There exists $\\bar{\\varepsilon}>0$ such that for all $0<\\varepsilon<\\bar{\\varepsilon}$, type $\\theta_{\\varepsilon}$, defined as\n\n$$\n\\theta_{\\varepsilon} \\simeq\\left(\\varepsilon^{n-1}, \\varepsilon^{n-2}, \\ldots, \\varepsilon^{2}, \\varepsilon, 1-\\sum_{k=1}^{n-1} \\varepsilon^{k}\\right),\n$$\n\nbelongs to $\\Theta$ and prices efficiently after observing direct signal $\\mathcal{I}\\left(\\theta_{\\varepsilon}\\right)$, where the approximation means being in an $\\varepsilon^{n}$-neighborhood.\n\nStart with value $v=v_{1}$. Since $\\theta_{\\varepsilon}$ attaches strictly positive probability to $v=v_{1}$ and prices efficiently, $\\mathcal{I}\\left(\\theta_{\\varepsilon}\\right)$ recommends $p=v_{1}$ at value $v_{1}$ with probability 1. For the recommendation to be incentive compatible, this recommendation must be sent with probability $O\\left(\\varepsilon^{k-1}\\right)$ at all values $v_{k}, 1<k \\leq n .{ }^{14}$ Proceed to value $v=v_{2}$. Because $\\theta_{\\varepsilon}$ attaches strictly positive probability to $v=v_{2}$ and prices efficiently, and $\\mathcal{I}\\left(\\theta_{\\varepsilon}\\right)$ recommends price $p=v_{1}$ with probability $O(\\varepsilon)$ at $v=v_{2}$, it must be that $\\mathcal{I}\\left(\\theta_{\\varepsilon}\\right)$ recommends price $p=v_{2}$ with probability $1-O(\\varepsilon)=O(1)$ at $v=v_{2}$. For the recommendation to be incentive compatible, it must be sent with probability $O\\left(\\varepsilon^{k-2}\\right)$ at all values $v_{k}, 2<k \\leq n$. Proceeding analogously for all higher values, we obtain that signal $\\mathcal{I}\\left(\\theta_{\\varepsilon}\\right)$ must take the following form:\n\n| $\\theta_{\\varepsilon}$ | $\\mathcal{I}\\left(\\theta_{\\varepsilon}\\right)$ | $p=v_{1}$ | $p=v_{2}$ | ... | $p=v_{n}$ |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| $\\varepsilon^{n-1}$ | $v=v_{1}$ | 1 | 0 | ⋯ | 0 |\n| $\\varepsilon^{n-2}$ | $v=v_{2}$ | $O(\\varepsilon)$ | $O(1)$ | ⋯ | 0 |\n| ⋯ | … | ⋯ | ⋯ | ⋱ | 0 |\n| $1-\\sum_{k=1}^{n-1} \\varepsilon^{k}$ | $v=v_{n}$ | $O\\left(\\varepsilon^{n-1}\\right)$ | $O\\left(\\varepsilon^{n-2}\\right)$ | ⋯ | $O(1)$ |\n\nWhen $\\varepsilon \\rightarrow 0, \\mathcal{I}\\left(\\theta_{\\varepsilon}\\right)$ converges to a fully informative signal. Since $\\varepsilon$ can be set arbitrarily small, incentive compatibility implies that each type $\\theta \\in \\Theta$ earns a maximal possible rent and thus is offered a fully informative signal. $\\square$","text_sha256":"1775bc833823922a466be8c43216851eb7a7a44fa867eadf31de748f836041bb"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0026","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Appendix: Proofs Omitted from the Main Text","text":"[^11]\n[^0]:    *Ichihashi: Queen's University, shotaichihashi@gmail.com. Smolin: Toulouse School of Economics, University of Toulouse Capitole and CEPR, alexey.v.smolin@gmail.com. For valuable suggestions and comments, we would like to thank Nageeb Ali, Ricardo Alonso, James Best, Teck Yong Tan, Jidong Zhou, and especially our discussant Daniele Condorelli, as well as seminar participants at Carnegie Mellon University (Tepper), University of California Irvine, Concordia University, Monash-NTU-RUC joint seminar, University of Illinois Urbana-Champaign, Penn State University, Queen's University, CETC 2022, EARIE 2022, Conference on Mechanism and Institution Design, 2nd Workshop on Contracts, Incentives and Information in Collegio Carlo Alberto, and TSE Digital Conference 2023. Smolin acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d'Avenir) program (grant ANR-17-EURE-0010).\n\n[^1]:    ${ }^{1}$ Relatedly, Roesler and Szentes (2017) and Deb and Roesler (2022) analyze the informational impact in a second-degree price discrimination setting.\n\n[^2]:    ${ }^{2}$ Given set $X$, we write $\\Delta(X)$ for the set of all probability distributions on $X$.\n    ${ }^{3}$ By Bayes' rule, this timing is equivalent to one in which the value is drawn before the type.\n\n[^3]:    ${ }^{4}$ By Bayes plausibility, some signal realization $s$ will cause type $\\theta_{2}$ to believe that value $H$ is even more likely than $\\theta_{2}$. To cause type $\\theta_{2}$ to price efficiently, signal realization $s$ must reveal that the value is $H$.\n\n[^4]:    ${ }^{5}$ Proposition 2 is reminiscent of the equivalence between the experiments and persuasion mechanisms of Kolotilin et al. (2017). However, there are important differences between the settings and the results. First, in our setting, the seller's private information is correlated with the value, which affects the seller's assessment and response to data. Second, we obtain a stronger equivalence result, which applies to allocation rules and not just to the seller's interim utility profile. In our setting, this stronger result is necessary for welfare analysis, because the seller's interim utility profile alone does not determine the buyer's surplus.\n    ${ }^{6}$ If $\\alpha(\\theta)$ and $\\beta(\\theta)$ are not right continuous, $\\pi$ employs their right-continuous modifications.\n    ${ }^{7}$ The condition $\\left(\\alpha\\left(\\theta_{3}\\right)-\\alpha\\left(\\theta_{2}\\right)\\right)\\left(\\beta\\left(\\theta_{2}\\right)-\\beta\\left(\\theta_{1}\\right)\\right) \\geq\\left(\\alpha\\left(\\theta_{2}\\right)-\\alpha\\left(\\theta_{1}\\right)\\right)\\left(\\beta\\left(\\theta_{3}\\right)-\\beta\\left(\\theta_{2}\\right)\\right)$ is the general definition for CDF $\\alpha$ dominating CDF $\\beta$ in the monotone likelihood ratio order (see Theorem 1.C. 5 of Shaked and Shanthikumar (2007)).\n\n[^5]:    ${ }^{8}$ Alonso and Camara (2016) use this property to analyze Bayesian persuasion with heterogeneous priors.\n\n[^6]:    ${ }^{9}$ This observation is general-see the proof in Appendix, which characterizes those flagging signals in closed form.\n\n[^7]:    ${ }^{10}$ In other words, third-party data may crowd out first-party data. A similar interplay between several sources of information is the main focus of Alonso and Zachariadis (2021) in an investment setting.\n\n[^8]:    ${ }^{11}$ This is a large class of protocols that are natural in applications. However, like a revelation principle, our results can be stated with respect to a broader class of protocols at the expense of additional notation.\n\n[^9]:    ${ }^{12}$ If the seller is uninformed, this setting is a special case of Ali, Lewis, and Vasserman (forthcoming).\n\n[^10]:    ${ }^{13}$ All conditions of Proposition 9 hold, for example, whenever $F$ has full support over $\\Delta(V)$.\n\n[^11]:    ${ }^{14}$ That is, the recommendation probability is bounded by $L \\cdot \\varepsilon^{k-1}$ for some fixed $L$.","text_sha256":"6c626388cc653429b728a173db202db2f1a9ae1ab661e6ad579225b752c398db"}
{"schema_version":"1.0","chunk_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03:0027","work_id":"alex-smolin:data-provision-to-an-informed-seller","paper_id":"alex-smolin:data-provision-to-an-informed-seller:2023-03-03","title":"Data Provision to an Informed Seller","authors":[{"name":"Shota Ichihashi","url":"https://shota2.github.io/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-03-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md","source_record":"https://arxiv.org/abs/2204.08723","doi":"https://doi.org/10.1016/j.geb.2025.06.002","citation":"Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Shota Ichihashi; Alex Smolin\n\n**Canonical citation:** Ichihashi, Shota, and Alex Smolin. “Data Provision to an Informed Seller.” Games and Economic Behavior 153 (2025): 131–144.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/data-provision-to-an-informed-seller.md\n\n**Source record:** https://arxiv.org/abs/2204.08723\n\n**Published record:** https://doi.org/10.1016/j.geb.2025.06.002\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"aa2890e5aa6920318a20f9a7be1851f8f80dafab994f6ee434486d2e167a46fc"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0001","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Laura Doval; Alex Smolin.\n> Canonical citation: Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"ebc0be96ebf80ad5ff91c11dc89456c1356478d5fae8e06244aa7ea5a954a930"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0002","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Persuasion and Welfare","text":"# Persuasion and Welfare\n\n**Authors:** Laura Doval; Alex Smolin\n\n**Manuscript date:** 2023-09-06\n\n#### Abstract\n\nInformation policies such as scores, ratings, and recommendations are increasingly shaping society's choices in high-stakes domains. We provide a framework to study the welfare implications of information policies on a population of heterogeneous individuals. We define and characterize the Bayes welfare set, consisting of the population's utility profiles that are feasible under some information policy. The Pareto frontier of this set can be recovered by a series of standard Bayesian persuasion problems, in which a utilitarian planner takes the role of the information designer. We provide necessary and sufficient conditions under which an information policy exists that Pareto dominates the no-information policy. We illustrate our results with applications to data leakage, price discrimination, and credit ratings.\n\nKeywords: Bayesian persuasion, information design, welfare economics, algorithms, information policies.\n\n[^0]","text_sha256":"101183b7dc70ef2a16e743e5d4bf63b60cf87d1e60c3264068aa66629177f442"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0003","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nInformation has increasingly become a tool for shaping society's choices in highstakes domains. Consider, for instance, the role of algorithms in making recommendations for bail (Angwin et al., 2016), recruiting (Raghavan et al., 2020; Li et al., 2020), health (Obermeyer et al., 2019), education (Kučak et al., 2018), and lending (Jagtiani and Lemieux, 2019), among others. The role of information as a policy instrument is not confined to the big data economy. Indeed, information policies in the form of scores and ratings have been in place long before algorithmic recommendations to determine school placement, promotions, and who receives credit. As society becomes reliant on information to guide decisions in policy-relevant domains, understanding the welfare implications of information-based policies becomes a first-order concern.\n\nIn this paper, we provide a framework to study the welfare impact of information policies in a population of heterogeneous agents. Formally, we study the following model. There is a unit mass population with types in a finite set, distributed according to a given prior distribution. We model an information policy as an information structure, which associates to each type a distribution over signals and hence, via Bayes' rule, a distribution over posterior beliefs (Kamenica and Gentzkow, 2011). Our primitive is a welfare function that represents for each (posterior) type distribution the welfare of individuals of a given type. Given the welfare function, each information structure induces a Bayes welfare profile, which describes for each type in the population their expected payoff under the information structure. We define the Bayes welfare set to be the set of all such profiles.\n\nWe characterize the Bayes welfare set and study its properties. The Bayes welfare set allows us to reduce society's choice of an information structure to the choice of a Bayes welfare profile. Instead of imposing properties that the information structure must satisfy, society's preferences over the welfare distribution in the population determine the properties of the chosen information structure. Our perspective thus complements that of the algorithmic fairness literature which remains agnostic about the population's payoffs, focusing instead on statistical properties of information structures, such as accuracy, parity, or fairness. It also complements that of the literature in Bayesian persuasion (Rayo and Segal, 2010; Kamenica and Gentzkow, 2011), which characterizes the (maximum) average welfare consistent with some information structure, but not necessarily the Bayes welfare profiles that give rise to the average welfare.\n\nTheorem 1 characterizes the Bayes welfare set via the convex hull of a vector-valued function. In doing so, we extend the geometric characterizations of Aumann and Maschler (1995) and Kamenica and Gentzkow (2011) of the feasible set of ex ante payoffs to the characterization of the Bayes welfare set. Whereas a Bayes welfare profile depends on the distribution over posteriors conditional on each type, we show it can be alter-\nnatively expressed as the unconditional expectation over posteriors of a truth-adjusted payoff function, where the adjustment is proportional to the posterior likelihood ratio of each type. Evaluated at a given type, the truth-adjusted welfare function allows us to characterize the welfare individuals of a given type may obtain under some information structure. In turn, interpreting the truth-adjusted payoff function as a vectorvalued function allows us to capture the across-type restrictions imposed by Bayes' rule and precisely characterize the Bayes welfare set.\n\nTheorem 2 characterizes the Pareto frontier of the Bayes welfare set. Points in the Pareto frontier are natural candidates for being the outcome of efficient bargaining over information structures or a social planner's choice. Theorem 2 shows the points in the Pareto frontier of the Bayes welfare set can be recovered by a series of standard Bayesian persuasion problems, in which a utilitarian planner takes the role of an information designer. We use Theorem 2 throughout the paper to characterize optimal information structures in specific applications. Leveraging Theorem 2, Corollary 3 provides a necessary and sufficient condition under which an information structure exists that Pareto dominates providing no information.\n\nTheorem 3 enriches our characterization in the case in which the welfare function is equal to the expectation of a one-dimensional random variable, the support of which we call the reputation vector. This special case constitutes a natural benchmark and is commonly used in the literature on career concerns (Holmström, 1999), social image (Bénabou and Tirole, 2006, Tirole, 2021), and policy prediction problems (Mullainathan, 2018). Theorem 3 shows a welfare profile belongs to the Bayes welfare set if and only if it can be represented as the product between the reputation vector and a completely positive matrix that satisfies a version of Bayes plausibility. ${ }^{1}$ In addition, we show that the Bayesian persuasion problems that characterize the relative boundary of the Bayes welfare set correspond to instances of the problem in Rayo and Segal (2010). It follows that the information structures that induce welfare profiles on the relative boundary of the Bayes welfare set can be characterized using the graph-theoretic approach in Rayo and Segal (2010). We leverage their approach in Proposition 3, where we show that the information structures that maximize the welfare of a given type in the population correspond to a noisy version of the priority mechanisms studied in the matching literature (e.g., Celebi and Flynn, 2022).\n\nFinally, we note that by interpreting our welfare function as an individual's typedependent payoff function, the Bayes welfare set is also the object of interest in more standard information design applications. For instance, the types may represent the private information of an informed principal who can commit to an information structure only after observing her type, as in Perez-Richet (2014) and Koessler and Skreta","text_sha256":"9469b67382b359ab6f659a72f2a7d251bae90cf69431d283615ffc481e298e65"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0004","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"[^1](Forthcoming). Similarly, in the study of mechanism design with limited commitment, Doval and Skreta (2022) describe the principal's mechanism as an information structure that must satisfy an informed agent's incentive constraints. Similar constraints appear in the studies of information design without commitment, as in Fréchette et al. (2022), Lipnowski and Ravid (2020), and Salamanca (2021), in the analysis of tests subject to participation constraints in Rosar (2017), and in the analysis of moral hazard in Saeedi and Shourideh (2020). Thus, the Bayes welfare set can be viewed as a unifying concept that underlies the incentive constraints the equilibrium information structure must satisfy. As we show in our first working paper version, Doval and Smolin (2021), our tools also open the door to the study of new problems in this literature.\n\nRelated Literature: Our work contributes to the literature on information design reviewed in the introduction. Starting from the work of Kamenica and Gentzkow (2011) and Rayo and Segal (2010), a series of papers investigate the limits imposed by common knowledge of Bayesian rationality (Aumann, 1987). Whereas the Bayesian persuasion literature studies the (maximum) average welfare that can be achieved under some information structure, we characterize instead the welfare profiles that are consistent with some information structure.\n\nWhereas ours is the first characterization of the Bayes welfare set, a small literature studies certain Bayes welfare profiles within applications. Assuming the information designer is an informed principal, Perez-Richet (2014) refers to the payoff profile induced by an information structure as an interim payoff and studies the informed principal's preferred Bayes welfare profile. Recently, Galperti et al. (2023) study the Bayes welfare profile that gives rise to the sender's maximum average payoff and relate it to the Lagrange multiplier in the Bayes plausibility constraint. Restricting attention to the case in which the welfare function is linear in beliefs, Saeedi and Shourideh (2020) characterize a subset of the Bayes welfare set that satisfies certain incentive compatibility constraints, thus obtaining a different characterization. Finally, as the analysis below makes clear, a Bayes welfare profile depends on the distribution over posteriors induced by an information structure conditional on each type. Whereas Levy et al. (2021) and Arieli et al. (2022) characterize the set of conditional distributions over posteriors consistent with the prior, we follow a complementary approach that allows us to carry only the (unconditional) distribution over posteriors induced by an information structure (see Claim 1).\n\nWe also contribute to the economics literature that studies algorithmic fairness. Mullainathan (2018), Kleinberg et al. (2018), and Rambachan et al. (2020) argue for letting the social planner's objective determine the properties of algorithms. Our analysis is also related to Liang et al. (2022). Starting from a fixed joint distribution over groups, covariates, and states, Liang et al. (2022) model an algorithm as taking actions directly as a func-\ntion of covariates and study the group error profiles in the fairness-accuracy frontier as the algorithm varies. Instead, we model an algorithm as an information structure that sends non-binding action recommendations to an unmodeled receiver, whose actions determine the population's welfare and characterize the set of all Bayes welfare profiles as we vary the algorithm.\n\nBy considering the welfare redistribution effects of information, our work joins the mechanism design literature that studies the role of markets in redistributing welfare (see, e.g., Dworczak et al., 2021, Akbarpour et al., Forthcoming, and Akbarpour et al., 2023). We complement this work, which typically assumes the planner can utilize transfers to achieve its objectives, by considering the role of information, which can be a powerful tool when the planner does not have access to transfers.\n\nFinally, our work contributes indirectly to the literature on higher-order beliefs. Indeed, when the welfare function is linear in beliefs as in Section 5, the welfare profile can be seen as a profile of second-order expectations. Starting with Samet (1998), a body of work uses Markov matrices to represent such higher-order beliefs and expectations of higher-order beliefs for a given information structure (see, e.g., Cripps et al., 2008; Golub and Morris, 2017). Instead, our result in Theorem 3 identifies the set of matrices that correspond to some information structure.\n\nIn lieu of an organizational paragraph, we summarize below the notation used throughout the paper:\n\nNotation: For ease of presentation, we sometimes find it convenient to denote a function from a set $\\Theta$ to $\\mathbb{R}$ as a vector in $\\mathbb{R}^{N}$, where $N$ is the cardinality of $\\Theta$. In this case, we reserve the italic notation $x$ for the function $x: \\Theta \\mapsto \\mathbb{R}$ and the upright notation x for the vector in $\\mathbb{R}^{N}$. Any vector $\\mathrm{x} \\in \\mathbb{R}^{N}$ is taken to be a column vector; we denote its $i^{\\text {th }}$ component by $\\mathrm{x}_{i}$ or $x\\left(\\theta_{i}\\right)$ interchangeably. If $\\mathrm{x} \\in \\mathbb{R}^{N}$ is a column vector, $\\mathrm{x}^{T}$ denotes its transpose. If $\\mathrm{x}, \\mathrm{y}$ are two vectors, $\\mathrm{x} * \\mathrm{y}$ denotes their Hadamard (elementwise) product and x/y denotes their Hadamard division. We denote by $\\mathrm{e} \\in \\mathbb{R}^{N}$ the vector with $\\mathrm{e}_{1}=\\cdots=\\mathrm{e}_{N}=1$. When we want to emphasize that $x$ is a random variable, we write it as $\\tilde{x}$.","text_sha256":"e20509c2ca3f4cfcc45a2a2b8db42ef9a2abc38c9e2feee83b870f46e97cb464"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0005","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"## 2 Model\n\nA unit mass population has types in a finite set, $\\Theta \\equiv\\left\\{\\theta_{1}, \\ldots, \\theta_{N}\\right\\}$. Letting $\\Delta(\\Theta)$ denote the set of probability distributions over $\\Theta$, we denote by $\\mu_{0} \\in \\Delta(\\Theta)$ the frequency of types in the population. We assume that $\\mu_{0}$ has full support. We denote by $\\Delta(\\Delta(\\Theta))$ the set of distributions over posteriors, and by $\\Delta_{\\mu_{0}}(\\Delta(\\Theta))$ the set of distributions over posteriors with mean equal to the prior $\\mu_{0}$.\n\nWelfare function: An individual's welfare depends on her type $\\theta$ and an (unmodeled) outside observer's belief about her type. We represent this by a welfare function $w: \\Delta(\\Theta) \\times \\Theta \\mapsto \\mathbb{R}$ that represents for each belief $\\mu$ and each type $\\theta$, the welfare of individuals of type $\\theta$ under belief $\\mu, w(\\mu, \\theta)$. We assume throughout that $w$ is bounded.\n\nA welfare profile is a vector $\\mathrm{w} \\in \\mathbb{R}^{N}$, where $\\mathrm{w}_{i}$ describes the welfare level of individuals with type $\\theta_{i}$. Any welfare profile w induces an ex ante welfare of $\\mu_{0}^{T} \\mathrm{w}=$ $\\sum_{i=1}^{N} \\mu_{0}\\left(\\theta_{i}\\right) \\mathrm{w}_{i}$. When we average a profile w using weights other than the prior $\\mu_{0}$, we refer to average welfare instead. We are interested in characterizing those welfare profiles that are induced by some information structure.\n\nInformation structures: An information structure $\\Pi=(\\pi, S)$ consists of a countable set of labels $S$, and a mapping $\\pi$, which associates to each type $\\theta$ a distribution over signals $\\pi(\\cdot \\mid \\theta) \\in \\Delta(S)$. Given an information structure $\\Pi$ and a signal realization $s \\in S$, the corresponding posterior belief $\\mu_{s} \\in \\Delta(\\Theta)$ is obtained by Bayes' rule whenever possible, and is given by\n\n$$\n\\mu_{s}(\\theta)=\\frac{\\mu_{0}(\\theta) \\pi(s \\mid \\theta)}{\\sum_{\\theta^{\\prime} \\in \\Theta} \\mu_{0}\\left(\\theta^{\\prime}\\right) \\pi\\left(s \\mid \\theta^{\\prime}\\right)} .\n$$\n\nThus, an information structure can be seen as inducing a distribution over posterior beliefs $\\left\\{\\mu_{s}: s \\in S\\right\\}$. In what follows, two such distributions are of interest: the distribution over posterior beliefs conditional on an individual's type-as induced by $\\pi(\\cdot \\mid \\theta)$ and the unconditional distribution over posterior beliefs-as induced by the prior $\\mu_{0}$ and the signal distribution. When taking expectations using these distributions, we use the notations $\\langle\\Pi \\mid \\theta\\rangle$ and $\\langle\\Pi\\rangle$ to denote the conditional and unconditional distributions over posterior beliefs, respectively.\n\nBayes welfare profiles: The welfare function $w$ together with an information structure, $\\Pi$, defines a welfare profile, $w_{\\Pi}: \\Theta \\mapsto \\mathbb{R}$, as\n\n$$\nw_{\\Pi}(\\theta) \\equiv \\mathbb{E}_{\\langle\\Pi \\mid \\theta\\rangle}[w(\\tilde{\\mu}, \\theta)]=\\sum_{s \\in S} \\pi(s \\mid \\theta) w\\left(\\mu_{s}, \\theta\\right) .\n$$\n\nThat is, for each type $\\theta, w_{\\Pi}(\\theta)$ describes the (expected) welfare of type- $\\theta$ individuals under information structure $\\Pi$. Note that in computing the welfare of type- $\\theta$ individuals, their type $\\theta$ enters twice: directly through the welfare function, $w(\\cdot, \\theta)$, and indirectly through the signal distribution, $\\pi(\\cdot \\mid \\theta)$.\n\nWe now present our two main objects of study:\nDefinition 1 (Bayes welfare profile). A welfare profile $\\mathrm{w} \\in \\mathbb{R}^{N}$ is a Bayes welfare profile if an information structure, $\\Pi$, exists such that for all types $\\theta_{i}, \\mathrm{w}_{i}=w_{\\Pi}\\left(\\theta_{i}\\right)$.\n\nDefinition 2 (Bayes welfare set). The Bayes welfare set is the set of all Bayes welfare profiles; that is,\n\n$$\n\\mathrm{W} \\equiv\\left\\{\\mathrm{w} \\in \\mathbb{R}^{N}: \\exists \\Pi \\text { s.t. } \\mathrm{w}_{i}=w_{\\Pi}\\left(\\theta_{i}\\right) \\forall i \\in\\{1, \\ldots, N\\}\\right\\} .\n$$\n\nThe Bayes welfare set W represents the utility possibility set in an economy where the allocations are given by information structures. As such, it describes the welfare effects that different information structures have for individuals with different types in applications such as grading schemes in the case of schooling (Ostrovsky and Schwarz, 2010), disclosure about job performance (Mukherjee, 2008), affirmative action in the case of college admissions or the job market, rating systems in the case of platforms (Saeedi and Shourideh, 2020), and market segmentations (Bergemann et al., 2015).\n\nThroughout, we illustrate our results using the following examples:\nExample 1 (Data leakage). Consumers concerned about how a third party may use their data wish to maximize the third party's uncertainty about their types. ${ }^{2}$ We formalize this as follows. There are two types of consumer, $\\Theta=\\left\\{\\theta_{A}, \\theta_{B}\\right\\}$. Letting $\\mu_{0}$ denote the frequency of consumers of type $\\theta_{B}$ (i.e., $\\mu_{0} \\equiv \\mu_{0}\\left(\\theta_{B}\\right)$ ), we assume that $\\mu_{0}=0.3$. When the third party believes the consumer's type is $\\theta_{B}$ with probability $\\mu \\in[0,1]$, a consumer experiences a welfare loss of $w(\\mu, \\theta)=-(\\mu-1 / 2)^{2}$. The consumer's welfare loss is minimal when the third party is maximally confused about the consumer's type (i.e., $\\mu=\\frac{1}{2}$ ). Instead, the consumer's welfare loss is maximal when the third party has precise information about the consumer's type (i.e., $\\mu \\in\\{0,1\\}$ ). ◇\n\nExample 2 (Price discrimination). An online marketplace makes algorithmic recommendations to a seller about what price the seller should set for consumers. There are two consumer types, $\\theta_{H}$ and $\\theta_{L}$. Assume $\\mu_{0} \\equiv \\mu_{0}\\left(\\theta_{L}\\right)=0.6$. Consumers can have one of three values for the seller's good: low (1), medium (2), and high (3). A consumer's type indexes their distribution over values, with $\\theta_{H}$-consumers being more likely to have high valuations, and $\\theta_{L}$-consumers being more likely to have low valuations. In particular, we assume the likelihoods of the values \\{1, 2, 3\\} are $\\left\\{\\frac{1}{5}, \\frac{1}{5}, \\frac{3}{5}\\right\\}$ and $\\left\\{\\frac{3}{5}, \\frac{1}{5}, \\frac{1}{5}\\right\\}$ for $\\theta_{H^{-}}$and $\\theta_{L^{-}}$-consumers, respectively.","text_sha256":"37c86d1f46e538101bb47905f98e83249265149d0ed8aa223cc34f34a55e5332"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0006","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"The platform can provide information only about the consumer's type and not about their value. Hence, the seller's price depends on the likelihood $\\mu$ the seller attaches to the consumer's type being $\\theta_{L}$. Assuming the seller breaks ties in favor of consumers,\n\n[^2]consumers' welfare as a function of the seller's belief $\\mu$ and their type $\\theta$ is as follows:\n$$\nw\\left(\\mu, \\theta_{H}\\right)=\\left\\{\\begin{array}{ll}\n0 & \\text { if } \\mu<1 / 2 \\\\\n3 / 5 & \\text { if } \\mu \\in[1 / 2,3 / 4) \\\\\n7 / 5 & \\text { if } \\mu \\in[3 / 4,1]\n\\end{array}, w\\left(\\mu, \\theta_{L}\\right)=\\left\\{\\begin{array}{ll}\n0 & \\text { if } \\mu<\\frac{1}{2} \\\\\n1 / 5 & \\text { if } \\mu \\in[1 / 2,3 / 4) \\\\\n3 / 5 & \\text { if } \\mu \\in[3 / 4,1]\n\\end{array} .\\right.\\right.\n$$ $\\square$\n\nExample 3 (Credit ratings). A credit agency makes lending decisions based on an applicant's perceived repayment probability. An applicant of type $\\theta$ repays loans with probability $\\rho(\\theta)$. The credit agency approves loans with probability proportional to the expected value of $\\rho$. A regulator wishes to maximize the probability that applicants of a given type $\\theta_{i}$ receive a loan by choosing the information on which the credit agency can condition its approval decision. $\\square$\n\nRemark 1 (Interpretation of the welfare function). The welfare function admits several interpretations. First, following Kamenica and Gentzkow (2011), the population's welfare may be determined by the actions taken by the outside observer after observing the realization of an information structure, as in Examples 2 and 3. Using the notation in that paper, denote by $v(a, \\theta)$ the utility of type- $\\theta$ individuals when the outside observer takes action $a \\in A$. If the outside observer takes action $a(\\mu)$ when her posterior belief is $\\mu$, type- $\\theta$ individuals obtain payoff $v(a(\\mu), \\theta) \\equiv w(\\mu, \\theta) .^{3}$ Second, the welfare function may also capture that the population's welfare may be driven by image or reputation concerns, as in Bénabou and Tirole (2006) and Tirole (2021), or psychological motives, as in Lipnowski and Mathevet (2018). Indeed, Example 1 admits a psychological interpretation under which an individual derives utility from keeping the outside observer, who is attempting to guess the individual's type, in suspense (cf. Ely et al., 2015).","text_sha256":"3627032e97f1fd9016bbae9e5c0995b5f5da70ec28a2f812926bca45235e5b38"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0007","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Characterization","text":"## 3 Characterization\n\nSection 3 presents our characterization of the Bayes welfare set via the convex hull of the graph of a vector-valued function, in the spirit of the belief-based approach of Kamenica and Gentzkow (2011).\n\nTruth-drifting: An apparent obstacle in following the belief approach in Kamenica and Gentzkow (2011) is that the elements of W are expressed in terms of expectations conditional on a given type $\\theta \\in \\Theta$, rather than unconditional expectations. Indeed, as shown in Francetich and Kreps (2014), conditional expectations do not satisfy the martingale\n\n[^3]property; rather, they drift toward the truth. More precisely, for any type $\\theta$ and for any information structure $\\Pi$, the expectation of the posterior probability of $\\theta$ conditional on $\\tilde{\\theta}=\\theta$ is higher than the prior probability of $\\theta$. That is, ${ }^{4}$\n$$\n\\mathbb{E}_{\\langle\\Pi \\mid \\theta\\rangle}\\left[\\frac{\\tilde{\\mu}(\\theta)}{\\mu_{0}(\\theta)}\\right]=\\sum_{s \\in S} \\pi(s \\mid \\theta) \\frac{\\mu_{s}(\\theta)}{\\mu_{0}(\\theta)} \\geq 1 .\n$$\nInstead of pursuing a characterization of conditional distributions of posteriors, we recover the belief approach in Kamenica and Gentzkow (2011) by studying a suitably modified welfare function.\n\nTruth-adjusted welfare: We show any element $\\mathrm{w} \\in \\mathrm{W}$ can be expressed as the unconditional expectation of an adjusted version of the welfare function. Indeed, define the truth-adjusted welfare function $\\hat{w}: \\Delta(\\Theta) \\times \\Theta \\mapsto \\mathbb{R}$ to be\n\n$$\n\\hat{w}(\\mu, \\theta) \\equiv \\frac{\\mu(\\theta)}{\\mu_{0}(\\theta)} w(\\mu, \\theta) .\n$$\n\nThat is, $\\hat{w}$ is the welfare function $w$ adjusted by the truth-drift $\\mu(\\theta) / \\mu_{0}(\\theta)$. For any given posterior belief $\\mu$, the likelihood ratio $\\mu(\\theta) / \\mu_{0}(\\theta)$ measures the representation of type $\\theta$ under $\\mu$ relative to its ex ante representation under $\\mu_{0}$.\n\nThe truth-adjusted welfare function combines the preferences of individuals of type $\\theta$ for a particular belief- $w(\\mu, \\theta)$-and the resource constraint-Bayes plausibility-in our economy, where information structures take the role of allocations. Indeed, the likelihoodratio adjustment $\\mu(\\theta) / \\mu_{0}(\\theta)$ captures the constraint that comes from Bayes plausibility. Intuitively, type- $\\theta$ individuals have an endowment equal to $\\mu_{0}(\\theta)$ that can be spread over different beliefs $\\mu(\\theta)$, and the Bayes plausibility constraint ensures this spread is done in a way that respects the budget. ${ }^{5}$\n\nExample 1 (continued). We illustrate the truth-adjusted welfare function in the context of Example 1. Figure 1 depicts the welfare function $w(\\mu, \\theta)$ (Figure 1a) and the truth-adjusted welfare function for $\\theta_{A}$ (Figure 1b) and $\\theta_{B}$ (Figure 1c). Whereas in this example welfare is assumed to be type-independent, the truth-adjusted welfare function is type-dependent. This natural consequence of the likelihood-ratio adjustment reflects that individuals of different types get to benefit differently from various induced beliefs and therefore, from the same information structure. $\\square$\n\nClaim 1 justifies our interest in the truth-adjusted welfare function $\\hat{w}$ :\n\n[^4]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: Truth-adjusted welfare function in Example 1\n\nClaim 1 (From conditional to unconditional expectations). For any information structure $\\Pi$ and any type $\\theta \\in \\Theta$, the following holds:\n\n$$\nw_{\\Pi}(\\theta)=\\mathbb{E}_{\\langle\\Pi \\mid \\theta\\rangle}[w(\\tilde{\\mu}, \\theta)]=\\mathbb{E}_{\\langle\\Pi\\rangle}[\\hat{w}(\\tilde{\\mu}, \\theta)] .\n$$\n\nThe proof of Claim 1 and of other results can be found in the appendix.\nClaim 1 implies the expectation of $w$ under $\\Pi$ conditional on type $\\theta$ can be expressed as the unconditional expectation of $\\hat{w}$ under $\\Pi$. Because the definition of $\\hat{w}$ does not depend on $\\Pi$, the distribution over posteriors induced by $\\Pi,\\langle\\Pi\\rangle$, is enough to determine the unconditional expectation of $\\hat{w}$ under $\\Pi$. Consequently, the analysis that follows relies on the unconditional distribution over posteriors induced by an information structure $\\Pi$, rather than on the family of conditional distributions over posteriors induced by $\\Pi$.\n\nClaim 1 can be obtained from the analysis in Alonso and Câmara (2016). ${ }^{6}$ Indeed, for a given type $\\theta$, one can interpret our model as one of Bayesian persuasion with heterogeneous priors in which the sender assigns probability 1 to state $\\theta$ and the receiver's prior belief is $\\mu_{0}$. Alonso and Câmara (2016, pp. 683-684) show the sender's payoff under an information structure $\\Pi$ can be equivalently obtained as the expectation over the distribution of the receiver's posterior beliefs induced by $\\Pi$ of an adjusted payoff function. Specialized to the case in which the sender assigns probability 1 to state $\\theta$ and the receiver's prior belief is $\\mu_{0}$, the truth-adjusted welfare function, $\\hat{w}(\\mu, \\theta)$, corresponds to the adjusted payoff function in Alonso and Câmara (2016).\n\nRelying on Claim 1, we can immediately characterize the range of welfare values that individuals of type $\\theta$ may obtain under some information structure, via the expected\n\n[^5]value of $\\hat{w}(\\cdot, \\theta)$ under a Bayes plausible distribution over posteriors (Aumann and Maschler, 1995; Kamenica and Gentzkow, 2011). Indeed, let $w_{*}(\\theta), w^{*}(\\theta)$ denote the minimum and maximum welfare individuals of type $\\theta$ can obtain under some information structure. That is, $w_{*}(\\theta)=\\inf \\{w(\\theta): \\mathrm{w} \\in \\mathrm{W}\\}$ and $w^{*}(\\theta)=\\sup \\{w(\\theta): \\mathrm{w} \\in \\mathrm{W}\\}$. We have the following:\n\nProposition 1 (Individually feasible welfare bounds). For any type $\\theta$, the following holds: ${ }^{7}$\n\n$$\nw_{*}(\\theta)=\\operatorname{vex} \\hat{w}\\left(\\mu_{0}, \\theta\\right), w^{*}(\\theta)=\\operatorname{cav} \\hat{w}\\left(\\mu_{0}, \\theta\\right) .\n$$","text_sha256":"58fa1cab654e85042a3d632edc6b69b70c0025290a613167af96e5fd6d93f345"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0008","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Characterization","text":"Proposition 1 follows from the main result in Kamenica and Gentzkow (2011). The vertical solid lines in Figures 1b and 1c illustrate the individually feasible welfare values for $\\theta_{A}$ and $\\theta_{B}$ in Example 1. An implication of Proposition 1 is that any Bayes welfare profile w satisfies that for all types $\\theta, w(\\theta) \\in\\left[\\operatorname{vex} \\hat{w}\\left(\\mu_{0}, \\theta\\right)\\right.$, cav $\\left.\\hat{w}\\left(\\mu_{0}, \\theta\\right)\\right]$.\n\nWhereas Proposition 1 characterizes what is individually feasible for each type in the population, it does not deliver the characterization of the Bayes welfare set. The reason is that it ignores the across-type restrictions imposed by Bayes' rule. For instance, the truth-adjusted welfare function (AW) highlights that only types on the support of belief $\\mu$ get to enjoy the payoff of inducing said belief. Similarly, inspection of Figures 1b and 1c shows $\\theta_{A}$ 's preferred information structure is no disclosure, whereas $\\theta_{B}$ would prefer some disclosure to no disclosure. In other words, the profile $\\left(w^{*}\\left(\\theta_{A}\\right), w^{*}\\left(\\theta_{B}\\right)\\right)$ is not jointly feasible.\n\nInstead, the characterization of the Bayes welfare set can be obtained by studying the convex hull of the graph of the vector-valued function $\\hat{\\mathrm{w}}, \\hat{\\mathrm{w}}: \\Delta(\\Theta) \\mapsto \\mathbb{R}^{N}$, where for each $i \\in\\{1, \\ldots, N\\}, \\hat{\\mathrm{w}}_{i}(\\mu) \\equiv \\hat{w}\\left(\\mu, \\theta_{i}\\right)$. Indeed, we have the following:\n\nTheorem 1 (Belief-based characterization). The Bayes welfare set W satisfies the following:\n\n$$\n\\mathrm{W}=\\left\\{\\mathrm{w} \\in \\mathbb{R}^{N}:\\left(\\mu_{0}, \\mathrm{w}\\right) \\in \\operatorname{co}(\\operatorname{graph} \\hat{\\mathrm{w}})\\right\\} .\n$$\n\nTheorem 1 provides a geometric characterization of the set W : it is the section at the prior of the convex hull of the graph of the truth-adjusted welfare function $\\hat{\\mathrm{w}}$. Relying on the result in Kamenica and Gentzkow (2011) that any Bayes plausible distribution over posteriors is the outcome of some information structure, ${ }^{8}$ Theorem 1 characterizes a more primitive object, the set of welfare profiles that can be generated by some\n\n[^6]information structure. Indeed, whereas the main result in Kamenica and Gentzkow (2011) would allow us to characterize the (maximal) ex ante welfare a population with welfare function $w$ can obtain, Theorem 1 characterizes the welfare profiles whose average leads to that welfare.\n\nCalculating the Bayes welfare set: Relying on Theorem 1, Figure 2 illustrates the construction of the Bayes welfare set in Example 1. Figure 2a depicts the convex hull of the graph of $\\hat{\\mathrm{w}}$. Applying Theorem 1, the resulting Bayes welfare set is the section of this convex hull at $\\mu_{0}=0.3$. The blue shaded area in Figure 2a depicts this section, which is represented in Figure 2b.\n\nFigure 2b illustrates which welfare profiles are jointly feasible under some information structure in Example 1. For instance, fully revealing or concealing individuals' types is always possible, so that the full and no-disclosure profiles, $\\mathrm{w}^{F D}$ and $\\mathrm{w}^{N D}$, are feasible. All Bayes welfare profiles Pareto dominate the full-disclosure profile, whereas no Bayes welfare profile Pareto dominates the no-disclosure one. As discussed above, simultaneously giving all individual types their maximum welfare $w^{*}(\\theta)$ is not possible, because $\\theta_{B}$ 's welfare is maximized by a policy that sometimes reveals an individual is of type $\\theta_{A}$.\n\nBecause the welfare function is continuous in Example 1, the Bayes welfare set is closed, but this property does not follow from the ongoing assumption that the welfare function is bounded. However, making assumptions other than that the welfare function is bounded may not be natural. To illustrate, consider the case in which the welfare function captures in reduced form that the welfare of the population is determined by the outside observer's actions after observing the realization of the information structure. As we explained in Remark 1, each selection from the outside observer's best-response correspondence induces $a$ welfare function and hence a corresponding Bayes welfare set. If one assumes-as we do in Example 2-the outside observer breaks ties in favor of the individuals, one would naturally obtain an upper-semicontinuous welfare function, but under adversarial tie-breaking, having a lower-semicontinuous welfare function would have made sense. As we illustrate in Section 4, for a fixed tie-breaking rule, the corresponding Bayes welfare set may not be closed (see, e.g., Figure 3b). When one instead considers the welfare implications of different selection rules, the analogue of the Bayes welfare set is the set of all profiles that are induced by some information structure and some selection from the best-response correspondence. As we show in Proposition A. 1 in the appendix, the characterization behind Theorem 1 delivers that this analogue of the Bayes welfare set is closed.\n\nCardinality: Theorem 1 has an immediate implication for the cardinality of the information structures that generate points in W :\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 2: Constructing the Bayes welfare set in Example 1\n\nCorollary 1. Let $\\mathrm{w} \\in \\mathrm{W}$. Then, an information structure $\\Pi$ with at most $2 N$ signals exists such that $\\mathrm{w}_{i}=w_{\\Pi}\\left(\\theta_{i}\\right)$ for all $i \\in\\{1, \\ldots, N\\}$.","text_sha256":"eb4560c790007feb91b81819f878c515e8da48b087fc707147733a6acd033384"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0009","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Characterization","text":"Corollary 1 stands in contrast to the result in Bayesian persuasion that finding an information structure that delivers the sender's maximal payoff and employs at most $N$ posteriors is always possible. There are two reasons behind this difference. First, Theorem 1 characterizes all welfare profiles consistent with some information structure so that without further assumptions on the truth-adjusted welfare function, we rely on Carathédory's theorem to obtain Corollary 1. Instead, Kamenica and Gentzkow (2011) characterize the sender's maximal payoff and assume the indirect utility function is upper-semicontinuous. For that reason, Kamenica and Gentzkow (2011) can rely on Fénchel-Bunt's theorem instead of Carathéodory to obtain an upper bound of $N$ instead of $N+1$. Second, we are interested not just in the payoff of one \"sender,\" but in the payoff of $N$, one for each type.\n\nIn Example 1, the upper bound in Corollary 1 is loose. Because the truth-adjusted welfare function is continuous, we can rely on Fénchel-Bunt's theorem to reduce the upper bound in Corollary 1 by 1. Furthermore, the Bayes welfare profiles on the boundary of W are induced by information structures with at most two signals. We explain the reason for this further reduction in Section 4, where we characterize the boundary of the Bayes welfare set and, in particular, its Pareto frontier.","text_sha256":"d57493e6ab3e5b7252d24f54002a0b737ca5f4c0644b9a0837b8ef0bd8380857"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0010","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 The Pareto frontier of the Bayes welfare set","text":"## 4 The Pareto frontier of the Bayes welfare set\n\nWe characterize in this section the Pareto frontier of W. Points in the Pareto frontier are natural candidates for being the outcome of efficient bargaining over information structures or a social planner's choice. Indeed, these points correspond to the solution of a utilitarian planner as we vary the weights the planner assigns to different types. Theorem 2 shows these points can be recovered as solutions to standard Bayesian persuasion problems, where an information designer takes the role of the utilitarian planner. Armed with this characterization, Corollary 3 provides a necessary and sufficient condition for the no-disclosure profile to be part of the Pareto frontier. In other words, it provides a necessary and sufficient condition for information disclosure to lead to a Pareto improvement relative to no disclosure.\n\nWe define the Pareto frontier of W to be the set of weak Pareto efficient Bayes welfare profiles. Formally,\n\n$$\n\\mathrm{W}_{\\mathrm{P}}=\\left\\{\\mathrm{w} \\in \\mathrm{~W}:\\left(\\nexists \\mathrm{w}^{\\prime} \\in \\mathrm{W}\\right) \\mathrm{w}^{\\prime}>\\mathrm{w}\\right\\} .\n$$\n\nBecause the Bayes welfare set W is convex, for any $\\mathrm{w} \\in \\mathrm{W}_{\\mathrm{P}}$, the separating hyperplane theorem implies a direction $\\lambda \\in \\mathbb{R}_{+}^{N} \\backslash\\{0\\}$ exists such that ${ }^{9}$\n\n$$\n\\begin{aligned}\n\\lambda^{T} \\mathrm{w}=\\max \\left\\{\\lambda^{T} \\mathrm{w}^{\\prime}: \\mathrm{w}^{\\prime} \\in \\mathrm{W}\\right\\} & =\\max \\left\\{\\lambda^{T} \\mathbb{E}_{\\tau}[\\hat{\\mathrm{w}}(\\tilde{\\mu})]: \\tau \\in \\Delta_{\\mu_{0}}(\\Delta(\\Theta))\\right\\} \\\\\n& =\\max \\left\\{\\mathbb{E}_{\\tau}\\left[\\lambda^{T} \\hat{\\mathrm{w}}(\\tilde{\\mu})\\right]: \\tau \\in \\Delta_{\\mu_{0}}(\\Delta(\\Theta))\\right\\} .\n\\end{aligned}\n$$\n\nThe first equality simply states that w is a maximizer of the support function of the Bayes welfare set in direction $\\lambda$. Instead, the second equality follows from Theorem 1. Indeed, Theorem 1 implies we can exchange the maximization over welfare profiles in W for a maximization over Bayes plausible distributions over posteriors. Note we can interchangeably talk about Pareto efficient Bayes welfare profiles and Pareto efficient information structures, and we do this in what follows.\n\nOnce we note we can restrict attention to directions $\\lambda \\in \\Delta(\\Theta)$, Equation 6 has two economic interpretations. First, consider the problem of a social planner who assigns weight $\\lambda(\\theta)$ to type $\\theta$ and wishes to maximize the weighted sum of utilities of each type. Under this interpretation, Equation 6 states that w is a solution to the social planner's problem. Second, we can interpret $\\lambda^{T} \\mathrm{w}$ as the expectation with respect to $\\theta$ of the welfare profile w under the measure $\\lambda$. In this case, Equation 6 implies w is the vector of interim payoffs of a sender with payoff function $w(\\mu, \\theta)$ and prior $\\lambda$. For instance, when $\\lambda=\\mu_{0}$, so that the sender's prior coincides with that of the outside observer, the above problem coincides with that of Kamenica and Gentzkow\n\n[^7](2011). Instead, whenever the direction $\\lambda$ is any element of $\\Delta(\\Theta)$, the above problem coincides with that considered by Alonso and Câmara (2016). ${ }^{10}$\n\nMoreover, Equation 6 has an important practical implication: any Pareto efficient Bayes welfare profile is induced by the solution to a supporting Bayesian persuasion problem. A supporting Bayesian persuasion problem is an instance of the model in Kamenica and Gentzkow (2011) in which the sender's indirect utility function equals\n\n$$\n\\hat{v}_{\\lambda}(\\mu)=\\lambda^{T} \\hat{\\mathrm{w}}(\\mu)=\\sum_{\\theta \\in \\Theta} \\lambda(\\theta) \\frac{\\mu(\\theta)}{\\mu_{0}(\\theta)} w(\\mu, \\theta)=\\sum_{\\theta \\in \\Theta} \\mu(\\theta) \\frac{\\lambda(\\theta)}{\\mu_{0}(\\theta)} w(\\mu, \\theta) .\n$$\n\nThis indirect utility is the product of the truth-adjusted welfare function and the Pareto weights, which capture the rate of substitution between the truth-adjusted welfare of different types. Theorem 2 summarizes the above discussion:\n\nTheorem 2 (Pareto frontier). The welfare profile $\\mathrm{w} \\in \\mathbb{R}^{N}$ is in the Pareto frontier of W if and only if a direction $\\lambda \\in \\Delta(\\Theta)$ exists such that w is the profile induced by an information structure that solves the supporting Bayesian persuasion problem with indirect utility function $\\hat{v}_{\\lambda}$.\n\nBecause W is convex, the characterization in Theorem 2 extends to all Bayes welfare profiles on the boundary of W, except that the supporting direction may no longer be in $\\Delta(\\Theta)$.\n\nTheorem 2 characterizes the Pareto frontier of W by connecting the solution of the utilitarian planner with weights $\\lambda$ to the Bayesian persuasion problem of the sender with indirect utility function $\\hat{v}_{\\lambda}$. Whereas Theorem 1 characterizes the Bayes welfare set via the convex hull of the graph of a vector-valued function, $\\hat{\\mathrm{w}}$, the results in Kamenica and Gentzkow (2011) imply the supporting Bayesian persuasion problems corresponding to Pareto efficient profiles can be solved by concavifying a real-valued function, $\\hat{v}_{\\lambda}$. Theorem 2 thus provides us with a tractable way of characterizing the Pareto efficient information structures and thus recover the Pareto efficient profiles (see, e.g., the analysis in Section 5). The ability to recover the Pareto frontier may be useful when maximizing non-utilitarian social welfare functions, such as those that correspond to a fairness-aware planner (e.g., Rawls' criterion or Epstein and Segal's quadratic social welfare function), or those that obtain from efficient bargaining (e.g., Nash bargaining). Even if such an objective would select a Pareto efficient profile, the supporting direction $\\lambda$ may only be identifiable after characterizing the solution, so that knowledge of the whole Pareto frontier may be important.\n\n[^8]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure (a) The convex hull of the graph of $\\hat{w}$\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure (b) The Bayes welfare set W","text_sha256":"b6b3263499d0ca1980def1c7fcd8dc6c7acfcc407aabd9b671cb16829966a140"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0011","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 The Pareto frontier of the Bayes welfare set","text":"Figure 3: Constructing the Bayes welfare set in Example 2. The diagonal dashed line is the $45^{\\circ}$ degree line. Blue dashed lines denote profiles on the boundary of the Bayes welfare set, which are not in the Bayes welfare set.\n\nTheorem 2 has yet another practical implication: when a (Pareto efficient) w is an extreme point of W , an information structure exists that employs at most $N$ signals and generates w. We record this result below:\n\nCorollary 2 (Extreme points of W). Let w be an extreme point of W . Then, an information structure $\\Pi$ with at most $N$ signals exists such that $\\mathrm{w}_{i}=w_{\\Pi}\\left(\\theta_{i}\\right)$ for all $i \\in\\{1, \\ldots, N\\}$.\n\nIn Example 1, all points in $\\mathrm{W}_{\\mathrm{P}}$ are extreme (Figure 2b). Corollary 2 then implies information structures that induce at most two posteriors are enough to characterize the Pareto frontier in Example 1.\n\nWe use Example 2 to illustrate properties of the Bayes welfare profiles on the Pareto frontier and why the bound in Corollary 2 applies only to extreme points on the Pareto frontier of W. In doing so, we highlight the difference between the value of the program in Equation 6 and the Bayes welfare profiles consistent with that value:\n\nExample 2 (continued). Figure 3 illustrates the convex hull of the graph of $\\hat{w}$ (Figure 3a) and the Bayes welfare set (Figure 3b) for the online marketplace example. We note the following features of the Pareto frontier of W in this example, depicted in black in Figure 3b. First, in contrast to Example 1, the full-disclosure profile, $\\mathrm{w}^{F D}$, is Pareto efficient, whereas the no-disclosure profile, $\\mathrm{w}^{N D}$, is not. Second, like in Example 1, there is a continuum of Bayes welfare profiles that are fair in the sense of equalizing welfare\nacross different consumer types. Whereas the points in the flat segment of the Pareto frontier to the right of $\\mathrm{w}=(0.6,0.6)$ maximize Rawls' criterion, only $\\mathrm{w}=(0.6,0.6)$ is both Pareto efficient and fair.\n\nThird, contrary to Example 1, not every point on the boundary of W is an extreme point. Consider, for instance, the points on the boundary in the direction $\\lambda=\\left(\\frac{2}{3}, \\frac{1}{3}\\right)$ in Figure 3b, which would be consistent with the platform being interested in promoting the participation of $\\theta_{H}$-consumers. It is possible to show that the points in the interior of this line are generated by information structures with three signals. ${ }^{11}$ Note all the points in that line lead to the same value in the Bayesian persuasion problem with indirect utility function $\\hat{v}_{\\lambda}$, namely, cav $\\hat{v}_{\\lambda}\\left(\\mu_{0}\\right)$. However, they correspond to different welfare profiles with different implications regarding how $\\theta_{H^{-}}$and $\\theta_{L}$-consumers share the payoff cav $\\hat{v}_{\\lambda}\\left(\\mu_{0}\\right)$. This highlights a benefit of the perspective we develop in this paper: insofar as one cares about the cross-sectional implications of different information structures, studying only the average welfare induced by a given information structure may not be sufficient.\n\nBecause the seller breaks ties in favor of the consumer when setting prices, the welfare function in Example 2 is upper-semicontinuous, but not continuous. Consequently, the Bayes welfare set is not closed in this example: The profiles corresponding to the blue dashed lines in Figure 3b can be induced by an information structure only when the seller breaks ties against the consumer. Note, however, that upper-semicontinuity of the welfare function guarantees that the supporting Bayesian persuasion problem attains a solution in all directions $\\lambda \\in \\Delta(\\Theta)$. Whereas in general upper-semicontinuity does not guarantee the Pareto frontier is closed, it ensures that all weakly monotone social welfare functions attain a maximum in the Bayes welfare set (see Proposition A.2). ◇\n\nWe close this section by noting that when W is closed, Theorem 2 provides an alternative characterization of the Bayes welfare set. Whereas points on the Pareto frontier are natural candidates for the choice of a social planner, points outside the Pareto frontier may be relevant if, for instance, the social planner faces constraints in their choice of a Bayes welfare profile. Because such constraints may rule out Pareto efficient and even boundary profiles, knowledge of the whole Bayes welfare set may be important. For this reason, we see the characterizations in both theorems as complementary.\n\n[^9]","text_sha256":"7039b6ee55f9ba4a662b6ed825c7e2068b1a454593d4a1df2d109db4dc0aa793"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0012","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.1 When is (no) information disclosure efficient?","text":"### 4.1 When is (no) information disclosure efficient?\n\nA natural question to ask is when providing some information is the efficient thing to do; in other words, when does an information structure exist that all types strictly prefer to no disclosure? When such an information structure exists, we say all types benefit from disclosure. Examples 1 and 2 offer an interesting contrast in this respect: because the no-disclosure profile $\\mathrm{w}^{N D}$ is not in the Pareto frontier in Example 2, all consumer types benefit from disclosure. Instead, the no-disclosure profile $\\mathrm{w}^{N D}$ is in the Pareto frontier in Example 1. Indeed, in Example 1 information disclosure necessarily hurts at least one of the types (in this case, $\\theta_{A}$ ).\n\nCorollary 3 provides a necessary and sufficient condition for all types to benefit from disclosure. To introduce this condition, recalling a definition from Kamenica and Gentzkow (2011) is useful: a sender with indirect utility function $\\hat{v}$ benefits from persuasion if the concavification of $\\hat{v}$ at the prior exceeds the value of $\\hat{v}$ at the prior; that is, $\\hat{v}\\left(\\mu_{0}\\right)<\\operatorname{cav} \\hat{v}\\left(\\mu_{0}\\right)$. We have the following:\n\nCorollary 3 (Efficiency of disclosure). All types benefit from disclosure if and only if, for all directions $\\lambda \\in \\Delta(\\Theta)$, a sender with indirect utility $\\hat{v}_{\\lambda}$ benefits from persuasion.\n\nIn other words, Corollary 3 states that the no-disclosure profile $\\mathrm{w}^{N D}$ is in the Pareto frontier of W if and only if a direction $\\lambda^{N D} \\in \\Delta(\\Theta)$ exists such that a sender with indirect utility $\\hat{v}_{\\lambda^{N D}}$ does not benefit from persuasion. Alternatively, a social planner with Pareto weights $\\lambda^{N D}$ would find no disclosure to be an optimal information structure. Recall Example 1: in that case, when $\\lambda=\\mu_{0}$, the indirect utility function $\\hat{v}_{\\mu_{0}}$ equals the strictly concave function $-(\\mu-1 / 2)^{2}$, so that no disclosure is optimal. Corollary A. 1 in Appendix A. 2 shows that this observation reflects a more general result: if for all $\\theta \\in \\Theta$ the welfare function takes the form $a(\\theta) w(\\mu)+b(\\theta)$, where $a(\\theta)>0$ and $w$ is concave, $\\mathrm{w}^{N D}$ is in the Pareto frontier of W.\n\nWhereas Corollary 3 provides a condition in terms of the indirect utility function $\\hat{v}_{\\lambda}$, Observation 1 provides conditions on the truth-adjusted welfare function $\\hat{w}$ under which all types do or do not benefit from disclosure: ${ }^{12}$\n\nObservation 1 (Disclosure benefits). The following hold:\n\n1. If, for all $\\theta \\in \\Theta, \\hat{w}(\\cdot, \\theta)$ is strictly convex in a neighborhood of the prior, all types benefit from disclosure.\n2. Instead, if a type $\\theta$ exists such that $\\hat{w}(\\cdot, \\theta)$ is concave in $\\mu$, no disclosure is Pareto efficient.\n\nAs in Kamenica and Gentzkow (2011), concavity and convexity properties of a pay-\n\n[^10]off function determine whether information disclosure is beneficial. In contrast to Kamenica and Gentzkow (2011), the concavity and convexity properties of the truthadjusted welfare function are what determine whether any given type can benefit from disclosure. This result can be clearly seen in Example 1, where the truth-adjusted welfare of $\\theta_{A}$ is concave around the prior and that of $\\theta_{B}$ is strictly convex around the prior, even though each type's welfare function is strictly concave. ${ }^{13}$\n\nTo understand the role of the convexity (concavity) properties of the truth-adjusted welfare function in determining the benefits from disclosure, note that information disclosure affects the welfare of a given type $\\theta$ through two channels: directly through its impact on the welfare function as in Kamenica and Gentzkow (2011) and indirectly through the truth-drift adjustment. The second channel is most easily seen in the case of binary types. In that case, the property in Equation TD ensures that the posterior belief drifts along a straight line toward the true type $\\theta$. When the welfare function of individuals of type $\\theta$ is convex and increasing in $\\mu(\\theta)$, both effects are positive, ensuring that individuals of type $\\theta$ benefit from disclosure. Observation 2 below summarizes this discussion:\n\nObservation 2 (Binary types). Let $N=2$. If for all $\\theta \\in \\Theta, w(\\mu, \\theta)$ is strictly convex in $\\mu$ and increasing in $\\mu(\\theta)$, both types benefit from disclosure; moreover, full disclosure is uniquely Pareto efficient. Instead, if for some $\\theta \\in \\Theta, w(\\mu, \\theta)$ is concave in $\\mu$ and decreasing in $\\mu(\\theta)$, no disclosure is Pareto efficient.","text_sha256":"d675dac7ab916ff9aa5c8e42297e6a315efc144ffbb1216553540fe2441a501c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0013","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Expected Reputation","text":"## 5 Expected Reputation\n\nWe now specialize our results to the case in which the welfare function equals the expectation of some one-dimensional variable of interest, such as an individual's productivity, quality, or trade value. This is a standard way to model reputation, image, or career concerns in economics (see, e.g., Holmström, 1999, Bénabou and Tirole, 2006). It also captures the class of prediction policy problems in Rambachan et al. (2020), in which a decision maker-our outside observer-selects a treatment (e.g., hiring, bail, loan-approval) on the basis of the prediction of an outcome of interest (e.g., productivity, recidivism, creditworthiness).\n\n[^11]Formally, we assume a reputation vector $\\rho \\in \\mathbb{R}^{N}$ exists such that for all $\\theta_{i}, \\theta_{j} \\in \\Theta$,\n\n$$\nw\\left(\\mu, \\theta_{i}\\right)=w\\left(\\mu, \\theta_{j}\\right)=\\mathbb{E}_{\\mu}[\\rho(\\theta)]=\\sum_{k=1}^{N} \\mu\\left(\\theta_{k}\\right) \\rho\\left(\\theta_{k}\\right)=\\mu^{T} \\rho .\n$$\n\nWe refer to $w$ as the individual's reputation. Thus, a Bayes welfare profile is a profile of expected reputations. Without loss of generality, we label $\\rho$ in increasing order, that is, $\\rho_{1} \\leq \\cdots \\leq \\rho_{N}$, so that types are labeled in increasing order of their values under $\\rho$.\n\nThe analysis in this section allows us to focus on the redistributive role of information. Indeed, all information structures lead to the same ex ante welfare. That is, for any information structure $\\Pi$ and the corresponding Bayes welfare profile w, the ex ante welfare is:\n\n$$\n\\mu_{0}^{T} \\mathrm{w}=\\sum_{\\theta \\in \\Theta} \\mu_{0}(\\theta) \\mathbb{E}_{\\langle\\Pi \\mid \\theta\\rangle}\\left[\\tilde{\\mu}^{T} \\rho\\right]=\\mathbb{E}_{\\langle\\Pi\\rangle}\\left[\\tilde{\\mu}^{T} \\rho\\right]=\\mu_{0}^{T} \\rho .\n$$\n\nHowever, as the results in this section illustrate, different information structures lead to different welfare profiles, so that the chosen information structure determines how the different types in the population share the ex ante welfare.\n\nWhen $w$ is as in Equation 8, we can provide an alternative characterization of the set W . From Section 3, it follows that $\\mathrm{w} \\in \\mathrm{W}$ if and only if we can find a Bayes plausible distribution over posteriors, $\\tau \\in \\Delta_{\\mu_{0}}(\\Delta(\\Theta))$, with finite support, such that\n\n$$\n\\mathrm{w}=\\mathbb{E}_{\\tau}[\\hat{\\mathrm{w}}(\\tilde{\\mu})]=\\mathbb{E}_{\\tau}\\left[\\frac{\\tilde{\\mu}}{\\mu_{0}}\\left(\\tilde{\\mu}^{T} \\rho\\right)\\right]=\\mathrm{D}_{0} \\mathbb{E}_{\\tau}\\left[\\tilde{\\mu} \\tilde{\\mu}^{T}\\right] \\rho,\n$$\n\nwhere $\\mathrm{D}_{0}$ denotes a diagonal matrix with $(i, i)$-th element equal to $1 / \\mu_{0}\\left(\\theta_{i}\\right)$.\nEquation 10 shows a Bayes welfare profile can be represented as the product of three terms: the reputation vector $\\rho$, the prior-normalizing matrix $\\mathrm{D}_{0}$, and the matrix $\\mathbb{E}_{\\tau}\\left[\\mu \\mu^{T}\\right]$. Furthermore, the matrix $\\mathbb{E}_{\\tau}\\left[\\mu \\mu^{T}\\right]$ satisfies the following two properties. First, it is a completely positive matrix (Berman, 1988): an $N \\times N$ matrix C is completely positive if it can be written as $\\sum_{m=1}^{M} \\mathrm{x}_{\\mathrm{m}} \\mathrm{x}_{\\mathrm{m}}^{T}$ for some finite collection of non-negative vectors $\\mathrm{x}_{\\mathrm{m}} \\in \\mathbb{R}_{+}^{N}$. ${ }^{14}$ Second, the rows of the matrix $\\mathbb{E}_{\\tau}\\left[\\mu \\mu^{T}\\right]$ add up to the prior: $\\mathbb{E}_{\\tau}\\left[\\mu \\mu^{T}\\right] \\mathrm{e}=\\mathbb{E}_{\\tau}\\left[\\mu\\left(\\mu^{T} \\mathrm{e}\\right)\\right]=\\mathbb{E}_{\\tau}[\\mu]=\\mu_{0}$. Theorem 3 shows these two properties are\n\n[^12]not only necessary but also sufficient and thus fully characterize the Bayes welfare set: ${ }^{15}$\n\nTheorem 3. Given the reputation vector $\\rho, \\mathrm{w} \\in \\mathrm{W}$ if and only if a completely positive matrix $\\mathrm{C} \\in \\mathbb{R}^{N \\times N}$ exists such that $\\mathrm{Ce}=\\mu_{0}$ and\n\n$$\n\\mathrm{w}=\\mathrm{D}_{0} \\mathrm{C} \\rho .\n$$\n\nPutting together the properties in Theorem 3, we obtain that any Bayes welfare profile w is the product of the reputation vector $\\rho$ and a matrix P , where $\\mathrm{P} \\equiv \\mathrm{D}_{0} \\mathrm{C}$ is the transition matrix of a reversible Markov chain with invariant distribution $\\mu_{0}$. That is, (i) $\\mu_{0}^{T} \\mathrm{P}=\\mu_{0}^{T}$, (ii) $\\mathrm{Pe}=\\mathrm{e}$, and (iii) P satisfies the detailed balance conditions: for all $i, j \\in N, \\mu_{0 i} \\mathrm{P}_{i j}=\\mu_{0 j} \\mathrm{P}_{j i}$. The first property captures the pure redistribution of welfare highlighted in Equation 9. The second property implies any Bayes welfare profile can be viewed as a garbled version of the full information profile $\\rho$. The third property delineates the limits of how payoffs can be redistributed by linking how much of $\\rho\\left(\\theta_{i}\\right)$ can be attributed to $\\theta_{j}$, and vice versa. Indeed, because P is the transition matrix of a reversible Markov chain, we obtain that there is mean reversion in the redistribution of payoffs across types. To see this, note that if $\\mathrm{w}=\\mathrm{P} \\rho \\in \\mathrm{W}$, then also $\\mathrm{Pw} \\in \\mathrm{W} .{ }^{16}$ Because $\\mu_{0}$ is the invariant distribution of P , we have that $\\mathrm{P}^{k} \\mathrm{w} \\rightarrow_{k \\rightarrow \\infty}$ $\\left(\\mu_{0}^{T} \\mathrm{w}\\right) * \\mathrm{e}=\\left(\\mu_{0}^{T} \\rho\\right) * \\mathrm{e}=\\mathrm{w}^{N D}$, where $\\mathrm{w}^{N D}$ is the no-disclosure profile.\n\nRemark 2 (Connections to the literature). Reversible Markov chains are prominent in the study of higher-order beliefs and expectations of higher-order beliefs (Samet, 1998; Cripps et al., 2008; Golub and Morris, 2017). Indeed, note that when the welfare function is linear and type independent, a Bayes welfare profile is a vector of second-order expectations: for any type $\\theta$ and any $\\mathrm{w} \\in \\mathrm{W}, w(\\theta)$ is the expectation under some information structure of the random variable $\\mu^{T} \\rho$ of an individual that knows $\\theta$. Whereas that literature takes the information structure as given and shows (sequences of) higher-order expectations can be obtained by iteratively applying the transition matrix of a reversible Markov chain, Theorem 3 identifies which transition matrices are consistent with some information structure and shows complete positivity is the key property they must satisfy.","text_sha256":"068bd804934c56c9599c21e7cd408a9f2a2633332230c40e5705f0fb9fc7a060"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0014","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Expected Reputation","text":"Theorem 3 also relates to the literature on majorization (Hardy et al., 1952): if all types are equally likely, P is doubly stochastic and $\\rho$ majorizes w . However, not any profile majorized by $\\rho$ is a Bayes welfare profile, because not all doubly stochastic matrices are symmetric, and hence, some do not satisfy the detailed balance conditions.\n\n[^13]We now show how Theorem 3 delivers a more general version of the truth-drifting property discussed in Section 3. This property has been obtained in different forms in the literature that studies the feasible evolution of beliefs (e.g., Francetich and Kreps, 2014, Hart and Rinott, 2020). Truth-drifting states that whereas an information structure can occasionally \"deceive\" the outside observer about an individual's true type, it cannot systematically do so. This property underlies the limits of using information as a tool to distribute welfare in the population.\n\nFormally, consider any event $X$ that is correlated with the types according to the conditional probability function $\\beta \\in[0,1]^{N}, \\beta_{i} \\equiv \\operatorname{Pr}\\left(X \\mid \\theta_{i}\\right)$, so that the prior probability of the event is $\\operatorname{Pr}(X)=\\mu_{0}^{T} \\beta .^{17}$ If all $\\beta_{i} \\in\\{0,1\\}$, the event effectively indicates a subset of types. More generally, the event may involve extraneous uncertainty, and the types may be only imperfectly informative about it. We show that if the event is true, the average posterior probability that the outside observer attaches to this event must be at least as large as the prior probability:\n\nClaim 2 (Truth drifting). For any event $X$ and information structure $\\Pi$,\n\n$$\n\\mathbb{E}_{\\Pi}[\\operatorname{Pr}(X \\mid s) \\mid X] \\geq \\operatorname{Pr}(X) .\n$$\n\nFrancetich and Kreps (2014) obtain this result, relying on the properties of KullbackLeibler divergence. ${ }^{18}$ Hart and Rinott (2020) obtain a version of Claim 2 in the special case of $X \\subseteq \\Theta$, relying on the monotone-likelihood ratio property. Instead, our proof of Claim 2, presented in Appendix A.3, builds on the property that the underlying matrix C is completely positive and thus necessarily positive semi-definite.\n\nBoundary information structures: Recall that Theorem 2 characterizes the boundary of the Bayes welfare set by means of supporting Bayesian persuasion problems. In the reputation model, Equation 9 implies the Bayes welfare set lies within a hyperplane with orthogonal vector $\\mu_{0}$. Thus, instead of studying the boundary of the Bayes welfare set, we focus on its relative boundary, which consists of all Bayes welfare profiles not in the relative interior of the Bayes welfare set. ${ }^{19}$ In a slight abuse of terminology, we refer to the profiles on the relative boundary of W and the information structures that induce them as boundary profiles and information structures, respectively.\n\n[^14]As we show next, the supporting Bayesian persuasion problems in the reputation model take a well-known structure. Indeed, fix a direction $\\lambda \\in \\mathbb{R}^{N} \\backslash\\{0\\}$ not collinear with $\\mu_{0}$ and consider the induced supporting Bayesian persuasion problem:\n\n$$\n\\begin{aligned}\n\\max _{\\tau \\in \\Delta_{\\mu_{0}}(\\Delta(\\Theta))} \\mathbb{E}_{\\tau}\\left[\\lambda^{T} \\hat{\\mathrm{w}}(\\mu)\\right] & =\\max _{\\tau \\in \\Delta_{\\mu_{0}}(\\Delta(\\Theta))} \\mathbb{E}_{\\tau}\\left[\\left(\\frac{\\lambda^{T}}{\\mu_{0}} \\mu\\right)\\left(\\rho^{T} \\mu\\right)\\right] \\\\\n& =\\max _{\\tau \\in \\Delta_{\\mu_{0}}(\\Delta(\\Theta))} \\mathbb{E}_{\\tau}\\left[\\mathbb{E}_{\\mu}\\left[\\frac{\\lambda(\\theta)}{\\mu_{0}(\\theta)}\\right] \\mathbb{E}_{\\mu}[\\rho(\\theta)]\\right]\n\\end{aligned}\n$$\n\nwhere the first equality uses the form of $w$ and the definition of $\\hat{w}$. Equation $\\mathrm{RS}_{\\lambda}$ shows that if an information structure $\\Pi$ delivers a profile w on the relative boundary of W, the information structure solves an instance of the information design problem in Rayo and Segal (2010). To be precise, Rayo and Segal (2010) consider the following problem. A sender owns a prospect, and his objective is that the receiver accepts it. When the sender's type is $\\theta$ and the receiver accepts the prospect, the sender and the receiver obtain a payoff $\\gamma(\\theta) \\equiv \\lambda(\\theta) / \\mu_{0}(\\theta)$ and $\\rho(\\theta) \\in[0,1]$, respectively. Instead, if the receiver rejects the prospect, the sender obtains a payoff of 0, whereas the receiver obtains a payoff $u$ distributed uniformly over [0,1] independently of $\\theta$. The sender chooses an information structure, $\\Pi$, without observing the realization of $u$. Thus, when $\\Pi$ induces a belief $\\mu$, the sender expects the receiver to accept the project with probability, $\\rho^{T} \\mu$. It follows that the last term in Equation $\\mathrm{RS}_{\\lambda}$ represents the sender's expected payoff when $\\tau$ is the distribution over posteriors induced by information structure $\\Pi$.\n\nProposition 2. (Boundary profiles) A welfare profile w is on the relative boundary of W if and only if a direction $\\lambda \\in \\mathbb{R}^{N} \\backslash\\{0\\}$ not collinear with $\\mu_{0}$ exists such that w is induced by an information structure that solves the program $R S_{\\lambda}$.\n\nProposition 2 allows us to rely on the approach of Rayo and Segal (2010) to characterize the shape of the information structures that achieve the boundary Bayes welfare profiles. This approach relies on a graphical representation of an information structure, in which the prospect values $\\{(\\gamma(\\theta), \\rho(\\theta)): \\theta \\in \\Theta\\}$ are the nodes (see Remark A. 1 in the appendix). The results in Rayo and Segal (2010) have immediate implications for the information structures that induce the boundary profiles of W:\n\nCorollary 4. In the reputation model, the following hold:\n\n1. An information structure that induces a boundary Bayes welfare profile in the direction $\\lambda$ does not pool types $\\theta_{i}$ and $\\theta_{j}$ whenever their ranking under the vector $\\lambda / \\mu_{0}$ and the reputation vector $\\rho$ is the same;\n2. The full- and no-disclosure profiles are on the relative boundary of W.","text_sha256":"846f4a24e65058f86f7a3e1c859896561c403268e5c0cd07624f0d49ed53b3f4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0015","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Expected Reputation","text":"The first part of Corollary 4 highlights a natural feature of optimal information provision in the reputation model in terms of the alignment of preferences of the social planner, captured by $\\lambda / \\mu_{0}$, and of the outside observer, captured by $\\rho$ : the planner should not pool any two types as long as the planner and the outside observer are in agreement about the types' relative ranking. The second part of Corollary 4 shows that the full- and no-disclosure profiles are on the boundary of the Bayes welfare set. Whereas this property holds in Examples 1 and 2, in which the welfare function is not linear, this property is not a general one, as we illustrate in Example A. 4 in the appendix.\n\nIndividual reputation bounds: We can further build on the graphical approach of Rayo and Segal (2010) to characterize the information structures that deliver maximal (or minimal) welfare to any given type. This exercise provides a rough way to bound the Bayes welfare set W and also suggests how information may be employed to boost (or dilute) the reputation of particular types in the population. In the context of our credit agency example, Example 3, the information structure that maximizes the welfare of individuals of type $\\theta_{i}$ maximizes the probability that individuals of type $\\theta_{i}$ obtain credit.\n\nFormally, given a target type $\\theta_{i}$, we want to solve the following problem:\n\n$$\n\\max _{\\mathrm{w} \\in \\mathrm{~W}} \\mathrm{w}_{i} .\n$$\n\n(i-MAX)\nProposition 3 below shows a particular class of information structures solves the problem $i$-MAX.\n\nDefinition 3 (Noisy priority). A $\\theta_{i}$-noisy-priority policy with threshold $k$ is an information structure $(\\pi, S)$ such that $S=\\Theta$, and the likelihood function $\\pi$ satisfies:\n\n1. If $j \\neq i, \\pi\\left(s=\\theta_{j} \\mid \\theta_{j}\\right)=1$,\n2. If $j<k, \\pi\\left(s=\\theta_{j} \\mid \\theta_{i}\\right)=0$, and\n3. If $j \\geq k, \\pi\\left(s=\\theta_{j} \\mid \\theta_{i}\\right)>0$.\n\nIn other words, a $\\theta_{i}$-noisy-priority policy pairwise pools the target type $\\theta_{i}$ with all types with indices above some threshold and separates all other types. A noisy-priority policy has an implementation akin to the priority mechanisms in the matching literature (Celebi and Flynn, 2022), and hence its name. A noisy-priority policy with threshold $k \\geq i$ can be implemented by first assigning a perfectly revealing score to each type equal to their index, and then prioritizing the target type $\\theta_{i}$ by increasing this type's score by a random number.\n\nProposition 3. The Bayes welfare profile that solves $i-M A X$ is induced by a $\\theta_{i}$-noisypriority policy with threshold $k \\geq i$.\n\nThe proof is in Appendix A.3. One part of Proposition 3 is straightforward: if one wishes to increase the reputation of $\\theta_{i}$, then $\\theta_{i}$ should be separated from all types with lower indices. What might be less obvious is that whenever $\\theta_{i}$ is pooled with some other type, $\\theta_{i}$ should be pooled with it pairwise. In a sense, pooling several types together redistributes the reputation from higher-quality types to lower-quality types. Pairwise pooling then allows the target type to obtain maximal reputation gains from any other type without sharing the gains with others. Finally, pairwise pooling with many types ensures no signal is overly \"muddled,\" which in turn ensures an overall high reputation for $\\theta_{i}$.\n\nBy simply reversing signs, Proposition 3 can be used to characterize the information structure that minimizes the expected reputation of individuals of type $\\theta_{i}$ : this information structure should pairwise pool the target type with types whose indices are below some threshold. Such adversarial pairwise pooling inflicts maximal reputation losses and can be viewed as a noisy-degrading policy.\n\nWe conclude this section by illustrating the Bayes welfare set in the context of Example 3 and highlight the importance of population heterogeneity as captured by the number of types.\n\nExample 3 (continued). Figure 4 illustrates the results of this section in the context of Example 3. Recall that in this case $\\rho(\\theta)$ denotes the probability that an individual of type $\\theta$ repays the loan, and hence, $w(\\mu, \\theta)$ is the expected repayment probability under belief $\\mu$. Like in Rayo and Segal (2010), we assume the credit agency has a uniform outside option. Assuming individuals wish to maximize the probability the lending agency approves the loan justifies that $\\rho^{T} \\mu$ corresponds to their welfare.\n\nFigure 4a depicts the individually feasible welfare profiles (dashed square) and the Bayes welfare set (blue line) in the case of $N=2$. Proposition 1 implies any payoff between $\\rho_{1}=0$ and $\\mu_{0}^{T} \\rho$ is feasible for $\\theta_{1}$, whereas any payoff between $\\mu_{0}^{T} \\rho$ and $\\rho_{2}=1$ is feasible for $\\theta_{2}$. As Figure 4a illustrates, the Cartesian product $\\left[\\rho_{1}, \\mu_{0}^{T} \\rho\\right] \\times$ $\\left[\\mu_{0}^{T} \\rho, \\rho_{2}\\right]$ is a rather lax bound in this example. In particular, the Cartesian product $\\left[\\rho_{1}, \\mu_{0}^{T} \\rho\\right] \\times\\left[\\mu_{0}^{T} \\rho, \\rho_{2}\\right]$ ignores that all Bayes welfare profiles satisfy $\\mu_{0}^{T} \\mathrm{w}=\\mu_{0}^{T} \\rho=0.5$ (Equation 9). In the case of $N=2$, adding this restriction is enough to pin down the Bayes welfare set. The reason is that by Theorem 3, all Bayes welfare profiles can be obtained by \"garbling\" the full-disclosure Bayes welfare profile, $\\rho$, and in the case of binary types, this garbling turns out to span a linear segment. Finally, note the structure of the Bayes welfare set implies a social planner with Pareto weights $\\lambda$ finds it optimal to provide no or full information, depending on whether the planner weighs the welfare of $\\theta_{1}$-individuals more than that of $\\theta_{2}$-individuals (that is, $\\lambda_{1} \\lessgtr \\lambda_{2}$ ).\n\nFigure 4b depicts the Bayes welfare set W in the case of $N=3$. In contrast to the binary-type case, the boundary of the Bayes welfare set is non-linear and features a\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 4: Expected reputation in Example 3. The blue color marks Bayes welfare set W . The dashed segments outline the Cartesian product of individual welfare sets.","text_sha256":"695b7e69fd516b0c4d11d6bc933b46c7877cc452a52b13f53380dbaf3e736243"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0016","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Expected Reputation","text":"continuum of extreme points. This example illustrates that the constraints imposed by Bayes plausibility are richer than the simple garbling constraint that characterizes the set when $N=2$. As we show in Appendix A.3, four classes of information structures span the boundary of the Bayes welfare set. The first two are the $\\theta_{1}$-noisy-priority and $\\theta_{3}$-noisy-degrading policies, which span the nonlinear segment of the boundary. The other two span the linear segments in Figure 4b and similar to the $\\theta_{3}$-noisy-priority and the $\\theta_{1}$-noisy-degrading policies, either maximize the loan-approval probability of $\\theta_{3}$-individuals, by separating them from $\\theta_{1}$ - and $\\theta_{2}$-individuals, or minimize the loanapproval probability of $\\theta_{1}$-individuals, by separating them from $\\theta_{2}$ - and $\\theta_{3}$-individuals. Unlike the $\\theta_{3}$-noisy-priority and the $\\theta_{1}$-noisy-degrading policies, the two classes of information structures that span the linear segments may pool together the types below $\\theta_{3}$ or above $\\theta_{1}$, respectively. Notably, for almost all boundary points, $\\theta_{2}$-individuals are pooled with individuals of some other type. Intuitively, $\\theta_{2}$-individuals exert pooling externalities on individuals of types $\\theta_{1}$ and $\\theta_{3}$. ${ }^{20}$ For instance, $\\theta_{2}$-individuals enable boosting the loan-approval probability of $\\theta_{1}$-individuals in the case of the $\\theta_{1}$-noisy-priority policy, but exert a negative externality on $\\theta_{3}$-individuals in the $\\theta_{3}$-noisy-degrading policy. $\\square$","text_sha256":"f1d53c0dbe3529d7c9c29f167299f618d60c0e33cccb03b3f05a32658a35cb13"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0017","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Conclusion","text":"## 6 Conclusion\n\n[^15]We provide a framework to study the potentially disparate impact of information policies in a population of heterogeneous individuals. Because information policies increasingly shape society's choices in high-stakes domains, the Bayes welfare set describes the limits of what society can achieve under such policies and the welfare trade-offs implied by the choice between different information policies. In the spirit of mechanism design and information design, our characterization of the Bayes welfare set provides a unifying tool to evaluate the welfare implications of different information policies across a wide array of objective functions.\n\nWe see several avenues worth exploring and left for future work. First, our model assumes any information can be provided about an individual's payoff-relevant type. However, this assumption does not necessarily hold in applications of interest in which an individual's type may include protected characteristics. In the online appendix, we extend our framework to accommodate limits on how much information can be disclosed about the individuals in the population. The analysis there, however, does not consider that these limits may be designed when a potentially malicious third party selects the information structure. Liang et al. (2022) consider this case in the context of decision-making algorithms, and we expect their insights to extend to the case of recommendations algorithms like the ones we consider.\n\nSecond, because the welfare function depends only on the first-order beliefs about an individual's type, our model only accounts for strategic interactions that follow the realization of a public signal (see, e.g., Laclau and Renou, 2017). Extending the analysis to account for general strategic interactions is worth exploring. Galperti et al. (2023), which studies the Bayes welfare profile that gives rise to the sender's maximum average payoff across all Bayes correlated equilibria, is a step in this direction.\n\nThird, motivated by recent policies, Tirole (2021) studies the use of information in the form of a social score to incentivize good behavior in the population. Whereas Tirole (2021) studies this question in the context of a parametric family of information structures, our initial explorations show the Bayes welfare set allows us to extend his results by allowing any information structure. More generally, information has been suggested as a substitute for monetary incentives, and the Bayes welfare set describes what can be achieved with information alone.\n\nFinally, whereas we characterize individuals' welfare as a function of their type, thinking of applications in which we care instead about the welfare of groups is natural. For instance, an individual's type could encompass their gender and their ability, and the social planner is concerned with the welfare different genders may obtain. This extension can inform the study of statistical discrimination, where recent work shows Bayesian persuasion tools can shed new light to this problem (Chambers and Echenique, 2021; Escudé et al., 2022; Deb and Renou, 2022). Whereas much of the existing litera-\nture focuses on statistical properties of discrimination, our work highlights the important aspect of economic welfare, offering a complementary perspective that enriches the ongoing dialogue between statistics and economics.","text_sha256":"be6d5a95b98403a51b33c0a54f77f13ff400d1f1ab4965a16caed96833bea1ee"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0018","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAkbarpour, M., E. Budish, P. Dworczak, and S. D. Kominers (2023): \"An Economic Framework for Vaccine Prioritization,\" The Quarterly Journal of Economics, qjad022.\n\nAkbarpour, M., P. Dworczak, and S. D. Kominers (Forthcoming): \"Redistributive AIlocation Mechanisms,\" Journal of Political Economy.\n\nAliprantis, C. D. and K. C. Border (2013): Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer-Verlag Berlin and Heidelberg GmbH \\& Company KG.\n\nAlonso, R. and O. Câmara (2016): \"Bayesian Persuasion with Heterogeneous Priors,\" Journal of Economic Theory, 165, 672-706.\n\nAngwin, J., J. Larson, S. Mattu, and L. Kirchner (2016): \"Machine Bias: There's Software Used across the Country to Predict Future Criminals. And It's Biased Against Blacks.\" ProPublica, 23, 77-91.\n\nArieli, I., Y. Babichenko, and F. Sandomirskiy (2022): \"Persuasion as Transportation,\" in Proceedings of the 23rd ACM Conference on Economics and Computation, 468.\n\nAumann, R. 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(2014): \"Interim Bayesian Persuasion: First Steps,\" American Economic Review, 104, 469-74.","text_sha256":"dd02143a0f650c91c053b8ff4eebd0e7fbbe1989bfe10a728b55d4b651d7ec71"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0019","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"Quigley, D. and A. Walther (2019): \"Contradiction-Proof Information Design,\" Working Paper.\n\nRaghavan, M., S. Barocas, J. Kleinberg, and K. Levy (2020): \"Mitigating Bias in Algorithmic Hiring: Evaluating Claims and Practices,\" in Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency, 469-481.\n\nRambachan, A., J. Kleinberg, J. Ludwig, and S. Mullainathan (2020): \"An Economic Perspective on Algorithmic Fairness,\" in AEA Papers and Proceedings, vol. 110, 91-95.\n\nRayo, L. and I. Segal (2010): \"Optimal Information Disclosure,\" Journal of Political Economy, 118, 949-987.\n\nRosar, F. (2017): \"Test Design Under Voluntary Participation,\" Games and Economic Behavior, 104, 632-655.\n\nSaeedi, M. and A. Shourideh (2020): \"Optimal Rating Design,\" arXiv preprint arXiv:2008.09529.\n\nSalamanca, A. (2021): \"The Value of Mediated Communication,\" Journal of Economic Theory, 192, 105191.\n\nSamet, D. (1998): \"Iterated Expectations and Common Priors,\" Games and Economic Behavior, 24, 131-141.\n\nSayin, M. O. and T. Başar (2021): \"Bayesian Persuasion with State-Dependent Quadratic Cost Measures,\" IEEE Transactions on Automatic Control, 67, 1241-1252.\n\nTirole, J. (2021): \"Digital Dystopia,\" American Economic Review, 111, 2007-48.","text_sha256":"dd2c0720d3bf8103a6c26a527fcf667a1d71054c8a96e7400ec5ca40a618fe8a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0020","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Omitted results and proofs from the main text","text":"## A Omitted results and proofs from the main text","text_sha256":"2f395a8eb83ffc206b249357f65882951209d8ecd2cc62a46fefdd05fa039aa4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0021","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 1 Omitted proofs from Section 3","text":"## A. 1 Omitted proofs from Section 3\n\nFor completeness, we include a proof of Claim 1, which follows from the analysis in Alonso and Câmara (2016, pp. 683-684), in present notation:\n\nProof of Claim 1. For an information structure $\\Pi$, let $\\operatorname{supp}\\langle\\Pi\\rangle$ denote the support of the distribution over posterior beliefs induced by $\\Pi$ and let $\\operatorname{Pr}_{\\Pi}(s)$ denote the unconditional probability of signal $s$ under the prior distribution $\\mu_{0}$, i.e., $\\operatorname{Pr}_{\\Pi}(s)=$ $\\sum_{\\theta \\in \\Theta} \\mu_{0}(\\theta) \\pi(s \\mid \\theta)$. Then, for a given type $\\theta$, their welfare under information structure $\\Pi$ can be written as follows:\n\n$$\n\\begin{aligned}\nw_{\\Pi}(\\theta)=\\mathbb{E}_{\\langle\\Pi \\mid \\theta\\rangle}[w(\\tilde{\\mu}, \\theta)] & =\\sum_{\\mu \\in \\operatorname{supp}\\langle\\Pi\\rangle} \\sum_{s \\in S: \\mu_{s}=\\mu} \\pi(s \\mid \\theta) w(\\mu, \\theta) \\\\\n& =\\sum_{\\mu \\in \\operatorname{supp}\\langle\\Pi\\rangle} \\sum_{s \\in S: \\mu_{s}=\\mu} \\operatorname{Pr}_{\\Pi}(s) \\frac{1}{\\mu_{0}(\\theta)} \\frac{\\mu_{0}(\\theta) \\pi(s \\mid \\theta)}{\\operatorname{Pr}_{\\Pi}(s)} w(\\mu, \\theta) \\\\\n& =\\sum_{\\mu \\in \\operatorname{supp}\\langle\\Pi\\rangle} \\sum_{s \\in S: \\mu_{s}=\\mu} \\operatorname{Pr}_{\\Pi}(s) \\frac{\\mu(\\theta)}{\\mu_{0}(\\theta)} w(\\mu, \\theta) \\\\\n& =\\sum_{\\mu \\in \\operatorname{supp}\\langle\\Pi\\rangle} \\sum_{s \\in S: \\mu_{s}=\\mu} \\operatorname{Pr}_{\\Pi}(s) \\hat{w}(\\mu, \\theta)=\\mathbb{E}_{\\langle\\Pi\\rangle}[\\hat{w}(\\tilde{\\mu}, \\theta)] .\n\\end{aligned}\n$$ $\\square$\n\nProof of Theorem 1. By definition, the point $\\left(\\mu_{0}, \\mathrm{w}\\right) \\in \\operatorname{co}(\\operatorname{graph} \\hat{\\mathrm{w}})$ if and only if a Bayes plausible distribution over posteriors $\\tau$ exists such that $\\mathbb{E}_{\\tau}[\\hat{\\mathrm{w}}(\\mu)]=\\mathrm{w}$. At the same time, the distribution over posteriors induced by an information structure, $\\langle\\Pi\\rangle$, is Bayes plausible, i.e., $\\mathbb{E}_{\\langle\\Pi\\rangle}[\\tilde{\\mu}]=\\mu_{0}$. The result follows from Claim 1. $\\square$\n\nOn closedness Proposition A. 1 collects the results discussed in Section 3. To introduce Proposition A.1, we first introduce the analogue of the Bayes welfare set when the population's welfare depends on the outside observer's actions and together with the information structure, we can flexibly choose a selection from the outside observer's best-response correspondence. Formally, upon observing the realization from\nthe information structure, the outside observer takes an action in a finite set $A$ to maximize their expected payoff. We denote the outside observer's utility by $u: A \\times \\Theta \\mapsto \\mathbb{R}$ and the welfare function by $v: A \\times \\Theta \\mapsto \\mathbb{R}$. Given an information structure, $\\Pi$, let $\\operatorname{supp}\\langle\\Pi\\rangle$ denote the support of the distribution over posteriors $\\langle\\Pi\\rangle$. For each $\\mu_{s} \\in \\operatorname{supp}\\langle\\Pi\\rangle$, let\n\n$$\n\\alpha\\left(\\mu_{s}\\right) \\in \\Delta\\left(\\arg \\max _{a \\in A} \\sum_{\\theta \\in \\Theta} \\mu_{s}(\\theta) u(a, \\theta)\\right) \\equiv \\Delta\\left(a^{*}(\\mu)\\right),\n$$\n\ndenote the outside observer's (possibly mixed) best response. Recall that the Theorem of the Maximum implies that the correspondence $a^{*}(\\mu)$ is upper-hemicontinuous.\n\nDenote by $F$ set of tuples $(\\Pi, \\alpha)$, such that selection $\\alpha$ satisfies Equation A.2. Each $(\\Pi, \\alpha) \\in F$ defines a welfare profile, $w_{\\Pi, \\alpha}: \\Theta \\mapsto \\mathbb{R}^{N}$, such that\n\n$$\nw_{\\Pi, \\alpha}(\\theta)=\\sum_{s \\in S} \\pi(s \\mid \\theta) \\sum_{a \\in A} \\alpha\\left(\\mu_{s}\\right)(a) v(a, \\theta)=\\mathbb{E}_{\\langle\\Pi \\mid \\theta\\rangle}\\left[\\sum_{a \\in A} \\alpha(\\tilde{\\mu})(a) v(a, \\theta)\\right] .\n$$\n\nThe analogue of the Bayes welfare set, which we denote by $\\mathrm{W}_{\\mathrm{BP}}$, is then\n\n$$\n\\mathrm{W}_{\\mathrm{BP}}=\\left\\{\\mathrm{w} \\in \\mathbb{R}^{N}:(\\exists(\\Pi, \\alpha) \\in F) \\text { s.t. } \\mathrm{w}_{i}=w_{\\Pi, \\alpha}\\left(\\theta_{i}\\right) \\forall i \\in\\{1, \\ldots, N\\}\\right\\} .\n$$\n\nWe have the following result:\nProposition A.1. The following hold:\n\n(a) If for each $\\theta \\in \\Theta$, the welfare function $w(\\cdot, \\theta)$ is continuous on $\\Delta(\\Theta)$, the Bayes welfare set W is closed.\n(b) The set $\\mathrm{W}_{\\mathrm{BP}}$ is closed.\n\nProof of Proposition A.1. The proof of part (a) is immediate and hence omitted. Consider then part (b). Define the analogue of the truth-adjusted welfare function $\\hat{v}(a, \\mu, \\theta)$ to be\n\n$$\n\\hat{v}(a, \\mu, \\theta)=\\frac{\\mu(\\theta)}{\\mu_{0}(\\theta)} v(a, \\theta) .\n$$\n\nThe same arguments as in Claim 1 imply that for any pair $(\\Pi, \\alpha) \\in F$ and all $\\theta \\in \\Theta$,\n\n$$\nw_{\\Pi, \\alpha}(\\theta)=\\mathbb{E}_{\\langle\\Pi\\rangle}\\left[\\sum_{a \\in A} \\alpha(\\tilde{\\mu})(\\theta) \\hat{v}(a, \\tilde{\\mu}, \\theta)\\right] .\n$$\n\nDefine the correspondence $\\hat{V}: \\Delta(\\Theta) \\rightrightarrows \\mathbb{R}^{N}$ as follows\n\n$$\n\\hat{V}(\\mu)=\\operatorname{co}\\left\\{\\hat{\\mathrm{v}}(a, \\mu): a \\in a^{*}(\\mu)\\right\\} .\n$$\n\nObserve that $\\hat{V}$ is a non-empty valued, convex-valued, and compact-valued correspondence. Furthermore, $\\hat{V}$ is upper-hemicontinuous. The first part follows immediately from noting that $\\hat{V}$ is the convex hull of finitely many vectors in $\\mathbb{R}^{N}-a^{*}(\\mu)$ is nonempty-and $\\hat{v}$ is bounded. Upper-hemicontinuity of $\\hat{V}$ follows from continuity of $\\hat{\\mathrm{v}}$ in $\\mu$ and upper-hemicontinuity of $a^{*}(\\mu)$. Consequently, the graph of $\\hat{V}$,\n\n$$\n\\operatorname{graph} \\hat{V}=\\left\\{(\\mu, \\mathrm{v}) \\in \\Delta(\\Theta) \\times \\mathbb{R}^{N}: \\mathrm{v} \\in \\hat{V}(\\mu)\\right\\},\n$$\n\nis closed.\nWe now show that similar to Theorem 1, the set $\\mathrm{W}_{\\mathrm{BP}}$ is the section at the prior of the convex hull of the graph of the correspondence $\\hat{V}$, that is\n\n$$\n\\mathrm{W}_{\\mathrm{BP}}=\\left\\{\\mathrm{w} \\in \\mathbb{R}^{N}:\\left(\\mu_{0}, \\mathrm{w}\\right) \\in \\operatorname{co}(\\operatorname{graph} \\hat{V})\\right\\} .\n$$","text_sha256":"a98646756f43a5aae493c95a80461987118d5df8999a834e4276ae4956ddf362"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0022","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 1 Omitted proofs from Section 3","text":"Clearly, if $\\mathrm{w} \\in \\mathrm{W}_{\\mathrm{BP}}$, then $\\left(\\mu_{0}, \\mathrm{w}\\right) \\in \\operatorname{co}(\\operatorname{graph} \\hat{V})$. To see that the opposite holds, let $\\left(\\mu_{0}, \\mathrm{w}\\right) \\in \\operatorname{co}($ graph $\\hat{V})$. Then, a finite collection $\\left(\\tau_{k}, \\mu_{k}, \\tilde{\\mathrm{v}}_{k}\\right)_{k=1}^{M}$ of non-negative weights, beliefs, and correspondence values exists such that $\\sum_{k=1}^{M} \\tau_{k}=1, \\sum_{k=1}^{M} \\tau_{k} \\mu_{k}=$ $\\mu_{0}, \\tilde{\\mathrm{v}}_{k} \\in \\hat{V}\\left(\\mu_{k}\\right)$ for all $k \\in\\{1, \\ldots, M\\}$, and\n\n$$\n\\mathrm{w}=\\sum_{k=1}^{M} \\tau_{k} \\tilde{\\mathrm{v}}_{k} .\n$$\n\nBecause for each $k \\in\\{1, \\ldots, M\\}, \\tilde{\\mathrm{v}}_{k} \\in \\hat{V}\\left(\\mu_{k}\\right)$, the definition of $\\hat{V}$ implies a finite collection of non-negative weights and actions, $\\left\\{\\alpha_{k, l}, a_{l}\\right\\}_{l=1}^{L_{k}}$, exists such that $\\sum_{l=1}^{L_{k}} \\alpha_{l, k}=$ 1 , for all $l \\in\\left\\{1, \\ldots, L_{k}\\right\\}, a_{l} \\in a^{*}\\left(\\mu_{k}\\right)$ and for all $i \\in\\{1, \\ldots, N\\}$,\n\n$$\n\\tilde{\\mathrm{v}}_{k, i}=\\sum_{l=1}^{L_{k}} \\alpha_{k, l} \\hat{v}\\left(a_{l}, \\mu_{k}, \\theta_{i}\\right) .\n$$\n\nDefine $\\Pi$ to be the information structure with signals $S=\\left\\{\\mu_{1}, \\ldots, \\mu_{M}\\right\\}$, and signal distribution $\\pi\\left(\\mu_{k} \\mid \\theta\\right)=\\left(\\mu_{k}(\\theta) / \\mu_{0}(\\theta)\\right) \\tau_{k}$. Furthermore, define $\\alpha$ so that for belief $\\mu_{k}$, $\\alpha\\left(\\mu_{k}\\right) \\in \\Delta\\left(a^{*}\\left(\\mu_{k}\\right)\\right)$ coincides with $\\left\\{\\alpha_{k, l}\\right\\}_{l=1}^{L_{k}}$. By construction, $(\\Pi, \\alpha) \\in F$. Equations A. 5 and A. 6 together imply that\n\n$$\n\\mathrm{w}=\\sum_{k=1}^{M} \\tau_{k} \\sum_{l=1}^{L_{k}} \\alpha_{k, l \\hat{\\mathrm{v}}}\\left(a_{l}, \\mu_{k}\\right)=\\mathbb{E}_{\\langle\\Pi\\rangle}\\left[\\sum_{a \\in A} \\alpha(\\tilde{\\mu})(a) \\hat{\\mathrm{v}}(a, \\mu)\\right] .\n$$\n\nFinally, Equation A. 4 allows us to conclude that the set $\\mathrm{W}_{\\text {BP }}$ is closed: it is the section at the prior of the convex hull of the graph of the correspondence $\\hat{V}$, which is closed. $\\square$","text_sha256":"863b6b2b658d79f31ffa8e197ec4d81024826657d43eaa3e89b8e472aaf271bf"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0023","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 2 Omitted proofs from Section 4","text":"## A. 2 Omitted proofs from Section 4\n\nProof of Theorem 2. To complete the proof of the first direction of Theorem 2, we provide the steps to show that if $\\mathrm{w} \\in \\mathrm{W}_{\\mathrm{P}}$, a direction $\\lambda \\in \\mathbb{R}_{+}^{N} \\backslash\\{0\\}$ exists such that\n\n$$\n\\lambda^{T} \\mathrm{w}=\\max \\left\\{\\lambda^{T} \\mathrm{w}^{\\prime}: \\mathrm{w}^{\\prime} \\in \\mathrm{W}\\right\\},\n$$\n\nThe opposite direction in Theorem 2 immediately follows.\nFix a Pareto efficient w and let $\\Gamma=\\left\\{\\mathrm{w}^{\\prime} \\in \\mathbb{R}^{N}: \\mathrm{w}^{\\prime} \\geq \\mathrm{w}\\right\\}$. Clearly, $\\Gamma$ is convex and int $\\Gamma$ is non-empty. Because w is Pareto efficient, int $\\Gamma \\cap \\mathrm{W}=\\emptyset$. By Minkowski's separating hyperplane theorem, a direction $\\lambda \\in \\mathbb{R}^{N} \\backslash\\{0\\}$ exists such that for all $\\mathrm{w}^{\\prime \\prime} \\in \\mathrm{W}$ and $\\mathrm{w}^{\\prime} \\in \\Gamma$,\n\n$$\n\\lambda^{T} \\mathrm{w}^{\\prime \\prime} \\leq \\lambda^{T} \\mathrm{w}^{\\prime} .\n$$\n\nBecause $\\mathrm{w} \\in \\Gamma$, we have that $\\lambda^{T} \\mathrm{w}^{\\prime \\prime} \\leq \\lambda^{T} \\mathrm{w}$ for all $\\mathrm{w}^{\\prime \\prime}$ in Bayes welfare set. Thus, $\\lambda^{T} \\mathrm{w}=\\max \\left\\{\\lambda^{T} \\mathrm{w}^{\\prime \\prime}: \\mathrm{w}^{\\prime \\prime} \\in \\mathrm{W}\\right\\}$ (cf. Equation 6). Similarly, because $\\mathrm{w} \\in \\mathrm{W}$, then we have that $\\lambda^{T} \\mathrm{w} \\leq \\lambda^{T} \\mathrm{w}^{\\prime}$ for all $\\mathrm{w}^{\\prime} \\in \\Gamma$.\n\nWe now show that $\\lambda \\geq 0$. Let $\\mathrm{t}_{i}$ denote the canonical vector that has a 1 in coordinate $\\theta_{i}$ and 0 otherwise. Then, $\\mathrm{w}+\\mathrm{t}_{i} \\in \\Gamma$, so that\n\n$$\n\\lambda^{T} \\mathrm{w} \\leq \\lambda^{T}\\left(\\mathrm{w}+\\mathrm{t}_{i}\\right) \\Rightarrow 0 \\leq \\lambda\\left(\\theta_{i}\\right) .\n$$\n\nBy definition, $\\lambda \\neq 0$ so that without loss of generality $\\lambda \\in \\Delta(\\Theta)$. $\\square$\n\nProposition A.2. Suppose $w(\\cdot, \\theta)$ is upper-semicontinuous for all $\\theta \\in \\Theta .{ }^{21}$ Suppose $\\mathrm{w}^{*}$ is a limit point of $\\mathrm{W}_{\\mathrm{P}}$. Then, a point $\\mathrm{w}^{* *} \\in \\mathrm{~W}_{\\mathrm{P}}$ exists such that $\\mathrm{w}^{* *} \\geq \\mathrm{w}^{*}$. Consequently, any weakly monotone social welfare function attains a solution in W .\n\nProof of Proposition A.2. Let $\\left(\\mathrm{w}_{n}\\right)_{n \\in \\mathbb{N}} \\subset \\mathrm{~W}_{\\mathrm{P}}$ be such that $\\mathrm{w}_{n} \\rightarrow \\mathrm{w}^{*}$ as $n \\rightarrow \\infty$. For each $n \\in \\mathbb{N}$ a direction $\\lambda_{n} \\in \\Delta(\\Theta)$ exists such that\n\n$$\n\\left(\\forall \\mathrm{w}^{\\prime} \\in \\mathrm{W}\\right) \\lambda_{n}^{T} \\mathrm{w}_{n} \\geq \\lambda_{n}^{T} \\mathrm{w}^{\\prime} .\n$$\n\nSince $\\Delta(\\Theta)$ is compact, then up to a subsequence $\\lambda_{n} \\rightarrow \\lambda_{*}$. Linearity of $\\lambda_{n}^{T} \\mathrm{w}^{\\prime}$ implies that taking limits on both sides of Equation A. 8 we obtain\n\n$$\n\\left(\\forall \\mathrm{w}^{\\prime} \\in \\mathrm{W}\\right) \\lambda_{*}^{T} \\mathrm{w}^{*} \\geq \\lambda_{*}^{T} \\mathrm{w}^{\\prime} .\n$$\n\n[^16]Thus, if $\\mathrm{w}^{*} \\in \\mathrm{~W}$, Theorem 2 implies that $\\mathrm{w}^{*} \\in \\mathrm{~W}_{\\mathrm{P}}$ and $\\lambda_{*} \\in \\Delta(\\Theta)$ is the direction that witnesses this.\n\nNow, for each $\\mathrm{w}_{n}$, a distribution over posteriors $\\tau_{n} \\in \\Delta_{\\mu_{0}}(\\Delta(\\Theta))$ exists such that $\\mathrm{w}_{n}=\\mathbb{E}_{\\tau_{n}}[\\hat{\\mathrm{w}}]$. Because $\\Delta_{\\mu_{0}}(\\Delta(\\Theta))$ is compact, we have that, up to a subsequence, $\\tau_{n} \\rightarrow \\tau^{*} \\in \\Delta_{\\mu_{0}}(\\Delta(\\Theta))$. Aliprantis and Border (2013, Theorem 15.5) implies $\\mathbb{E}_{\\tau}[\\hat{\\mathrm{w}}]$ is upper-semicontinuous as a function of $\\tau$, thus for all $i \\in\\{1, \\ldots, N\\}$ we have that\n\n$$\n\\mathbb{E}_{\\tau^{*}}\\left[\\hat{\\mathrm{w}}_{i}\\right] \\geq \\lim _{n \\rightarrow \\infty} \\mathbb{E}_{\\tau_{n}}\\left[\\hat{\\mathrm{w}}_{i}\\right]=\\lim _{n \\rightarrow \\infty} \\mathrm{w}_{n, i}=\\mathrm{w}_{i}^{*} .\n$$\n\nLet $\\mathrm{w}^{* *} \\equiv \\mathbb{E}_{\\tau^{*}}[\\hat{\\mathrm{w}}]$ and note that it is an element of W that dominates $\\mathrm{w}^{*}$ coordinateby-coordinate. Moreover, because $\\lambda_{*} \\in \\Delta(\\Theta)$, Equation A. 10 implies that $\\lambda_{*}^{T} \\mathrm{w}^{* *} \\geq$ $\\lambda_{*}^{T} \\mathrm{w}^{*}$. This, together with Equation A.9, implies $\\mathrm{w}^{* *} \\in \\mathrm{~W}_{\\mathrm{P}}$, which completes the proof. $\\square$\n\nThe proof of the statements in Observation 1 follows from the following result:\nCorollary A. 1 ((No) Benefit from disclosure). The following hold:\n\n1. All types benefit from disclosure if for all $\\theta \\in \\Theta, \\hat{w}(\\cdot, \\theta)$ is strictly convex in a neighborhood of the prior. Furthermore, if $\\hat{w}(\\cdot, \\theta)$ is everywhere strictly convex, then full disclosure is uniquely Pareto efficient.\n2. No disclosure is Pareto efficient if either\n    (a) a type $\\theta \\in \\Theta$ exists such that $\\hat{w}(\\mu, \\theta)$ is concave in $\\mu$, or\n    (b) a vector $a \\in \\mathbb{R}_{+}^{N}$ and a concave function $w: \\Delta(\\Theta) \\mapsto \\mathbb{R}$ exist such that for all $\\theta \\in \\Theta$, the welfare function is given by $a(\\theta) w(\\mu)+b(\\theta)$.\n\nProof of Corollary A.1. Consider first the conditions in part 1. Toward a contradiction, suppose that $\\mathrm{w}^{N D}$ is in the Pareto frontier. Then, by Corollary 3 a direction $\\tilde{\\lambda} \\in \\Delta(\\Theta)$ exists such that no disclosure is a solution to the supporting Bayesian persuasion problem in direction $\\tilde{\\lambda}$. However, under the conditions in part 1, the indirect utility function is strictly convex in a neighborhood of the prior for all directions $\\lambda \\in \\Delta(\\Theta)$, contradicting that no disclosure is a solution to the supporting Bayesian persuasion problem in some direction $\\tilde{\\lambda}$ in $\\Delta(\\Theta)$. Furthermore, when $\\hat{w}(\\cdot, \\theta)$ is strictly convex, so is $\\hat{v}_{\\lambda}$ for all $\\lambda \\in \\Delta(\\Theta)$ and the value of full disclosure strictly dominates that of any other information structure.","text_sha256":"9fde32097c80398dedddcfbc54ed0de1cf4d8b23085563dad88b2cb0d426ef0f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0024","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 2 Omitted proofs from Section 4","text":"The proof of part 2 follows from Theorem 2 by looking at the solution of the supporting Bayesian persuasion problem in certain directions. Under the conditions in part 2a, no disclosure is a solution to the supporting Bayesian persuasion problem in direction $\\lambda \\in \\Delta(\\Theta)$ such that $\\lambda(\\theta)=1$ and $\\lambda\\left(\\theta^{\\prime}\\right)=0$ for $\\theta^{\\prime} \\neq \\theta$. Instead, under the conditions\nin part 2b, no disclosure is a solution to the supporting Bayesian persuasion problem in direction $\\lambda(\\theta)=\\mu_{0}(\\theta) / a(\\theta)$. $\\square$\n\nProof of Observation 2. The result follows from Corollary 3. For the first part, the condition ensures $\\mu(\\theta) w(\\mu, \\theta)$ is strictly convex for all $\\theta \\in\\left\\{\\theta_{1}, \\theta_{2}\\right\\}$, so the indirect utility function $\\hat{v}_{\\lambda}$ is strictly convex for any $\\lambda \\in \\Delta(\\Theta)$ and full disclosure is the unique solution to the supporting Bayesian persuasion problem. For the second part, the condition ensures $\\mu(\\theta) w(\\mu, \\theta)$ is weakly concave for type $\\theta$, and the result follows from looking at the direction $\\lambda$ such that $\\lambda(\\theta)=1$ and $\\lambda\\left(\\theta^{\\prime}\\right)=0$ for $\\theta^{\\prime} \\neq \\theta$. $\\square$","text_sha256":"232d5b8561fdc0763509a047a4ca685c05d5228f268df60490acbf179cb40f85"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0025","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 3 Omitted proofs in Section 5","text":"## A. 3 Omitted proofs in Section 5\n\nProof of Theorem 3. As explained in the main text, necessity follows from noting\n\n$$\n\\mathrm{w} \\in \\mathrm{~W} \\Rightarrow \\mathrm{w}=\\mathrm{D}_{0} \\sum_{m=1}^{M} \\alpha_{m} \\mu_{m} \\mu_{m}^{T} \\rho \\equiv \\mathrm{D}_{0} \\mathrm{C} \\rho,\n$$\n\nwhere $M \\leq 2 N$ follows from Corollary 1. C is completely positive because it is the convex combination of rank-one non-negative matrices, $\\mu_{m} \\mu_{m}^{T}$. That $\\mathrm{Ce}=\\mu_{0}$ follows from the martingale property of beliefs.\n\nFor sufficiency, consider $\\mathrm{w}=\\mathrm{D}_{0} \\mathrm{C} \\rho$, for some completely positive matrix C, such that $\\mathrm{Ce}=\\mu_{0}$. Then, $\\left\\{\\mathrm{x}_{1}, \\ldots, \\mathrm{x}_{\\mathrm{M}}\\right\\} \\subseteq \\mathbb{R}_{+}^{N}$ exist such that\n\n$$\n\\mathrm{C}=\\sum_{m=1}^{M} \\mathrm{x}_{\\mathrm{m}} \\mathrm{x}_{\\mathrm{m}}^{T} .\n$$\n\nLet $\\sqrt{\\alpha_{m}}=\\sum_{j=1}^{N} \\mathrm{x}_{\\mathrm{m} j}$ and note $\\mathrm{x}_{\\mathrm{m}} /\\left(\\sqrt{\\alpha_{m}}\\right) \\equiv \\mu_{m} \\in \\Delta(\\Theta)$.\n\n$$\n\\mathrm{C}=\\sum_{m=1}^{M} \\alpha_{m}\\left(\\frac{\\mathrm{x}_{\\mathrm{m}}}{\\sqrt{\\alpha_{m}}}\\right)\\left(\\frac{\\mathrm{x}_{\\mathrm{m}}}{\\sqrt{\\alpha_{m}}}\\right)^{T}=\\sum_{m=1}^{M} \\alpha_{m} \\mu_{m} \\mu_{m}^{T} .\n$$\n\nIt remains to show $\\sum_{m=1}^{M} \\alpha_{m}=1$ and $\\sum_{m=1}^{M} \\alpha_{m} \\mu_{m}=\\mu_{0}$. Note that for all $i \\in$ $\\{1, \\ldots, N\\}$,\n\n$$\n(\\mathrm{Ce})_{i}=\\sum_{m=1}^{M} \\alpha_{m} \\mu_{m i} \\sum_{j=1}^{N} \\mu_{m j}=\\sum_{m=1}^{M} \\alpha_{m} \\mu_{m i}=\\mu_{0}\\left(\\theta_{i}\\right) .\n$$\n\nFurthermore,\n\n$$\n\\sum_{i=1}^{N} \\mu_{0}\\left(\\theta_{i}\\right)=1=\\sum_{i=1}^{N} \\sum_{m=1}^{M} \\alpha_{m} \\mu_{m i}=\\sum_{m=1}^{M} \\alpha_{m} .\n$$\n\nThus, an information structure exists that generates the distribution over posteriors $\\left\\{\\alpha_{m}, \\mu_{m}\\right\\}_{m=1}^{M}$. Therefore, $\\mathrm{w} \\in \\mathrm{W}$. $\\square$\n\nProof of Claim 2. If $\\operatorname{Pr}(X)=0$, the statement is trivial. If $\\operatorname{Pr}(X)>0$, denote by $\\mathrm{P}_{\\mathrm{i}}$. the i-th row of the matrix $\\mathrm{P} \\equiv \\mathrm{D}_{0} \\mathrm{C}$, presented as a row-vector. By Bayes' rule, $\\operatorname{Pr}(X)=\\mu_{0}^{T} \\beta$ and $\\operatorname{Pr}\\left(\\theta_{i} \\mid X\\right)=\\left(\\mu_{0 i} \\beta_{i}\\right) /\\left(\\mu_{0}^{T} \\beta\\right)$, so\n\n$$\n\\begin{gathered}\n\\mathbb{E}_{\\Pi}\\left[\\operatorname{Pr}(X \\mid s) \\mid \\theta_{i}\\right]=\\sum_{j=1}^{N} \\mathbb{E}_{\\Pi}\\left[\\operatorname{Pr}\\left[\\theta_{j} \\mid s\\right] \\mid \\theta_{i}\\right] \\operatorname{Pr}\\left(X \\mid \\theta_{j}\\right)=\\mathrm{P}_{\\mathrm{i} \\cdot} \\beta \\\\\n\\mathbb{E}_{\\Pi}[\\operatorname{Pr}(X \\mid s) \\mid X]=\\sum_{i=1}^{N} \\operatorname{Pr}\\left(\\theta_{i} \\mid X\\right) \\mathbb{E}_{\\Pi}\\left[\\operatorname{Pr}(X \\mid s) \\mid \\theta_{i}\\right]=\\sum_{i=1}^{N} \\frac{\\mu_{0 i} \\beta_{i}}{\\mu_{0}^{T} \\beta} \\operatorname{Pi}_{\\mathrm{i} \\cdot \\beta} .\n\\end{gathered}\n$$\n\nHence, the truth-drifting condition can be restated as:\n\n$$\n\\sum_{i=1}^{N} \\frac{\\mu_{0 i} \\beta_{i}}{\\mu_{0}^{T} \\beta} P_{i \\cdot} \\beta \\geq \\mu_{0}^{T} \\beta .\n$$\n\nDefine $\\hat{\\mathrm{C}} \\equiv \\mathrm{PD}_{0}=\\mathrm{D}_{0} \\mathrm{CD}_{0}$. By Theorem 3, $\\hat{\\mathrm{C}}$ is a completely positive matrix such that $\\hat{\\mathrm{C}} \\mu_{0}=\\mathrm{e}$ and $\\mu_{0}^{T} \\hat{\\mathrm{C}} \\mu_{0}=1$. Hence, the truth-drifting condition can be restated in a matrix form as:\n\n$$\n\\left(\\frac{\\mu_{0} * \\beta}{\\mu_{0}^{T} \\beta}\\right)^{T} \\hat{\\mathrm{C}}\\left(\\frac{\\mu_{0} * \\beta}{\\mu_{0}^{T} \\beta}\\right) \\geq \\mu_{0}^{T} \\hat{\\mathrm{C}} \\mu_{0} .\n$$\n\nThe term $\\zeta \\equiv\\left(\\mu_{0} * \\beta\\right) /\\left(\\mu_{0}^{T} \\beta\\right)$ is an element of the simplex $\\Delta(\\Theta)$, equal to $\\mu_{0}$ when $\\beta=$ e. Hence, showing that $\\mu_{0}$ is a minimizer of a quadratic form $\\zeta^{T} \\hat{\\mathrm{C}} \\zeta$ among all $\\zeta \\in \\Delta(\\Theta)$ is enough to prove the result. Noting that we can rely on the Lagrangian approach, at $\\zeta=\\mu_{0}$, the derivative of the quadratic form is collinear to e and hence, collinear to the space $\\Delta(\\Theta)$. Thus, first-order conditions are satisfied. At the same time, $\\hat{\\mathrm{C}}$ is completely positive and thus positive semi-definite. Thus, second-order conditions are satisfied. The result follows. $\\square$\n\nExample A. $4\\left(\\mathrm{w}^{N D}\\right.$ and $\\mathrm{w}^{F D}$ not on the boundary of W). Consider the case of binary types, $\\Theta=\\left\\{\\theta_{1}, \\theta_{2}\\right\\}$. Denote by $\\mu \\in[0,1]$ the probability of type $\\theta_{2}$ and let $\\mu_{0}=1 / 2$. Consider the following welfare function:\n\n$$\nw\\left(\\mu, \\theta_{1}\\right)=\\frac{\\sin (2 \\pi \\mu)}{2(1-\\mu)}, w\\left(\\mu, \\theta_{2}\\right)=\\frac{\\sin (4 \\pi \\mu)}{2 \\mu},\n$$\n\nwith $w\\left(1, \\theta_{1}\\right)$ and $w\\left(0, \\theta_{2}\\right)$ defined by continuity as equal to $-\\pi$ and $2 \\pi$, respectively. Given this welfare function, the truth-adjusted welfare function is\n\n$$\n\\hat{w}\\left(\\mu, \\theta_{1}\\right)=\\sin (2 \\pi \\mu), \\hat{w}\\left(\\mu, \\theta_{2}\\right)=\\sin (4 \\pi \\mu) .\n$$\n\nThe corresponding indirect utility in the supporting Bayesian persuasion problem in the direction $\\lambda=\\left(\\lambda_{1}, \\lambda_{2}\\right)$ is equal to\n\n$$\n\\hat{v}_{\\lambda}(\\mu)=\\lambda_{1} \\sin (2 \\pi \\mu)+\\lambda_{2} \\sin (4 \\pi \\mu) .\n$$\n\nFor any $\\lambda \\in \\mathbb{R}^{2} \\backslash\\{0\\}, \\hat{v}_{\\lambda}\\left(\\mu_{0}\\right)=\\hat{v}_{\\lambda}(1 / 2)=\\hat{v}_{\\lambda}(0)=\\hat{v}_{\\lambda}(1)=0$. Hence, both full disclosure and no disclosure results in zero payoff. At the same time, for any such $\\lambda$, $\\hat{v}_{\\lambda}(\\mu)$ is a non-constant continuous function anti-symmetric around $\\mu=1 / 2$. Hence, it achieves strictly positive values on [0, 1] and $\\operatorname{cav} \\hat{v}_{\\lambda}\\left(\\mu_{0}\\right)>0$ so that optimal disclosure outperforms both full disclosure and no disclosure. Theorem 2-extended to all boundary points-implies that $\\mathrm{w}^{N D}$ and $\\mathrm{w}^{F D}$ are not on the boundary of W.\n\nThe proofs of Proposition 3 and Corollary 4 rely on the graph-theoretic approach in Rayo and Segal (2010), the main properties of which we summarize in Remark A.1:","text_sha256":"bab6bb9244ee10126305b025e13c46718ce0c7a0decbfb747381ea1a33e854c9"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0026","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 3 Omitted proofs in Section 5","text":"Remark A. 1 (Rayo and Segal, 2010). Rayo and Segal (2010) propose the following graphical depiction of an information structure, II. Given a direction $\\lambda$, let the prospect values $\\left(\\frac{\\lambda\\left(\\theta_{i}\\right)}{\\mu_{0}\\left(\\theta_{i}\\right)}, \\rho\\left(\\theta_{i}\\right)\\right)=\\left(\\gamma_{i}, \\rho_{i}\\right)$ for $i=1, \\ldots, N$ be vertices of a graph in $\\mathbb{R}^{2}$. Connect the points $\\left(\\gamma_{i}, \\rho_{i}\\right)$ and $\\left(\\gamma_{j}, \\rho_{j}\\right)$ by an edge if and only if a signal $s$ exists such that $\\pi\\left(s \\mid \\theta_{j}\\right) \\pi\\left(s \\mid \\theta_{i}\\right)>0$. The set of types that have positive probability under $s$ is called the pooling set of signal $s$.\n\nLemmas 2-5 in Rayo and Segal (2010) establish that under any optimal information structure, the following hold:\n\n(a) the posterior expectations of the prospect values induced by any two signals are ranked (in vector order), that is, for any two signals $s, s^{\\prime}$, either $\\left(\\mathbb{E}_{\\mu_{s}}[\\gamma(\\tilde{\\theta})], \\mathbb{E}_{\\mu_{s}}[\\rho(\\tilde{\\theta})]\\right) \\geq$ $\\left(\\mathbb{E}_{\\mu_{s^{\\prime}}}[\\gamma(\\tilde{\\theta})], \\mathbb{E}_{\\mu_{s^{\\prime}}}[\\rho(\\tilde{\\theta})]\\right)$ or the opposite inequality holds.\n(b) prospects appear in the support of some signal only if they lie on a straight line with non-positive slope,\n(c) if the pooling segments ${ }^{22}$ of two signals do not lie on the same line, they can intersect only if they share an endpoint, and\n(d) if two prospect values are ranked and appear in the support of two signals, then the posterior expectations induced by these signals are ranked in the same way.\n\nProof of Proposition 3. By the arguments presented in the main text, any optimal information structure solves the instance of the problem of Rayo and Segal (2010) in which the prospect values are ( $0, \\rho\\left(\\theta_{j}\\right)$ ) for $j \\neq i$ and $\\left(1, \\rho\\left(\\theta_{i}\\right)\\right)$ for the sender and for the receiver, respectively.\n\n[^17]Given the structure of the prospect values in our problem, the property in part (b) implies that $\\theta_{i}$ is never pooled with lower-index types. Furthermore, whenever it is pooled with some type, it is pairwise pooled. The property in part (c) implies that whenever $\\theta_{i}$ is pooled with some type $\\theta_{j}$, then no types $\\theta_{k}, \\theta_{l}$ with $k<j<l$ can be pooled. Together with the property in part (d), this observation implies that whenever $\\theta_{i}$ is pooled with some type $\\theta_{j}$, it is also pooled with all types $\\theta_{k}$ with $k>j$. Moreover, as $\\theta_{i}$ is pooled with increasingly higher-index types, the corresponding posterior expectations increase in vector order, which means that higher signals induce higher reputation yet have a relatively higher proportion of $\\theta_{i}$ (if all types are equally likely, then the probability of pooling $\\theta_{i}$ with $\\theta_{j}$ increases in $j$ ).\n\nIt is left to show that the threshold type-the lowest type with which $\\theta_{i}$ is pooled-is not pooled with any type of lower index. However, because the threshold type is of higher index than $\\theta_{i}$, such pooling could clearly be improved by pooling the threshold type exclusively with $\\theta_{i}$. $\\square$\n\nCalculations for Example 3. Define the following parameterized family of information structures (rows correspond to types and columns to signals):\n\n$$\n\\begin{aligned}\n& \\Pi_{1}(\\alpha, \\beta)=\\left(\\begin{array}{ccc}\n\\alpha & 1-\\alpha & 0 \\\\\n1-\\beta & \\beta & 0 \\\\\n0 & 0 & 1\n\\end{array}\\right), \\quad \\Pi_{2}(\\alpha)=\\left(\\begin{array}{cc}\n\\alpha & 1-\\alpha \\\\\n1 & 0 \\\\\n0 & 1\n\\end{array}\\right), \\\\\n& \\Pi_{3}(\\beta)=\\left(\\begin{array}{cc}\n1 & 0 \\\\\n0 & 1 \\\\\n\\beta & 1-\\beta\n\\end{array}\\right), \\quad \\Pi_{4}(\\alpha, \\beta)=\\left(\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & \\alpha & 1-\\alpha \\\\\n0 & 1-\\beta & \\beta\n\\end{array}\\right) .\n\\end{aligned}\n$$\n\nNote information structures $\\Pi_{1}(1,1)$ and $\\Pi_{4}(1,1)$ coincide and correspond to full disclosure. Likewise, information structures $\\Pi_{2}(0)$ and $\\Pi_{3}(1)$ both correspond to full pooling of types $\\theta_{1}$ and $\\theta_{3}$. Information structures $\\Pi_{2}$ and $\\Pi_{3}$ are the $\\theta_{1}$-noisy-priority policy and the $\\theta_{3}$-noisy-degrading policies, respectively. Like the $\\theta_{3}$-noisy-priority policy, $\\Pi_{1}$ separates $\\theta_{3}$ from $\\theta_{1}$ and $\\theta_{2}$, but unlike the $\\theta_{3}$-noisy-priority policy, it allows for $\\theta_{1}$ and $\\theta_{2}$ to be pooled, which does not affect $\\theta_{3}$ 's expected reputation. Similarly, $\\Pi_{4}$ separates $\\theta_{1}$ from $\\theta_{2}$ and $\\theta_{3}$ like the $\\theta_{1}$-noisy-degrading policy, but unlike this policy, it allows for $\\theta_{2}$ and $\\theta_{3}$ to be pooled.\n\nBy Proposition 2, any solution to the supporting Bayesian persuasion problem solves the instance of the problem of Rayo and Segal (2010) with prospect values $\\left\\{\\left(\\gamma_{i}, \\rho_{i}\\right):\\right.$ $i \\in\\{1,2,3\\}\\}$ for the sender and for the receiver, respectively. For simplicity, we assume that the types are strictly ranked under $\\rho$, i.e., $\\rho_{1}<\\rho_{2}<\\rho_{3}$.\n\nThe property in part (b) in Remark A. 1 implies that if $\\left(\\gamma_{i}, \\rho_{i}\\right)<\\left(\\gamma_{j}, \\rho_{j}\\right)$, then types $\\theta_{i}$ and $\\theta_{j}$ are never pooled (cf. Corollary 4). We can then immediately establish the properties of the boundary information structures in the following cases:\n\n- If $\\gamma_{1}<\\gamma_{2}<\\gamma_{3}$, then the uniquely optimal information structure is full disclosure.\n- If $\\gamma_{2}<\\gamma_{1}<\\gamma_{3}$, then any optimal information structure separates type $\\theta_{3}$ and belongs to class $\\Pi_{1}(\\alpha, \\beta)$.\n- If $\\gamma_{1}<\\gamma_{3}<\\gamma_{2}$, then any optimal information structure separates type $\\theta_{1}$ and belongs to class $\\Pi_{4}(\\alpha, \\beta)$.\n- If $\\gamma_{2}<\\gamma_{3}<\\gamma_{1}$, then an optimal information structure never pools types $\\theta_{2}$ and $\\theta_{3}$ and belongs to class $\\Pi_{2}(\\alpha)$.\n- If $\\gamma_{3}<\\gamma_{1}<\\gamma_{2}$, then an optimal information structure never pools types $\\theta_{1}$ and $\\theta_{2}$ and belongs to class $\\Pi_{3}(\\beta)$.","text_sha256":"77342236eea1e8ca727150b8a00b455d4d95f59107fbeff8e27a50ae5f381798"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0027","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A. 3 Omitted proofs in Section 5","text":"In the remaining case $\\gamma_{3}<\\gamma_{2}<\\gamma_{1}$, no two prospects are ranked. However, by the property in part (a) in Remark A.1, the induced posterior expectations are necessarily ranked. Hence, if all three prospects lie on a straight line, then no disclosure is optimal. In contrast, if the three prospects do not lie on a straight line, then an optimal information structure separates either types $\\theta_{1}$ and $\\theta_{2}$ or types $\\theta_{2}$ and $\\theta_{3}$, and thus belongs to either class $\\Pi_{2}(\\alpha)$ or to class $\\Pi_{3}(\\beta)$.\n\nFinally, it is easy to see that by the same arguments, an optimal information structure for the cases in which $\\gamma_{i}=\\gamma_{j}$ for some $i$ and $j$ belongs to one of the same four classes of information structures.\n\nKnowing the classes of boundary information structures, we can plot the Bayes welfare set in Example 3 by direct calculation. $\\square$","text_sha256":"3c9cabbde5b2ea1a5939fed1eae335a14a281fca749e886a9eb7205746ea04f1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0028","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Online Appendix","text":"## Online Appendix","text_sha256":"bbd7674e759fe83b2c4b5027283e27b84576d5c364e5ff9d4e211700e71c962f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0029","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B Data Limits","text":"## B Data Limits\n\nThe analysis in the paper assumes that the information structure can arbitrarily condition on an individual's payoff-relevant type. However, this assumption does not necessarily hold in many applications of interest. For instance, regulation may prevent the disclosure of protected characteristics, such as gender or race. Thus, when $\\theta$ encompasses such characteristics, considering information structures that respect these restrictions is natural.\n\nIn this section, we extend our analysis by removing this assumption. Formally, we consider the following extension of the model in Section 2. Together with the individuals' types, we are given a data source that is potentially informative about these types. The data source has realizations in a finite set $D \\equiv\\left\\{d_{1}, \\ldots, d_{M}\\right\\}$. We describe the joint distribution over payoff-relevant types and data via the prior distribution on $\\Theta, \\mu_{0}$, and a system of conditional probabilities $\\left\\{\\nu_{0}(\\cdot \\mid \\theta): \\theta \\in \\Theta\\right\\}$, describing the distribution of the data source $d$ conditional on the types $\\theta$. We let $\\eta_{0} \\in \\Delta(D)$ denote the induced marginal distribution on $D .{ }^{23}$ The model in Section 2 corresponds to the case in which $\\Theta=D$ and $\\nu_{0}(d \\mid \\theta)=\\mathbb{1}[d=\\theta]$.\n\nWe assume information can be provided to the outside observer only about datasource realizations, but not an individual's type. Formally, an information structure $\\Pi=(\\pi, S)$ consists of a countable set of labels $S$ and a mapping $\\pi$, which associates to each data-source realization, $d$, a distribution over signals $\\pi(\\cdot \\mid d) \\in \\Delta(S)$. Given an information structure $\\Pi$ and a signal realization $s \\in S$, updated beliefs about $\\theta$ depend only on the updated belief about the realization of $d$. Indeed,\n\n$$\n\\mu_{s}(\\theta)=\\sum_{d \\in D} \\frac{\\mu_{0}(\\theta) \\nu_{0}(d \\mid \\theta)}{\\eta_{0}(d)} \\eta_{s}(d),\n$$\n\nwhere $\\eta_{s}$ is the marginal on $D$ of the updated joint belief on $\\Theta \\times D$. It follows that we can define the welfare function as depending on beliefs about $d$ rather than about $\\theta$. That is, we can define the function $w_{\\dagger}: \\Delta(D) \\times \\Theta \\mapsto \\mathbb{R}$ as follows:\n\n$$\nw_{\\dagger}(\\eta, \\theta)=w(\\mu(\\eta), \\theta),\n$$\n\nwhere the function $\\mu(\\eta)$ is determined by Equation B.1.\nGiven an information structure ( $\\pi, S$ ), the welfare of an individual of type $\\theta$ is\n\n$$\nw_{\\Pi}(\\theta) \\equiv \\mathbb{E}_{\\langle\\Pi \\mid \\theta\\rangle}\\left[w_{\\dagger}(\\tilde{\\eta}, \\theta)\\right]=\\sum_{s \\in S} \\sum_{d \\in D} \\nu_{0}(d \\mid \\theta) \\pi(s \\mid d) w_{\\dagger}\\left(\\eta_{s}, \\theta\\right),\n$$\n\n[^18]and the Bayes welfare set continues to be defined as the set of Bayes welfare profiles.\nWe now show the analysis in the main text extends verbatim. Indeed, by the same arguments as in Section 3, the welfare of an individual of type $\\theta$ under information structure $\\Pi=(\\pi, S)$ can be written as:\n$$\nw_{\\Pi}(\\theta)=\\mathbb{E}_{\\langle\\Pi \\mid \\theta\\rangle}\\left[w_{\\dagger}(\\tilde{\\eta}, \\theta)\\right]=\\mathbb{E}_{\\langle\\Pi\\rangle}\\left[\\hat{w}_{\\dagger}(\\tilde{\\eta}, \\theta)\\right],\n$$\nwhere the truth-adjusted welfare function $\\hat{w}_{\\dagger}$ now takes the form:\n$$\n\\hat{w}_{\\dagger}(\\eta, \\theta)=\\sum_{d \\in D} \\nu_{0}(d \\mid \\theta) \\frac{\\eta(d)}{\\eta_{0}(d)} w_{\\dagger}(\\eta, \\theta) .\n$$\nBy separating the variable on which welfare is conditioned on-the payoff-relevant types, $\\theta$-from the variable about which information is provided-the data source, $d-$Equation B. 4 allows us to provide further insight into the truth-adjusted welfare function in the model in Section 2. Indeed, note the likelihood correction is based on the variable $d$, highlighting that it corresponds to the variable about which information is provided. Similar to before, we can interpret the likelihood-ratio adjustment as describing that each data-source realization $d$ has a budget $\\eta_{0}(d)$ to be distributed across different (data) posteriors $\\eta$. Unlike the analysis before, individuals of type $\\theta$ only own a fraction $\\nu_{0}(d \\mid \\theta)$ of this ratio.\n\nEquation B. 3 implies Theorem 1 immediately extends to this setting:\nTheorem B.1. The Bayes welfare set W satisfies the following:\n\n$$\n\\mathrm{W}=\\left\\{\\mathrm{w} \\in \\mathbb{R}^{N}:\\left(\\eta_{0}, \\mathrm{w}\\right) \\in \\operatorname{co}\\left(\\operatorname{graph} \\hat{\\mathrm{w}}_{\\dagger}\\right)\\right\\} .\n$$\n\nIn what follows, we explore how the Bayes welfare set changes as we change the informativeness of the data source. Intuitively, we would expect that the Bayes welfare set shrinks as data becomes less precise. Proposition B. 1 below shows that this intuition holds when the notion of less precise coincides with the notion of garbling in Blackwell (1953).\n\nFormally, given the distribution of payoff-relevant types $\\mu_{0} \\in \\Delta(\\Theta)$, we wish to understand the effect of different data sources, as described by data-source realizations $D^{\\prime}$ and conditional probability systems $\\left\\{\\nu_{0}^{\\prime}(\\cdot \\mid \\theta) \\in \\Delta\\left(D^{\\prime}\\right): \\theta \\in \\Theta\\right\\}$. Following Blackwell (1953), we say $\\left(D^{\\prime}, \\nu_{0}^{\\prime}\\right)$ is a garbling of ( $D, \\nu_{0}$ ) if a stochastic matrix $G: D \\mapsto \\Delta\\left(D^{\\prime}\\right)$ exists such that for every data-type pair $\\left(d^{\\prime}, \\theta\\right)$,\n\n$$\n\\nu_{0}^{\\prime}\\left(d^{\\prime} \\mid \\theta\\right)=\\sum_{d \\in D} G\\left(d^{\\prime} \\mid d\\right) \\nu_{0}(d \\mid \\theta)\n$$\n\nLet $\\mathrm{W}\\left(\\mu_{0}, w, D, \\nu_{0}\\right)$ denote the Bayes welfare set for prior type distribution $\\theta$ and welfare function $w$, as we vary the (informativeness of the) data source $\\left(D, \\nu_{0}\\right)$. We then have the following:\n\nProposition B. 1 (Data Comparison). $\\mathrm{W}\\left(\\mu_{0}, w, D^{\\prime}, \\nu_{0}^{\\prime}\\right) \\subseteq \\mathrm{W}\\left(\\mu_{0}, w, D, \\nu_{0}\\right)$ for all welfare functions $w$ and type distributions $\\mu_{0}$ if and only if $\\left(D^{\\prime}, \\nu_{0}^{\\prime}\\right)$ is a garbling of $\\left(D, \\nu_{0}\\right)$.","text_sha256":"ec3b90b75c99e27d815dcdd224bb17f056fa90886de4af61d4f3f976ea3ac6be"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0030","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B Data Limits","text":"Proof of Proposition B.1. One direction is straightforward: if $\\left(D^{\\prime}, \\nu_{0}^{\\prime}\\right)$ is a garbling of $\\left(D, \\nu_{0}\\right)$, any distribution of signals conditional on payoff-relevant types induced by some information structure under data source ( $D^{\\prime}, \\nu_{0}^{\\prime}$ ) is feasible under data source $\\left(D, \\nu_{0}\\right)$. Consequently, any welfare profile that can be induced by some information structure under ( $D^{\\prime}, \\nu_{0}^{\\prime}$ ) can be induced under ( $D, \\nu_{0}$ ).\n\nTo obtain the other direction, toward a contradiction, assume $\\left(D^{\\prime}, \\nu_{0}^{\\prime}\\right)$ is not a garbling of $\\left(D, \\nu_{0}\\right)$. Then, by Blackwell (1953), a prior $\\mu_{0} \\in \\Delta(\\Theta)$ and a payoff function $u:$ $A \\times \\Theta \\rightarrow \\mathbb{R}$ exist such that a decision maker with utility $u$ derives strictly greater value from having access to $\\left(D^{\\prime}, \\nu_{0}^{\\prime}\\right)$ than to $\\left(D, \\nu_{0}\\right)$. That is, letting $U(\\mu)$ denote the decision maker's indirect utility, $\\max _{a \\in A} \\mathbb{E}_{\\mu}[u(a, \\theta)]$, we have that:\n\n$$\n\\sum_{\\theta \\in \\Theta} \\mu_{0}(\\theta) \\mathbb{E}_{\\left(D^{\\prime}, \\nu_{0}^{\\prime}\\right)}[U(\\mu) \\mid \\theta]>\\sum_{\\theta \\in \\Theta} \\mu_{0}(\\theta) \\mathbb{E}_{\\left(D, \\nu_{0}\\right)}[U(\\mu) \\mid \\theta] .\n$$\n\nConsider now the Bayes welfare sets given the welfare function $w(\\mu, \\theta)=U(\\mu)$ and prior distribution $\\mu_{0}$, under data sources $\\left(D, \\nu_{0}\\right)$ and $\\left(D^{\\prime}, \\nu_{0}^{\\prime}\\right)$. We have that\n\n$$\n\\begin{aligned}\n\\sum_{\\theta \\in \\Theta} \\mu_{0}(\\theta) \\mathbb{E}_{\\left(D^{\\prime}, \\nu_{0}^{\\prime}\\right)}[U(\\mu) \\mid \\theta] & =\\max _{\\mathrm{w} \\in \\mathrm{~W}\\left(\\cdot, D^{\\prime}, \\nu_{0}^{\\prime}\\right)} \\sum_{\\theta \\in \\Theta} \\mu_{0}(\\theta) \\mathrm{w}(\\theta) \\\\\n& \\leq \\max _{\\mathrm{w} \\in \\mathrm{~W}\\left(\\cdot, D, \\nu_{0}\\right)} \\sum_{\\theta \\in \\Theta} \\mu_{0}(\\theta) \\mathrm{w}(\\theta)=\\sum_{\\theta \\in \\Theta} \\mu_{0}(\\theta) \\mathbb{E}_{\\left(D, \\nu_{0}\\right)}[U(\\mu) \\mid \\theta],\n\\end{aligned}\n$$\n\nwhere the equalities follow because the maximal ex ante payoff is obtained by having full access to available data, and the inequality follows from the assumption that $\\mathrm{W}\\left(\\mu_{0}, w, D^{\\prime}, \\nu_{0}^{\\prime}\\right) \\subseteq \\mathrm{W}\\left(\\mu_{0}, w, D, \\nu_{0}\\right)$ for all $w$ and $\\mu_{0}$. Comparing Equations B. 6 and B. 7 leads to the desired contradiction and the result follows. $\\square$\n\nRemark B. 1 (When to blind an algorithm). Proposition B. 1 stands in contrast with the recommendation in the algorithmic fairness literature to \"blind\" algorithms to sensitive inputs such as race or gender. Indeed, having taken into account the impact of the outside observer's incentives in the population's welfare, allowing the information structure to condition on the individuals' payoff-relevant types leads to the largest Bayes welfare set, thereby (weakly) increasing the value of any social welfare function that is used to choose what information structure to implement.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure B.1: Noisy data in the online marketplace. Bayes welfare sets for different values of $\\sigma \\in\\{0.55,0.7,0.85,1\\}$.\n\nThere may be other reasons outside our model that could justify blinding the information structure to the individuals' types. For instance, Liang et al. (2022) show that when an agent different from the social planner selects a decision-making algorithm, the social planner may prefer to restrict the inputs into the agent's algorithm. Even though in our model algorithms are information structures that send non-binding recommendations, a similar result would hold in our setting.\n\nWe conclude this section with an example that illustrates the following two points. First, whereas Proposition B. 1 shows that less precise data sources limit the ability to generate and distribute welfare via information, the example shows that this effect is not uniform across individuals of different types. Second, the Bayes welfare set may collapse to the no-disclosure Bayes welfare profile for data sources that are strictly more informative than no information in the Blackwell order.\n\nExample B. 2 (Example 2 continued; Noisy Data). Suppose the online marketplace only has access to a noisy estimate of the consumer's type, perhaps from past purchases or undeleted cookies. We model this as a data source that reveals a consumer's type with a fixed precision $\\sigma \\in[1 / 2,1]: D=\\left\\{d_{1}, d_{2}\\right\\}$ and $\\nu_{0}\\left(d_{i} \\mid \\theta_{i}\\right)=\\sigma$. When $\\sigma=1$, the data source is perfectly informative about a consumer's type; when $\\sigma=1 / 2$, the data source is pure noise. More generally, if $\\sigma<\\sigma^{\\prime}$, the data source that corresponds to $\\sigma$ is a garbling of the data source that corresponds to $\\sigma^{\\prime}$.\n\nFigure B. 1 illustrates the Bayes welfare set W for different precision values. Three features are worth noting. First, in line with Proposition B.1, Bayes welfare sets resulting from data sources with lower precision are subsets of those with higher precision.\n\nWhen $\\sigma=1$, the Bayes welfare set naturally coincides with the one in Figure 3b in the main text. Second, at high values of $\\sigma$, lower data precision has asymmetric effects across types: it decreases the maximal payoff of $\\theta_{L}$-consumers without affecting their minimal payoff, yet it increases the minimal payoff of $\\theta_{H}$-consumers without affecting their maximal payoff. Indeed, for sufficiently low values of $\\sigma$, the unique Pareto efficient information structure is the one that maximizes the payoff of $\\theta_{H}$-consumers. That is, in this example, lower data precision benefits $\\theta_{H}$-consumers. Finally, whereas it is immediate that the Bayes welfare set coincides with the no-disclosure profile $\\mathrm{w}^{N D}$ when $\\sigma=1 / 2$, the Bayes welfare set actually collapses to this point at $\\sigma=3 / 5$ : Once $\\sigma<3 / 5$, generating Bayes plausible distributions over posteriors with support outside the interval $[1 / 2,3 / 4)$ is not possible, and on this interval, $w$ is constant. This feature highlights that an incrementally more informative data source may have a discontinuous impact on welfare redistribution possibilities. $\\square$","text_sha256":"aa07ac3ee2ea272fe88708ee1013aec114e7e29f9ea19ea0827c6a6299ec21e5"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0031","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B Data Limits","text":"[^0]:    *We thank the Editor, Emir Kamenica, and three anonymous referees for feedback that has greatly improved this paper. For valuable suggestions and comments, we would like to thank Ricardo Alonso, Odilon Câmara, Navin Kartik, Elliot Lipnowski, Antonio Penta, Jean Tirole, and Kai Hao Yang, as well as seminar participants at Toulouse School of Economics, Columbia, Bonn Winter Theory Workshop 2021, Warwick Theory Workshop 2022, Stony Brook 2022, ESSET 2022, EEA-ESEM 2022, and Virtual Seminars in Economic Theory. We thank Shunsuke Matsuno for excellent research assistance. Smolin acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d'Avenir) program (grant ANR-17-EURE-0010).\n    ${ }^{\\dagger}$ Columbia Business School and CEPR. E-mail: laura.doval@columbia.edu.\n    ${ }^{\\ddagger}$ Toulouse School of Economics and CEPR. E-mail: alexey.v.smolin@gmail.com.\n\n[^1]:    ${ }^{1}$ A matrix $\\mathrm{C} \\in \\mathbb{R}^{N \\times N}$ is completely positive if non-negative vectors $\\mathrm{c}_{1}, \\ldots, \\mathrm{c}_{K} \\in \\mathbb{R}_{+}^{N}$ exist such that $\\mathrm{C}=\\sum_{i=1}^{K} \\mathrm{c}_{i} \\mathrm{c}_{i}^{T}$ (Berman, 1988).\n\n[^2]:    ${ }^{2}$ Because maximizing uncertainty can sometimes be accomplished by providing some information, this notion of data privacy differs from an unambiguous preference for no information disclosure.\n\n[^3]:    ${ }^{3}$ Note that the welfare function $w$ differs from the sender's indirect utility function in Kamenica and Gentzkow (2011), usually denoted by $\\hat{v}$. The indirect utility function is the expectation under $\\mu$ of the welfare function, $w(\\mu, \\cdot)$. That is, $\\hat{v}(\\mu)=\\sum_{\\theta \\in \\Theta} \\mu(\\theta) v(a(\\mu), \\theta)=\\sum_{\\theta \\in \\Theta} \\mu(\\theta) w(\\mu, \\theta)$.\n\n[^4]:    ${ }^{4}$ Claim 2 provides a more general version of this result based on Theorem 3.\n    ${ }^{5}$ Formally, for any information structure, $\\Pi$, and type $\\theta \\in \\Theta, \\mathbb{E}_{\\langle\\Pi\\rangle}\\left[\\tilde{\\mu}(\\theta) / \\mu_{0}(\\theta)\\right]=1$.\n\n[^5]:    ${ }^{6}$ Rosar (2017) and Quigley and Walther (2019) similarly observe that the distribution over posteriors conditional on an individual's type can be written in terms of the modified unconditional distribution.\n\n[^6]:    ${ }^{7}$ For a real-valued function $f, \\operatorname{cav} f$ denotes the smallest concave function that dominates $f$ and vex $f$ denotes the highest convex function dominated by $f$ (Hiriart-Urruty and Lemaréchal, 2004).\n    ${ }^{8}$ See also Aumann and Maschler (1995) and Rayo and Segal (2010).\n\n[^7]:    ${ }^{9}$ The maximum is attained because by definition $\\mathrm{w} \\in \\mathrm{W}$. As we show in Appendix A.2, that $\\lambda \\in$ $\\mathbb{R}_{+}^{N} \\backslash\\{0\\}$ follows from $\\mathrm{w} \\in \\mathrm{W}_{\\mathrm{P}}$.\n\n[^8]:    ${ }^{10}$ Thus, one can always interpret the heterogeneous priors model in Alonso and Câmara (2016) as a model in which the sender and the receiver share the same prior, but the sender assigns weights different than those under the prior $\\mu_{0}$ to each of his possible types.\n\n[^9]:    ${ }^{11}$ For instance, the midpoint $\\mathrm{w}=(3 / 4,1 / 2)$ can only be generated by an information structure that employs at least three signals. One such information structure is given by\n\n    $$\n    \\begin{array}{c|ccc}\n    \\theta_{H} & 1 / 4 & 3 / 8 & 3 / 8 \\\\\n    \\theta_{L} & 0 & 1 / 4 & 3 / 4\n    \\end{array} .\n    $$\n\n[^10]:    ${ }^{12}$ Appendix A. 2 provides weaker conditions under which all types (do not) benefit from disclosure.\n\n[^11]:    ${ }^{13}$ For an even starker example, consider $\\Theta=\\left\\{\\theta_{1}, \\theta_{2}\\right\\}$ and $w\\left(\\mu, \\theta_{1}\\right)=-\\mu^{2}-2 \\mu+3, w\\left(\\mu, \\theta_{2}\\right)=$ $-\\mu^{2}+4 \\mu$, where $\\mu \\equiv \\mu\\left(\\theta_{2}\\right)$. For each type, the welfare function is concave; however, for any prior distribution, the truth-adjusted welfare functions are globally convex. As a result, the unique Pareto efficient information structure is full disclosure.\n\n[^12]:    ${ }^{14}$ Completely positive matrices have been studied extensively as they play an important role in optimization theory, machine learning, and other applications (Berman and Shaked-Monderer, 2003). A completely positive matrix is symmetric and positive-semidefinite, with positive elements; for $N \\leq 4$, the converse is also true.\n\n[^13]:    ${ }^{15}$ An analogous characterization appears in concurrent work by Sayin and Başar (2021), who discuss the computational advantages of working with completely positive matrices to solve information design problems.\n    ${ }^{16} \\mathrm{P}^{2} \\mathrm{e}=\\mathrm{Pe}=\\mathrm{e}$ and $\\mathrm{P}^{2}=\\mathrm{D}_{0} \\mathrm{C}^{\\prime}$, where $\\mathrm{C}^{\\prime} \\equiv \\mathrm{CD}_{0} \\mathrm{C}$ is completely positive because C is symmetric.\n\n[^14]:    ${ }^{17}$ For concreteness, $X$ can be seen as a subset of $\\Theta \\times[0,1]$ equipped with a probability measure that agrees with $\\mu_{0}$ on $\\Theta$ (Green and Stokey, 1978; Gentzkow and Kamenica, 2017).\n    ${ }^{18}$ In a setting in which information respects the state space's ordinal structure, Kartik et al. (2021) formalize the sense in which the drift toward the truth is stronger for more Blackwell informative information structures.\n    ${ }^{19}$ Recall that the relative interior of a set $X$ is the interior of $X$ within its affine hull, which is the set of all affine combinations of elements in $X$.\n\n[^15]:    ${ }^{20}$ We follow the terminology in Galperti et al. (2023), who highlight that individuals with certain types are valuable precisely because of the possibility of pooling them with other types.\n\n[^16]:    ${ }^{21}$ Taking $\\Theta$ to be a compact Polish space, we endow the set of Borel probability measures on $\\Theta, \\Delta(\\Theta)$, and on $\\Delta(\\Theta), \\Delta(\\Delta(\\Theta))$, with the weak* topology, so they are also compact Polish (Aliprantis and Border, 2013, Theorems 15.11 and 15.12).\n\n[^17]:    ${ }^{22}$ By part (b), the pooling set of a signal lies on a segment.\n\n[^18]:    ${ }^{23}$ Although we should index $\\eta_{0}$ by $\\left(\\mu_{0},\\left(\\nu_{0}(\\cdot \\mid \\theta)\\right)_{\\theta \\in \\Theta}\\right)$, we omit this dependence to simplify notation.","text_sha256":"8546af6aa3ab7546540bb12cb807b69706422de0939289ab1b496b1b16ab1a50"}
{"schema_version":"1.0","chunk_id":"alex-smolin:persuasion-and-welfare:2023-09-06:0032","work_id":"alex-smolin:persuasion-and-welfare","paper_id":"alex-smolin:persuasion-and-welfare:2023-09-06","title":"Persuasion and Welfare","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2023-09-06","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md","source_record":"https://arxiv.org/abs/2109.03061","doi":"https://doi.org/10.1086/729067","citation":"Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Laura Doval; Alex Smolin\n\n**Canonical citation:** Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md\n\n**Source record:** https://arxiv.org/abs/2109.03061\n\n**Published record:** https://doi.org/10.1086/729067\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"359b8a84e1950bdc650e1adf0585c5a689c00bbe50a8a672246ec23a66cac4da"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0001","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Daniel F. Garrett; George Georgiadis; Alex Smolin; Balázs Szentes.\n> Canonical citation: Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"9f5d5bdec325aa69fe4a2e3963d86a8a7fc9a1196bfddbafd12ed0b87abbc41d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0002","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Optimal Technology Design","text":"# Optimal Technology Design\n\n**Authors:** Daniel F. Garrett; George Georgiadis; Alex Smolin; Balázs Szentes\n\n**Manuscript date:** 2023-01\n\n#### Abstract\n\nThis paper considers a moral hazard model with agent limited liability. Prior to interacting with the principal, the agent designs the production technology, which is a specification of his cost of generating each output distribution. After observing the production technology, the principal offers a payment scheme and then the agent chooses a distribution over outputs. We show that there is an optimal design involving only binary distributions (i.e., the cost of any other distribution is prohibitively high), and we characterize the equilibrium technology defined on the binary distributions. Notably, the equilibrium payoff of both players is $1 / e$.\n\nJEL classification: D82, D86\nKeywords: moral hazard, limited liability, contract theory\n\n[^0]","text_sha256":"023ca6e933a8ebce909bcc92805c11993f5c9890f5b2ff9e48e6da02e0b31dc1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0003","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nA central result in contract theory is that agency rents are a key source of economic welfare. When analyzing environments with asymmetric information, most microeconomic models take the determinants of these agency frictions as given. In hidden-information models, for example, the distribution of types, which determines information rents, is typically treated as exogenous. Similarly, in principal-agent problems with hidden actions, the production technology available to the agent, which governs the principal's cost of implementing various actions, is usually part of the model description. However, if an agent's payoff depends on agency frictions, then he is likely to pursue generating these frictions in a way that enhances his payoff. The goal of this paper is to reconsider the standard limited-liability moral hazard problem and understand how an agent might maximize rents by optimally designing the production technology.\n\nFor a potential application where such a problem may arise, consider an entrepreneur who is starting a business, and will eventually need venture capital backing to grow it. Prior to contracting with venture capitalists, he must make a host of choices pertaining to the product, the business model, the product market strategy, and so on. ${ }^{1}$ If the venture capitalist has strong bargaining power, the entrepreneur benefits from making choices that exacerbate the moral hazard problem to increase agency rents. Even if there were more profitable alternatives, they may not be considered in the contractual negotiations if the venture capitalist is unaware of them.\n\nIdentifying the agent-optimal technology turns out to be useful even in those environments where the agent has no meaningful way of influencing the production technology and, from his viewpoint, it is given exogenously. Indeed, we are able to express predictions regarding surplus sharing in principal-agent models with limited liability that are robust to the set of available technologies. Specifically, we characterize the entire set of payoff combinations in such models which can arise for some production technology. ${ }^{2}$ The agent-optimal technology corresponds to an extreme point in this set at which the agent's payoff is maximal. This exercise is similar in spirit to that of Bergemann et al. (2015), who characterize the set of consumer and seller payoffs in a model of third-degree price discrimination for some information of the seller. ${ }^{3}$\n\nIn the baseline setup, we consider a risk-neutral agent who can choose a production technology\n\n[^1](or \"project\") before interacting with a principal. ${ }^{4}$ A production technology specifies the agent's cost of each output distribution with support contained in [0, 1]. That is, the only restriction on the available projects is that output is uniformly bounded. Such a bound may represent a physical constraint and is normalized to one. After observing the agent's project, the principal offers a wage contract, which is a mapping from output realizations to monetary compensation. We assume that the agent has limited liability and hence the payment must be non-negative. Finally, the agent chooses an output distribution at a cost determined by his first-stage choice.\n\nIn order to focus on the incentives to generate agency rents we consider a particularly stylized model which abstracts from a number of forces that may have first-order importance in applications. Perhaps most importantly, the agent in our model does not incur any costs in choosing a production technology. In practice, developing a project is likely to require a substantial amount of irreversible investment. In fact, the necessity of such investments may prevent the technology from being renegotiated at the contracting stage. The reason is that, even if both parties are aware of production technologies that are more profitable than the one put forward by the agent, they would not be implemented if modifying the agent's project is too expensive. ${ }^{5}$ Since the agent chooses the production technology before the contracting stage, our model is a hold-up problem. The expense of modifications could also prevent the agent from secretly adjusting the technology after contracting to make the chosen output distribution less costly, helping to make the agent's commitment to the technology credible. ${ }^{6}$\n\nWe also recognize that, in practice, the agent's ability to shape the production technology may be severely limited. For example, some output distributions may be impossible to generate. The constraints on the set of available technologies are likely to be important determinants of the optimal design. However, even when the space of projects is restricted, our main result holds as long as this space includes the agent-optimal technology. Moreover, in the Discussion Section, we demonstrate that our analysis can accommodate certain moment constraints on the domain of production technologies.\n\nOur first main result is that the optimal project involves only binary distributions on $\\{0,1\\} .^{7}$\n\n[^2]In other words, the cost of all other distributions can be assumed to be so high that the principal never wants to implement them, and the agent would never choose them irrespective of the payment scheme. This means that the equilibrium project can be thought of as a task which yields a positive payoff only if completed. The production technology specifies the cost of each probability of completion. The principal's wage contract can be viewed as a bonus paid for project completion, with payments set to zero otherwise.","text_sha256":"87ead872c38c58dcb555e3cd81ad116f3bac41496beec5658d48fe0cf3b52f9b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0004","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"Let us explain the optimality of binary projects. Just like in standard moral hazard problems, output plays a dual role in our model. On the one hand, it is the principal's revenue, and on the other hand, it is an informative signal about the output distribution chosen by the agent, which is used by the principal to incentivize the agent. By the Informativeness Principle, if this signal is made less informative, incentivizing the agent becomes more expensive. The key observation is that each binary distribution with support \\{0, 1\\} can be viewed as a garbling of a distribution with the same mean. Consider now a transformation of each project so that, if the agent incurs a cost of a distribution, output is distributed according to the binary distribution with the same mean. This means that the agent's cost of inducing a given level of expected output remains the same in the transformed project but the principal's cost of implementing it goes up. In this sense, such a transformation exacerbates the moral hazard problem. We show how this observation can be used to replace any project with a binary one for which the agent's payoff is at least as high.\n\nOur second main result is a full characterization of the optimal binary project. In this project, the cost of completing the task with probability $1 / e$ is zero. That is, even if the agent incurs no cost, project completion can be achieved with probability $1 / e$. In equilibrium, the principal offers a bonus which induces the agent to complete the project with probability one. Furthermore, the principal is indifferent between offering this bonus and anything less than that. This indifference condition pins down the cost to the agent of any probability of success between $1 / e$ and one. Since the marginal cost of the success probability is less than one and the maximal output is produced surely, the equilibrium is ex-post efficient. That is, given the equilibrium production technology, the allocation is efficient. It turns out that the optimal project yields an equal split of surplus: both the principal and the agent earn payoff $1 / e$.\n\nThe first-best social surplus in our model is one since projects that can generate output one at no cost are feasible. Of course, the agent does not choose such a project because then the principal could achieve the maximal output without making any payment. To earn rents, the agent designs the technology so that generating high expected output is artificially costly. In fact, since the\nequilibrium output is one with probability one, the only source of distortion induced by the optimal design relates to this cost. This might be considered a form of \"cost padding\", different from others identified in the literature. ${ }^{8}$\n\nAs mentioned above, an identifying feature of the optimal project is that the principal is indifferent between implementing a large range of completion probabilities. Let us explain the economic reasoning behind this feature. Note that, since the agent receives the bonus offered by the principal with the probability of completion, her marginal benefit at each completion probability is the bonus. Hence, at the agent's optimal completion probability, the marginal cost equals the bonus. This means that the principal's expected payment to implement a given completion probability is increasing in the marginal cost at that probability, so lowering this marginal cost makes it more attractive to the principal to implement the completion probability. However, if the principal strictly prefers to implement the equilibrium probability of completion to implementing some smaller probabilities, then the marginal costs at these smaller probabilities can be lowered without affecting the principal's equilibrium choice. Since the cost of a completion probability is the integral of the marginal costs of smaller probabilities, such a modification of the project decreases the agent's total cost and thus increases his overall payoff (while the principal is still willing to offer the same bonus that implements the equilibrium completion probability). The agent can improve any project in this way unless the principal is indifferent between implementing any completion probability which has a positive marginal cost.\n\nWe demonstrate that our main results remain valid even if the agent is risk averse. ${ }^{9}$ In particular, the search for an optimal project can still be restricted to the set of binary projects. Moreover, the optimal binary project is still ex-post efficient; that is, the principal implements completion probability one. The optimal binary project is still characterized by the requirement that the principal must be indifferent between offering the equilibrium bonus and anything less than that. Of course, the equilibrium payoffs of the principal and the agent are no longer $1 / e$ and they depend on the agent's concave utility function.\n\nDespite the fact that our model is stylized, we believe that there are a number of lessons emerging from our analysis which may be useful in applications. First, a general message is that the agent in a technology design problem will often commit to a project that is not cost-efficient. By padding costs, he may be able to earn additional rents. Second, there is reason to expect that, where possible,\n\n[^3]an agent's optimal technology will be binary. That is, the project essentially specifies a task and the agent either completes it or fails to solve it. More generally, the agent is likely to favor designs where the information content of output regarding his effort is limited. The reason is that this makes it more expensive for the principal to induce the target level of effort. A third observation that we expect to generalize is the importance of principal incentive constraints in the design of the agent's project. The agent's design must dissuade the principal from choosing ungenerous incentive schemes that give the agent little rent. Where such incentive constraints are slack, however, it may be possible for the agent to redesign the project to lower his equilibrium cost.","text_sha256":"152f83a714f9252015df455e10711d92045a6ae1458a15482e210cb214a0bb2a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0005","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"Finally, we characterize the payoff combinations that can arise in limited-liability moral hazard problems for some exogenously given technology where output is restricted to the interval [0, 1]. This follows by first identifying the largest payoff the agent can achieve as a function of a given profit of the principal. The domain of this function is the interval [0, 1] because the principal can always guarantee a nonnegative profit by offering zero wage and she cannot get more than the maximal output. This function is shown to be strictly concave and zero at the boundaries of its domain. We argue that a payoff profile can be generated by some production technology if and only if it lies weakly below this curve.\n\nThe rest of the paper is as follows. Next we discuss related literature. Section 2 introduces the model, Section 3 provides our characterization of the agent's optimal project, and Section 4 provides discussion and extensions, including the characterization of the set of possible player payoffs. Appendix A provides proofs not given in the main text and Appendix B discusses the uniqueness of the agent's optimal project. The Online Appendix solves the agent's project design problem when the agent is risk averse.\n\nLiterature. - The limited liability model of moral hazard for a risk-neutral agent is a staple of introductory courses on contract theory, where a restriction to binary output (which emerges endogenously in our setting) is often made for tractability. A classic reference for limited-liability moral hazard is Innes (1990), who demonstrates the optimality of simple debt contracts in a model with a continuum of outputs. More recent treatments of moral hazard with limited liability include Poblete and Spulber (2012) for a model with a continuum of outputs and Ollier and Thomas (2013) for a model with binary output, but complicated by the presence of adverse selection. ${ }^{10}$\n\nWhile the models discussed above feature a risk-neutral agent with limited liability, the al-\n\n[^4]ternative friction commonly explored is risk aversion. Seminal work for the moral hazard model with a risk-averse agent includes Mirrlees (1976), Holmstrom (1979), Grossman and Hart (1983), Rogerson (1985) and MacLeod (2003); see Bolton and Dewatripont (2005) and Holmstrom (2017) for comprehensive treatments, and Georgiadis (2022) for a review. More recently, a strand of this literature has focused on models where the agent can shape the entire distribution of output under different assumptions about the cost of distributions (Hebert, 2018; Bonham and Riggs-Cragun, 2021, Georgiadis et al. , 2022, and Mattsson and Weibull, 2022). The focus of the moral hazard literature, then, has been on contract design taking the agent's technology as given. Our paper departs from this approach by viewing the production technology as a choice of the agent, raising the problem of technology design. We are unaware of this kind of problem being posed elsewhere in the moral hazard literature.\n\nThe question of project design is also related to work on how the primitive contractual environment affects payoffs in moral hazard problems. A relevant example in our context is the development of the Informativeness Principle by Holmstrom (1979), which was later refined for instance by Chaigneau et al. (2019). These papers clarify how additional information about the agent's action can reduce agency costs for the principal.\n\nAnother related paper is Condorelli and Szentes (2020) who study the problem of optimally generating information rents in the context of a bilateral trade model. Before interacting with the seller, the buyer can choose the distribution of her valuation for the seller's good. This choice is observed by the seller before she makes a take-it-or-leave-it offer. It turns out that the equilibrium distribution generates a unit-elastic demand, that is, it makes the seller indifferent between setting any price on its support. This is reminiscent of our optimal binary project which makes the principal indifferent across a range of bonuses. ${ }^{11}$ This similarity might explain why, when the buyer is restricted to choose distributions with support in the interval [0, 1], the equilibrium payoffs of the buyer and the seller are also $1 / e$. We further elaborate on the relationship between these two results in Section 4. ${ }^{12}$\n\n[^5]","text_sha256":"0f716620b7ea692b94bd82ab0318d32b20b3c117d0f23b0969a283ad56c9d9cf"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0006","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"## 2 Model\n\nWe consider a game between a principal (she) and an agent (he), which proceeds as follows. In the first stage, the agent chooses a cost function $c: \\mathcal{F} \\rightarrow \\mathbb{R}_{+} \\cup\\{+\\infty\\}$, where $\\mathcal{F}$ denotes the set of CDFs with support on $[0,1]$. We refer to such a function $c$ as a project. Then, after observing $c$, the principal offers a payment scheme $w:[0,1] \\rightarrow \\mathbb{R}_{+}$, which is restricted to be Borel-measurable. ${ }^{13}$ Finally, after observing the offered payment scheme, the agent chooses a distribution $F \\in \\mathcal{F}$, and output is realized according to $F .{ }^{14}$ If the realized output is $x$ then the agent's and principal's payoffs are $w(x)-c(F)$ and $x-w(x)$, respectively. Both parties are expected payoff maximizers.\n\nNotation.- For each $F \\in \\mathcal{F}$, let $\\mu_{F}$ denote the expected value of $F$, that is, $\\mu_{F}=\\int_{0}^{1} x d F(x)$. The set of projects and the set of Borel-measurable payment schemes are denoted by $\\mathcal{C}$ and $\\mathcal{W}$, respectively. We refer to a triple $(c, w, F) \\in \\mathcal{C} \\times \\mathcal{W} \\times \\mathcal{F}$ as an outcome. Let $U$ and $\\Pi$ denote the expected payoffs of the agent and the principal defined on the outcomes; that is,\n\n$$\nU(c, w, F)=\\int_{0}^{1} w(x) d F(x)-c(F), \\text { and } \\Pi(w, F)=\\int_{0}^{1}[x-w(x)] d F(x)\n$$\n\nEquilibrium in a Project.-Loosely speaking, we wish to call a pair $(w, F) \\in(\\mathcal{W}, \\mathcal{F})$ an equilibrium in project $c \\in \\mathcal{C}$ if it satisfies the following two requirements. First, the distribution $F$ is incentive compatible (or a best response) for the agent given project $c$ and the payment schedule $w$. Second, the payment scheme $w$ is incentive compatible for the principal given project $c$. Formal statements of the incentive compatibility constraints and the equilibrium definition are provided below.\n\nOptimal Projects. - We define the project $c$ to be optimal if there is an equilibrium $(w, F)$ in $c$ such that the agent's payoff in outcome $(c, w, F)$ is larger than in any outcome $\\left(c^{\\prime}, w^{\\prime}, F^{\\prime}\\right)$ such that $\\left(w^{\\prime}, F^{\\prime}\\right)$ is an equilibrium in $c^{\\prime}$. Thus, we assess the optimality of a project $c$ that induces multiple equilibria by considering those which give the highest payoff to the agent. This is in line with the approach prevalent in mechanism design, where the designer is permitted to pick the most favorable equilibrium.\n\nIncentive Compatibility.- We say that choosing $F$ is incentive compatible for the agent in the\n\n[^6]subgame $(c, w)$ if\n$$\nU(c, w, F) \\geq U\\left(c, w, F^{\\prime}\\right) \\text { for all } F^{\\prime} \\in \\mathcal{F} \\text {. }\n$$\nTo describe the principal's incentive constraint is harder because the agent may not have a best response in a subgame generated by a pair $(c, w)$. In turn, this can make it difficult to assess the profitability of certain deviations. To circumvent this problem, we define the value of the agent, $u(c, w)$, in each subgame $(c, w)$, by\n$$\nu(c, w) \\equiv \\sup _{F \\in \\mathcal{F}} U(c, w, F)\n$$\nWe aim to define the value of the principal in a subgame $(c, w)$ by reference to sequences of distributions along which the agent's payoff converges to his value. In general, there may be many such sequences, potentially generating different limit payoffs to the principal. Let $\\mathbf{F}^{c, w}$ denote the set of sequences of distributions $\\left(F_{n}\\right)$ along which the agent's payoff converges to $u(c, w)$. Formally, $\\left(F_{n}\\right) \\in \\mathbf{F}^{c, w}$ if and only if $\\lim _{n \\rightarrow \\infty} U\\left(c, w, F_{n}\\right)=u(c, w)$. Then, the principal's value in subgame $(c, w)$ is given as\n$$\n\\pi(c, w) \\equiv \\sup \\left\\{\\limsup _{n \\rightarrow \\infty} \\Pi\\left(w, F_{n}\\right):\\left(F_{n}\\right) \\in \\mathbf{F}^{c, w}\\right\\} .\n$$\nEvaluating the principal's value by reference to the supremum is again in the spirit of the approach prevalent in mechanism design, where the principal is permitted to pick the most favorable best response of the agent. The principal's incentive compatibility constraint guaranteeing that she offers payment schedule $w$ in project $c$ can be stated as follows: for all $w^{\\prime} \\in \\mathcal{W}$,\n$$\n\\pi(c, w) \\geq \\pi\\left(c, w^{\\prime}\\right) .\n$$\nThat is, the principal cannot gain by deviating to a payment schedule $w^{\\prime}$, whether or not the agent has a best response to $w^{\\prime}$.\n\nEquipped with the incentive compatibility constraints, we are ready to define equilibria formally.\nDefinition. The pair $(w, F)$ is said to be an equilibrium in project $c$ if\n\n(i) the distribution $F$ satisfies (1),\n(ii) the payment scheme $w$ satisfies (2), and\n(iii) $\\Pi(w, F)=\\pi(c, w)$.\n\nObserve that Part (iii) requires that the distribution $F$ indeed generates the principal's value in\nsubgame $(c, w)$.\nBinary Projects and Linear Contracts.- As mentioned in the Introduction, binary projects play an important role in our analysis. Next, we formally define these projects. We call a distribution in $\\mathcal{F}$ binary if its support is contained in $\\{0,1\\}$. For each $\\mu \\in[0,1]$, let $B_{\\mu}$ denote the binary CDF which specifies an atom of size $\\mu$ at one. Note that the mean of $B_{\\mu}$ is also $\\mu$. Let $\\mathcal{B}$ denote the set of binary distributions; that is, $\\mathcal{B}=\\left\\{B_{\\mu}: \\mu \\in[0,1]\\right\\}$. We call a project $c$ binary if $c(F)=+\\infty$ whenever $F \\notin \\mathcal{B}$.\n\nIn each binary project, the principal always finds it optimal to offer a compensation scheme which pays zero if output is zero. So, the optimal payment scheme can be summarized by a single bonus, $b$, which is paid to the agent if output is one. If the project is binary, the wage at output $x \\notin\\{0,1\\}$ is irrelevant, so without loss of generality, such a wage contract can be assumed to be linear, denoted by $w_{b}$, that is, $w_{b}(x)=b x$ for all $x$.\n\nIf the principal offers a linear contract, $w_{b}$, the output distribution affects the payoffs of the agent and the principal only through its mean. That is, whether or not the project is binary,\n\n$$\nU\\left(c, w_{b}, F\\right)=\\mu_{F} b-c(F), \\text { and } \\Pi(w, F)=\\mu_{F}(1-b) .\n$$\n\nThis implies that, if the principal offers a linear contract to which the agent has a best response, this best response must involve a distribution which is the least costly among those with the same mean.","text_sha256":"e6a64172c3d8ecd999fce10744eddfe14533b62b4a37b935fc76e75d7269b348"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0007","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Main Results","text":"## 3 Main Results\n\nThis section is devoted to our two main results. In the next section, we show that it suffices to restrict attention to binary projects and, in Section 3.2, we fully characterize an optimal binary project.","text_sha256":"9af4b767c72f4db5de577770348f84e684c6662dbc08560c50cecd9c1fd693eb"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0008","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.1 Binary Projects","text":"### 3.1 Binary Projects\n\nIn this section, we fix a project $c^{*}$ and an equilibrium $\\left(w^{*}, F^{*}\\right)$ in $c^{*}$. Our aim is to construct a binary project $\\widetilde{c}$ and an equilibrium $(\\widetilde{w}, \\widetilde{F})$ in $\\widetilde{c}$ so that the outcome $(\\widetilde{c}, \\widetilde{w}, \\widetilde{F})$ Pareto dominates the outcome $\\left(c^{*}, w^{*}, F^{*}\\right)$. Since $c^{*}$ can be an optimal project, this result implies that there exists an optimal project in the class of binary projects.\n\nAs explained in the Introduction, the key observation for this result is that an output realization not only determines the principal's payoff, but also serves as an informative signal about the agent's action. If this signal is made less informative in the sense of Blackwell, incentivizing the agent becomes harder for the principal. To see how an output distribution can be made less informative, consider the following garbling: instead of observing output $x$, the principal observes output one with probability $x$ and output zero otherwise. That is, the garbling of each $F \\in \\mathcal{F}$ is $B_{\\mu_{F}}$, so the expected output is unaffected. In fact, $B_{\\mu_{F}}$ is the least informative garbling of $F$, as the same transformation can be applied to any other garbling of $F$ which would again result in $B_{\\mu_{F}}$. So, if the principal could contract only on the realization of $B_{\\mu_{F}}$ but not on that of $F$, her wage cost of implementing $F$ would increase. We next explain how this observation can be used to transform the project $c^{*}$ to a binary one which is more beneficial for the agent.\n\nThe idea behind the construction of the binary project, $\\widetilde{c}$, is as follows. First, define the agent's cost of any binary distribution to be the cost of the cheapest distribution in project $c^{*}$ with the same mean. In this binary project, the principal's wage cost of attaining any level of expected output is higher than in project $c^{*}$. In fact, the wage cost of generating $\\mu_{F^{*}}$ may be so high that the principal prefers to implement a distribution with a lower mean, thus saving on payments to the agent. In this case, the payoffs of both parties can be lower. Therefore, we further modify the binary project by reducing the agent's cost of $B_{\\mu_{F^{*}}}$ so that the principal can implement it at exactly the same wage cost as that of $F^{*}$ in $c^{*}$.\n\nBefore stating the main result of this section, let us introduce an additional piece of notation. Note that the expected payment in outcome $\\left(c^{*}, w^{*}, F^{*}\\right)$ is $\\mathbb{E}_{F^{*}}\\left[w^{*}\\right]$. If $\\mu_{F^{*}}>0$ (as must be the case if the outcome is optimal for the agent), we can define $b^{*}$ to equal $\\mathbb{E}_{F^{*}}\\left[w^{*}\\right] / \\mu_{F^{*}}$. We can then observe that ${ }^{15}$\n\n$$\n\\mathbb{E}_{F^{*}}\\left[w^{*}\\right]=\\mu_{F^{*}} b^{*}=\\mathbb{E}_{F^{*}}\\left[w_{b^{*}}\\right]=\\mathbb{E}_{B_{\\mu_{F^{*}}}}\\left[w_{b^{*}}\\right] .\n$$\n\nThat is, the expected payment induced by the pair $\\left(w^{*}, F^{*}\\right)$ is the same as that induced by $\\left(w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)$.\n\nProposition 1. Suppose that $\\left(w^{*}, F^{*}\\right)$ is an equilibrium in project $c^{*}$ with $\\mu_{F^{*}}>0$. Then there exists a binary project, $\\widetilde{c}$, such that\n\n(i) $\\left(w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)$ is an equilibrium in $\\widetilde{c}$,\n(ii) $U\\left(c^{*}, w^{*}, F^{*}\\right) \\leq U\\left(\\widetilde{c}, w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)$, and\n\n[^7](iii) $\\Pi\\left(w^{*}, F^{*}\\right)=\\Pi\\left(w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)$.\n\nLet us describe the binary project $\\widetilde{c}$ and the main arguments in the proof of the proposition. It turns out that, in this binary project, the agent's rent can be ensured by making it hard for the principal to dissuade the agent from deviating downwards (i.e., to distributions with lower means). Upwards deviations need not play a role, so we specify the agent's cost of each $B_{\\mu}$ with $\\mu>\\mu_{F^{*}}$ to be infinity throughout the construction. We now explain the two steps of constructing $\\widetilde{c}$ from $c^{*}$ in more detail. In the first step, we take the agent's cost of $B_{\\mu_{F^{*}}}$ to also be infinite. For each $\\mu<\\mu_{F^{*}}$, we specify the cost of $B_{\\mu}$ to be the cost of the cheapest distribution in project $c^{*}$ with expectation $\\mu .^{16}$ We then prove that, in order to achieve any expected output, the principal must make a higher expected payment in this binary project than in $c^{*}$. In the second step, we redefine the agent's cost of $B_{\\mu_{F^{*}}}$ so that the principal's wage cost of implementing $B_{\\mu_{F^{*}}}$ is exactly $\\mathbb{E}_{F^{*}}\\left[w^{*}\\right]=\\mathbb{E}_{B_{\\mu_{F^{*}}}}\\left[w_{b^{*}}\\right]$, thus obtaining the project $\\widetilde{c}$. We show that the agent's cost of $B_{\\mu_{F^{*}}}$ in $\\widetilde{c}$ is less than $c^{*}\\left(F^{*}\\right)$. This means that, by Equation (4), Parts (ii) and (iii) of the proposition are satisfied.\n\nWe now explain how to obtain Part (i). Note that, by our choice of the agent's cost of the distribution $B_{\\mu_{F^{*}}}$ in project $\\widetilde{c}$, the agent best responds to $w_{b^{*}}$ by choosing $B_{\\mu_{F^{*}}}$. Therefore, to prove that $\\left(w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)$ is an equilibrium in this project, we need to demonstrate only that offering $w_{b^{*}}$ is incentive compatible for the principal. By construction, if the principal wants to implement $B_{\\mu_{F^{*}}}$, she offers payment schedule $w_{b^{*}}$. She therefore receives a payoff of $\\Pi\\left(w^{*}, F^{*}\\right)$. As explained, attaining any other expected output $\\mu\\left(\\mu \\neq \\mu_{F^{*}}\\right)$ is more expensive for the principal in $\\widetilde{c}$ than in $c^{*}$. Therefore, since the principal found it optimal to implement $F^{*}$ in $c^{*}$, she optimally chooses to implement $B_{\\mu_{F^{*}}}$ in $\\widetilde{c}$ by offering $w_{b^{*}}$; that is, $w_{b^{*}}$ is incentive compatible in $\\widetilde{c}$.\n\nTowards the first step described above, let us define the binary project, $\\widehat{c}$, as follows:\n\n$$\n\\widehat{c}\\left(B_{\\mu}\\right)= \\begin{cases}\\inf \\left\\{c^{*}(F): \\mu_{F}=\\mu\\right\\} & \\text { if } \\mu<\\mu_{F^{*}} \\\\ \\infty & \\text { otherwise }\\end{cases}\n$$","text_sha256":"546cb3507ca9e1329c9033da0ddfaaaa6446a036f604c9614e83ae83a17aa55a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0009","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.1 Binary Projects","text":"We next formalize the aforementioned implication of the Informativeness Principle; in particular, we demonstrate that the principal is worse off in $\\widehat{c}$ than in $c^{*}$. In fact, we show that, from the principal's point of view, the transformed project $\\widehat{c}$ is worse than being restricted to linear contracts in $c^{*}$ in the sense that each contract $w_{b}$ generates weakly more profit to the principal in project $c^{*}$ than in $\\widehat{c}$.\n\n[^8]Lemma 1. For all $b \\in[0,1], \\pi\\left(\\widehat{c}, w_{b}\\right) \\leq \\pi\\left(c^{*}, w_{b}\\right)$.\nProof. See the Appendix. QED\n\nLet us illustrate the argument behind the proof of this lemma for the case where the principal's value in subgame $\\left(c^{*}, w_{b}\\right), \\pi\\left(c^{*}, w_{b}\\right)$, is generated by a best response of the agent. Since the agent's expected payment generated by the linear contract $w_{b}$ depends only on the expected output, he chooses a distribution only if it is the cheapest among those with the same mean. Therefore, the agent's value in subgame ( $c^{*}, w_{b}$ ) is\n\n$$\n\\sup _{\\mu \\in[0,1]}\\left\\{\\mu b-\\inf \\left\\{c^{*}(F): F \\in \\mathcal{F}, \\mu_{F}=\\mu\\right\\}\\right\\} .\n$$\n\nThis is the same problem as the one which determines the agent's value in the subgame $\\left(\\widehat{c}, w_{b}\\right)$, except that in the latter, the domain is effectively restricted to be $\\left[0, \\mu_{F^{*}}\\right)$. Suppose now that $F$ is incentive compatible in $\\left(c^{*}, w_{b}\\right)$ and generates the principal's value. That is, $\\mu_{F}$ solves the problem in (5) and $\\pi\\left(c^{*}, w_{b}\\right)=\\Pi\\left(w_{b}, F\\right)$. If $\\mu_{F}<\\mu_{F^{*}}$, then $\\mu_{F}$ also solves the agent's problem with the restricted domain, implying that $B_{\\mu_{F}}$ is incentive compatible in $\\left(\\widehat{c}, w_{b}\\right)$. In this case, $\\pi\\left(\\widehat{c}, w_{b}\\right)=\\mu_{F}(1-b)=\\pi\\left(c^{*}, w_{b}\\right)$. If $\\mu_{F} \\geq \\mu_{F^{*}}$, then the principal's value is at least $\\mu_{F^{*}}(1-b)$ in the subgame $\\left(c^{*}, w_{b}\\right)$, so $\\pi\\left(c^{*}, w_{b}\\right) \\geq \\mu_{F^{*}}(1-b) \\geq \\pi\\left(\\widehat{c}, w_{b}\\right)$, where the second inequality holds because, in project $\\widehat{c}$, the agent never chooses a distribution which has mean larger than $\\mu_{F^{*}}$.\n\nWe are now ready to define project $\\widetilde{c}$. Our aim is to modify $\\widehat{c}$ at $B_{\\mu_{F^{*}}}$ so that $\\left(w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)$ is an equilibrium in project $\\widetilde{c}$. On the one hand, this requires the cost of $B_{\\mu_{F^{*}}}$ to be sufficiently small to guarantee that $B_{\\mu_{F^{*}}}$ is a best response to $w_{b^{*}}$. On the other hand, this cost cannot be too small, for otherwise $B_{\\mu_{F^{*}}}$ could be implemented with a bonus smaller than $b^{*}$. Therefore, we specify the cost of $B_{\\mu_{F^{*}}}$ to be the largest cost at which the agent still best responds to $w_{b^{*}}$ by choosing $B_{\\mu_{F^{*}}}$. This cost, denoted by $\\bar{c}$, satisfies $\\mu_{F^{*}} b^{*}-\\bar{c}=\\sup \\left\\{\\mu b^{*}-\\widehat{c}\\left(B_{\\mu}\\right)\\right\\}$. The binary project $\\widetilde{c}$ is defined as follows:\n\n$$\n\\widetilde{c}(F)= \\begin{cases}\\bar{c} & \\text { if } F=B_{\\mu_{F^{*}}}, \\\\ \\widehat{c}(F) & \\text { if } F \\neq B_{\\mu_{F^{*}}} .\\end{cases}\n$$\n\nNext, we demonstrate that the outcome $\\left(\\widetilde{c}, w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)$ Pareto dominates $\\left(c^{*}, w^{*}, F^{*}\\right)$. To this end, we first argue that the cost of $B_{\\mu_{F^{*}}}$ in project $\\widetilde{c}$ is weakly smaller than $c^{*}\\left(F^{*}\\right)$. Suppose, for a contradiction, that $\\bar{c}>c^{*}\\left(F^{*}\\right)$. Then,\n\n$$\n\\mu_{F^{*}} b^{*}-c^{*}\\left(F^{*}\\right)>\\mu_{F^{*}} b^{*}-\\bar{c}=\\sup \\left\\{\\mu b^{*}-\\widehat{c}\\left(B_{\\mu}\\right)\\right\\}=\\sup \\left\\{\\mu_{F} b^{*}-c^{*}(F): F \\in \\mathcal{F}, \\mu_{F}<\\mu_{F^{*}}\\right\\},\n$$\n\nwhere the two equalities follow from the definitions of $\\bar{c}$ and $\\widehat{c}$, respectively. By continuity, this chain implies the existence of $b<b^{*}$ such that\n\n$$\n\\mu_{F^{*}} b-c^{*}\\left(F^{*}\\right)>\\sup \\left\\{\\mu_{F} b-c^{*}(F): F \\in \\mathcal{F}, \\mu_{F}<\\mu_{F^{*}}\\right\\} .\n$$\n\nThis means that offering the linear contract $w_{b}$ in project $c^{*}$ provides the principal with a value at least $\\mu_{F^{*}}(1-b)$, which is strictly more than her equilibrium payoff, $\\Pi\\left(w^{*}, F^{*}\\right)=\\mu_{F^{*}}\\left(1-b^{*}\\right)$. This contradicts the incentive compatibility of $w^{*}$ in $c^{*}$.\n\nWe are now ready to show that the agent is weakly better off in the outcome $\\left(\\widetilde{c}, w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)$ than in $\\left(c^{*}, w^{*}, F^{*}\\right)$. Indeed,\n\n$$\nU\\left(c^{*}, w^{*}, F^{*}\\right)=\\mu_{F^{*}} b^{*}-c^{*}\\left(F^{*}\\right) \\leq \\mu_{F^{*}} b^{*}-\\widetilde{c}\\left(B_{\\mu_{F^{*}}}\\right)=U\\left(\\widetilde{c}, w_{b^{*}}, B_{\\mu_{F^{*}}}\\right),\n$$\n\nwhere the equalities follow from (4) and the inequality follows from $\\widetilde{c}\\left(B_{\\mu_{F^{*}}}\\right)=\\bar{c} \\leq c^{*}\\left(F^{*}\\right)$. Also note that (4) implies that the principal's payoffs are the same in these two outcomes:\n\n$$\n\\Pi\\left(w^{*}, F^{*}\\right)=\\mu_{F^{*}}-\\mathbb{E}_{F^{*}}\\left[w^{*}\\right]=\\mu_{F^{*}}\\left(1-b^{*}\\right)=\\Pi\\left(w_{b^{*}}, B_{\\mu_{F^{*}}}\\right) .\n$$\n\nWe defined $\\widetilde{c}\\left(B_{\\mu_{F^{*}}}\\right)$ so that the payment schedule $w_{b^{*}}$ implements $B_{\\mu_{F^{*}}}$ in project $\\widetilde{c}$. Next, we confirm that the principal cannot implement $B_{\\mu_{F^{*}}}$ in project $\\widetilde{c}$ with any payment schedule $w_{b}$ such that $b<b^{*}$. This means that the principal's value from offering $w_{b}$ in project $\\widetilde{c}$ is the same as in project $\\widehat{c}$.\n\nLemma 2. For all $b \\in\\left[0, b^{*}\\right), \\pi\\left(\\widetilde{c}, w_{b}\\right)=\\pi\\left(\\widehat{c}, w_{b}\\right)$.\nProof. See the Appendix. $\\square$","text_sha256":"8a2d1b442b991b5a868e0656b8fd4b6ebea6b862abb3f3c053130b926d0ddbec"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0010","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.1 Binary Projects","text":"Let us illustrate the argument of the proof for the case where the agent has a best response to $w_{b^{*}}$ in project $\\widehat{c}$, say $B_{\\mu^{\\prime}}$ (with $\\mu^{\\prime}<\\mu_{F^{*}}$ ); that is, $\\mu^{\\prime} b^{*}-\\widehat{c}\\left(B_{\\mu^{\\prime}}\\right)=\\sup \\left\\{\\mu b^{*}-\\widehat{c}\\left(B_{\\mu}\\right)\\right\\}$. By the definition of $\\widetilde{c}$, this means that $\\mu^{\\prime} b^{*}-\\widetilde{c}\\left(B_{\\mu^{\\prime}}\\right)=\\mu_{F^{*}} b^{*}-\\widetilde{c}\\left(B_{\\mu_{F^{*}}}\\right)$. Since $\\mu^{\\prime}<\\mu_{F^{*}}$, this equality implies that, for all $b \\in\\left[0, b^{*}\\right), \\mu^{\\prime} b-\\widetilde{c}\\left(B_{\\mu^{\\prime}}\\right)>\\mu_{F^{*}} b-\\widetilde{c}\\left(B_{\\mu_{F^{*}}}\\right)$, implying that $B_{\\mu_{F^{*}}}$ is not incentive compatible in $\\left(\\widetilde{c}, w_{b}\\right)$. Since the projects $\\widetilde{c}$ and $\\widehat{c}$ are identical on the rest of their domains, the statement of the lemma follows.\n\nFinally, we are ready to prove Proposition 1.\nProof of Proposition 1. Observe that the outcome $\\left(\\widetilde{c}, w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)$ satisfies Parts (ii) and (iii)\nof the proposition by Equations (6) and (7), respectively. Therefore, we only need to establish Part (i); that is, we need to argue that $\\left(w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)$ is an equilibrium in project $\\widetilde{c}$. By the definition of $\\bar{c}, B_{\\mu_{F^{*}}}$ is incentive compatible in the subgame $\\left(\\widetilde{c}, w_{b^{*}}\\right)$. Next, we prove that $w_{b^{*}}$ is incentive compatible in $\\widetilde{c}$.\n\nIf $b>b^{*}$, then\n\n$$\n\\pi\\left(\\widetilde{c}, w_{b}\\right) \\leq \\mu_{F^{*}}(1-b)<\\mu_{F^{*}}\\left(1-b^{*}\\right)=\\Pi\\left(w_{b^{*}}, B_{\\mu_{F^{*}}}\\right),\n$$\n\nwhere the first inequality follows from $\\widetilde{c}\\left(B_{\\mu}\\right)=\\infty$ for all $\\mu>\\mu_{F^{*}}$, and the second inequality is implied by $b>b^{*}$.\n\nIf $b<b^{*}$, then\n\n$$\n\\pi\\left(\\widetilde{c}, w_{b}\\right) \\leq \\pi\\left(c^{*}, w_{b}\\right) \\leq \\Pi\\left(w^{*}, F^{*}\\right)=\\Pi\\left(w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)\n$$\n\nwhere the first inequality follows from Lemmas 1 and 2, the second one follows from $\\left(w^{*}, F^{*}\\right)$ being an equilibrium outcome in project $c^{*}$, and the equality is implied by the definition of $b^{*}$. $\\square$","text_sha256":"14e20ddefb4795c3702328be5bd4eab6edc2fa328f6a5080c1c0e0424607d9c5"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0011","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.2 Optimal Project","text":"### 3.2 Optimal Project\n\nEach binary project, $c$, can be described by specifying the cost of each probability of success through a function $C:[0,1] \\rightarrow \\mathbb{R}_{+}$. In particular, we set $C(\\mu)=c\\left(B_{\\mu}\\right)$ for all $\\mu$, and we keep in mind that the cost of a non-binary distribution is infinity. Recall that, in binary projects, it is without loss of generality to restrict attention to bonus contracts, $w_{b}$, where the agent is paid $b$ if output is one. Finally, the agent's choice of distribution $B_{\\mu}$ can be identified by its mean, $\\mu$ (equivalently, by its \"completion probability\"). In what follows, we describe each binary outcome $\\left(c, w_{b}, B_{\\mu}\\right)$ by a triple $(C, b, \\mu)$.\n\nWe are ready to state the main result of this section.\n\nProposition 2. There is an optimal binary project, $C^{*}$, and an agent-optimal equilibrium in $C^{*}$, $\\left(b^{*}, \\mu^{*}\\right)$, such that\n\n(i) $C^{* \\prime}(\\mu)=1-1 /(e \\mu)$ if $\\mu \\geq 1 / e$ and zero otherwise,\n(ii) $b^{*}=1-1 / e$, and\n(iii) $\\mu^{*}=1$.\n\nProposition 2 describes an optimal binary project in terms of the marginal costs of the possible completion probabilities. Of course, adding a fixed cost has no impact on incentives, so $C^{*}(0)=0$.\n\nWe explain below that the functional form of the marginal cost in Part (i) is pinned down by the requirement that the principal be indifferent between implementing a large range of completion probabilities. To this end, we sketch an argument for Proposition 2 that restricts the agent to choose cost functions $C$ that are non-decreasing, differentiable, and such that equilibrium can be determined using the first-order approach. The formal proof in the Appendix uses an envelope type argument to determine the agent's payoff and does not rely on any such restrictions.\n\nWe view the agent's problem of finding an optimal binary project as a maximization problem subject to incentive compatibility constraints. Our argument can be understood in terms of backward induction. We first determine a condition relating the principal's choice of bonus $b$ to the agent's optimal choice of completion probability $\\mu$. We can then determine a condition on the project $C$ for the principal to implement a given completion probability $\\mu$. Finally, we consider optimizing the agent's payoff over the project $C$ and the completion probability $\\mu$ to be implemented by the principal, subject to the constraints determined in the previous steps.\n\nLet us then describe the agent's incentive constraint in a subgame $(C, b)$. Note that, at this stage, the agent's problem is to solve $\\max _{\\widehat{\\mu} \\in[0,1]}\\{\\widehat{\\mu} b-C(\\widehat{\\mu})\\}$. Then the requirement on the reward $b$ ensuring the agent chooses a given completion probability $\\mu$ can be described by the first-order condition\n\n$$\nb=C^{\\prime}(\\mu) .\n$$\n\nWe now turn to the incentive constraint of the principal. In project $C$, the principal's problem is\n\n$$\n\\max _{\\mu \\in[0,1], b \\in \\mathbb{R}_{+}} \\mu(1-b)\n$$\n\nsubject to the constraint that $\\mu$ and $b$ satisfy Equation (8). Plugging the constraint into the maximand, the principal's problem can be expressed solely in terms of the completion probability she wants to implement; that is, $\\max _{\\mu \\in[0,1]} \\mu\\left(1-C^{\\prime}(\\mu)\\right)$. So, in project $C$, the principal's choice of $\\mu$ must satisfy\n\n$$\n\\mu\\left(1-C^{\\prime}(\\mu)\\right) \\geq \\widetilde{\\mu}\\left(1-C^{\\prime}(\\widetilde{\\mu})\\right),\n$$\n\nfor all $\\widetilde{\\mu} \\in[0,1]$.\nWe are now ready to state the project selection problem. As anticipated above, we include the completion probability $\\mu$ as a choice variable alongside the cost function $C$. This is important because the principal may be indifferent between implementing various completion probabilities, generating different payoffs for the agent. The interpretation of including $\\mu$ as a choice is that,\nafter designing the project, the agent makes a recommendation to the principal regarding which $\\mu$ to implement. The agent's first-stage problem can be written\n\n$$\n\\begin{aligned}\n& \\max _{C, \\mu, b} \\mu b-C(\\mu) \\\\\n& \\text { s.t. (8) and (9). }\n\\end{aligned}\n$$\n\nAlternatively, plugging the constraint (8) into the maximand, it can be written as\n\n$$\n\\begin{aligned}\n& \\max _{C, \\mu} \\mu C^{\\prime}(\\mu)-C(\\mu) \\\\\n& \\text { s.t. (9). }\n\\end{aligned}\n$$\n\nNow let us explain that, if $(\\widehat{C}, \\widehat{\\mu})$ solves this problem, the constraint (9) must bind at each $\\widetilde{\\mu}<\\widehat{\\mu}$ such that $\\widehat{C}^{\\prime}(\\widetilde{\\mu})>0$. To understand this observation, notice first that the agent's cost of $\\widehat{\\mu}$ must be given by $\\widehat{C}(\\widehat{\\mu})=\\int_{0}^{\\widehat{\\mu}} \\widehat{C}^{\\prime}(\\widetilde{\\mu}) d \\widetilde{\\mu}$, since $\\widehat{C}(0)=0 .{ }^{17}$ Hence, reducing the marginal cost $\\widehat{C}^{\\prime}(\\widetilde{\\mu})$ for any $\\widetilde{\\mu}<\\widehat{\\mu}$ implies a reduction in the cost $\\widehat{C}(\\widehat{\\mu})$. However, the possibility to reduce the marginal costs $\\widehat{C}^{\\prime}(\\widetilde{\\mu})$ is restricted by the constraint (9) evaluated at $(C, \\mu)=(\\widehat{C}, \\widehat{\\mu})$. We can conclude that, for the optimum $(\\widehat{C}, \\widehat{\\mu})$, the constraint (9) binds for all $\\widetilde{\\mu}<\\widehat{\\mu}$ as long as $\\widehat{C}^{\\prime}(\\widetilde{\\mu})>0$, implying that the principal is indifferent between implementing any completion probability below $\\widehat{\\mu}$ which has a strictly positive marginal cost.\n\nOur next aim is to use this indifference condition to reduce the agent's problem in (10) to a twodimensional problem by replacing the domain of projects by the principal's possible equilibrium payoffs. To this end, note that if $(\\widehat{C}, \\widehat{\\mu})$ solves (10), then $\\widehat{C}^{\\prime}$ can be expressed in terms of the principal's equilibrium payoff, $\\widehat{\\pi} \\equiv \\widehat{\\mu}\\left(1-\\widehat{C}^{\\prime}(\\widehat{\\mu})\\right)$. Indeed, the binding constraint (9) evaluated at $(C, \\mu)=(\\widehat{C}, \\widehat{\\mu})$ can be written as\n\n$$\n\\widehat{C}^{\\prime}(\\widetilde{\\mu})=\\left\\{\\begin{array}{cc}\n0 & \\text { if } \\widetilde{\\mu}<\\widehat{\\pi} \\\\\n1-\\frac{\\widehat{\\pi}}{\\mu} & \\text { if } \\widetilde{\\mu} \\in[\\widehat{\\pi}, \\widehat{\\mu}] .\n\\end{array}\\right.\n$$\n\nConsequently, the agent's problem in (10) can be rewritten as","text_sha256":"e296151b8d92961e16c84c030885f02e063f449629b043fe44cbc1280e85792c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0012","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.2 Optimal Project","text":"$$\n\\max _{\\widehat{\\mu}, \\widehat{\\pi} \\in[0,1]} \\widehat{\\mu}\\left(1-\\frac{\\widehat{\\pi}}{\\widehat{\\mu}}\\right)-\\int_{\\widehat{\\pi}}^{\\widehat{\\mu}}\\left(1-\\frac{\\widehat{\\pi}}{\\widetilde{\\mu}}\\right) d \\widetilde{\\mu}\n$$\n\n[^9]To conclude Proposition 2, observe that\n\n$$\n\\widehat{\\mu}\\left(1-\\frac{\\widehat{\\pi}}{\\widehat{\\mu}}\\right)-\\int_{\\widehat{\\pi}}^{\\widehat{\\mu}}\\left(1-\\frac{\\widehat{\\pi}}{\\widetilde{\\mu}}\\right) d \\widetilde{\\mu}=\\int_{\\widehat{\\pi}}^{\\widehat{\\mu}} \\frac{\\widehat{\\pi}}{\\widetilde{\\mu}} d \\widetilde{\\mu}=\\widehat{\\pi}[\\log \\widehat{\\mu}-\\log \\widehat{\\pi}],\n$$\n\nwhich is maximized at $\\widehat{\\mu}=1$ and $\\widehat{\\pi}=1 / e$. This explains how we obtain Parts (ii) and (iii) of the proposition. In particular, the principal's profit in an optimal outcome is $\\pi^{*}=\\mu^{*}\\left(1-b^{*}\\right)=1 / e$, and since $\\mu^{*}=1$, we have $b^{*}=1-1 / e$. Finally, note that evaluating the right-hand side of (11) at $\\widehat{\\mu}=1$ and $\\widehat{\\pi}=1 / e$ yields Part (i) of the proposition.\n\nFinally, we compute the payoffs of the agent and the principal in the outcome $\\left(C^{*}, b^{*}, \\mu^{*}\\right)$. As mentioned above, the principal's equilibrium payoff is $\\mu^{*}\\left(1-b^{*}\\right)=1 / e$. The agent's payoff is pinned down by evaluating (13) at $\\left(\\mu^{*}, \\pi^{*}\\right)=(1,1 / e)$, also yielding $1 / e$.","text_sha256":"cdbbc076f42f8c893599ac54c7638debe50882bbee03e14a639d43359a1c6b90"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0013","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Discussion","text":"## 4 Discussion\n\nPayoff Possibility Set.-We can use our results to characterize the set of possible payoff combinations which can arise in principal-agent models for some arbitrary production technology. To be more specific, we still consider an environment where the agent is risk neutral and has limited liability. However, the production technology is exogenously given and satisfies the constraint that the largest output cannot exceed one. Moreover, the agent has an outside option we take to be zero. We wish to characterize those payoff profiles which can arise in equilibrium for some production technology. We next show that there exists a production technology where the principal receives $\\widehat{\\pi}$ and the agent receives $\\widehat{u}$ if and only if $\\widehat{\\pi} \\in[0,1]$ and $\\widehat{u} \\in[0,-\\widehat{\\pi} \\log \\widehat{\\pi}]$; see Figure 1 for illustration.\n\nFirst note that the principal's payoff, $\\widehat{\\pi}$, must be between zero and one. The reason is that the contract $w_{0}$ guarantees at least zero profit. Since output is less than one, limited liability implies that the profit cannot exceed one. Next, we identify the frontier of the payoff possibility set. That is, we compute the largest payoff the agent can get if the principal's payoff is $\\widehat{\\pi}$. By Proposition 1, we know that this largest payoff is achieved in a binary project. Furthermore, recall that in Section 3.2 we rewrote the problem of designing the optimal binary project as a maximization problem with respect to the equilibrium probability of success, $\\widehat{\\mu}$, and the principal's equilibrium profit, $\\widehat{\\pi}$; see (12). So the problem of designing the agent-optimal binary technology which generates $\\widehat{\\pi}$ can be reduced to a similar maximization problem except $\\widehat{\\pi}$ is treated as a parameter instead of a choice variable. By Equation (13), the agent's maximal payoff is $-\\widehat{\\pi} \\log \\widehat{\\pi}$.\n\nSince the agent's outside option is zero, his equilibrium payoff cannot be negative. It remains to\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: The shaded area represents the set of payoffs for the principal and the agent that arise for some technology of the agent. The players' payoffs under the optimal project characterized in Proposition 2 are illustrated by the red dot.\n\nargue that given the principal's payoff, $\\widehat{\\pi}$, for each $\\widehat{u} \\in[0,-\\widehat{\\pi} \\log \\widehat{\\pi})$, there is a production technology so that the equilibrium payoff profile is $(\\widehat{\\pi}, \\widehat{u})$. To do so, consider the production technology generating $(\\widehat{\\pi},-\\widehat{\\pi} \\log \\widehat{\\pi})$ and add a fixed cost of $-\\widehat{\\pi} \\log \\widehat{\\pi}-\\widehat{u}$. That is, the agent's cost of each distribution is increased by this quantity. Adding this fixed cost does not change the agent's incentives, but it lowers his payoff to $\\widehat{u}$.\n\nCompetition Among Agents.- As is standard in the literature, we maintained the assumption that the principal has full bargaining power and makes a take-it-or-leave-it offer to the agent. The principal's strong bargaining power is often motivated by fierce competition among the agents, which is usually left unmodeled. At first glance, such a justification might be questioned in our setting because competition could limit the rent an agent can earn from contracting with any principal. It is therefore of interest to examine which projects may arise in markets with imperfect competition among agents.\n\nOne way to understand competition among agents is to consider embedding our model into a standard search and matching framework in which one-to-one matches occur over time. Each\nsteady state equilibrium in such a model is associated with a continuation value of an unmatched principal. So, when an agent designs a project, he must bear in mind that any principal prefers to remain unmatched whenever her value from contracting with the agent is less than the continuation value from further search. As before, the agent's problem of designing his equilibrium project can be understood in terms of maximizing the expression in Equation (13) by choice of the completion probability and principal expected profits. The difference, however, is that the principal profit, $\\widehat{\\pi}$, is bounded from below by a constant which makes a principal indifferent between contracting with the agent and remaining unmatched. The solution to that problem has features similar to those of the optimal project described by Proposition 2. In particular, the equilibrium completion probability is one and the principal is indifferent between offering the equilibrium bonus and anything less than that.\n\nOther Constraints on the Technology.- While we believe there are a range of situations where the agent can determine the technology in advance of contracting, in reality the agent's choice of projects would typically be more limited than in our setting where the only constraint is the bounds on output. Such constraints on technology design may ultimately be responsible for the key properties of the selected project. An example is where noise in the environment prevents distributions that put mass at the extremes of the possible output values, ruling out binary projects. Note also that imposing additional constraints on the set of available technologies is related to the possibility that some designs are more costly than others, e.g. binary technologies might be feasible but too costly to set up. We have treated in this paper the case where the agent is only subject to output bounds, and where all designs are costless to the agent. This was intended as a natural first step and helps shed light on the forces at play in the agent's design problem, while future work could examine the choice of projects given different constraints or costly technology choice.","text_sha256":"4956f4401c46698f7f02a26e587897d671a01020e9de76debb0ca9004dccdc7a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0014","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Discussion","text":"In spite of these observations, the analysis in the paper does already allow determination of the agent's optimal project in some situations where he is more constrained. For instance, suppose that the choice of project $c$ is restricted to come from a set that contains $c^{*}$, a project determined to maximize the agent's payoff in our setting with only bounds on output. Then the same project $c^{*}$ remains optimal in the more constrained setting. An example is where there is, in addition to the output bounds, a lower bound $L(\\mu)$ on the cost of generating mean output $\\mu$. Recall that $C^{*}$ is the optimal binary project defined in Proposition 2. Provided $L(\\mu) \\leq C^{*}(\\mu)$ for all $\\mu \\in[0,1]$, then $C^{*}$ remains an optimal project for the agent.\n\nThere are also cases where constraints on project selection bind, but the optimal project follows\nclosely from our analysis. Suppose that, in addition to output being bounded between zero and one, mean output is restricted to be no greater than some $m \\in(0,1)$. This constraint does not affect any of the arguments needed to obtain Proposition 1, so there is an optimal project for the agent which is binary. ${ }^{18}$ Also, the agent's maximal payoff in an equilibrium where the probability of output one is $\\widehat{\\mu}$ and the principal earns payoff $\\widehat{\\pi}$ is still given by the expression in Equation (13), following the arguments in the proof of Proposition 2. An optimal project is then defined by Equation (11) with $\\widehat{\\mu}=m$ and $\\widehat{\\pi}=m / e$. There is an equilibrium given this binary project where the principal offers a bonus payment $1-1 / e$ for output one (and pays zero otherwise), and the agent chooses probability $m$ of output one. We have thus seen that, in the mean-constrained version of the problem, the agent's equilibrium probability of a high output is less than one.\n\nRisk Aversion.- Our results can be generalized to the case where the agent is risk averse. To explain it more formally, let us modify our model so that, if the agent receives payment $w$ and chooses $F$ at cost $c(F)$, then his payoff is $v(w)-c(F)$, where $v$ is an increasing, concave and continuously differentiable function. It can be shown that, even in this case, binary projects remain optimal. More precisely, the following version of the statement of Proposition 1 remains valid. For each project $c^{*}$ and any equilibrium $\\left(w^{*}, F^{*}\\right)$ in $c^{*}$, there is a binary project $\\widetilde{c}$ with an equilibrium $\\left(\\widetilde{w}, B_{1}\\right)$ such that the outcome $\\left(\\widetilde{c}, \\widetilde{w}, B_{1}\\right)$ Pareto dominates $\\left(c^{*}, w^{*}, F^{*}\\right)$. The characterization of optimal binary projects follows the same steps as for the risk-neutral case. The marginal cost of any completion probability $\\mu>\\pi^{*}$ is $v\\left(1-\\left[\\pi^{*} / \\mu\\right]\\right)$, where $\\pi^{*}$ is the equilibrium payoff of the principal. These claims are proved in the Online Appendix.\n\nRelation to Another Hold-up Problem.- As mentioned in the Introduction, Condorelli and Szentes (2020) study a different hold-up problem in the context of bilateral trade. In their model, the buyer first chooses the distribution of her valuation for the seller's good. Then the seller, after observing the value distribution, makes a take-it-or-leave-it price offer. It turns out that, when the support of any value distribution must be in $[0,1]$, the buyer's equilibrium CDF, $F^{*}$, has the same functional form as $C^{* \\prime}$ described in Proposition 2. More precisely, $F^{*}(v)=1-1 /(e v)$ on $(1 / e, 1)$. In what follows, we attempt to illuminate this similarity and the relationships between the two models by transforming the buyer's problem into a strategically equivalent one and show that the transformed problem is identical to the constrained maximization problem (10).\n\nFirst note that by choosing a value distribution, $F$, the buyer determines his demand curve,\n\n[^10]$1-F$. That is, the probability of trade at price $p$ is $1-F(p)$. The idea behind the transformation is that we replace the buyer's choice set by inverse demand curves and assume that the seller sets a quantity instead of a price, i.e., the probability of trade. Since there is a bijection between demand curves and inverse demand curves and the price is pinned down by inverse demand curve for each quantity, the transformed model is strategically equivalent to that of Condorelli and Szentes (2020). Since both the buyer's willingness-to-pay and the probability of trade are in [0, 1], the domain as well as the range of the inverse demand curve must also be in [0,1]. Let $\\mathcal{P}$ denote the set of such inverse demand functions. So, if the buyer chooses $P \\in \\mathcal{P}$ and the seller sets $Q$ then the price is $P(Q)$. Next, we explain that the buyer's problem can be written as\n$$\n\\begin{aligned}\n& \\max _{q \\in[0,1], P \\in \\mathcal{P}} \\int_{0}^{Q}(P(x)-P(Q)) d x \\\\\n& \\text { s.t. } Q P(Q) \\geq \\tilde{Q} P(\\tilde{Q}) \\text { for all } \\tilde{Q} \\in[0,1]\n\\end{aligned}\n$$\nNote that the objective function is the familiar expression for the buyer's payoff from Consumer Theory, for a given inverse demand curve $P$ and quantity $Q$. In addition, the constraint guarantees that when the buyer's inverse demand curve is $P$, the seller indeed prefers to set $Q$ to any smaller quantity. Now, observe that replacing $P$ by $1-C^{\\prime}$ in this problem yields (10), that is, the two problems are equivalent. This means that the buyer-optimal inverse demand curve, $P^{*}$, is given by $P^{*}(Q)=1-C^{* \\prime}(Q)$, where $C^{*}$ is characterized Proposition 2, that is, $P^{*}(Q)=1 /[e Q]$ on $[1 / e, 1]$. Finally, notice that for each inverse demand curve $P$, the buyer's value distribution is determined by the following equation: $P(1-F(v))=v$. Coincidently, applying this formula for $P^{*}$ yields $F^{*}(v)=1-1 /(e v)$ on $(1 / e, 1)$, confirming the main result in Condorelli and Szentes (2020).","text_sha256":"3e23097c2545e10de75b9274adda1c545016e8450902734404a64703e6b8b036"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0015","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Discussion","text":"Our discussion about the similarity of Proposition 2 to the main result in Condorelli and Szentes (2020) has followed the heuristic argument presented in Section 3.2. This assumes the applicability of the first-order approach to derive that the agent's bonus is equal to his marginal cost, i.e., to derive Equation (8). Importantly, the validity of this approach must be verified, and hence the proof of Proposition 2 in the appendix relies on a different argument, one which has no parallel in Condorelli and Szentes.","text_sha256":"b6118e589fb2104d8116a4b926aa031b3ca9b06fa31e97a56de6b13c261463e0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0016","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAverch, H. and Johnson, L.L., 1962. Behavior of the firm under regulatory constraint. American Economic Review, 52(5), pp.1052-1069.\n\nBergemann, D. and Schlag, K., 2011. Robust monopoly pricing. Journal of Economic Theory, 146(6), pp.2527-2543.\n\nBergemann, D., Brooks, B. and Morris, S., 2015. The limits of price discrimination. American Economic Review, 105(3), pp.921-57.\n\nBolton, P. and Dewatripont, M., 2005. Contract theory. MIT Press.\nBonham, J. and Riggs-Cragun, A., 2021. Contracting on what firm owners value. Working Paper.\nCarroll, G., 2015. Robustness and linear contracts. American Economic Review, 105(2), pp. 536-563.\nChaigneau, P., Edmans, A. and Gottlieb, D., 2019. The informativeness principle without the firstorder approach. Games and Economic Behavior, 113, pp.743-755.\n\nCondorelli, D. and Szentes, B., 2020. Information design in the holdup problem. Journal of Political Economy, 128(2), pp.681-709.\n\nGans, J. S., Stern, S. and Wu, J., 2019. Foundations of entrepreneurial strategy. Strategic Management Journal, 40(5), pp.736-756.\n\nGarrett, D., 2021. Payoff implications of incentive contracting. Theoretical Economics, 16(4), pp. 1281-1312.\n\nGeorgiadis, G. 2022. Contracting with Moral Hazard: A Review of Theory \\& Empirics.\nGeorgiadis, G., Ravid, D. and Szentes, B., 2022. Flexible Moral Hazard Models. Working Paper.\nGrossman, S.J. and Hart, O.D., 1983. An analysis of the principal-agent problem. Econometrica, 51(1), pp.7-46.\n\nHebert, B. 2018. Moral Hazard and the Optimality of Debt. Review of Economic Studies, 85 (4), pp. 2214-2252.\n\nHolmstrom, B., 1979. Moral hazard and observability. Bell Journal of Economics, 10(1), pp.74-91.\n\nHolmstrom, B., 2017. Pay for performance and beyond. American Economic Review, 107(7), pp.1753-77.\n\nInnes, R.D., 1990. Limited liability and incentive contracting with ex-ante action choices. Journal of Economic Theory, 52(1), pp.45-67.\n\nJewitt, I., Kadan, O. and Swinkels, J., 2008. Moral hazard with bounded payments. Journal of Economic Theory, 143(1), pp.59-82.\n\nLaux, C., 2001. Limited-liability and incentive contracting with multiple projects. RAND Journal of Economics, 32(3), pp.514-526.\n\nMacLeod, W.B., 2003. Optimal contracting with subjective evaluation. American Economic Review, 93(1), pp.216-240.\n\nMattsson, Lars-Göran and Jorgen Weibull. 2022. An Analytically Solvable Principal- Agent Model. Available at SSRN 4252495.\n\nMilgrom, P. and Segal, I., 2002. Envelope theorems for arbitrary choice sets. Econometrica, 70(2), pp.583-601.\n\nMirrlees, J.A., 1976. The optimal structure of incentives and authority within an organization. Bell Journal of Economics, 7(1), pp. 105-131.\n\nOllier, S. and Thomas, L., 2013. Ex post participation constraint in a principal-agent model with adverse selection and moral hazard. Journal of Economic Theory, 148(6), pp.2383-2403.\n\nOrtner, J. and Chassang, S., 2018. Making corruption harder: Asymmetric information, collusion, and crime. Journal of Political Economy, 126(5), pp.2108-2133.\n\nPerez-Richet, E. and Skreta, V., 2018. Test design under falsification. Working Paper.\nPoblete, J. and Spulber, D., 2012. The form of incentive contracts: agency with moral hazard, risk neutrality, and limited liability. RAND Journal of Economics, 43(2), pp.215-234.\n\nRoesler, A.K. and Szentes, B., 2017. Buyer-optimal learning and monopoly pricing. American Economic Review, 107(7), pp.2072-80.\n\nRogerson, W.P., 1985. The first-order approach to principal-agent problems. Econometrica, 53(6), pp.1357-1367.","text_sha256":"d188c07780b8cea7fbe2a8f099fb0d1aa78a8f3b513d155397e80d4fb8ec7ddf"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0017","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Appendix A: Omitted proofs","text":"## Appendix A: Omitted proofs","text_sha256":"8a818c6f6f4b7da291306b879e5ec0e0fd19ba46587633f7cbe8afbfca924fbf"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0018","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proofs of results in Section 3.1","text":"## Proofs of results in Section 3.1\n\nProof of Lemma 1. Consider a sequence $\\left(\\mu_{n}\\right)$, with $\\mu_{n} \\in\\left[0, \\mu_{F^{*}}\\right)$ for all $n$, such that\n\n$$\nu\\left(\\widehat{c}, w_{b}\\right)=\\lim _{n \\rightarrow \\infty} U\\left(\\widehat{c}, w_{b}, B_{\\mu_{n}}\\right) \\text { and } \\pi\\left(\\widehat{c}, w_{b}\\right)=\\lim _{n \\rightarrow \\infty} \\Pi\\left(w_{b}, B_{\\mu_{n}}\\right) .\n$$\n\nFor each $k \\in \\mathbb{N}$, there exists $n_{k}$ such that, for all $\\mu \\in[0,1]$,\n\n$$\n\\mu_{n_{k}} b-\\hat{c}\\left(B_{\\mu_{n_{k}}}\\right)+\\frac{1}{k} \\geq \\mu b-\\hat{c}\\left(B_{\\mu}\\right) .\n$$\n\nEquivalently, for all $k$, and all $\\mu \\in\\left[0, \\mu_{F^{*}}\\right)$,\n\n$$\n\\mu_{n_{k}} b-\\inf \\left\\{c^{*}(F): \\mu_{F}=\\mu_{n_{k}}\\right\\}+\\frac{1}{k} \\geq \\mu b-\\inf \\left\\{c^{*}(F): \\mu_{F}=\\mu\\right\\} .\n$$\n\nFor each $k$, we can pick a distribution $F_{n_{k}}$ with mean $\\mu_{n_{k}}$ such that\n\n$$\nc^{*}\\left(F_{n_{k}}\\right)<\\inf \\left\\{c^{*}(F): \\mu_{F}=\\mu_{n_{k}}\\right\\}+\\frac{1}{k} .\n$$\n\nThen, for all $k$ and all $F \\in \\mathcal{F}$ with mean $\\mu_{F} \\in\\left[0, \\mu_{F^{*}}\\right)$,\n\n$$\n\\mu_{n_{k}} b-c^{*}\\left(F_{n_{k}}\\right)+\\frac{2}{k}>\\mu_{F} b-c^{*}(F) .\n$$\n\nThere are then two cases. In the first, the inequality (14) holds for all $k$ and all $F \\in \\mathcal{F}$ (not only those $F$ with $\\mu_{F}<\\mu_{F^{*}}$ ). Then\n\n$$\nu\\left(c^{*}, w_{b}\\right)=\\lim _{k \\rightarrow \\infty} U\\left(c^{*}, w_{b}, F_{n_{k}}\\right)\n$$\n\nand hence\n\n$$\n\\pi\\left(c^{*}, w_{b}\\right) \\geq \\lim _{k \\rightarrow \\infty} \\Pi\\left(w_{b}, F_{n_{k}}\\right)=\\lim _{k \\rightarrow \\infty} \\Pi\\left(w_{b}, B_{\\mu_{n_{k}}}\\right)=\\pi\\left(\\widehat{c}, w_{b}\\right)\n$$\n\nas desired. In the second, the inequality (14) fails to hold for some $k$ and some $F \\in \\mathcal{F}$ with $\\mu_{F} \\geq \\mu_{F^{*}}$, which implies\n\n$$\nu\\left(c^{*}, w_{b}\\right)=\\sup \\left\\{\\mu_{F} b-c^{*}(F): F \\in \\mathcal{F}\\right\\}>\\sup \\left\\{\\mu_{F} b-c^{*}(F): F \\in \\mathcal{F}, \\mu_{F}<\\mu_{F^{*}}\\right\\} .\n$$\n\nThis means that there is a sequence of distributions in $\\mathcal{F}$ along which the agent's payoff converges to his value $u\\left(c^{*}, w_{b}\\right)$ and for which every distribution has mean at least $\\mu_{F^{*}}$. By the definition of the principal's value, we have\n\n$$\n\\pi\\left(c^{*}, w_{b}\\right) \\geq \\mu_{F^{*}}(1-b) \\geq \\pi\\left(\\widehat{c}, w_{b}\\right),\n$$\n\nwhere the second inequality follows because any distribution with mean at least $\\mu_{F^{*}}$ is assigned an infinite cost in the project $\\widehat{c}$. $\\square$\n\nProof of Lemma 2. Let us fix $b \\in\\left[0, b^{*}\\right)$. We first show that $w_{b}$ does not implement $B_{\\mu_{F^{*}}}$ in $\\left(\\widetilde{c}, w_{b}\\right)$. Suppose for a contradiction that $B_{\\mu_{F^{*}}}$ satisfies the agent's incentive constraint in $\\left(\\widetilde{c}, w_{b}\\right)$, that is,\n\n$$\n\\mu_{F^{*}} b-\\bar{c} \\geq \\sup _{\\mu<\\mu_{F^{*}}}\\left\\{\\mu b-\\widetilde{c}\\left(B_{\\mu}\\right)\\right\\} .\n$$\n\nTherefore,\n\n$$\n\\bar{c} \\leq-\\sup _{\\mu<\\mu_{F^{*}}}\\left\\{\\left(\\mu-\\mu_{F^{*}}\\right) b-\\widetilde{c}\\left(B_{\\mu}\\right)\\right\\} \\leq-\\sup _{\\mu<\\mu_{F^{*}}}\\left\\{\\left(\\mu-\\mu_{F^{*}}\\right) b^{*}-\\widetilde{c}\\left(B_{\\mu}\\right)\\right\\}=\\bar{c},\n$$\n\nwhere the first inequality is just the previous displayed inequality rearranged, the second inequality follows from $b<b^{*}$, and the equality is the definition of $\\bar{c}$. Since the farthest left term and the farthest right term are equal in the previous chain, all inequalities must be equalities. Note that the second inequality is an equality only if the supremum in Equation (15) is approached along a sequence of $\\mu^{\\prime}$ s converging to $\\mu_{F^{*}}$. Since $\\widetilde{c}\\left(B_{\\mu}\\right)=\\widehat{c}\\left(B_{\\mu}\\right)$ whenever $\\mu \\neq \\mu_{F^{*}}$, and since $\\widehat{c}\\left(B_{\\mu}\\right)=\\infty$ for $\\mu \\geq \\mu_{F^{*}}$, it follows that the supremum of $\\mu b-\\hat{c}\\left(B_{\\mu}\\right)$ is approached by the same sequence. Hence, $\\pi\\left(\\widehat{c}, w_{b}\\right)=\\mu_{F^{*}}(1-b)$. We can conclude that\n\n$$\n\\pi\\left(c^{*}, w_{b}\\right) \\geq \\pi\\left(\\widehat{c}, w_{b}\\right)=\\mu_{F^{*}}(1-b)>\\mu_{F^{*}}\\left(1-b^{*}\\right)=\\Pi\\left(w^{*}, F^{*}\\right),\n$$\n\nwhere the first inequality follows from Lemma 1, the strict inequality is implied by $b<b^{*}$ and the second equality follows from the definition of $b^{*}$. This inequality implies that $w^{*}$ is not incentive compatible in project $c^{*}$, a contradiction.\n\nSince $w_{b}$ does not implement $B_{\\mu_{F^{*}}}$ in $\\widetilde{c}, U\\left(\\widetilde{c}, w_{b}, B_{\\mu_{F^{*}}}\\right)<u\\left(\\widetilde{c}, w_{b}\\right)$. We next show that $\\mathbf{F}^{\\widetilde{c}, w_{b}}=$ $\\mathbf{F}^{\\widehat{c}, w_{b}} .{ }^{19}$ Note that, for each $\\left(F_{n}\\right) \\in \\mathbf{F}^{\\widetilde{c}, w_{b}} \\cup \\mathbf{F}^{\\widehat{c}, w_{b}}$, there exists $K \\in \\mathbb{N}$ such that $F_{k} \\neq B_{\\mu_{F^{*}}}$\n\n[^11]if $k>K$. If $\\left(F_{n}\\right) \\in \\mathbf{F}^{\\widehat{c}, w_{b}}$, it follows from $\\widehat{c}\\left(B_{\\mu_{F^{*}}}\\right)=\\infty$. If $\\left(F_{n}\\right) \\in \\mathbf{F}^{\\widetilde{c}, w_{b}}$, it is implied by $U\\left(\\widetilde{c}, w_{b}, B_{\\mu_{F^{*}}}\\right)<u\\left(\\widetilde{c}, w_{b}\\right)$. Since $\\widetilde{c}(F)=\\widehat{c}(F)$ whenever $F \\neq B_{\\mu_{F^{*}}}$, this means that, for each $\\left(F_{n}\\right) \\in \\mathbf{F}^{\\widetilde{c}, w_{b}} \\cup \\mathbf{F}^{\\widehat{c}, w_{b}}$,\n$$\n\\lim _{n \\rightarrow \\infty} U\\left(\\widetilde{c}, w_{b}, F_{n}\\right)=\\lim _{n \\rightarrow \\infty} U\\left(\\widehat{c}, w_{b}, F_{n}\\right),\n$$\nimplying that $\\mathbf{F}^{\\widetilde{c}, w_{b}}=\\mathbf{F}^{\\widehat{c}, w_{b}}$. Consequently,\n$$\n\\begin{aligned}\n\\pi\\left(\\widetilde{c}, w_{b}\\right) & \\equiv \\sup \\left\\{\\limsup _{n \\rightarrow \\infty} \\Pi\\left(w_{b}, F_{n}\\right):\\left(F_{n}\\right) \\in \\mathbf{F}^{\\widetilde{c}, w_{b}}\\right\\} \\\\\n& =\\sup \\left\\{\\limsup _{n \\rightarrow \\infty} \\Pi\\left(w_{b}, F_{n}\\right):\\left(F_{n}\\right) \\in \\mathbf{F}^{\\widehat{c}, w_{b}}\\right\\}=\\pi\\left(\\widehat{c}, w_{b}\\right) .\n\\end{aligned}\n$$\n\nQED","text_sha256":"77e314a7bcce4066698072517e6d4e773178f6050f273df73e9c132965dd739e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0019","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proof of Proposition 2","text":"## Proof of Proposition 2\n\nThis section proves Proposition 2. We begin by considering an arbitrary binary project $C$. Let us drop the dependence on $C$ and write the value for the agent when the bonus is $b$ as $u(b)$. The value is given by\n\n$$\nu(b)=\\sup _{\\mu \\in[0,1]}\\{b \\mu-C(\\mu)\\}\n$$\n\nNote that $u$ is non-decreasing. Moreover, as the upper envelope of linear functions, it is convex and hence continuous.\n\nLet $\\Gamma(b)$ be the set of values $\\mu$ such that there is a sequence $\\left(\\mu_{n}\\right)$ with $\\mu_{n} \\rightarrow \\mu$ and $b \\mu_{n}-C\\left(\\mu_{n}\\right) \\rightarrow$ $u(b)$. Take $\\bar{\\mu}(b)=\\max \\Gamma(b)$, and note that the maximum is attained. Similarly, let $\\underline{\\mu}(b)$ be the minimum of $\\Gamma(b)$ (also attained). Note that, if the principal offers a bonus $b \\in[0,1]$ in project $C$, then she obtains value $\\bar{\\mu}(b)(1-b)$.\n\nFor any $b \\geq 0$, let $u_{+}^{\\prime}(b)$ be the right derivative of $u$ at $b$. For any $b>0$, let $u_{-}^{\\prime}(b)$ be the left derivative of $u$ at $b$. We next show a result that is analogous to Theorem 1 of Milgrom and Segal (2002), but adjusted for the possibility that the agent's payoff $u(b)$ is not attained by values $\\mu \\in \\Gamma(b)$.\n\nLemma 3. For all $b \\geq 0, u_{+}^{\\prime}(b) \\geq \\bar{\\mu}(b)$. For all $b>0, u_{-}^{\\prime}(b) \\leq \\underline{\\mu}(b)$.\nProof. Consider a sequence $\\left(\\mu_{n}\\right)$ with $\\mu_{n} \\rightarrow \\bar{\\mu}(b)$ and $b \\mu_{n}-C\\left(\\mu_{n}\\right) \\rightarrow u(b)$. Then, for any $b^{\\prime}>b$, we have\n\n$$\n\\left(b^{\\prime}-b\\right) \\bar{\\mu}(b)=\\lim _{n \\rightarrow \\infty}\\left\\{b^{\\prime} \\mu_{n}-C\\left(\\mu_{n}\\right)\\right\\}-\\lim _{n \\rightarrow \\infty}\\left\\{b \\mu_{n}-C\\left(\\mu_{n}\\right)\\right\\} \\leq u\\left(b^{\\prime}\\right)-u(b) .\n$$\n\nDividing by $b^{\\prime}-b$ and taking limits as $b^{\\prime}$ approaches $b$ from above yields $u_{+}^{\\prime}(b) \\geq \\bar{\\mu}(b)$.\nLet $b>0$ and consider a sequence $\\left(\\mu_{n}\\right)$ with $\\mu_{n} \\rightarrow \\underline{\\mu}(b)$ and $b \\mu_{n}-C\\left(\\mu_{n}\\right) \\rightarrow u(b)$. For any $b^{\\prime}<b$, we have\n\n$$\n\\left(b-b^{\\prime}\\right) \\underline{\\mu}(b)=\\lim _{n \\rightarrow \\infty}\\left\\{b \\mu_{n}-C\\left(\\mu_{n}\\right)\\right\\}-\\lim _{n \\rightarrow \\infty}\\left\\{b^{\\prime} \\mu_{n}-C\\left(\\mu_{n}\\right)\\right\\} \\geq u(b)-u\\left(b^{\\prime}\\right) .\n$$\n\nDividing by $b-b^{\\prime}$ and taking limits as $b^{\\prime}$ approaches $b$ from below yields $u_{-}^{\\prime}(b) \\leq \\underline{\\mu}(b)$.\nQED\nWe can further use the convexity of $u$ to determine its right derivative in terms of the completion probability attainable with a given bonus.\n\nLemma 4. For all $b \\geq 0, u_{+}^{\\prime}(b)=\\bar{\\mu}(b)$.\nProof. Fix $b \\geq 0$ and suppose for a contradiction that $u_{+}^{\\prime}(b)>\\bar{\\mu}(b)$. By convexity of $u$ and the previous lemma\n\n$$\n\\bar{\\mu}(b)<u_{+}^{\\prime}(b) \\leq u_{-}^{\\prime}\\left(b^{\\prime}\\right) \\leq \\underline{\\mu}\\left(b^{\\prime}\\right)\n$$\n\nfor all $b^{\\prime}>b$. For each $n \\in \\mathbb{N}$, let\n\n$$\nb_{n} \\in\\left(b, b+\\frac{1}{n}\\right)\n$$\n\nand let $\\mu_{n} \\in\\left[\\frac{\\bar{\\mu}(b)+u_{+}^{\\prime}(b)}{2}, 1\\right]$ and such that\n\n$$\nb_{n} \\mu_{n}-C\\left(\\mu_{n}\\right)>u\\left(b_{n}\\right)-\\frac{1}{n}\n$$\n\n(that such a choice is possible follows because $\\frac{\\bar{\\mu}(b)+u_{+}^{\\prime}(b)}{2}<u_{-}^{\\prime}\\left(b_{n}\\right) \\leq \\underline{\\mu}\\left(b_{n}\\right)$ for all $n$ ). Consider a subsequence ( $b_{n_{k}}$ ) such that $\\mu_{n_{k}} \\rightarrow \\mu^{*} \\geq \\frac{\\bar{\\mu}(b)+u_{+}^{\\prime}(b)}{2}$ for some $\\mu^{*}$. We have\n\n$$\n\\lim \\left\\{b \\mu_{n_{k}}-C\\left(\\mu_{n_{k}}\\right)\\right\\}=\\lim \\left\\{b_{n_{k}} \\mu_{n_{k}}-C\\left(\\mu_{n_{k}}\\right)\\right\\}=\\lim u\\left(b_{n_{k}}\\right)=u(b)\n$$\n\nwhere the final equality follows by continuity of $u$. The fact that $\\mu^{*}>\\bar{\\mu}(b)$ contradicts the definition of $\\bar{\\mu}(b)$.\n\nQED\n\nNote now that, because $u$ is convex, it is absolutely continuous and hence differentiable almost everywhere. This means that\n\n$$\nu(b)=u(0)+\\int_{0}^{b} \\bar{\\mu}(s) d s\n$$\n\nIt is immediate from the agent's problem that we must have $u(0) \\leq 0$; i.e., the agent cannot obtain a strictly positive payoff if the bonus is set to zero.\n\nConsider now a project $C$ with an equilibrium in which the principal offers bonus $\\widehat{b}$ for project completion, the agent chooses completion probability $\\widehat{\\mu}$, and therefore the principal's payoff is given by $\\widehat{\\pi}=\\widehat{\\mu}(1-\\widehat{b})$. Note that, if $(C, \\widehat{b}, \\widehat{\\mu})$ is an optimal outcome for the agent, then we must have $\\widehat{\\mu}>0$. Incentive compatibility of the principal offering bonus $\\widehat{b}$ requires that, for all $b$,\n\n$$\n\\begin{aligned}\n\\widehat{\\pi} & \\geq \\bar{\\mu}(b)(1-b) \\\\\n& =u_{+}^{\\prime}(b)(1-b) .\n\\end{aligned}\n$$\n\nHence, if the agent is to get positive rent in outcome $(C, \\widehat{b}, \\widehat{\\mu})$, we must have also $\\widehat{\\pi}>0$. Assume from now on that $\\widehat{\\mu}, \\widehat{\\pi}>0$.\n\nNow let us determine the highest agent value, across projects $C$, that can occur for an equilibrium in which the principal offers bonus $\\widehat{b}$ for completion and the agent chooses completion probability $\\widehat{\\mu}$. Consider then the problem of maximizing the agent's equilibrium payoff\n\n$$\nu(\\widehat{b})=u(0)+\\int_{0}^{\\widehat{b}} u_{+}^{\\prime}(b) d b\n$$\n\nby choice of convex function $u: \\mathbb{R}_{+} \\rightarrow \\mathbb{R}$ satisfying (i) $u(0) \\leq 0$, and (ii) $\\widehat{\\pi} \\geq u_{+}^{\\prime}(b)(1-b)$ for all $b$. The first requirement reflects the above observation that the agent cannot obtain a positive payoff if the bonus is zero. The second condition is a re-statement of Condition (17). Any solution to this problem involves $u(0)=0$ and\n\n$$\nu_{+}^{\\prime}(b)=\\frac{\\widehat{\\pi}}{1-b}\n$$\n\nfor all $b \\in[0, \\widehat{b}]$. In other words, the constraint (ii), or equivalently (17), holds with equality over $b \\in[0, \\widehat{b}]$ (in which case, the principal must obtain payoff $\\widehat{\\pi}$ from all such bonuses $b$ ). The agent's value function is therefore given on $[0, \\widehat{b}]$ by\n\n$$\nu(b)=\\int_{0}^{b} \\frac{\\widehat{\\pi}}{1-z} d z .\n$$\n\nNow, recall that $\\widehat{\\pi}=\\widehat{\\mu}(1-\\widehat{b})$, or $\\widehat{b}=1-\\frac{\\widehat{\\pi}}{\\mu}$. The agent's equilibrium payoff can then be written as","text_sha256":"62dd8f1869fd882ff0eed8f1701fb80d7391e563c454af4046745840efba024a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0020","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proof of Proposition 2","text":"$$\n\\int_{0}^{1-\\frac{\\widehat{\\pi}}{\\mu}} \\frac{\\widehat{\\pi}}{1-z} d z=[-\\widehat{\\pi} \\log (1-z)]_{0}^{1-\\frac{\\widehat{\\pi}}{\\mu}}=\\widehat{\\pi}(\\log (\\widehat{\\mu})-\\log (\\widehat{\\pi}))\n$$\n\nwhich is the expression given in Equation (13) (hence establishing also the one in Equation (12)). As explained in the main text, this payoff is maximized across feasible equilibrium values of $\\widehat{\\mu}$ and $\\widehat{\\pi}$ by $\\widehat{\\mu}=1$ and $\\widehat{\\pi}=\\frac{1}{e}$. The corresponding equilibrium bonus must be $\\widehat{b}=1-1 / e$.\n\nNote then that, if the project is $C^{*}$ as given in the proposition (with $C^{*}(0)=0$, as explained in the main text), and the principal offers any $b \\in[0,1-1 / e]$, the agent best responds by choosing $\\mu$ such that\n\n$$\nb=1-\\frac{1}{e \\mu},\n$$\n\ni.e. $\\mu=\\frac{1}{e(1-b)}$. All such bonuses therefore generate profit $1 / e$ for the principal. Hence, it is indeed an equilibrium of project $C^{*}$ for the principal to offer bonus $b^{*}=1-1 / e$, and the agent to choose completion probability equal to $\\mu^{*}=1$. This completes the proof of the proposition. $\\square$","text_sha256":"f844cbd3d200ec44b8c4aa71af3e9e50abeddb599e68349c366bb9e539063517"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0021","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Appendix B: Discussion of uniqueness of optimal projects","text":"## Appendix B: Discussion of uniqueness of optimal projects\n\nThe optimal project, $C^{*}$, described in Proposition 2 is not unique. To see this, note that $C^{*}$ can be arbitrarily modified at completion probabilities strictly below $1 / e$. Since $C^{*}(1 / e)=0$, any of these probabilities are weakly dominated by $1 / e$, so the modified project is still optimal. The goal of this appendix is to argue that optimal projects differ only in non-essential ways.\n\nUniqueness of the Optimal Binary Project.- Recall that the optimal project $C^{*}$ in Proposition 2 is determined via the principal's incentive constraint, (9), evaluated now at $(C, \\mu)=\\left(C^{*}, \\mu^{*}\\right)$ and taken to hold with equality for completion probabilities $\\widetilde{\\mu}$ above $\\pi^{*}$. This means that the principal can also generate her equilibrium payoff by setting any bonus smaller than $b^{*}$. Appendix A showed that these conclusions are valid for each optimal binary project. We also showed that, in any optimal binary project, the agent's optimal payoff is determined by an equilibrium which satisfies Parts (ii) and (iii) of the proposition. We now state formally these observations. We abuse notation and write the players' values, $u$ and $\\pi$, as functions of a binary outcome.\n\nRemark 1. In any optimal binary project $C$,\n\n(i) $\\left(b^{*}, \\mu^{*}\\right)=(1-1 / e, 1)$ is an agent-optimal equilibrium,\n(ii) for all $b \\in[0,1-1 / e], u(C, b)=\\int_{0}^{b} 1 /[e(1-z)] d z$, and\n(iii) for all $b \\in[0,1-1 / e], \\pi(C, b)=1 / e$.\n\nProof of Remark 1. The proof of Remark 1 recalls the proof of Proposition 2 in Appendix A.\n\nPart (i) of the remark follows because the expression in Equation (19) represents the agent's highest possible expected payoff in any project in which the completion probability is $\\widehat{\\mu}$ and the principal's expected payoff is $\\widehat{\\pi}$. Moreover, it is uniquely maximized by $\\widehat{\\mu}=\\mu^{*}=1$ and $\\widehat{\\pi}=\\pi^{*}=1 / e$. Part (ii) of Remark 1 then follows from Equation (18). Part (iii) of Remark 1 follows because the constraint (17) holds with equality in an optimal binary project over the relevant range of bonus payments. That is, because\n\n$$\n\\pi^{*}=\\bar{\\mu}(b)(1-b)\n$$\n\nfor all $b \\in[0,1-1 / e]$, where recall $\\bar{\\mu}(b)(1-b)$ coincides with the principal's value $\\pi(C, b)$ in any subgame $(C, b)$. $\\square$\n\nQED\n\nWe now argue that Remark 1 implies optimal binary projects are \"close\" to uniquely determined. We begin with a remark which compares any optimal binary project to the project $C^{*}$ of Proposition 2.\n\nRemark 2. In any optimal binary project $C$,\n\n(i) for all $\\mu \\in[0,1], C(\\mu) \\geq C^{*}(\\mu)$, and\n(ii) for all $\\mu \\in[1 / e, 1]$, there is a sequence $\\left(\\mu_{n}\\right)$ with $\\mu_{n} \\rightarrow \\mu$ and $C\\left(\\mu_{n}\\right) \\rightarrow C^{*}(\\mu)$.\n\nProof of Remark 2. To show Part (i) of Remark 2, consider the optimal project $C^{*}$ in Proposition 2. Notice that, as we reduce the bonus $b$ from $b^{*}$ to zero, the agent's best response in the subgame $\\left(C^{*}, b\\right)$ decreases continuously from one to zero. Suppose now that $C$ is a project with $C\\left(\\mu^{\\prime}\\right)<C^{*}\\left(\\mu^{\\prime}\\right)$ for some $\\mu^{\\prime}$, and let $b^{\\prime} \\in\\left(0, b^{*}\\right]$ be the bonus that implements $\\mu^{\\prime}$ in project $C^{*}$. Then we have\n\n$$\nu\\left(C, b^{\\prime}\\right) \\geq b^{\\prime} \\mu^{\\prime}-C\\left(\\mu^{\\prime}\\right)>b^{\\prime} \\mu^{\\prime}-C^{*}\\left(\\mu^{\\prime}\\right)=u\\left(C^{*}, b^{\\prime}\\right)=\\int_{0}^{b^{\\prime}} 1 /[e(1-z)] d z\n$$\n\nThe first equality follows because $\\mu^{\\prime}$ is a best response in the subgame ( $C^{*}, b^{\\prime}$ ). The second equality follows because $C^{*}$ is an optimal project, and by Part (ii) of Remark 1. Hence, by Part (ii) of Remark 1, $C$ is not an optimal project.\n\nTo show Part (ii) of Remark 2, fix an optimal binary project $C$ and consider a completion probability $\\mu \\in[1 / e, 1]$. By Part (iii) of Remark 1, it can be attained by a bonus $b=1-1 /(\\mu e) \\in$ $[0,1-1 / e]$. Formally, we mean that there exists a sequence $\\left(\\mu_{n}\\right)$ convergent to $\\mu$ with $\\mu_{n} b-C\\left(\\mu_{n}\\right) \\rightarrow$ $u(C, b)$. Note that the bonus $b$ also implements $\\mu$ in project $C^{*}$. By Part (ii) of Remark 1, the\nagent's value in subgame ( $C, b$ ) is the same as in subgame ( $C^{*}, b$ ); that is, $u(C, b)=u\\left(C^{*}, b\\right)$. In particular, considering the aforementioned sequence $\\left(\\mu_{n}\\right)$, we have\n\n$$\n\\mu_{n} b-C\\left(\\mu_{n}\\right) \\rightarrow u\\left(C^{*}, b\\right)=\\mu b-C^{*}(\\mu),\n$$\n\nimplying $\\lim _{n \\rightarrow \\infty} C\\left(\\mu_{n}\\right)=C^{*}(\\mu)$, which is what we wanted to show. $\\square$\n\nWe argue that, in spite of possible differences between any optimal binary project $C$ and the project $C^{*}$ of Proposition 2, such projects can be viewed almost equivalently from the agent's perspective. First, note that Remark 2 admits that some completion probabilities in $[1 / e, 1]$ may be more costly under $C$ than $C^{*}$. In this case, however, there are arbitrarily close probabilities which are as affordable to the agent as in $C^{*}$, or for which the difference in costs is negligible (this follows by Part (ii) of Remark 2). In addition, it turns out that, for any optimal binary project $C$, the specification of costs for probabilities below $1 / e$ is irrelevant. The reason is that, using Part (ii) of Remark 2, the agent can generate a completion probability at least $1 / e$ at negligible cost. ${ }^{20}$ Hence, the agent's value can be approached by completion probabilities at least $1 / e$ irrespective of the bonus.\n\nWe now formalize further the equivalence of optimal binary projects $C$ on $[1 / e, 1]$. Note first that if $C$ is restricted to be continuous, then $C$ is unique on $[1 / e, 1]$ and equal to $C^{*}$ on this interval by Part (ii) of Remark 2. For cost functions $C$ that are not continuous, we now demonstrate formally that the completion probabilities $\\mu$ that are relevant for the agent's problem are only those for which the cost is close to the one given by $C^{*}$.\n\nTo achieve our goal we let, for any $\\varepsilon>0$,\n\n$$\nP(\\varepsilon) \\equiv\\left\\{(\\mu, C(\\mu)): \\mu \\in[1 / e, 1], C(\\mu)-C^{*}(\\mu) \\leq \\varepsilon\\right\\} .\n$$","text_sha256":"267c23b70cf2338384f525569131622b10050016380b567d0afcbd5d1938a837"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0022","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Appendix B: Discussion of uniqueness of optimal projects","text":"Suppose the agent can choose only probabilities $\\mu$ with $(\\mu, C(\\mu)) \\in P(\\varepsilon)$, and suppose the associated costs are $C(\\mu)$. Then $P(\\varepsilon)$ describes the agent's technology in project $C$ but after removing his ability to choose probabilities $\\mu$ that we anticipate being redundant, either because they are less than $1 / e$ or because their costs exceed $C^{*}(\\mu)$ by more than $\\varepsilon$. By Parts (i) and (ii) of Remark 2, for a fixed bonus payment $b$, the agent's value is the same for the restricted technology $P(\\varepsilon)$ as if\n\n[^12]the agent could choose any completion probability with a cost specified by $C$. In fact, the feature of $P(\\varepsilon)$ that is relevant in determining the agent's value is the lower boundary of its closure. For any $\\varepsilon>0$, any $\\mu \\in[1 / e, 1]$, we have\n$$\n\\min \\{y:(\\mu, y) \\in \\operatorname{cl}(P(\\varepsilon))\\}=C^{*}(\\mu),\n$$\nwhich again can be seen directly from Parts (i) and (ii) of Remark 2. ${ }^{21}$ This demonstrates a further sense of equivalence between $C$ and $C^{*}$.\n\nUniqueness beyond binary projects. - The above discussion describes a qualified sense in which optimal binary projects are uniquely determined. Still, the possibility of optimal but non-binary projects may also be a source of non-uniqueness. Nonetheless, we show that properties analogous to those described above continue to hold, even among non-binary projects.\n\nWe first explain that output is one in any agent-optimal equilibrium of any optimal project. To this end, let $c^{*}$ be an optimal project and $\\left(w^{*}, F^{*}\\right)$ an agent-optimal equilibrium in $c^{*}$. Proposition 1 applied to the optimal outcome $\\left(c^{*}, w^{*}, F^{*}\\right)$ implies that the corresponding binary outcome $\\left(\\widetilde{c}, w_{b^{*}}, B_{\\mu_{F^{*}}}\\right)$ is also optimal. Then, Part (i) of Remark 1 implies that we must have $\\mu_{F^{*}}=1$, and hence $F^{*}=B_{1}$.\n\nNext we argue that output realizations in $(0,1)$ are redundant in the sense that replacing these realizations by output zero has no impact on equilibrium behavior. To this end, we first show that it can be assumed that $w^{*}(x)=0$ for all $x<1$. The intuition is that if the principal wants to implement output one, she should not reward the agent for any other output realization by offering a positive payment. To state it formally, let us define the payments scheme, $\\rho_{b}$, for each $b$ such that $\\rho_{b}(1)=b$ and $\\rho_{b}(x)=0$ for $x \\neq 1$. Observe that replacing $w^{*}$ by $\\rho_{w^{*}(1)}$ makes choosing $B_{1}$ no less attractive to the agent, so $\\left(\\rho_{w^{*}(1)}, B_{1}\\right)$ is also an agent-optimal equilibrium in $c^{*}$.\n\nProvided that the compensation scheme is $\\rho_{w^{*}(1)}$, when the agent is contemplating choosing a distribution, all that matters is the probability that output is one. So, moving all the probability mass from $(0,1)$ to zero has no impact on the agent's choice of a distribution. Moreover, these new distributions generate smaller expected outputs, hence the principal still prefers to implement $B_{1}$. To make these claims precise, let us define a binary project $\\widetilde{C}$ such that $\\widetilde{C}(\\mu)=\\inf \\left\\{c^{*}(F): \\Delta(F)=\\mu\\right\\}$, where $\\Delta(F)$ denotes the atom at one specified by $F .{ }^{22}$ Given the payment schedule $\\rho_{w^{*}(1)}$, the change in cost function from $c^{*}$ to $\\widetilde{C}$ does not affect the agent's\n\n[^13]willingness to choose completion probability one. We conclude that the binary project $\\widetilde{C}$ is optimal and $\\left(w^{*}(1), 1\\right)$ is an agent-optimal equilibrium in $\\widetilde{C}$.\n\nTo give a further sense in which the results in the previous section are robust to the considerations of non-binary projects, we state the following.\n\nRemark 3. If $c^{*}$ is an optimal project and $\\left(w^{*}, F^{*}\\right)$ is an agent-optimal equilibrium in $c^{*}$, then\n\n(i) $F^{*}=B_{1}$,\n(ii) $\\Pi\\left(w^{*}, F^{*}\\right)=U\\left(c^{*}, w^{*}, F^{*}\\right)=1 / e$,\n(iii) $u\\left(c^{*}, \\rho_{b}\\right)=\\int_{0}^{b} 1 /[e(1-z)] d z$ for all $b \\in[0,1-1 / e]$, and\n(iv) $\\pi\\left(c^{*}, \\rho_{b}\\right)=1 / e$ for all $b \\in[0,1-1 / e]$.\n\nWe have already established Part (i) and that the binary outcome $\\left(\\widetilde{C}, w^{*}(1), 1\\right)$ is agent-optimal. So, by Part (iii) of Remark 1, $w^{*}(1)=1-1 / e$, and the principal's payoff in project $c^{*}$ must be $1 / e$, establishing Part (ii). By the construction of $\\widetilde{C}, u\\left(c^{*}, \\rho_{b}\\right)$ is equal to $u(\\widetilde{C}, b)$ for all $b$. Optimality of $\\widetilde{C}$ and Part (ii) of Remark 1 then imply Part (iii) of Remark 3. Part (iii) of Remark 1 implies $\\pi(\\widetilde{C}, b)=1 / e$ for any $b \\in[0,1-1 / e]$ and hence $\\pi\\left(c^{*}, \\rho_{b}\\right) \\geq 1 / e$. Part (ii) of Remark 3 then implies $\\pi\\left(c^{*}, \\rho_{b}\\right)=1 / e$ for all $b \\in[0,1-1 / e]$ establishing Part (iv).\n\nAn interpretation of Remark 3 is that the conclusions reached in Remark 1 remain valid even when projects are not restricted to be binary. For instance, consider any optimal project $c^{*}$ and an optimal binary project $C$. The players' values are the same across both projects for any payment schedule $\\rho_{b}, b \\in[0,1-1 / e]$, as follows from Parts (iii) and (iv) of Remark 3. This can be compared to Parts (ii) and (iii) of Remark 1.","text_sha256":"aa4e6efb5f12f7e8fac950966a249b511c35ba360106370e24ee09c3deec969e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0023","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Appendix B: Discussion of uniqueness of optimal projects","text":"[^0]:    *We are grateful to Alessandro Bonatti, Florian Ederer, Uli Hege, Harry Di Pei, Luis Rayo, Phil Reny, Dan Spulber, Jean Tirole, as well as to participants at several seminars and conferences for helpful comments. In the initial stages of this project, D. Garrett was based at Toulouse School of Economics, University of Toulouse Capitole. He is grateful for funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 714147). D. Garrett and A. Smolin are grateful for funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d'Avenir) program (grant ANR-17-EURE-0010).\n    ${ }^{\\dagger}$ D. Garrett: University of Essex, Wivenhoe Park, Colchester CO4 3SQ, U.K., mailto:d.garrett@essex.ac.uk; G. Georgiadis: Kellogg School of Management, Northwestern University, Evanston, IL 60208, U.S.A., mailto:ggeorgiadis@kellogg.northwestern.edu; A. Smolin: Toulouse School of Economics, University of Toulouse Capitole, 31080 Toulouse Cedex 06, France, mailto:alexey.v.smolin@gmail.com; B. Szentes: Department of Economics, London School of Economics, London, WC2A 2AE, U.K., mailto:b.szentes@lse.ac.uk.\n\n[^1]:    ${ }^{1}$ A sizable literature on entrepreneurship examines the decisions of entrepreneurs when developing a business. See Gans et al. (2019) for a recent review.\n    ${ }^{2}$ To be precise, we characterize this set as a function of the largest possible output realization.\n    ${ }^{3}$ See also Garrett (2021) for a related exercise in a setting with moral hazard and adverse selection.\n\n[^2]:    ${ }^{4}$ We extend our analysis to environments where the agent is risk averse in the Online Appendix.\n    ${ }^{5}$ Gans et al. (2019) argue that many of the decisions an entrepreneur takes during business development cannot be revisited or renegotiated.\n    ${ }^{6}$ Another consideration that may rule out secret deviations by the agent is monitoring by the principal. The principal may not permit the agent to adjust the project initially proposed, especially if she finds it costly to evaluate all the implications of the adjustments.\n    ${ }^{7}$ While there is some multiplicity of optimal projects, we show in Appendix B that all optimal projects share the same essential attributes, so we write informally of \"the\" optimal project.\n\n[^3]:    ${ }^{8}$ For example, Averch and Johnson (1962) observe that a regulated firm has incentives to inflate capital costs.\n    ${ }^{9}$ See the Online Appendix.\n\n[^4]:    ${ }^{10}$ Also related are Carroll (2015) who studies a principal facing ambiguity with respect to the agent's technology, Laux (2001) who considers agents with multiple projects, and Jewitt et al. (2008) who explore general bounds on payments beyond limited liability.\n\n[^5]:    ${ }^{11}$ Such an indifference argument has appeared in the contexts of incentivizing monitoring (Ortner and Chassang (2018)), optimal testing (Perez-Richet and Skreta (2018)) and monopoly pricing in the presence of ambiguity (Bergemann and Schlag (2011)).\n    ${ }^{12}$ Also related, Roesler and Szentes (2017) consider a setting where signals inform an otherwise uninformed buyer of his value, and asks which signal structure yields the highest information rent.\n\n[^6]:    ${ }^{13}$ Non-negativity of payments encodes the limited-liability constraint.\n    ${ }^{14}$ That the agent can choose any distribution on [0, 1] departs from much of the moral hazard literature, where the output distribution is parameterized by a one-dimensional variable called \"effort\". We view the agent's chosen distribution as synonymous with his action, an approach also taken for instance by Carroll (2015).\n\n[^7]:    ${ }^{15}$ Recall that $w_{b^{*}}(x)=b^{*} x$ for all $x$.\n\n[^8]:    ${ }^{16}$ We take the infimum in case no cheapest distribution exists.\n\n[^9]:    ${ }^{17}$ Note that $\\widehat{C}(0)=0$ follows because, if $\\widehat{C}(0)>0$, we could reduce all costs by this amount, keeping the players' incentives unchanged, but increasing the agent's equilibrium payoff.\n\n[^10]:    ${ }^{18}$ Given a possibly non-binary equilibrium distribution $F^{*}$, we showed that agent upward deviations to distributions with means above $\\mu_{F^{*}}$ could be ignored in the analysis (recall the construction of the binary project $\\widetilde{c}$ ).\n\n[^11]:    ${ }^{19}$ Recall that $\\left(F_{n}\\right) \\in \\mathbf{F}^{c, w}$ if and only if $\\lim _{n \\rightarrow \\infty} U\\left(c, w, F_{n}\\right)=u(c, w)$.\n\n[^12]:    ${ }^{20}$ Formally, there is a sequence $\\left(\\mu_{n}\\right)$ that approaches $1 / e$ from above, and for which $C\\left(\\mu_{n}\\right) \\rightarrow 0$. This follows from Part (ii) of Remark 2 and the continuity of $C^{*}$.\n\n[^13]:    ${ }^{21}$ Here, \" $\\operatorname{cl}(\\cdot)$ \" refers to the closure of the set.\n    ${ }^{22}$ That is, $\\Delta(F)=F(1)-\\lim _{x \\nearrow 1} F(x)$.","text_sha256":"d93a83233c7cc83d9d3c244f814c09153ee5c71289bab433be9419b311a8c962"}
{"schema_version":"1.0","chunk_id":"alex-smolin:optimal-technology-design:2023-01:0024","work_id":"alex-smolin:optimal-technology-design","paper_id":"alex-smolin:optimal-technology-design:2023-01","title":"Optimal Technology Design","authors":[{"name":"Daniel F. Garrett","url":"https://sites.google.com/site/dfgarrett/"},{"name":"George Georgiadis","url":"https://www.kellogg.northwestern.edu/faculty/georgiadis/index.html"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"},{"name":"Balázs Szentes","url":"https://www.hkubs.hku.hk/people/balazs-szentes/"}],"manuscript_date":"2023-01","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/optimal-technology-design.md","source_record":"https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf","doi":"https://doi.org/10.1016/j.jet.2023.105621","citation":"Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Daniel F. Garrett; George Georgiadis; Alex Smolin; Balázs Szentes\n\n**Canonical citation:** Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/optimal-technology-design.md\n\n**Source record:** https://alexsmolin.com/files/optimal-technology-design-working-paper.pdf\n\n**Published record:** https://doi.org/10.1016/j.jet.2023.105621\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"7dabbe0620204ed26ef2449c065e4f268ac71bb3580d1d6762500366de2f5ba7"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0001","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Alex Smolin.\n> Canonical citation: Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"13e7d73275324d7bea56153b0fd417c9e6147c4181756a11a4c93b535bd4ff31"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0002","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Disclosure and Pricing of Attributes","text":"# Disclosure and Pricing of Attributes\n\n**Authors:** Alex Smolin\n\n**Manuscript date:** 2022-08-08\n\n#### Abstract\n\nA monopolist sells an object characterized by multiple attributes. A buyer can be one of many types, differing in their willingness to pay for each attribute. The seller can provide arbitrary attribute information in the form of a statistical experiment. To screen different types, the seller offers a menu of options that specify information prices, experiments, and object prices.\n\nI characterize revenue-maximizing menus. All experiments belong to a class of linear disclosure rules. An optimal menu may be nondiscriminatory and qualitatively depends on the structure of buyer heterogeneity. The analysis highlights the importance of demand microstructure and the benefits of information control in trade settings.\n\nKeywords: advertising, attributes, bilateral trade, demand transformation, information design, intermediaries, mechanism design, multidimensional disclosure, persuasion\n\nJEL Codes: D11, D42, D82, D83, L15\n\n[^0]","text_sha256":"69abfff72f669c77cc1224cb29c730df393efd98551b82e8dc1ce1102f3ee310"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0003","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nIn many important markets, sellers have considerable control over the information available to their buyers. Business brokers can control the extent of the firm investigation and documentation that they supply, recruiting platforms can decide what parts of a job candidate's profile to reveal to employers, and producers can decide what features of their consumer goods to advertise. In all of these markets, the products (i.e., a business, meeting with a job candidate, and consumer good) are characterized by multiple attributes that appeal to different types of buyers. To maximize revenue, sellers need to understand what attribute information to provide, how to price their products, and whether and how to price the information provided. These questions require unification of information and mechanism design paradigms to allow for joint control over information and monetary incentives.\n\nAs a concrete example, consider the operation of Ziprecruiter.com, a major online recruiting platform. The platform facilitates matching job seekers with employers: employers subscribe to the platform to advertise their open vacancies and obtain access to a large database of resumes. ${ }^{1}$ The recruitment market features substantial heterogeneity on both sides. Candidate profiles vary in many attributes, including work experience, education levels, and standardized test scores. The employers belong to distinct types, such as tech start-ups, chain stores, and government agencies. Naturally, different types of employers are looking for different attributes in their candidates.\n\nZiprecruiter.com has access to a large amount of data about prospective candidates and, by programming its algorithms, it can commit to coarsening the data or denying access to some attributes. Moreover, the platform can price both information, through upfront fees, and the decision to contact a job candidate, through contact fees. Indeed, the platform currently employs a nonlinear pricing scheme for subscriptions, which varies in the breadth of information provided and the ability to contact preferred candidates (Dubé and Misra (2019)). My goal is to study the trade-offs that the platform faces and to evaluate the allocation distortions introduced by its information control.\n\nIn this paper, I develop a framework to study information disclosure and the pricing of multiattribute products. I consider a monopolist seller who has an indivisible object for sale to a single buyer and aims to maximize her revenue. The object has several attributes, and the buyer is uncertain about their values. The buyer's valuation for the object is linear in attributes. The strengths of the preferences are the buyer's private information and constitute the buyer's type. The seller controls the pricing and attribute information available to the buyer.\n\n[^1]Both the object and the information about its attributes are valuable for a buyer, and I allow the seller to price them separately. In particular, the seller offers a menu of options that differ in their informativeness. Each option consists of an information price, paid upfront, attribute information, and a price for the object. The attribute information is modeled as an arbitrary statistical experiment informative about attributes. Information control enables price discrimination. By varying the information price, the experiment, and the object price, the seller can screen buyer types. I illustrate the qualitative features of multiattribute disclosure and pricing in Section 3.\n\nI emphasize that the model formulation implies the seller is not privately informed and cannot condition the price on the outcome of the experiment. In other words, the seller can provide the buyer access to attribute information about the product but cannot condition the price on the outcome of this information. This kind of information provision is referred to in the economic literature as \"private disclosure\" and has been widely used. ${ }^{2}$ One common motivation for the privacy of disclosure is that it may be hard for the seller to assess the impact of information on the buyer. As such, the analysis is directly applicable to the settings of intermediaries who organize sales of goods and services and in which the buyer has more expertise in assessing the product information than the intermediary. However, the analysis also provides insights into the more traditional buyer-seller settings, such as advertising, as long as the seller does not condition the price on the outcome of the information provided.\n\nThe general revenue-maximization problem features information design and multidimensional screening. As such, it entails two methodological challenges. First, the class of stochastic experiments is large since the underlying uncertainty covers a continuum of possible states, each having multiple dimensions. To understand the distortions driven by the information design, it is important to determine the structure of optimal experiments. Second, in the absence of a single-dimensional structure, it is not clear which incentive constraints are relevant for optimal design. This difficulty is further exacerbated by the distinct feature of information according to which different buyer types can respond differently to the same signal. I progress in both directions in turn.\n\nIn Section 4, I study the design of disclosure rules. ${ }^{3}$ Providing disclosure serves two functions. First, it swings the buyer's expectations and may persuade him to purchase the object at a higher price. Second, providing several disclosure options may facilitate screening because different buyer types prefer learning about different aspects of the object. Theorem 1 shows that an optimal way to combine these two functions is through a specific class\n\n[^2]of experiments-linear disclosure rule. A linear disclosure rule informs whether a linear combination of attributes is above or below a specified threshold. This disclosure guides the allocation and can be seen as informing the buyer about the valuation of a virtual type that can differ from the demanding type to account for incentive constraints. ${ }^{4}$ Notably, this result requires no assumptions about the distributions of types or attributes and, as such, can be generalized to arbitrary valuation functions.","text_sha256":"80f7373d0a506d27161e724244fee687580181c55e3349717a6a4de026b3a46b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0004","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"In addition, I establish an analog of the \"no distortion at the top\" property, which is common in mechanism design settings: If gains from trade are commonly known to be positive, then some type is provided with no information and always purchases the object.\n\nIn Section 5, I study optimal pricing mechanisms. In Theorem 2, I establish that if all buyer types value the same, always positive, attribute, then no information is optimally provided and the seller posts a single price for the object. The intuition behind this result lies in the product structure of the buyer's valuation. When all types value the same attribute, any disclosure realization simply scales their valuations and the corresponding demand curve. Even if the seller could condition the price on this realization, she would charge scaled prices, serve the same types, and obtain scaled revenue. By the martingale property of Bayesian expectations, the seller can obtain the same revenue by providing no information. ${ }^{5}$\n\nThe case of several attributes is qualitatively different because types can be differentiated not only vertically but also horizontally. Therefore, an optimal allocation may depend on attribute realizations: the seller should aim to allocate the object to types who value the realization the most. To guide the allocation, she should provide some attribute information.\n\nI formalize this intuition in the setting in which each type values one of many independent attributes (Section 5.2) but the seller does not know which attribute or the strength of the preference. I show that an optimal menu features free-of-charge partial disclosure and no price discrimination. Information is not priced since the payment can be backloaded into the object price. Price dispersion is not profitable, because it implies that the seller could extract more surplus on some items by simultaneously lowering their object prices and changing their informational content. The optimal menu admits nondiscriminatory implementation-posting a single price for the object and informing the buyer whether the object is sufficiently good along each attribute.\n\nIn the limit case, in which there is only one type per attribute, in the optimal menu the seller effectively persuades each type to purchase the object at a fixed price separately by\n\n[^3]informing him whether his valuation is sufficiently high. In Section 5.3, I show that similar mechanisms are optimal in a broad range of settings, as long as the types can be seen as belonging to distinct cohorts-with low valuation correlation across them.\n\nI conclude the analysis with a discussion in Section 6. First, I discuss what product information must be priced and when. Second, I emphasize that attribute information can rotate the demand curve locally and can justify attribute shrouding. Finally, I indicate how a multiattribute framework complements the existing disclosure frameworks, and I emphasize the importance of explicitly modeling demand microstructure, i.e., how the consumer valuation is formed, in the settings with information control.\n\nRelated Literature This paper is about information provision and pricing. One strand of the related literature focuses on nondiscriminatory mechanisms-in which a seller commits to a single disclosure rule. Lewis and Sappington (1994) introduce these mechanisms in a setting where a buyer has no prior information. They show that within a simple parameterized class, an optimal disclosure rule is extreme: either full or no disclosure. Bergemann and Pesendorfer (2007) further observe that if there is common knowledge of positive trade gains, then no disclosure dominates any other possible disclosure rule because it allows the seller to extract the full expected surplus. ${ }^{6}$ Johnson and Myatt (2006) extend the analysis to settings in which the buyer has prior information. They focus on disclosure rules that correspond to the global rotation of a demand curve and show, once again, that extreme disclosure rules are optimal. My paper contributes to this literature by showing that if the product has several attributes, then a partial disclosure can dominate both full and no disclosure, even if there is common knowledge of positive trade gains (Sections 4.4, 6.3).\n\nAt the same time, when the buyer has private information, it is natural to study discriminatory mechanisms and how they can be used to screen buyer types. In an influential paper, Eső and Szentes (2007) study settings in which the attribute and the buyer's type enter the valuation \"additively.\" In such settings, under certain distributional assumptions, the seller optimally provides full disclosure. However, Li and Shi (2017) show that the seller should withhold some information if the types represent private information about the object. ${ }^{7}$ Section 6.3 contains a detailed discussion related to these two papers.\n\nAll of the works described above operate in single-dimensional settings. Under complete object information, when comparing any two objects, all buyer types agree on the ranking.\n\n[^4]However, in practice, many products are multidimensional, with different attributes that appeal to different buyers. In this paper, I demonstrate that these settings can be successfully studied and lead to qualitatively different results. Despite the richness of the attribute space, optimal experiments belong to a tractable class of linear disclosure rules (Section 4.4). Optimal mechanisms feature partial disclosure but can be remarkably simple (Section 5.2).\n\nThis paper builds on several existing frameworks. The multiattribute buyer's valuation follows the characteristic model of Lancaster (1966). An unrestricted search for an optimal disclosure rule is a defining feature of the Bayesian persuasion literature (Rayo and Segal (2010), Kamenica and Gentzkow (2011)). ${ }^{8}$ The screening analysis builds on the mechanism design machinery of Myerson $(1981,1982)$ and Kolotilin, Mylovanov, Zapechelnyuk, and Li (2017). Finally, information design with screening and monetary transfers has already appeared in my previous work (Bergemann, Bonatti, and Smolin (2018)). There, the seller can price only information-the buyer's action is not contractable. In contrast, in this paper, the seller can price both services and as a result, in many settings, provides the information free of charge.","text_sha256":"ee423b9a98ae5549674bd8d6dc0f1226433742cb417e17bae06a26e38ee47d5c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0005","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"## 2 Model\n\nA buyer (he) decides whether to buy a single indivisible object from a seller (she). The object has a finite number $J$ of characteristics or attributes. The attribute values constitute an attribute vector $x=\\left(x_{1}, \\ldots, x_{J}\\right) \\in X=\\mathbb{R}^{J}$. The buyer's preferences toward each attribute constitute the buyer's type $\\theta=\\left(\\theta_{1}, \\ldots, \\theta_{J}\\right) \\in \\Theta \\subseteq \\mathbb{R}^{J}$. The ex post buyer's valuation for the object is: ${ }^{9}$\n\n$$\nv(\\theta, x)=\\theta \\cdot x=\\sum_{j=1}^{J} \\theta_{j} x_{j} .\n$$\n\nThe buyer's utility is quasilinear in transfers. The seller maximizes her revenue.\nPrior Information Attributes are distributed over $X$ according to a cumulative distribution function $G$. The buyer and seller are symmetrically informed about the attributes. The type space $\\Theta$ can be finite or infinite. The buyer's type is his privately known preferences, which are uncorrelated with attributes. From the seller's perspective, the types are distributed according to a cumulative distribution function $F$. Until Section 5, I do not impose any structural assumptions on the attribute and type distributions. The only technical\n\n[^5]requirement is that the ex ante expectations of all attributes are finite.\nInformation Disclosure The seller can disclose attribute information to the buyer. This information is modeled as a statistical experiment $E=(S, \\pi)$ that consists of a signal set $S$ and a likelihood function: ${ }^{10}$\n$$\n\\pi: X \\rightarrow \\Delta(S) .\n$$\n\nThe experiment can be arbitrarily informative about the attributes. That is, the experiment can provide no information, or no disclosure, $\\underline{E} \\triangleq(\\underline{S}, \\underline{\\pi})$, with $\\underline{S}$ being a singleton; it can fully reveal attributes, or provide full disclosure, $\\bar{E} \\triangleq(\\bar{S}, \\bar{\\pi})$, with $\\bar{S}=X$ and $\\bar{\\pi}(x)$ placing a probability of 1 on $s=x$; or it can provide partial information.\nSelling Mechanism For the environments in which the designer can control players' private information, it is not generally known what class of mechanisms one can look at without loss of generality. To make progress while avoiding trivialities, I follow Eső and Szentes (2007) and Li and Shi (2017) and focus on a class of menu mechanisms, so that the seller designs a menu of items, $i \\in \\mathcal{I}$ :\n\n$$\nM=(r(i), E(i), p(i))_{i \\in \\mathcal{I}} .\n$$\n\nEach item consists of an experiment $E(i)$ and two tariffs $r(i) \\geq 0$ and $p(i) \\geq 0$. The first tariff captures the price of information-the upfront payment made to observe the signal of experiment $E(i)$, irrespective of whether the buyer decides to purchase the object later. The second tariff captures the price of the object, paid only if the trade occurs. Effectively, the menu is a collection of call options that differ in monetary terms and information disclosure, designed to screen different buyer types. ${ }^{11}$ This class of mechanisms provides a natural and rich framework to study how information disclosure and pricing interact in design problems.\n\nThe timing is analogous to that of Courty and Li (2000) and is as follows. The seller posts a menu $M$. The attribute vector $x$ and the buyer's type $\\theta$ are realized. If the buyer refuses to participate, then the players obtain zero payoffs. Otherwise, the buyer chooses an item $i \\in \\mathcal{I}$ and pays the corresponding price $r(i)$. Next, the buyer observes a signal $s$ from the experiment $E(i)$ and decides whether to buy the object at the price $p(i)$. Finally, the payoffs are realized. The timing is illustrated in Figure 1.\n\nThe timing implies that the seller commits to a menu before the realization of the attributes $x$ and the type $\\theta$. The attributes $x$ and signals $s$ are not contractible, corresponding to the setting of \"private disclosure\" (Li and Shi (2017)). ${ }^{12}$ Sales are deterministic-a pay-\n\n[^6]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: Timeline of the selling mechanism.\n\nment guarantees a transaction. Sequential interactions between players are excluded, so belief-elicitation schemes and scoring rules are not available. ${ }^{13}$\n\nMy goal is to characterize a revenue-maximizing menu, i.e., a menu upon which the seller cannot strictly improve by offering another menu.","text_sha256":"a1a6ca3c2052d79264e20e3f15413ea18e4bb7faa342b7929509ac7ecb112a32"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0006","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Illustrative Example","text":"## 3 Illustrative Example\n\nI begin by illustrating the workings of disclosure and pricing through a simple example. There are two attributes that are uniformly and independently distributed, $J=2, x_{1} \\sim U[0,1]$, $x_{2} \\sim U[0,2]$. There is a continuum of buyer types split into two cohorts. Types $\\theta_{1} \\in \\Theta_{1}$ value only the first attribute, i.e., each $\\theta_{1}$ is of the form $\\left(\\theta_{11}, 0\\right)$, whereas types $\\theta_{2} \\in \\Theta_{2}$ value only the second attribute, i.e., each $\\theta_{2}$ is of the form $\\left(0, \\theta_{22}\\right)$. Each cohort is equally likely, and the marginal type distributions are uniform over [0, 2]. The sets of attributes and types are illustrated in Figure 2.\n\nPerhaps the simplest way to think about the impact of attribute disclosure is in terms of the demand curves that the seller faces and how these curves are affected by the release of information. To this end, consider a simple class of mechanisms in which the seller provides some information free of charge and follows it with posting a single object price. These mechanisms can be viewed as marketing strategies that combine pricing with informative, persuasive advertising.\n\nIf the seller provides no disclosure, then the expectation of the first and second attributes stay at their prior values of 1/2 and 1, respectively. As such, expected valuations of types in the first cohort are distributed uniformly over the interval [0, 1] and those in the second cohort over the interval [0, 2]. These valuations translate into a piecewise-linear demand curve with a kink at price $p=1$, as illustrated in Figure 3. Given this demand, the optimal no-disclosure price is $p_{n}=2 / 3$ and it generates revenue of 1/3.\n\n[^7]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 2: Illustrative example: Attributes are distributed uniformly over the gray rectangle (left); types are distributed uniformly over the L-shaped segment (right).\n\nIn contrast, if the seller provides full disclosure, then the valuation of each type stochastically changes according to the realization of a relevant attribute. As a result, a valuation of a first-cohort type with intensity $\\theta_{11}$ is distributed uniformly over $\\left[0, \\theta_{11}\\right]$, and a valuation of a second-cohort type with intensity $\\theta_{22}$ is distributed uniformly over $\\left[0,2 \\theta_{22}\\right]$. The demand curve aggregates these valuations across all types and is presented in Figure 3. Compared to the benchmark of no disclosure, the demand decreases at lower prices and increases at higher prices. The change is determined by the microstructure of the consumer types and the spread of their valuations. Roughly, one can think of this change as being driven by the types with intermediate ex ante valuations who, instead of remaining uninformed, learn that the attributes are too low, decreasing the demand at lower prices, or too high, increasing the demand at higher prices. The overall effect for the seller is negative: The optimal full-disclosure price is $p_{f} \\simeq 0.82$ and it leads to revenue $0.28<1 / 3$.\n\nTo increase revenue, the seller should provide partial disclosure. Consider an experiment that reveals whether the first attribute is above some threshold $\\alpha_{0}$ but provides no information about the second attribute. This disclosure affects only valuations of first-cohort types $\\theta_{1}$ and can be viewed as transforming this cohort into two others. One new cohort captures valuations of types $\\theta_{1}$ who observed that the first attribute is below the threshold; these valuations are uniformly distributed over an interval of $\\left[0,2 \\mathbb{E}\\left[x_{1} \\mid x_{1}<\\alpha_{0}\\right]\\right]$. Another new cohort captures valuations of types $\\theta_{1}$ who observed that the first attribute is above the threshold; these valuations are uniformly distributed over an interval of $\\left[0,2 \\mathbb{E}\\left[x_{1} \\mid x_{1} \\geq \\alpha_{0}\\right]\\right]$. As a result, the demand curve is piece-wise linear with two kinks. Relative to no disclosure,\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 3: Attribute disclosure and demand transformation. Left: demand curves under no disclosure and full disclosure. Right: demand curves under no disclosure and optimal disclosure. Vertical lines indicate revenue-maximizing prices.\n\ndemand decreases at low prices, increases at medium prices, and remains the same at high prices. If the disclosure threshold is chosen optimally, $\\alpha_{0}^{*} \\simeq 0.27$, then this demand transformation benefits the seller. The optimal price is $p_{\\text {opt }} \\simeq 0.80$, resulting in an approximate revenue of $0.35>1 / 3$ (Figure 3). Intuitively, this disclosure targets types with lower ex ante valuations, i.e., those in the first cohort, and persuades them to buy at a higher price.\n\nThe effectiveness of partial disclosure raises the question of whether the seller can increase revenue even further if she employs a discriminatory menu with upfront payments, offering a range of experiments at different prices. After all, the cohort structure presents a clear opportunity to screen types from different cohorts by offering information about different attributes. Perhaps surprisingly, the answer is negative, as I show in Section 5.2. The simple partial-disclosure mechanism presented here is optimal.\n\nIn the example that we have just studied, the attributes were independently distributed and each type cared about a single attribute. As a result, an optimal disclosure rule provided information only about one attribute. In a more general case, one can expect an optimal disclosure rule to be richer-providing information about several attributes simultaneously. In the next section, I show how such a disclosure rule should optimally be constructed.","text_sha256":"bcc0628712ef91f5dbaffec6af20a6c8d39b12f2c1b85e103e240744eb13ec4f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0007","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Design of Disclosure Rules","text":"## 4 Design of Disclosure Rules\n\nI proceed with studying the optimal menu design in general settings. I begin by discussing the buyer's incentives and formalizing his choice in an arbitrary menu. I use this formalization to approach the design problem in two consecutive steps. First, I identify the class of optimal disclosure rules without explicitly characterizing a pricing scheme; this step is presented in the current section. Second, I derive optimal pricing details and complete the characterization of optimal menus in several leading settings; that step is presented in Section 5.","text_sha256":"cc2735fb081cef2da8a7225f0b501f4b1e07e19a00c1ba038d3d968f3c478920"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0008","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.1 Buyer's Problem","text":"### 4.1 Buyer's Problem\n\nConsider the buyer's incentives when he chooses an item from a given menu. Let his type be $\\theta$. If he chooses option $i$, then he pays the upfront price $r(i)$. Then, a signal $s$ is realized according to the likelihood function $\\pi(E(i))$, leading to the interim valuation:\n\n$$\nV(i, s, \\theta) \\triangleq \\mathbb{E}[v(\\theta, x) \\mid E(i), s] .\n$$\n\nFinally, the buyer decides whether to buy the object and does so optimally if and only if $V(i, s, \\theta)-p(i)$ is greater than 0. By integrating over signal realizations, I can define the resulting (total) trade probability as:\n\n$$\nQ(i, \\theta) \\triangleq \\operatorname{Pr}(V(i, s, \\theta)-p(i) \\geq 0 \\mid E(i)) .\n$$\n\nThe corresponding indirect utility of choosing option $i$ can be written as:\n\n$$\nU(i, \\theta)=-r(i)+\\mathbb{E}[\\max \\{0, V(i, s, \\theta)-p(i)\\} \\mid E(i)],\n$$\n\nType $\\theta$ chooses an option with the largest indirect utility. Naturally, types seek information that most fits their interests. This feature gives the seller the opportunity to discriminate among the types by carefully designing the menu.","text_sha256":"a1188bec192cb401292ac786607e2b838829912b6a85c347923f07ad00bf7b45"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0009","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.2 Responsive Menus","text":"### 4.2 Responsive Menus\n\nThe seller's problem lies at the intersection of the mechanism and information design because the seller can both control the information available to the buyer and charge monetary transfers. In principle, she can offer complex experiments in an attempt to better discriminate among types; however, I show that an optimal class of experiments is simple and tractable.\n\nI begin approaching the seller's problem by binding the size of the optimal menus and signal sets. First, there is no need to have more items than there are types, so I can focus on direct menus:\n\n$$\nM=(r(\\theta), E(\\theta), p(\\theta)),\n$$\n\nwhich effectively ask the buyer his type and assign the experiment and the tariffs as functions of his report. Second, I can bound the size of the signal sets. For a given direct mechanism $M$, I call an experiment $E(\\theta)$ responsive if $S(\\theta)=\\left\\{s^{+}, s^{-}\\right\\}$, and type $\\theta$, when choosing this experiment, purchases the object if and only if $s=s^{+}$. I call the menu responsive if all of its experiments are responsive.\n\nProposition 1. (Responsive Menus)\nThe outcome of every menu can be replicated by a direct and responsive menu.\nProof. Detailed proofs of all formal statements can be found in the Appendix. $\\square$\n\nProposition 1 imposes an elementary structure on the exchange of information between the seller and the buyer. The buyer should inform the seller about his preferences, and the seller should provide a recommendation about whether to buy the object. The proof is analogous to the argument of the revelation principle of Myerson (1982). If the menu contains nonresponsive experiments, then the seller can replace them with responsive experiments that replicate the behavior of truth-telling types. After this modification, truth telling delivers the same payoff as before. Dishonesty, however, becomes weakly less appealing (Blackwell (1953)).\n\nIn a responsive menu, every experiment $E$ is characterized by its trade function:\n\n$$\nq(x) \\triangleq \\operatorname{Pr}\\left(s^{+} \\mid E, x\\right) .\n$$\n\nThe function defines a probability of the trade recommendation for each attribute realization. The probability of the no-trade recommendation is then the complimentary $1-q(x)$. A responsive menu features a collection of experiments, one per each buyer's type, that corresponds to a collection of trade functions. With slight abuse of notation, I refer to the trade function of type $\\theta$ as $q(\\theta, x)$.","text_sha256":"aeae01a9a5c0b4396a6eaf852eefed470f763530656ce9872df1fe6e3932d11b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0010","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.3 Seller's Problem","text":"### 4.3 Seller's Problem\n\nProposition 1 enables the association of each experiment with its trade function (8). Vice versa, any trade function $q: X \\rightarrow[0,1]$ determines a responsive experiment that recommends trading with probability $q(x)$. As such, the seller's problem can be written in a standard\nmechanism design form as a maximization of the expected revenue over the tariff and trade functions:\n\n$$\n\\max _{(r(\\theta), q(\\theta, x), p(\\theta))} \\int_{\\theta \\in \\Theta}\\left(r(\\theta)+p(\\theta) \\int_{x \\in X} q(\\theta, x) \\mathrm{d} G(x)\\right) \\mathrm{d} F(\\theta),\n$$\n\nsubject to the incentive-compatibility constraints and individual rationality constraints. The seller's revenue obtained from a particular type consists of the upfront payment $r(\\theta)$ and, if the buyer decides to purchase the object, the object price $p(\\theta)$. The incentive-compatibility constraints require that, for all $\\theta, \\theta^{\\prime} \\in \\Theta$ :\n\n$$\n\\int_{x \\in X}(\\theta \\cdot x-p(\\theta)) q(\\theta, x) \\mathrm{d} G(x)-r(\\theta) \\geq \\int_{x \\in X}\\left(\\theta \\cdot x-p\\left(\\theta^{\\prime}\\right)\\right) \\sigma\\left(q\\left(\\theta^{\\prime}, x\\right), k\\right) \\mathrm{d} G(x)-r\\left(\\theta^{\\prime}\\right),\n$$\n\nwhere $\\sigma\\left(q\\left(\\theta^{\\prime}, x\\right), k\\right)$ is a deviation function equal to $q\\left(\\theta^{\\prime}, x\\right), 1-q\\left(\\theta^{\\prime}, x\\right)$, 1, and 0 for $k=1, \\ldots, 4$. These constraints ensure that each type prefers truth telling over all doubledeviating strategies: misreporting and following the recommendations, \"swapping\" the buying decisions, always buying, or never buying. Deviations from $\\theta$ to $\\theta$ are included and ensure that the types are obedient on-path after truth telling.\n\nThe individual-rationality constraints require that, for all $\\theta \\in \\Theta$ :\n\n$$\n\\int_{x \\in X}(\\theta \\cdot x-p(\\theta)) q(\\theta, x) \\mathrm{d} G(x)-r(\\theta) \\geq 0,\n$$\n\nso that the seller cannot force the buyer to purchase an item from the menu.\nSeveral challenges are involved in this problem. First, the seller maximizes over a large class of all functions from a multidimensional space $X$. Second, it is a priori not clear what kinds of deviations are binding and, hence, relevant for the design problem: the buyer's type has no single-dimensional structure, and there is an additional multiplicity of constraints caused by double deviations. ${ }^{14}$\n\nThe following observation is crucial to address the experimental complexity: only two coarse statistics, not the entire trade function, matter for the revenue-maximizing problem. Namely, for a given responsive experiment $E$, the associated trade function $q$ achieves the\n\n[^8]attribute surplus and the (total) trade probability: ${ }^{15}$\n$$\n\\begin{aligned}\n& \\mathcal{X}(q) \\triangleq \\int_{x \\in X} x q(x) d G(x) \\in \\mathbb{R}^{J} \\\\\n& \\mathcal{Q}(q) \\triangleq \\int_{x \\in X} q(x) d G(x) \\in[0,1]\n\\end{aligned}\n$$\n\nThe formulations (9), (10), and (11) reveal that, due to the linearity of integration, these statistics are the only economically relevant parameters of the problem. A change in the trade function $q(\\theta, \\cdot)$ that does not affect the attribute surplus and the trade probability affects neither the buyer's incentives nor the seller's revenue. Accordingly, the seller can maximize directly over attribute surpluses and trade probabilities. In what follows, I refer to $\\mathcal{X}(q(\\theta, \\cdot))$ and $\\mathcal{Q}(q(\\theta, \\cdot))$ as $\\mathcal{X}(\\theta)$ and $\\mathcal{Q}(\\theta)$, respectively.\n\nNot all attribute surpluses and trade probabilities can be achieved by some trade function. At one extreme, if the trade probability is nil, then the trade never occurs, $q(\\cdot) \\equiv 0$, so the attribute surpluses must also be nil. At the other extreme, if the trade probability is 1, then the trade always occurs, $q(\\cdot) \\equiv 1$, so the attribute surplus is equal to its ex ante expectation $\\mathbb{E}[x]$. Intermediate values of trade probability provide more freedom to choose attribute surpluses because the seller can select the regions in which the trade recommendations are sent. The corresponding feasibility set $\\mathcal{F} \\subseteq \\mathbb{R}^{J+1}$, which is generated by all measurable trade functions, is:\n\n$$\n\\mathcal{F} \\triangleq\\{(\\mathcal{X}(q), \\mathcal{Q}(q)) \\mid q: X \\rightarrow[0,1]\\} .\n$$\n\nThe set $\\mathcal{F}$ is convex, and its shape is determined by the attribute distribution $G$.\nThese observations reduce the search to the following problem:\n\n$$\n\\max _{\\{r(\\theta), \\mathcal{X}(\\theta), \\mathcal{Q}(\\theta), p(\\theta)\\}} \\int_{\\theta \\in \\Theta}(r(\\theta)+\\mathcal{Q}(\\theta) p(\\theta)) \\mathrm{d} F(\\theta)\n$$\n\nsubject to incentive-compatibility constraints: $\\forall \\theta, \\theta^{\\prime} \\in \\Theta$,\n\n$$\n\\begin{aligned}\n& \\theta \\cdot \\mathcal{X}(\\theta)-\\mathcal{Q}(\\theta) p(\\theta)-r(\\theta) \\geq \\theta \\cdot \\mathcal{X}\\left(\\theta^{\\prime}\\right)-\\mathcal{Q}\\left(\\theta^{\\prime}\\right) p\\left(\\theta^{\\prime}\\right)-r\\left(\\theta^{\\prime}\\right), \\\\\n& \\theta \\cdot \\mathcal{X}(\\theta)-\\mathcal{Q}(\\theta) p(\\theta)-r(\\theta) \\geq \\theta \\cdot\\left(\\mathbb{E}[x]-\\mathcal{X}\\left(\\theta^{\\prime}\\right)\\right)-\\left(1-\\mathcal{Q}\\left(\\theta^{\\prime}\\right)\\right) p\\left(\\theta^{\\prime}\\right)-r\\left(\\theta^{\\prime}\\right), \\\\\n& \\theta \\cdot \\mathcal{X}(\\theta)-\\mathcal{Q}(\\theta) p(\\theta)-r(\\theta) \\geq \\theta \\cdot \\mathbb{E}[x]-p\\left(\\theta^{\\prime}\\right)-r\\left(\\theta^{\\prime}\\right), \\\\\n& \\theta \\cdot \\mathcal{X}(\\theta)-\\mathcal{Q}(\\theta) p(\\theta)-r(\\theta) \\geq-r\\left(\\theta^{\\prime}\\right),\n\\end{aligned}\n$$\n\n[^9]the individual-rationality constraints: $\\forall \\theta \\in \\Theta$,\n$$\n\\theta \\cdot \\mathcal{X}(\\theta)-\\mathcal{Q}(\\theta) p(\\theta)-r(\\theta) \\geq 0,\n$$\nand the feasibility constraints: $\\forall \\theta \\in \\Theta$,\n$$\n(\\mathcal{X}(\\theta), \\mathcal{Q}(\\theta)) \\in \\mathcal{F} .\n$$\n\nEven though the seller sells a single object, information disclosure allows her to control the multidimensional attribute surpluses $\\mathcal{X}(\\theta)$ at the time of a purchase. Moreover, the surpluses directly affect the feasible trade probability in a nonlinear fashion.","text_sha256":"4085eec611d506a09fdfbe96870e0048613b4fed999e1600d6d840bab810cba3"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0011","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.4 Optimal Disclosure","text":"### 4.4 Optimal Disclosure\n\nI begin by observing a special feature of a responsive experiment that always recommends the buyer to buy and, as such, provides no information about attributes. If all attributes are strictly positive, $X \\subseteq \\mathbb{R}_{++}^{J}$, then this experiment is a unique maximizer of the attribute surplus along all dimensions. If all types are strictly positive, $\\Theta \\subseteq \\mathbb{R}_{++}^{J}$, then this experiment is also a unique maximizer of the trade surplus.\n\nProposition 2. (No Disclosure)\nIf all attributes and types are strictly positive, $X \\subseteq \\mathbb{R}_{++}^{J}$ and $\\Theta \\subseteq \\mathbb{R}_{++}^{J}$, and the number of types is finite, then in any optimal menu, some type buys the object with probability one. That is, no disclosure, $\\underline{E}$, is part of any optimal responsive menu.\n\nProposition 2 is consistent with the \"no distortion at the top\" property, common in mechanism design problems: there is a type that is optimally served an efficient allocation. However, recall that a responsive experiment only recommends allocation, and the buyer always has an option to disobey. Hence, it is important that no disclosure also provides minimal information to the buyer and, thus, maximally limits the scope of deviation. No disclosure arises in an optimal mechanism because it maximizes efficiency and minimizes incentive costs simultaneously.\n\nFurthermore, Proposition 2 emphasizes the distinctive feature of the seller's problem, which combines information and mechanism design. In a typical information design problem, the payoff structure is exogenously fixed and, unless the receiver's indirect utility is concave everywhere, disclosure appears in the optimal mechanism for some prior distributions. Indeed, if the prices were fixed and the buyer's preferences had sufficiently low intensity, then the seller would have to provide some information to persuade the buyer to buy the object.\n\nIn contrast, when the seller has control over monetary incentives, she can compensate for the lack of information with lower prices and does find it optimal to do so.\n\nTo provide a further understanding of optimal experiments, it is useful to understand the general properties of the feasibility set $\\mathcal{F}$. To this end, I define a key class of experiments.\n\nDefinition 1. (Linear Disclosure)\nA responsive experiment $E$ is a linear disclosure if, for some coefficients $\\alpha=\\left(\\alpha_{1}, \\ldots, \\alpha_{J}\\right) \\in$ $\\mathbb{R}^{J}$ and $\\alpha_{0} \\in \\mathbb{R}$ not all being equal to zero, its trade function is:\n\n$$\nq(x)= \\begin{cases}1, & \\text { if } \\alpha \\cdot x>\\alpha_{0}, \\\\ 0, & \\text { if } \\alpha \\cdot x<\\alpha_{0} .\\end{cases}\n$$\n\nA linear disclosure informs the buyer whether a linear combination of attributes is above or below a specified threshold. A linear disclosure assigns probability one to some signal everywhere outside of the defining hyperplane, $\\left\\{x \\mid \\alpha \\cdot x=\\alpha_{0}\\right\\}$, on which is can possibly randomize. In the case of a single attribute, a linear disclosure corresponds to a binary monotone partition of the attribute space.\n\nA linear disclosure can be viewed as a \"reference\" disclosure that informs the buyer whether some virtual type $\\hat{\\theta}=\\alpha$ would like to buy the object at price $p=\\alpha_{0}$. If attributes are always positive and independently distributed, a linear disclosure admits additional interpretations. If elements of the coefficient vector $\\alpha$ are positive, this disclosure can be viewed as a \"level\" disclosure. Observing a \"trade\" recommendation uniformly increases attribute expectation, whereas observing a \"no-trade\" recommendation uniformly decreases it. In contrast, if the elements of a coefficient vector $\\alpha$ have different signs, then a linear disclosure can be viewed as a \"comparative\" disclosure between the attribute groups of different signs. A \"trade\" recommendation increases the attribute expectations in one group and decreases them in the other group.\n\nNote that the likelihood function of a linear disclosure is not restricted on the defining hyperplane. Furthermore, the hyperplane does not exist for $\\alpha \\equiv 0$ and $\\alpha_{0}$ being strictly positive or negative. Those disclosure rules correspond to never-trade and always-trade uninformative experiments.\n\nLemma 1. (Feasibility)\nThe feasibility set $\\mathcal{F}$ is compact and convex. Any linear disclosure achieves some boundary point of $\\mathcal{F}$. Any boundary point of $\\mathcal{F}$ is achieved by some linear disclosure.\n\nTo prove this central result, I first show that $\\mathcal{F}$ is compact as a continuous image of a compact set. Second, I show that $\\mathcal{F}$ is convex because a convex combination of trade\nfunctions achieves a convex combination of attribute surpluses and trade probabilities. Then, I appeal to the supporting hyperplane theorem to show that a given trade function achieves a boundary point if and only if it maximizes a linear combination of attribute surpluses and trade functions. Any such trade function corresponds to a linear disclosure.\n\nIn general multidimensional screening problems, one cannot be sure that all optimal bundles can be found at the boundary of a feasibility set. However, the current problem is an exception. To this end, say that an allocation $(\\mathcal{X}(\\theta), \\mathcal{Q}(\\theta))_{\\theta \\in \\Theta}$ is implementable if there exist tariff functions $r(\\theta), p(\\theta)$ such that each buyer's type $\\theta \\in \\Theta$ reports his type truthfully.\n\nLemma 2. (Implementability)\nFor any implementable allocation $(\\mathcal{X}(\\theta), \\mathcal{Q}(\\theta))_{\\theta \\in \\Theta}$ there exists an allocation $\\left(\\mathcal{X}(\\theta), \\mathcal{Q}^{\\prime}(\\theta)\\right)_{\\theta \\in \\Theta}$ such that (1) it can be implemented with the same revenue and the same payoffs for all types and (2) for all $\\theta \\in \\Theta,\\left(\\mathcal{X}(\\theta), \\mathcal{Q}^{\\prime}(\\theta)\\right)$ is on the boundary of $\\mathcal{F}$ and $\\mathcal{Q}^{\\prime}(\\theta) \\leq \\mathcal{Q}(\\theta)$.","text_sha256":"3dcf2eed8c7b5e6dcb4e68288919f8d93052b3008a338d88d62911569cd124f0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0012","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.4 Optimal Disclosure","text":"Lemma 2 clarifies that the buyer incentive structure leads the seller to minimize trade probability whenever it maintains a trade surplus. If $(\\mathcal{X}(\\theta), \\mathcal{Q}(\\theta))$ lies in the interior of $\\mathcal{F}$, then the seller can reduce the total trade probability while keeping the attribute surplus the same. Such change scales up the attribute expectation conditional on the trade recommendation without affecting the trade surplus. If the seller accompanies this change with a revenue-preserving increase in the object price, then the on-path payoff of type $\\theta$ remains the same. However, a higher object price renders deviations to this type's item less appealing.\n\nTheorem 1. (Optimal Disclosure)\nThere exists an optimal responsive menu in which every experiment is a linear disclosure.\nThe theorem is an immediate corollary of Lemmas 1 and 2 and does not require any assumptions about the attribute or type distributions. To appreciate this result, it is instructive to compare the allocation distortions driven by monopoly power in cases of complete and incomplete information about the object.\n\nFirst, consider the situation in which the object's attributes are commonly known to be $x_{0}$ so that there is no scope for information control. If the seller could observe the type, she would allocate the object efficiently, selling it if and only if $v(\\theta) \\geq 0$, and would extract full surplus. If the seller could not observe the type, she could attempt to screen by designing a menu of items varying in sale probabilities and prices. This screening is not beneficial as famously resolved by Myerson (1981). Each type $\\theta$ is assigned a virtual valuation $\\hat{v}(\\theta)$ and, under standard regularity conditions, an object is sold if and only if the virtual valuation is positive:\n\n$$\n\\hat{v}(\\theta) \\geq 0 .\n$$\n\nThis allocation is typically inefficient since the virtual valuation differs from the true valuation.\n\nCompare this scenario to the current situation in which the object's attributes are uncertain. If the seller could observe the type $\\theta$, then, according to Bergemann and Pesendorfer (2007), she would inform type $\\theta$ whether his valuation is positive, charge a maximal acceptable price, and extract the full surplus. If the seller could not observe the type, she could design a menu varying in information content and prices. By Theorem 1, the optimal allocation distortion would be remarkably similar to the case of complete information about the object. Each type $\\theta$ is assigned a virtual type $\\hat{\\theta}(\\theta)$. An object is sold when the buyer's virtual valuation is above a specified threshold, possibly with randomization on the boundary:\n\n$$\n\\hat{v}(\\theta)=\\hat{\\theta}(\\theta) \\cdot x \\geq \\alpha_{0}(\\theta) .\n$$\n\nThis allocation is also typically inefficient but is now with an additional distortion since the threshold may differ from 0.\n\nUniqueness Theorem 1 establishes the existence of an optimal responsive menu with each experiment being a linear disclosure. One might wonder whether there exist optimal responsive menus with non-linear disclosure rules. The proof argument does not preclude this possibility: although the adjustment to linear disclosure strictly relaxes constraints (17) and (18), it preserves the constraint (16); therefore, hypothetically, the relaxed constraints may be not exploited for additional revenue. It can be shown that with only two types, this relaxation can be exploited; thus, linear disclosure is uniquely optimal. With many types, the answer is less clear as the structure of incentive constraints is more complex.\n\nAt the same time, observe that the adjustment to linear disclosure strictly decreases the trade probability. Consequently, when the seller faces trading costs or, equivalently, attaches some value to the object, however small, linear disclosures are uniquely optimal.\n\nGeneral Payoffs The arguments behind Theorem 1 might seem to heavily rely on the linearity of the buyer's valuation function. However, Theorem 1 places no structural assumptions on the attribute distribution. This crucial feature allows for extending the optimal disclosure characterization beyond linear environments by carefully defining the relevant attributes. In particular, consider a general valuation function $v(\\theta, x)$ and define auxiliary attributes to coincide with the valuations of different buyer types. In this auxiliary formulation, each type's valuation is linear in the relevant attribute, and Theorem 1 applies.\n\nCorollary 1. (General Payoffs)\nLet $X$ be an arbitrary attribute set, $v(\\theta, x)$ be a general valuation function, and $|\\Theta|<\\infty$.\n\nThen, there exists an optimal menu in which every experiment has a linear form, i.e., for any experiment, there exist $\\alpha: \\Theta \\rightarrow \\mathbb{R}$ and $\\alpha_{0} \\in \\mathbb{R}$, not all zeros, such that:\n\n$$\nq(x)= \\begin{cases}1, & \\text { if } \\sum_{\\theta \\in \\Theta} \\alpha(\\theta) v(\\theta, x)>\\alpha_{0}, \\\\ 0, & \\text { if } \\sum_{\\theta \\in \\Theta} \\alpha(\\theta) v(\\theta, x)<\\alpha_{0} .\\end{cases}\n$$\n\nNote the difference between the definitions of a linear form (25) and a linear disclosure (22). A linear disclosure operates in a space of attributes and can be specified independently of a buyer. A linear form, in contrast, operates in the space of valuations of different buyer types. The richer the buyer heterogeneity is, the more complex the linear form can be. However, in the case of linear payoffs the linear form always reduces to a linear disclosure.\n\nCorollary 1 allows for a characterization of the classes of optimal disclosure rules in general environments with preferences that allow for bliss points or risk aversion. To illustrate, consider the case of location payoffs with the buyer's type capturing his bliss point in the attribute space $X \\subseteq \\mathbb{R}^{J}, \\Theta \\subseteq \\mathbb{R}^{J}, v_{0}>0$, and:\n\n$$\nv(\\theta, x)=v_{0}-(x-\\theta)^{2} .\n$$","text_sha256":"1aab97758304e335ab7a002dfa91dfd942f4f7b2f3393eb5d0d7aa82a5983350"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0013","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.4 Optimal Disclosure","text":"Let there be two types $\\theta_{1}, \\theta_{2} \\in \\mathbb{R}^{J}$. Assume that $X$ is bounded and for all $x \\in X$, the types' valuations are positive. Optimal disclosure rules can be identified as follows. First, Proposition 2 can be applied to establish that one type is offered no disclosure and always buys. Second, by Corollary 1, the other type is offered a linear form (25) that informs whether a linear combination of valuations $v\\left(\\theta_{1}, x\\right)$ and $v\\left(\\theta_{2}, x\\right)$ is above or below a specified threshold. Generically, this experiment is a neighborhood disclosure: it informs whether the attribute vector is sufficiently close to a virtual type $\\hat{\\theta}$ located on the line that connects $\\theta_{1}$ and $\\theta_{2}$.","text_sha256":"66b7c69f51e4413982d4d624cbe1f9df10472a4de44fa91ec4daa03b71fc946b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0014","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Design of Pricing Mechanisms","text":"## 5 Design of Pricing Mechanisms\n\nI proceed by studying pricing in the revenue-maximizing mechanisms. I identify a general class of optimal pricing mechanisms in the case of a single attribute. With many attributes, I am able to identify key trade-offs and characterize optimal mechanisms for specific classes of buyer types.\n\nFrom now on, I assume that all types and attributes are positive, $X \\subseteq \\mathbb{R}_{+}^{J}, \\Theta \\subseteq \\mathbb{R}_{+}^{J}$. In this scenario, it is commonly known that there are positive gains from trade. It makes it possible to ignore the efficiency role of disclosure and to focus solely on its screening effects.","text_sha256":"6ad1770a145e3398b3e99d12091d87e2ef1273f2b0afac5fa022e940e2fabba3"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0015","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.1 Single Attribute","text":"### 5.1 Single Attribute\n\nI begin with the basic case of a single attribute, $J=1, X \\subseteq \\mathbb{R}_{+}$. The buyer's type is one dimensional, $\\Theta \\subseteq \\mathbb{R}_{+}$, and the buyer's ex post valuation is\n\n$$\nv(\\theta, x)=\\theta x .\n$$\n\nThis setting features only vertical type heterogeneity. I establish that providing no attribute information is optimal in this case. The argument starts by considering a more beneficial setting for the seller, in which she can condition payment and allocation directly on the attribute realization, as do Eső and Szentes (2007). In this case, the revelation principle applies, and I can focus on the direct mechanisms in which all payments are front loaded: the buyer reports his type $\\theta$, pays the upfront payment $r(\\theta)$, and the trade occurs with probability $q(x, \\theta)$. The relevant variable is the single-dimensional attribute surplus:\n\n$$\n\\mathcal{X}(\\theta)=\\int_{x \\in X} x q(x, \\theta) \\mathrm{d} G(x)\n$$\n\nwhich can be anywhere between 0 and $\\mathbb{E}[x]$. I can then rewrite the seller's problem as:\n\n$$\n\\begin{aligned}\n& \\max _{r(\\theta), 0 \\leq \\mathcal{X}(\\theta) \\leq \\mathbb{E}[x]} \\int_{\\theta \\in \\Theta} r(\\theta) \\mathrm{d} F(\\theta), \\\\\n& \\text { s.t. } \\theta \\mathcal{X}(\\theta)-r(\\theta) \\geq \\theta \\mathcal{X}\\left(\\theta^{\\prime}\\right)-r\\left(\\theta^{\\prime}\\right), \\quad \\forall \\theta, \\theta^{\\prime} \\in \\Theta, \\\\\n& \\theta \\mathcal{X}(\\theta)-r(\\theta) \\geq 0, \\quad \\forall \\theta \\in \\Theta .\n\\end{aligned}\n$$\n\nThis problem is analogous to those of Myerson (1981) and Riley and Zeckhauser (1983), with the attribute surplus replacing the allocation probability. The optimal allocation $\\mathcal{X}(\\theta)$ is a step function equal to 0 for $\\theta<\\theta^{*}$ and to $\\mathbb{E}[x]$ for $\\theta \\geq \\theta^{*}$. The optimal upfront payment $r(\\theta)$ is equal to 0 for $\\theta<\\theta^{*}$ and to $r^{*}=\\theta^{*} \\mathbb{E}[x]$ for $\\theta \\geq \\theta^{*}$.\n\nThe argument concludes by noting that the optimal mechanism can be implemented by providing no disclosure and charging a price of $r^{*}$ for the object. This posted price mechanism is feasible in the original problem with private disclosure and is therefore also optimal there.\n\nTheorem 2. (Single Attribute)\nIf $J=1, X \\subseteq \\mathbb{R}_{+}$, and $\\Theta \\subseteq \\mathbb{R}_{+}$, then an optimal menu is a posted price mechanism with no disclosure, i.e., $r(\\theta) \\equiv 0, E(\\theta) \\equiv \\underline{E}$.\n\nSimple intuition underlies the optimality of no disclosure if the seller can only use a nondiscriminatory mechanism that consists of a single experiment followed by a posted price. Consider an arbitrary disclosure rule. Any signal realization $s$ scales the demand propor-\ntionally to the attribute expectation $\\mathbb{E}[x \\mid s]$. If the seller could observe this realization, she would optimally charge a scaled price and obtain scaled revenue. Importantly, the induced allocation would not depend on the realization $s$. Since any expectation is a martingale, the seller would serve the same population at, on average, the same price. The seller can do equally well using a posted price with no disclosure.\n\nAlthough intuitive, this argument does not consider discriminatory schemes with upfront payments. Theorem 2 confirms that no disclosure is optimal, even if the seller can use those schemes. Notably, this result requires no assumptions on the type or attribute distributions beyond the common knowledge of positive trade gains.\n\nRemark 1. (Attribute Index) The same argument can be applied to the case of many attributes, $J>1$, if the attributes and the types enter the valuation function through onedimensional indices:\n\n$$\nv(\\theta, x)=\\psi(\\theta) \\phi(x),\n$$\n\nfor $\\psi, \\phi: \\mathbb{R}^{J} \\rightarrow \\mathbb{R}_{+}$. For example, the result applies if all types belong to a ray $\\Theta=\\left\\{\\beta \\theta_{0}\\right\\}_{\\beta \\in \\mathbb{R}_{+}}$ for some direction vector $\\theta_{0} \\in \\mathbb{R}_{+}^{J}$. In this case, the indices can be defined as $\\psi(\\theta)=\\beta(\\theta)$ and $\\phi(x)=\\theta_{0} \\cdot x$. $\\square$\n\nRemark 2. (Uniqueness) The no-disclosure mechanism might not be uniquely optimal. In fact, the analysis of Eső and Szentes (2007) can be applied to show that, if the type distribution $F$ has a monotone hazard rate property, then full disclosure is also optimal. ${ }^{16}$ However, it must be accompanied by a complex structure of upfront payments and object prices. $\\square$","text_sha256":"c0cc8388e487041b2a6ca3c40b4edb1835e939cac5e942a8a154293ddf1366b3"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0016","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.2 Single-Minded Buyer","text":"### 5.2 Single-Minded Buyer\n\nI proceed with the case of multiple product attributes, $J \\geq 2$. This case is qualitatively different because it may feature horizontal heterogeneity across buyer types: type $\\theta$ may have a higher value for the object than type $\\theta^{\\prime}$ for some attribute realization, yet a lower value than type $\\theta^{\\prime}$ for another attribute realization. Since the seller should aim to allocate to object to the buyers who are willing to pay the most, she may need to provide attribute information through disclosure. To address such settings, I introduce and study a tractable type structure which allows to capture both vertical and horizontal heterogeneity of tastes.\n\nI call a type single minded if he values only one attribute. For a generic single-minded type, the vector $\\theta$ places a positive weight on only one dimension:\n\n$$\n\\theta=\\left(0, \\ldots, 0, \\theta_{j}, 0, \\ldots, 0\\right) .\n$$\n\n[^10]Thus, single-minded types allow for a simpler notation. I can represent the types by $J$ attribute cohorts $\\Theta_{j}$ such that all types within the same cohort value the same attribute. I slightly abuse the notation and let the type subscript identify the attribute cohort and the type value identify the valuation intensity, so that $\\Theta_{j} \\subseteq \\mathbb{R}_{+}$and\n\n$$\nv_{j}\\left(\\theta_{j}, x\\right)=\\theta_{j} x_{j} \\quad \\forall j, \\theta_{j} \\in \\Theta_{j} .\n$$\n\nI denote the frequency of a cohort $\\Theta_{j}$ by $f\\left(\\Theta_{j}\\right)$ and the cumulative type distribution within the cohort by $F_{j}\\left(\\theta_{j}\\right)$.\n\nA buyer is single minded if all types $\\theta \\in \\Theta$ are single minded and the attribute values are independently distributed such that $x_{j} \\sim G_{j}$ and $G(x)=\\times_{j} G_{j}\\left(x_{j}\\right) .{ }^{17}$ The independence requirement is substantive. Without it, any buyer can be viewed as being single minded by redefining the attributes as in the proof of Corollary 1.\n\nIf the buyer is single minded, then the seller knows that the buyer values only one of many independent attributes but does not know which one or the strength of the preference. Valuations of any two types are either perfectly correlated or independent.\n\nRemark 3. (Customizable Good) The setting of a single-minded buyer resembles a unitdemand multi-good monopolist problem in which each buyer type values only one of the goods. ${ }^{18}$ One difference between these problems is that I allow the seller to provide additional information to the buyer. Another difference is that in the multi-good monopolist problem a type who deviates across cohorts is guaranteed to obtain no value; therefore, the problem can effectively be separated into several single-good problems. In contrast, in my setting, when deviating across cohorts, the buyer can at least obtain the ex ante value of the relevant attribute, which links the problems together. At the same time, the setting of a single-minded buyer admits an interpretation of a sale of a customizable good. In this interpretation, the good that admits several possible configurations and the seller can inform the buyer about them. The buyer values only a single configuration and can set the configuration freely but only one time after the purchase. The question of optimal design can then be translated into what configuration information the seller provides and how this choice interacts with pricing. $\\square$\n\nIf the buyer is single minded, then the class of optimal experiments can be narrowed. Because attributes are independently distributed, a type $\\theta_{j} \\in \\Theta_{j}$ values only information about attribute $j$. This observation suggests an optimal way to screen single-minded types: if the buyer reports type $\\theta_{j} \\in \\Theta_{j}$, then the seller should provide information only about\n\n[^11]attribute $j$. Providing any other information would make misreporting more appealing without adding value for truth telling. At the same time, a linear disclosure informative only about attribute $j$ is a binary monotone partition defined on this attribute.\n\nProposition 3. (Directional Disclosure)\nIf the buyer is single minded, then there exists an optimal menu such that an experiment $E_{j}\\left(\\theta_{j}\\right)$ is a binary monotone partition of attribute $j$.\n\nIt follows that an optimal experiment $E_{j}\\left(\\theta_{j}\\right)$ can be characterized by its threshold $\\alpha_{0 j}\\left(\\theta_{j}\\right)$ so that it informs the buyer whether attribute $j$ is above or below this threshold. Incentive compatibility requires the buyer to purchase at higher attributes; thus, the attribute surplus can be written as:\n\n$$\n\\mathcal{X}_{j}\\left(\\theta_{j}\\right)=\\int_{\\alpha_{0 j}\\left(\\theta_{j}\\right)}^{\\infty} x_{j} \\mathrm{~d} G_{j}\\left(x_{j}\\right)\n$$\n\nThe attribute surplus can take any value between 0 and $\\mathbb{E}\\left[x_{j}\\right]$. The corresponding total trade probability can be written as an increasing and convex function $\\mathcal{Q}_{j}\\left(\\mathcal{X}_{j}\\right)$.\n\nIn what follows, I assume that that all attribute cohorts admit an upper bound, $\\Theta_{j}=$ $\\left[0, \\bar{\\theta}_{j}\\right]$, and $\\theta_{j}$ are continuously distributed over $\\Theta_{j}$ according to distribution function $F_{j}$ with the monotone hazard rate property. The seller's problem is to design the tariff and the attribute surplus functions, $r_{j}\\left(\\theta_{j}\\right), p_{j}\\left(\\theta_{j}\\right)$, and $\\mathcal{X}_{j}\\left(\\theta_{j}\\right)$ to maximize her objectives and can be written as follows (cf. Kolotilin et al. (2017)):","text_sha256":"37731f32447ccc7046f839b156af0ad405587a649027164351e91eebdf3c4572"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0017","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.2 Single-Minded Buyer","text":"$$\n\\begin{gathered}\n\\max _{\\left\\{r_{j}\\left(\\theta_{j}\\right), \\mathcal{X}_{j}\\left(\\theta_{j}\\right), p_{j}\\left(\\theta_{j}\\right)\\right\\}} \\sum_{j=1}^{J} f\\left(\\Theta_{j}\\right) \\int_{\\theta_{j} \\in \\Theta_{j}}\\left(r_{j}\\left(\\theta_{j}\\right)+\\mathcal{Q}_{j}\\left(\\theta_{j}\\right) p_{j}\\left(\\theta_{j}\\right)\\right) \\mathrm{d} F_{j}\\left(\\theta_{j}\\right) \\\\\n\\text { s.t. } \\theta_{j} \\mathcal{X}_{j}\\left(\\theta_{j}\\right)-p_{j}\\left(\\theta_{j}\\right) \\mathcal{Q}_{j}\\left(\\theta_{j}\\right)-r_{j}\\left(\\theta_{j}\\right) \\geq\\left(\\theta_{j} \\mathcal{X}_{j}\\left(\\theta_{j}^{\\prime}\\right)-p_{j}\\left(\\theta_{j}^{\\prime}\\right)\\right) \\mathcal{Q}_{j}\\left(\\theta_{j}^{\\prime}\\right)-r_{j}\\left(\\theta_{j}^{\\prime}\\right), \\forall j, \\theta_{j}, \\theta_{j}^{\\prime} \\in \\Theta_{j}, \\\\\n\\theta_{j} \\mathcal{X}_{j}\\left(\\theta_{j}\\right)-p_{j}\\left(\\theta_{j}\\right) \\mathcal{Q}_{j}\\left(\\theta_{j}\\right)-r_{j}\\left(\\theta_{j}\\right) \\geq \\theta_{j} \\mathbb{E}\\left[x_{j}\\right]-p_{k}\\left(\\theta_{k}\\right)-r_{k}\\left(\\theta_{k}\\right), \\forall j, k, \\theta_{j} \\in \\Theta_{j}, \\theta_{k} \\in \\Theta_{k}, \\\\\n\\theta_{j} \\mathcal{X}_{j}\\left(\\theta_{j}\\right)-p_{j}\\left(\\theta_{j}\\right) \\mathcal{Q}_{j}\\left(\\theta_{j}\\right)-r_{j}\\left(\\theta_{j}\\right) \\geq 0, \\mathcal{X}_{j}\\left(\\theta_{j}\\right) \\geq \\mathcal{Q}_{j}\\left(\\theta_{j}\\right) \\mathbb{E}\\left[x_{j}\\right], \\forall j, \\theta_{j} \\in \\Theta_{j},\n\\end{gathered}\n$$\n\nwhere $\\mathcal{Q}_{j}\\left(\\theta_{j}\\right) \\equiv \\mathcal{Q}_{j}\\left(\\mathcal{X}_{j}\\left(\\theta_{j}\\right)\\right) \\forall j, \\theta_{j} \\in \\Theta_{j}$. This problem resembles a collection of onedimensional mechanism design problems, one per attribute, with the following important differences. First, each item in this problem features both horizontal and vertical components. The upfront payments $r_{j}\\left(\\theta_{j}\\right)$ are purely vertical-all types value them the same. In contrast, the attribute surpluses $\\mathcal{X}_{j}\\left(\\theta_{j}\\right)$ are only valuable to types from cohort $\\Theta_{j}$. The object prices $p_{j}$ are mixed since they are paid only if the type decides to trade. Second, the problem features nonlinear terms $p_{j}\\left(\\theta_{j}\\right) \\mathcal{Q}_{j}\\left(\\mathcal{X}_{j}\\left(\\theta_{j}\\right)\\right)$ and $\\mathcal{Q}_{j}\\left(\\mathcal{X}_{j}\\right)$.\n\nBecause of these differences, I cannot apply standard mechanism design techniques. Instead, I solve the problem via a sequence of simplifications. In the first step, I observe that\nusing upfront payments is detrimental. For any $r_{j}\\left(\\theta_{j}\\right)>0$, the seller can reduce the transfer and increase $p_{j}\\left(\\theta_{j}\\right)$ while keeping the total expected transfer $r_{j}\\left(\\theta_{j}\\right)+\\mathcal{Q}_{j}\\left(\\theta_{j}\\right) p_{j}\\left(\\theta_{j}\\right)$ the same. This change does not affect the utilities of truth-telling types or the seller's revenue; however, it renders misreporting less appealing. Intuitively, by shifting the expected transfer toward the object price, the seller better discriminates against the types who would always purchase the object.\n\nIn the second step, I use standard mechanism design arguments to show that incentive compatibility within the same cohort implies that $\\mathcal{X}_{j}\\left(\\theta_{j}\\right)$ is nondecreasing in $\\theta_{j}$. That is, higher types must trade with a higher probability but lower conditional expectations of the relevant attribute. Moreover, the expected transfers can be derived from the attribute surplus functions. This allows to rewrite the seller's problem solely as a choice over attribute surplus functions $\\mathcal{X}_{j}\\left(\\theta_{j}\\right)$.\n\nLemma 3. The seller's problem can be written as\n\n$$\n\\begin{aligned}\n& \\quad \\max _{\\left\\{\\mathcal{X}_{j}\\left(\\theta_{j}\\right)\\right\\}_{j=1}^{J}} \\sum_{j=1}^{J} f\\left(\\Theta_{j}\\right) \\int_{0}^{\\bar{\\theta}_{j}}\\left(\\theta_{j}-\\frac{1-F_{j}\\left(\\theta_{j}\\right)}{f_{j}\\left(\\theta_{j}\\right)}\\right) \\mathcal{X}_{j}\\left(\\theta_{j}\\right) \\mathrm{d} F_{j}\\left(\\theta_{j}\\right) \\\\\n& \\text { s.t. } \\quad \\mathcal{X}_{j}\\left(\\theta_{j}\\right) \\text { is non }- \\text { decreasing }, \\mathcal{X}_{j}\\left(\\theta_{j}\\right) \\in\\left[0, \\mathbb{E}\\left[x_{j}\\right]\\right], \\\\\n& \\quad \\int_{0}^{\\bar{\\theta}_{j}} \\mathcal{X}_{j}\\left(\\theta_{j}\\right) d \\theta_{j} \\geq \\bar{\\theta}_{j} \\mathbb{E}\\left[x_{j}\\right]-\\underline{p}\\left(\\mathcal{X}_{1}(\\cdot), \\ldots, \\mathcal{X}_{J}(\\cdot)\\right) \\quad \\forall j=1, \\ldots, J .\n\\end{aligned}\n$$\n\nThe objective function and the monotonicity constraints capture the incentive-compatibility constraints within each attribute cohort. The integral constraints capture the incentivecompatibility constraints between different cohorts. In particular, these constraints require that the highest type within each cohort not want to purchase the object at the minimal price present in the menu $\\underline{p}$.\n\nIn an optimal menu, the minimal price must be offered to the highest types. Toward a contradiction, assume that for some cohort $\\Theta_{j}$, a neighborhood of $\\bar{\\theta}_{j}$ is not offered the minimal price. Then, these types are not imposing externalities on other cohorts through the integral constraint. Moreover, to not go for the lowest price, these types should be offered some disclosure so $\\mathcal{X}_{j}\\left(\\theta_{j}\\right)<\\mathbb{E}\\left[x_{j}\\right]$. It leads to a contradiction: the seller could marginally increase $\\mathcal{X}_{j}\\left(\\theta_{j}\\right)$ for these types, improving the revenue.\n\nThis observation allows for the stating of a relaxed problem in which the monotonicity and integral constraints are dropped but all high types are required to be offered a given price. If the type distributions have the monotone hazard rate property, then the solution to the problem is a collection of single-step functions. The solution corresponds to only one item per attribute cohort. It satisfies the original constraints and therefore solves the original problem.\n\nTheorem 3. (Optimal Menu, Single-Minded Buyer)\nIf the buyer is single minded and the type distributions have the monotone hazard rate property, then in an optimal responsive menu for all $j=1, \\ldots, J$ and $\\theta_{j} \\in \\Theta_{j}: r\\left(\\theta_{j}\\right)=0$, $p\\left(\\theta_{j}\\right)=p$, and $E_{j}\\left(\\theta_{j}\\right)=E_{j}$ where $E_{j}$ is a binary monotone partition of $x_{j}$.","text_sha256":"93f9c7b595c35474807d122493dbda87882e36db4c2b85574077763c70542f09"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0018","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.2 Single-Minded Buyer","text":"I emphasize the simplicity of both pricing and disclosure components of the optimal mechanism. With regard to pricing, the price for the object is the same for all types and the price of information is nil. With regard to disclosure, the mechanism admits a nondiscriminatory indirect implementation: the seller can simply inform the buyer whether each attribute is above the corresponding threshold and post a fixed price for the object. Because each type values only one attribute and the attributes are independent, each type will use only information about a relevant attribute when deciding whether to buy the object. This simplicity is not a consequence of some exogenous requirement but rather is a feature of a revenue-maximizing mechanism.\n\nFurthermore, my analysis also provides a partial characterization in the case of general distributions $F_{j}$. The first statement of Theorem 3 remains the same-upfront payments are not used with a single-minded buyer. However, the second and the third statements must be modified since an optimal menu may feature limited price discrimination. In particular, the arguments of Samuelson (1984) can be applied to limit the number of optimal items to two per cohort. That is, the highest types are still offered the unique minimal lowest price but per each attribute cohort, there could be one more item that targets lower types.","text_sha256":"97c6bb6004f4e405e662a1a101018ffe0915d86601052550bdaf5bc55911a553"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0019","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.3 Differentiated Types and Separate Persuasion","text":"### 5.3 Differentiated Types and Separate Persuasion\n\nA notable limit case of the previous section is the case in which each attribute cohort is a singleton, $\\Theta_{j}=\\left\\{\\theta_{j}\\right\\}$. In this case, there is no vertical within-attribute heterogeneity, and the number of types equals the number of attributes $|\\Theta|=J$. Any two different types $\\theta, \\theta^{\\prime} \\in \\Theta$ are orthogonal to one another as vectors in $\\mathbb{R}^{J}$; accordingly, this case can be viewed as the setting of orthogonal types. Without loss of generality, all type intensities can be set equal to one, $\\theta_{j} \\equiv 1$ for all $j$.\n\nBy the arguments analogous to those in the previous section, an optimal mechanism is a free-of-charge disclosure followed by a single posted price. Finding optimal experiments is straightforward. The seller should provide minimal information sufficient to convince the buyer to make a purchase at the price posted. If the type $\\theta_{j}$ is ex ante sufficiently optimistic, $\\mathbb{E}\\left[x_{j}\\right] \\geq p$, then the seller should provide no attribute information, $E_{j}=\\underline{E}$. Otherwise, the seller should increase the type's expectation up to the object price.\n\nThe corresponding optimal mechanism is illustrated in Figure 4. The mechanism is simple\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 4: Optimal mechanism in the case of orthogonal types. Green indicates attribute regions in which a purchase recommendation is sent for the types with partial disclosure. Blue indicates the types' rent conditional on a trade, for the types with no disclosure. Attributes are ordered by increasing ex ante expectations.\n\nand features a \"no distortions at the top, no rents at the bottom\" property: All types with the ex ante valuation above the optimal price always buy the object, whereas all other types are indifferent to participating in the mechanism. In this way, \"the top\" and \"the bottom\" are not single types, as is typical in mechanism design problems, but are two type classes that partition the type space.\n\nThe case of orthogonal types admits a clean illustration of an optimal mechanism and, importantly, suggests a generalization of the current analysis beyond a single-minded buyer. Toward this generalization, consider arbitrary attribute distribution $G(x)$ and valuation function $v(\\theta, x)$. For a given price $p$, define a separate persuasion mechanism $M^{S P}(p)$ as a direct menu in which $r^{S P}(\\theta)=0, p^{S P}(\\theta)=p$, and $E^{S P}(\\theta)$ recommends to trade when $v(\\theta, x)$ is above a threshold; the threshold is chosen so that $\\mathbb{E}\\left[v(\\theta, x) \\mid E^{S P}(\\theta), s^{+}\\right]=$ $\\max \\{p, \\mathbb{E}[v(\\theta, x)]\\}$. That is, the seller fixes an object price and for each type provides minimal valuation information to persuade him to make a purchase.\n\nThis mechanism is one with discriminatory disclosure that mimics the optimal menu for orthogonal types. If a type reports truthfully, then he is either left with no rents or is provided with no information; in both cases, the experiment brings no value to him. The mechanism may be incentive compatible or not-it depends on whether some types may benefit from information offered to other types. Denote with $p^{*}$ a price that maximizes the revenue, if the buyer is assumed to report his type truthfully.\n\nTheorem 4. (Separate Persuasion)\nIf the mechanism $M^{S P}\\left(p^{*}\\right)$ is incentive compatible, then it is an optimal mechanism.\n\nThis proposition is based on the observation that a separate persuasion mechanism solves a relaxation of the original problem, in which misreporting types are required to always purchase the object. If incentive compatible, the mechanism satisfies the relaxed constraints and also solves the original problem. ${ }^{19}$\n\nLet me highlight the significance of Theorem 4. A priori, there is no reason to expect the separate persuasion mechanism to be optimal since it does not use all of the flexibility available to the seller: it does not price information and it does not vary the price of the object. Furthermore, the mechanism does not extract the full surplus because it generally induces inefficient allocation. Rather, the result builds on and generalizes the analysis of orthogonal types.\n\nTheorem 4 can be used to find optimal mechanisms in environments in which the types are sufficiently differentiated. For the following statement, let the valuation function be linear (1) and the attributes be independently and continuously distributed.\n\nCorollary 2. (Differentiated Types)\nFix the number of types $J$ and their respective frequencies. Let $p^{*}$ be a uniquely optimal price in a separate persuasion mechanism for some orthogonal types $\\left(\\hat{\\theta}_{1}, \\ldots, \\hat{\\theta}_{J}\\right)$. If $p^{*} \\neq$ $\\mathbb{E}\\left[v\\left(\\hat{\\theta}_{j}, x\\right)\\right]$ for all $j$, then there exists $\\varepsilon>0$ such that for any type profile $\\left(\\theta_{1}, \\ldots, \\theta_{J}\\right)$ with $\\left\\|\\theta_{j}-\\hat{\\theta}_{j}\\right\\| \\leq \\varepsilon$ for all $j$ a separate persuasion mechanism is optimal.\n\nAccording to the corollary, separate persuasion mechanisms are generically optimal even if the types value several attributes as long as they place most of the weight on distinct attributes. In these cases, the seller does not benefit from price discrimination but does benefit from information discrimination.\n\nNote the structure of optimal disclosure when the types place weights on several attributes, however small. Even if the attributes are independent, providing information about each of them separately is not optimal. Instead, the disclosure leans towards the tastes of a reported type. At the same time, the informational content regarding a given attribute decreases as the type attaches less weight to it; thus, if the tastes are sufficiently concentrated, then providing information only about the leading attribute is approximately optimal.\n\n[^12]","text_sha256":"656c1ed4fa9fd621253ac3aa9498c3a0e4a44fe19de346e8dffede0362fcf2aa"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0020","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Discussion","text":"## 6 Discussion","text_sha256":"861fee8be6fb28b605f6eb9d296dcfd0c4de9bc1c2b1a367fe0a42cd3d74822c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0021","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6.1 Pricing of Product Information","text":"### 6.1 Pricing of Product Information\n\nIn the setting of Section 5, the seller cannot benefit from pricing the information that she provides-upfront payments can always be set to zero. Similarly, in reality, it is uncommon to price product information. This fact raises a question regarding why it might be not beneficial to price product information separately from the product itself.\n\nProposition 4. (Payment Backloading)\nIn any given responsive menu, $r(\\theta)$ can be decreased, and $p(\\theta)$ can be increased without any loss of revenue when (i) $\\mathcal{Q}(\\theta) \\leq 1 / 2$ or (ii) no type is willing to act contrary to both recommendations if being offered the experiment $E(\\theta)$.\n\nProposition 4 holds for any valuation function. To understand the logic behind it, take an item of type $\\theta$ in a responsive menu and consider the effect of backloading its payment, i.e., reducing the information price and increasing the object price to preserve the expected payment of type $\\theta$. By construction, this backloading preserves the incentives of the types who, if offered the item of type $\\theta$, would like to follow the recommendations. At the same time, it changes the incentives of the types who would like to disobey recommendations because the object price is paid only if the object is actually purchased. In particular, the backloading relaxes the incentives of the types who would purchase the object more frequently than $\\theta$ when faced with $E(\\theta)$. Hence, the incentives are clearly relaxed for the types who always purchase the object, because $1 \\geq \\mathcal{Q}(\\theta)$; but also, whenever $\\mathcal{Q}(\\theta) \\leq 1 / 2$, the incentives are relaxed for the types who would like to act contrary to recommendations because in that case $1-\\mathcal{Q}(\\theta) \\geq \\mathcal{Q}(\\theta)$. The incentives of the types who never purchase the object are captured by their individual rationality. As such, under the conditions of the proposition, the payments can be backloaded without any loss of revenue. In fact, this argument reveals that the second condition of the proposition need only hold for the types along the binding constraints.\n\nBy Proposition 4, a given experiment might need to be priced only if it induces a relatively frequent trade and some types would like to mismatch its recommendations. That is, a priced experiment should bring \"bad news\" by infrequently informing the buyer that the product is not worth its price. Moreover, this information should lead to a disagreement, with some type using it in the opposite fashion. In some cases, one can exclude the latter possibility.\n\nCorollary 3. (No Information Pricing)\nIn an optimal menu, the price of an experiment $E$ can be set to zero if (i) $J>1$, attributes\nare independent, $v(\\theta, x)=\\theta \\cdot x, \\Theta \\subseteq \\mathbb{R}_{+}^{J}$, and $E$ is a linear disclosure with $\\alpha \\in \\mathbb{R}_{+}^{J}$ or (ii) $J=1, v(\\theta, x)$ is increasing in $x$ for all $\\theta \\in \\Theta$ and $E$ is a binary monotone partition.\n\nIn the cases of Corollary 3, the types agree on the ranking of their interim valuations after different signals of the experiment $E$. Therefore, no type likes to mismatch decisions with recommendations. The first case indicates that only comparative disclosures may need to be priced. The second case explains why binary monotone partitions, frequent in economic analyses, need not be priced.\n\nAt the same time, examples can be constructed in a general setting with sufficiently opposed types such that, in an optimal menu, the swapping constraint binds and the information should be priced to deter mismatching deviations.","text_sha256":"5542a06396df5d956734a6359f855670a07af3f7357985ae1efc7d9e95a5bf66"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0022","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6.2 Demand Transformation and Shrouded Attributes","text":"### 6.2 Demand Transformation and Shrouded Attributes\n\nAs I discussed in Section 3, one way to think about attribute disclosure is in terms of its impact on the demand curve that the seller faces. This particular effect was studied by Johnson and Myatt (2006). They restrict their attention to disclosures that spread type valuations \"uniformly.\" Such disclosures translate into global rotations of the demand curve. The authors show that in many settings, the optimal global rotations are extreme and correspond to either no disclosure or full disclosure: no disclosure is associated with a mass market characterized by a low price and high demand; full disclosure is associated with a niche market characterized by a high price and low demand.\n\nIn contrast, I show that attribute disclosure can rotate the demand curve locally (recall Figure 3). The local rotations correspond to partial disclosures that target specific types and, hence, affect the demand curve over a particular price segment. They can outperform full and no disclosure in both mass and niche markets. Multiple attributes are required for this result-recall that no disclosure remains optimal in a one-dimensional framework.\n\nThis idea of targeted disclosure provides additional justification for selective advertising and attribute shrouding, ubiquitous in practice (Gabaix and Laibson (2006)). Even if customers are perfectly rational, the seller may have incentives to suppress information about some attributes while providing information about the others. Intuitively, different attributes can appeal to different customer cohorts. At a given price, some cohorts may have to be persuaded to purchase the product while others may not have to be.\n\nFor example, the customer base of a smartphone company could consist of two main groups: high-value customers, who are primarily interested in reliability and the quality of customer service; and low-value customers, who are primarily interested in entertainment features such as the screen size and camera performance. A smartphone price will optimally\nbalance the respective cohort valuations. In the absence of additional information, the price will be acceptable for the high-value customers but not for the low-value customers. The advertising campaign could thus optimally focus on the entertainment features to persuade the low-value cohort, while suppressing the information about reliability and services.","text_sha256":"87b00e558105f0f66e87ef02a373a8bedcafd439cc7986b4110243fd47330ba0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0023","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6.3 Alternative Disclosure Settings","text":"### 6.3 Alternative Disclosure Settings\n\nThe multiattribute disclosure setting complements the existing one-dimensional models of disclosure and pricing by Eső and Szentes (2007) and Li and Shi (2017). The main difference, however, lies not in the multidimensionality per se but rather in what kinds of information the seller can provide.\n\nEső and Szentes (2007) study discriminatory mechanisms in a valuation-rank framework, in which disclosure corresponds to statements such as \"Your valuation is in your y-th percentile,\" with y being the same for all types. The authors obtain two main qualitative results: first, they show that full information disclosure is generally optimal; and second, they show that the seller cannot benefit from conditioning the price on the disclosure realization. ${ }^{20}$\n\nFormally, their seller informs the buyer about an orthogonal shock $\\xi(\\theta)$, defined as the type's valuation percentile. By construction, these percentiles are uniformly distributed:\n\n$$\n\\xi(\\theta) \\sim U[0,1] \\quad \\forall \\theta \\in \\Theta .\n$$\n\nThe implicit assumption of the valuation-rank framework is that these shocks are equal, i.e., $\\xi(\\theta) \\equiv \\xi\\left(\\theta^{\\prime}\\right)$ for all $\\theta, \\theta^{\\prime} \\in \\Theta$. However, despite having the same distribution, the shocks $\\xi(\\theta)$ are generally different random variables. This observation is crucial and is particularly evident in the multiattribute setting.\n\nConsider the following example. Let there be two independently distributed attributes $J=2, x_{1} \\sim U[0,1], x_{2} \\sim U[0,2]$. Let there be two equally likely types: $\\theta_{1}=(1,0)$ and $\\theta_{2}=(0,1)$. The corresponding orthogonal shocks are $\\xi\\left(\\theta_{1}\\right)=x_{1}$ and $\\xi\\left(\\theta_{2}\\right)=x_{2} / 2$. Both $\\xi\\left(\\theta_{1}\\right)$ and $\\xi\\left(\\theta_{2}\\right)$ are uniformly distributed on $[0,1]$. However, they depend on different attributes and are thus independent from each other.\n\nConsequently, neither of the qualitative results of Eső and Szentes (2007) holds in this example. The optimal full-disclosure mechanism can be calculated to be $r_{1}=r_{2}=1 / 2$, $p_{1}=p_{2}=0$ : anticipating information revelation, the seller would prefer to effectively sell the object in advance. The corresponding revenue is 1/2. The seller can do strictly better by providing partial private disclosure. Building on the results of Section 5.2, it can be shown\n\n[^13]that an optimal mechanism informs the buyer, free of charge, whether the first attribute is above or below 1/2 and then follows with a posted object price of 3/4. This mechanism obtains revenue 9/16 > 1/2.\n\nAt the same time, the seller could further improve the revenue if she could condition the price directly on the disclosure realization. Consider the following mechanism. The seller provides full disclosure, observes the attributes, and chooses the price optimally given the realized valuation distribution. The corresponding revenue is:\n\n$$\n\\Pi=\\int_{0}^{1} \\int_{0}^{2} \\frac{1}{2} \\max \\left\\{\\min \\left\\{x_{1}, x_{2}\\right\\}, \\frac{\\max \\left\\{x_{1}, x_{2}\\right\\}}{2}\\right\\} \\mathrm{d} x_{1} \\mathrm{~d} x_{2}=\\frac{29}{48}>\\frac{9}{16} .\n$$\n\nThis example also emphasizes the difference between the multiattribute setting and the setting of Li and Shi (2017). These authors study common value settings in which the types represent private information about the object. In their settings, information disclosure can be seen as a valuation-level disclosure that corresponds to statements such as \"Your valuation is above x,\" with x being the same for all types. However, in general, attribute information affects the valuation of different types differently according to the valuation function. In the example above, any experiment informative only about attribute $x_{i}$ affects the valuation of only type $\\theta_{i}$. Consequently, attribute information cannot be modeled as an experiment that informs the buyer directly about his valuation. This modeling would misrepresent the buyer incentives in terms of choice across experiments.\n\nOverall, this discussion emphasizes the importance of explicitly modeling demand microstructure, i.e., how the consumer valuation is formed, in trade settings with information control.","text_sha256":"a94dc3db677f39cffe931f49c0e40f99d61d8c28ea5149aadf8d79be6d1e4c25"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0024","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"7 Conclusion","text":"## 7 Conclusion\n\nI studied a monopolist who sells a multiattribute object to a privately informed buyer and showed that the seller can benefit from the disclosure of attribute information. The benefit comes through two channels. First, disclosure can be used as a screening device, leveraging the preferences of different buyer types for learning about different aspects of the object. Second, disclosure can lift the buyer's expectations and persuade him to buy the object at a higher price. Both channels are important. However, I show that in many settings screening is not beneficial and information should be disclosed partially and free of charge. In those settings, the choice of information content is more important than the choice of its pricing.\n\nIn this paper, I deliberately focused on the simplest model of pricing and information control. In practice, additional details might be important and should be considered. The\nseller could be restricted in the kinds of information that she may provide. The buyer might feature heterogeneity in his ability to process data. The market could involve imperfect competition. Each of these extensions can be approached within the multiattribute disclosure framework that I have outlined.","text_sha256":"3b72be7542c4a17e829c15c7d8f07e83e7fcfc73f7674951551b5b3f750000fb"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0025","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"8 Appendix","text":"## 8 Appendix\n\nProof of Proposition 1. Consider any menu $M=(r(i), E(i), p(i))_{i \\in \\mathcal{I}}$. For any type $\\theta, M$ induces the allocation distribution $\\mu(\\theta): X \\rightarrow \\Delta(A), A=$ \\{buy, not buy $\\}$, the expected upfront payment $\\hat{r}(\\theta)$, and the expected object payment, conditional on a trade, $\\hat{p}(\\theta)$. Consider a direct responsive menu $M^{\\prime}=\\left(r^{\\prime}(\\theta), E^{\\prime}(\\theta), p^{\\prime}(\\theta)\\right)$ with $r^{\\prime}(\\theta)=\\hat{r}(\\theta)$, $p^{\\prime}(\\theta)=\\hat{p}(\\theta)$, and $E^{\\prime}(\\theta)=(A, \\mu(\\theta))$. If all types are truthful and obedient, then $M^{\\prime}$ results in the same allocation distribution and the same expected payments as $M$. At the same time, any deviation under $M^{\\prime}$ is available to the buyer under $M$. Therefore, reporting truthfully and following the recommendations is incentive-compatible under $M^{\\prime}$.\n\nProof of Proposition 2. Toward a contradiction, assume that a responsive menu $M=$ $(r(\\theta), \\mathcal{X}(\\theta), \\mathcal{Q}(\\theta), p(\\theta))_{\\theta \\in \\Theta}$ is optimal, yet no type buys the object with probability one. Construct a new menu $M^{\\prime}$ as follows. Select a type $\\bar{\\theta}$ with the highest expected payment $\\bar{T}=r(\\bar{\\theta})+p(\\bar{\\theta}) \\mathcal{Q}(\\bar{\\theta})$. Since $\\Theta$ is finite, this type exists. Change this type's item to no disclosure followed by an object price as follows:\n\n$$\n\\left(r^{\\prime}(\\bar{\\theta}), \\mathcal{X}^{\\prime}(\\bar{\\theta}), \\mathcal{Q}^{\\prime}(\\bar{\\theta}), p^{\\prime}(\\bar{\\theta})\\right)=(0, \\mathbb{E}[x], 1, \\bar{T}+\\bar{\\theta} \\cdot(\\mathbb{E}[x]-\\mathcal{X}(\\bar{\\theta}))) .\n$$\n\nKeep all other items the same. In this menu, type $\\bar{\\theta}$ chooses the new item and always buys the object. This strategy gives him exactly the same payoff as that of the original menu:\n\n$$\n\\bar{\\theta} \\cdot \\mathcal{X}^{\\prime}(\\bar{\\theta})-p^{\\prime}(\\bar{\\theta})=\\bar{\\theta} \\cdot \\mathcal{X}(\\bar{\\theta})-p(\\bar{\\theta}) \\mathcal{Q}(\\bar{\\theta})-r(\\bar{\\theta}) .\n$$\n\nAs $X \\subseteq \\mathbb{R}_{++}^{J}$, the uninformative experiment achieves a maximal attribute surplus, $\\mathbb{E}[x]=$ $\\int_{x \\in X} x d G>\\mathcal{X}(\\theta)$. As $\\Theta \\subseteq \\mathbb{R}_{++}^{J}$, the new expected payment from type $\\bar{\\theta}$ is strictly higher than that in the original menu, $p^{\\prime}(\\bar{\\theta})>\\bar{T}$.\n\nThe new menu $M^{\\prime}$ is not necessarily direct. The no disclosure item may be attractive to some types other than $\\bar{\\theta}$. However, the only profitable strategy under no disclosure is always buying. Such a deviation would only increase the seller's profit, as $\\bar{T}$ was chosen to be the highest expected payment. Accordingly, menu $M^{\\prime}$ brings strictly greater revenue than menu $M$, which is a contradiction.\n\nProof of Lemma 1. Set $\\mathcal{F}$ is an image of a set of measures dominated by the prior distribution. That set of measures is compact in the weak* topology. Moreover, the corresponding map is continuous by the dominated convergence theorem, since the ex-ante attribute expectations exist. Therefore, $\\mathcal{F}$ is compact.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 5: Feasibility set $\\mathcal{F}$ of attribute surplus $\\mathcal{X}=\\left(\\mathcal{X}_{1}, \\mathcal{X}_{2}\\right)$ and trade probability $\\mathcal{Q}$ for the case of two attributes distributed uniformly over a unit square $X=[0,1]^{2}$.\n\nSet $\\mathcal{F}$ is convex as a linear image of a convex set. Indeed, take any two points $\\left(\\mathcal{X}_{1}, \\mathcal{Q}_{1}\\right),\\left(\\mathcal{X}_{1}, \\mathcal{Q}_{2}\\right) \\in$ $\\mathcal{F}$ and $\\gamma \\in[0,1]$. By construction, there exist trade functions $q_{1}, q_{2}: X \\rightarrow[0,1]$ that generate these two points. Then, the function $q_{3} \\triangleq \\gamma q_{1}+(1-\\gamma) q_{2}$ is an admissible trade function that generates the attribute surplus:\n\n$$\n\\begin{aligned}\n\\mathcal{X}_{3}(\\theta) & =\\int_{x \\in X} x q_{3}(x) \\mathrm{d} G(x)=\\int_{x \\in X} x\\left(\\gamma q_{1}(x)+(1-\\gamma) q_{2}(x)\\right) \\mathrm{d} G(x) \\\\\n& =\\gamma \\int_{x \\in X} x q_{1}(x) \\mathrm{d} G(x)+(1-\\gamma) \\int_{x \\in X} x q_{2}(x) \\mathrm{d} G(x)=\\gamma \\mathcal{X}_{1}(\\theta)+(1-\\gamma) \\mathcal{X}_{2}(\\theta)\n\\end{aligned}\n$$\n\nThe same argument can be applied to the trade probability. Thus, $\\left(\\mathcal{X}_{3}, \\mathcal{Q}_{3}\\right)$ is a convex combination of $\\left(\\mathcal{X}_{1}, \\mathcal{Q}_{1}\\right)$ and $\\left(\\mathcal{X}_{2}, \\mathcal{Q}_{2}\\right)$, and it belongs to the feasibility set $\\mathcal{F}$. Figure 5 illustrates the feasibility set for the case of two uniformly and independently distributed attributes.\n\nSince $\\mathcal{F}$ is a finite-dimensional closed set, the supporting hyperplane theorem (Rockafellar (1970), Theorem 11.6, Corollary 11.6.1, p. 100) can be applied. ${ }^{21}$ A point $(\\hat{\\mathcal{X}}, \\hat{\\mathcal{Q}})$ belongs to the boundary of $\\mathcal{F}$ if and only if there are coefficients $\\left(\\lambda, \\lambda_{0}\\right)$, not all zero, such that:\n\n$$\n(\\hat{\\mathcal{X}}, \\hat{\\mathcal{Q}}) \\in \\arg \\max _{(\\mathcal{X}, \\mathcal{Q}) \\in \\mathcal{F}} \\lambda \\cdot \\mathcal{X}+\\lambda_{0} \\mathcal{Q} .\n$$\n\n[^14]By the definition of $\\mathcal{F}$, the trade function $\\hat{q}$ that generates the point $(\\hat{\\mathcal{X}}, \\hat{\\mathcal{Q}})$ is such that:\n\n$$\n\\begin{aligned}\n\\hat{q}(x) & \\in \\arg \\max _{q: X \\rightarrow[0,1]} \\lambda \\cdot \\int_{x \\in X} x q(x) \\mathrm{d} G(x)+\\lambda_{0} \\int_{x \\in X} q(x) \\mathrm{d} G(x)= \\\\\n& \\in \\arg \\max _{q: X \\rightarrow[0,1]} \\int_{x \\in X}\\left(\\lambda \\cdot x+\\lambda_{0}\\right) q(x) \\mathrm{d} G(x)\n\\end{aligned}\n$$\n\nThe integral is maximized pointwise. Any maximizer of it is a linear disclosure (22) with coefficients $\\alpha=\\lambda$ and $\\alpha_{0}=\\lambda_{0}$.\n\nProof of Lemma 2. The seller's problem can be written in terms of attribute surpluses and trade probabilities (15). Consider any profile $(r(\\theta), \\mathcal{X}(\\theta), \\mathcal{Q}(\\theta), p(\\theta))_{\\theta \\in \\Theta}$ that satisfies constraints (16), (17), (18), (19), (20). Define functions $\\mathcal{Q}^{\\prime}: \\Theta \\rightarrow[0,1]$ and $p^{\\prime}: \\Theta \\rightarrow \\mathbb{R}_{+}$as:","text_sha256":"ef3407db1e851d31320c69a404867ba6714d3f41c9d2011edc99f681e9846593"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0026","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"8 Appendix","text":"$$\n\\begin{aligned}\n\\mathcal{Q}^{\\prime}(\\theta) & =\\min _{Q:(\\mathcal{X}(\\theta), Q) \\in \\mathcal{F}} Q, \\\\\np^{\\prime}(\\theta) & = \\begin{cases}\\frac{\\mathcal{Q}(\\theta) p(\\theta)}{\\mathcal{Q}^{\\prime}(\\theta)}, & \\text { if } \\mathcal{Q}^{\\prime}(\\theta)>0, \\\\\np(\\theta), & \\text { if } \\mathcal{Q}^{\\prime}(\\theta)=0 .\\end{cases}\n\\end{aligned}\n$$\n\nIntuitively, these functions are perturbations of the original mechanism that minimize trade probability while keeping the expected revenue fixed. By Lemma 1, $\\mathcal{F}$ is compact, so for each $\\theta \\in \\Theta, \\mathcal{Q}^{\\prime}(\\theta)$ is well defined and $\\left(\\mathcal{X}(\\theta), \\mathcal{Q}^{\\prime}(\\theta)\\right)$ belongs to the boundary of $\\mathcal{F}$. At the same time, by the measurable maximum theorem (Aliprantis and Border (2006), Thm. 18.19) $\\mathcal{Q}^{\\prime}$, and hence $p^{\\prime}$, is a measurable function. By construction, for any type $\\theta \\in \\Theta, \\mathcal{Q}^{\\prime}(\\theta) \\leq \\mathcal{Q}(\\theta), p^{\\prime}(\\theta) \\geq p(\\theta)$, and $\\mathcal{Q}^{\\prime}(\\theta) p^{\\prime}(\\theta)=\\mathcal{Q}(\\theta) p(\\theta)$. As such, allocation $\\left(\\mathcal{X}(\\theta), \\mathcal{Q}^{\\prime}(\\theta)\\right)_{\\theta \\in \\Theta}$ is implementable via tariff functions $r(\\theta), p^{\\prime}(\\theta)$ and satisfies all conditions of the lemma. The result follows.\n\nProof of Theorem 1. The seller's problem (15) can be seen as a maximization of a continuous function over a compact set. Therefore, an optimal menu exists. By Lemma 2, there exists an optimal menu with all allocations located on the boundary of the feasibility set $\\mathcal{F}$. By Lemma 1, such allocations are achieved by linear disclosures.\n\nProof of Corollary 1. Define an auxiliary attribute $x_{\\theta}^{\\prime}$ as the valuation of a type $\\theta$, $x_{\\theta}^{\\prime} \\triangleq v(\\theta, x)$. By construction, the valuation of each type can be defined as $v^{\\prime}\\left(\\theta, x^{\\prime}\\right)=x_{\\theta}^{\\prime}$. This instance is a special case of the formulation (1). Thus, Theorem 1 applies and there exists an optimal menu with every experiment in it being a linear disclosure of auxiliary\nattributes $x^{\\prime}$ :\n\n$$\nq\\left(x^{\\prime}\\right)= \\begin{cases}1, & \\text { if } \\sum_{\\theta \\in \\Theta} \\alpha_{\\theta} x_{\\theta}^{\\prime}>\\alpha_{0}, \\\\ 0, & \\text { if } \\sum_{\\theta \\in \\Theta} \\alpha_{\\theta} x_{\\theta}^{\\prime}<\\alpha_{0},\\end{cases}\n$$\n\nfor $\\alpha \\in \\mathbb{R}^{|\\Theta|}, \\alpha_{0} \\in \\mathbb{R}$, not all zeros. In the original formulation, these are linear forms.\n\nCalculations behind Location Payoffs Example. Consider a linear form. If $\\alpha_{1}+\\alpha_{2} \\neq$ 0, then the sum can be normalized to equal 1. The linear form can be rewritten as:\n\n$$\nq(x)= \\begin{cases}1, & \\text { if }-\\left(x-\\left(\\alpha_{1} \\theta_{1}+\\alpha_{2} \\theta_{2}\\right)\\right)^{2} \\gtrless \\alpha_{0}^{\\prime}, \\\\ 0, & \\text { if }-\\left(x-\\left(\\alpha_{1} \\theta_{1}+\\alpha_{2} \\theta_{2}\\right)\\right)^{2} \\lessgtr \\alpha_{0}^{\\prime},\\end{cases}\n$$\n\nwith $\\alpha_{0}^{\\prime}=-v_{0}+\\alpha_{0}+\\alpha_{1} \\alpha_{2}\\left(\\theta_{1}-\\theta_{2}\\right)^{2}$ and the inequality sign depending on the sign of the original $\\alpha_{1}+\\alpha_{2}$. This is a neighborhood disclosure with $\\hat{\\theta}=\\alpha_{1} \\theta_{1}+\\alpha_{2} \\theta_{2}$ and $\\alpha_{1}+\\alpha_{2}=1$.\n\nIf $\\alpha_{1}=\\alpha_{2}=0$, then the linear form provides no disclosure and, as $X$ is bounded, is equivalent to a neighborhood disclosure for a sufficiently large $\\left|\\alpha_{0}\\right|$.\n\nIf $\\alpha_{1}+\\alpha_{2}=0$ and $\\alpha_{1} \\neq 0$, then the linear form is a linear disclosure:\n\n$$\nq(x)= \\begin{cases}1, & \\text { if }\\left(\\theta_{1}-\\theta_{2}\\right) \\cdot x \\gtrless \\alpha_{0}^{\\prime}, \\\\ 0, & \\text { if }\\left(\\theta_{1}-\\theta_{2}\\right) \\cdot x \\lessgtr \\alpha_{0}^{\\prime},\\end{cases}\n$$\n\nwith $\\alpha_{0}^{\\prime}=\\alpha_{0} /\\left(2 \\alpha_{1}\\right)+\\left(\\theta_{1}^{2}-\\theta_{2}^{2}\\right) / 2$ and the inequality sign depending on the sign of $\\alpha_{1}$. However, the proof of Lemma 1 established that the attribute surplus and probability achieved by a linear form with parameters $\\left(\\alpha_{1}, \\alpha_{2}, \\alpha_{0}\\right)$ correspond to a boundary point of $\\mathcal{F}$ in the auxiliary attributes, supported by the hyperplane orthogonal to the vector $\\left(\\alpha_{1}, \\alpha_{2}, \\alpha_{0}\\right)$. If $\\theta_{1} \\neq \\theta_{2}$, then $\\mathcal{F}$ has a strict interior. Thus, the set of boundary points supported by hyperplanes with $\\alpha_{1}+\\alpha_{2}=0$ has a measure of zero.\n\nProof of Theorem 2. The argument is given in the text. The only difference from the standard problems is that $\\mathcal{X}$ can take values in $[0, \\mathbb{E}[x]]$, not in $[0,1]$. However, this difference does not affect the extreme nature of the solution.\n\nLemma 4. (Directional Decomposition)\nLet $\\left(x_{1}, \\ldots, x_{J}\\right)$ be $J$ attributes independently distributed over $X \\subseteq \\mathbb{R}^{J}$ according to distributions $G_{1}, \\ldots, G_{J}$. Let $E=(S, \\pi)$ be an arbitrary experiment. Let $(\\mu(s, E), \\operatorname{Pr}(s, E))$ be the belief distribution induced by $E$ so that $\\mu(s, E)$ is a distribution over $X$ conditional on $s$ given $E$. Denote by $\\mu_{j}(s, E)$ the $j$ th marginal distribution of $\\mu(s, E)$. Then, there exist ex-\nperiments $\\left\\{E_{j}\\right\\}_{j=1}^{J}$ such that: $E_{j}=\\left(S, \\pi_{j}\\right)$ induces a belief distribution $\\left(\\mu\\left(s, E_{j}\\right), \\operatorname{Pr}\\left(s, E_{j}\\right)\\right)$ with $\\mu\\left(s, E_{j}\\right)=\\left(\\mu_{j}(s, E), G_{-j}\\right)$ and $\\operatorname{Pr}\\left(s, E_{j}\\right)=\\operatorname{Pr}(s, E)$ for all $s \\in S$.\n\nProof. The proof is constructive. Introduce dummy variables $\\left(x_{1}^{\\prime}, \\ldots, x_{J}^{\\prime}\\right)$ which are distributed as $\\left(x_{1}, \\ldots, x_{J}\\right)$ but are drawn independently of them. For a given $j$, construct $E_{j}$ as an experiment that provides information about the vector $\\left(x_{j}, x_{-j}^{\\prime}\\right)$ according to $\\pi$. By construction, $E_{j}$ induces the same marginal distribution of beliefs about attribute $j$. However, since $\\left(x_{1}, x_{1}^{\\prime}, \\ldots, x_{J}, x_{J}^{\\prime}\\right)$ are independent, it provides no information about other attributes. $\\square$","text_sha256":"aa045b07adfe3e262470f91c7d3b793ebbad4ef1a987b02095aed62f8ae32ccb"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0027","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"8 Appendix","text":"Proof of Proposition 3. Consider an arbitrary responsive experiment $E_{j}\\left(\\theta_{j}\\right)$. By Lemma 4, there exists a linear disclosure $E_{j}^{\\prime}\\left(\\theta_{j}\\right)$ such that $\\mathcal{X}_{j}^{\\prime}\\left(\\theta_{j}\\right)=\\mathcal{X}_{j}\\left(\\theta_{j}\\right), \\mathcal{Q}\\left(E_{j}^{\\prime}\\left(\\theta_{j}\\right)\\right)=\\mathcal{Q}\\left(E_{j}\\left(\\theta_{j}\\right)\\right)$, and $\\mathcal{X}_{k}^{\\prime}\\left(\\theta_{j}\\right)=\\mathbb{E}\\left[x_{k}\\right]$ for all $k \\neq j$. Replacing $E_{j}\\left(\\theta_{j}\\right)$ with $E_{j}^{\\prime}\\left(\\theta_{j}\\right)$ does not change the incentive compatibility within cohort $\\Theta_{j}$, but by Blackwell's Theorem, it relaxes the incentive compatibility of other cohorts. By Theorem 1, the result follows.\n\nProof of Lemma 3. Define the expected transfer function as $T_{j}\\left(\\theta_{j}\\right) \\triangleq \\mathcal{Q}_{j}\\left(\\theta_{j}\\right) p_{j}\\left(\\theta_{j}\\right)$. I can use standard one-dimensional arguments within each cohort to establish the connection between the attribute surplus and the expected transfer function:\n\n$$\nT_{j}\\left(\\theta_{j}\\right)=\\theta_{j} \\mathcal{X}_{j}\\left(\\theta_{j}\\right)-\\int_{0}^{\\bar{\\theta}_{j}} \\mathcal{X}_{j}(z) \\mathrm{d} z .\n$$\n\nIndividual rationality and incentive compatibility within each cohort are satisfied by construction. However, deviations between different cohorts impose additional constraints,\n\n$$\n\\int_{0}^{\\bar{\\theta}_{j}} \\mathcal{X}_{j}\\left(\\theta_{j}\\right) d \\theta_{j} \\geq \\bar{\\theta}_{j} \\mathbb{E}\\left[x_{j}\\right]-\\underline{p},\n$$\n\nwhere $\\underline{p}$ is the minimal object price in the menu $\\left\\{\\mathcal{X}_{j}\\right\\}_{j=1}^{J}$. The deviations from all other types $\\theta_{j} \\in \\Theta$ follow because the indirect utility function is convex and grows slower than $\\theta_{j} \\mathbb{E}\\left[x_{j}\\right]$. Applying double integration to the objective function completes the derivation.\n\nProof of Theorem 3. The argument in the text establishes that all high types are offered the minimal price. The optimal mechanism should then solve the problem (36) with the additional constraints that all high types are offered the same fixed price $\\underline{p}^{*}$ and are served\nthe fixed attribute surplus $\\mathcal{X}_{j}^{*}\\left(\\bar{\\theta}_{j}\\right)$. These constraints can be written as:\n\n$$\n\\begin{aligned}\n\\int_{0}^{\\bar{\\theta}_{j}} \\mathcal{X}_{j}\\left(\\theta_{j}\\right) d \\theta_{j} & =\\mathcal{X}_{j}\\left(\\bar{\\theta}_{j}\\right)-\\underline{p}^{*} \\mathcal{Q}_{j}\\left(\\mathcal{X}_{j}\\left(\\bar{\\theta}_{j}\\right)\\right) \\\\\n\\mathcal{X}_{j}\\left(\\bar{\\theta}_{j}\\right) & =\\mathcal{X}_{j}^{*}\\left(\\bar{\\theta}_{j}\\right)\n\\end{aligned}\n$$\n\nConsider a relaxed problem with the original integral constraints and the monotonicity constraints dropped. In this problem, by Luenberger (1969) (Chapter 8, Theorem 1), there exist Lagrange multipliers $\\left\\{\\lambda_{j}\\right\\}$ such that the optimal $\\mathcal{X}_{j}\\left(\\theta_{j}\\right)$ maximize the Lagrange function:\n\n$$\n\\mathcal{L} \\sim \\sum_{j=1}^{J} f\\left(\\Theta_{j}\\right) \\int_{0}^{\\bar{\\theta}_{j}}\\left(\\theta_{j}-\\frac{1-F_{j}\\left(\\theta_{j}\\right)}{f_{j}\\left(\\theta_{j}\\right)}-\\lambda_{j}\\right) \\mathcal{X}_{j}\\left(\\theta_{j}\\right) \\mathrm{d} F_{j}\\left(\\theta_{j}\\right)\n$$\n\nover a domain $\\mathcal{X}_{j}\\left(\\theta_{j}\\right) \\in\\left[0, \\mathcal{X}^{*}\\left(\\bar{\\theta}_{j}\\right)\\right]$. If all type distributions have the monotone hazard rate property, then the integrands increase in $\\theta_{j}$. Therefore, the optimal $\\mathcal{X}_{j}\\left(\\theta_{j}\\right)$ are bang-bang: $\\mathcal{X}_{j}\\left(\\theta_{j}\\right)=0$ for $\\theta_{j}<\\theta_{j}^{*}, \\mathcal{X}_{j}\\left(\\theta_{j}\\right)=\\mathcal{X}^{*}\\left(\\bar{\\theta}_{j}\\right)$ for $\\theta_{j}>\\theta_{j}^{*}$. This solution corresponds to a single item per each attribute cohort and satisfies the relaxed constraints.\n\nProof of Theorem 4. I begin by characterizing the optimal mechanism in the case of orthogonal types, since formally it is not covered by 3. The class of orthogonal types features particularly tractable incentive constraints. If type $\\theta_{j}$ misreports, then he is offered an experiment tailored to another orthogonal type that is hence not informative about attribute $j$. Thus, the type has no reason to act on the experiment realization, and the tightest incentive-compatibility constraint is one in which he always buys. All others can be dropped. The seller's problem can be written as:\n\n$$\n\\begin{array}{ll}\n& \\max _{\\left\\{r_{j}, \\mathcal{X}_{j}, p_{j}\\right\\}_{j=1}^{J}} \\sum_{j=1}^{J} f\\left(\\theta_{j}\\right)\\left(r_{j}+\\mathcal{Q}_{j} p_{j}\\right) \\\\\n\\text { s.t. } & \\mathcal{X}_{j}-p_{j} \\mathcal{Q}_{j}-r_{j} \\geq \\mathbb{E}\\left[x_{j}\\right]-p_{k}-r_{k}, \\forall j, k=1, \\ldots, J \\\\\n& \\mathcal{X}_{j}-p_{j} \\mathcal{Q}_{j}-r_{j} \\geq 0, \\mathcal{X}_{j} \\in\\left[0, \\mathbb{E}\\left[x_{j}\\right]\\right], \\mathcal{Q}_{j}=\\mathcal{Q}_{j}\\left(\\mathcal{X}_{j}\\right), \\forall j=1, \\ldots, J .\n\\end{array}\n$$\n\nI show that the solution to this problem does not feature price discrimination. First, by the arguments of Theorem 3, upfront payments can be without loss of revenue set to zero, $r_{j} \\equiv 0$. Second, consider an arbitrary solution to the seller's problem such that $r_{j} \\equiv 0$ and define $\\underline{p}=\\min _{j}\\left\\{p_{j}\\right\\}$. Toward the contradiction, assume that $p_{j}>\\underline{p}$ for some $j$.\n\nIf $\\mathbb{E}\\left[x_{j}\\right] \\geq \\underline{p}$, then the incentive-compatibility constraint is binding. Hence, $\\mathcal{Q}_{j} p_{j}=$ $\\mathcal{X}_{j}-\\mathbb{E}\\left[x_{j}\\right]+\\underline{p}$. For small $\\varepsilon>0$, consider a modified mechanism with $\\mathcal{X}_{j}^{\\prime}=\\mathcal{X}_{j}+\\varepsilon, \\mathcal{Q}_{j}^{\\prime} p_{j}^{\\prime}=$\n$\\mathcal{X}_{j}^{\\prime}-\\mathbb{E}\\left[x_{j}\\right]+\\underline{p}$. Because $\\mathcal{Q}_{j}\\left(\\mathcal{X}_{j}\\right)$ is continuous, the mechanism remains incentive compatible yet brings higher revenue, which is a contradiction. Moreover, note that as $\\mathcal{X}_{j}^{\\prime}>\\mathcal{X}_{j}, \\alpha_{0 j}^{\\prime}<$ $\\alpha_{0 j}$; hence, $\\mathcal{X}_{j}^{\\prime} / \\mathcal{Q}_{j}^{\\prime}=\\mathbb{E}\\left[x_{j} \\mid x_{j} \\geq \\alpha_{0 j}^{\\prime}\\right]<\\mathbb{E}\\left[x_{j} \\mid x_{j} \\geq \\alpha_{0 j}\\right]=\\mathcal{X}_{j} / \\mathcal{Q}_{j}$ and $\\mathcal{Q}_{j}^{\\prime}>\\mathcal{Q}_{j}$. Thus, $p_{j}^{\\prime}<p_{j}$.","text_sha256":"c794635cf358cac416d0c99f05c8a6cf8bcde38c7110becf198640113be83ca9"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0028","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"8 Appendix","text":"If $\\mathbb{E}\\left[x_{j}\\right]<\\underline{p}$, then the individual-rationality constraint is binding. Hence, $\\mathcal{Q}_{j} p_{j}=\\mathcal{X}_{j}$. For small $\\varepsilon>0$, consider the modified mechanism with $p_{j}^{\\prime}=p_{j}-\\varepsilon, \\mathcal{X}_{j}^{\\prime} / \\mathcal{Q}_{j}^{\\prime}=\\mathcal{X}_{j} / \\mathcal{Q}_{j}-\\varepsilon$. The mechanism remains incentive compatible yet brings higher revenue, which is a contradiction.\n\nNow, consider the optimal disclosure for a given object price. According to feasibility and individual rationality, $\\mathcal{X}_{j} / \\mathcal{Q}_{j} \\geq \\max \\left\\{p, \\mathbb{E}\\left[x_{j}\\right]\\right\\}$. If $\\mathcal{X}_{j} / \\mathcal{Q}_{j}>\\max \\left\\{p, \\mathbb{E}\\left[x_{j}\\right]\\right\\}$, then for small $\\varepsilon>0$, the mechanism with $\\mathcal{X}_{j}^{\\prime} / \\mathcal{Q}_{j}^{\\prime}=\\mathcal{X}_{j} / \\mathcal{Q}_{j}-\\varepsilon$ is incentive compatible and increases trade probability, $\\mathcal{Q}_{j}^{\\prime}>\\mathcal{Q}_{j}$, and consequently, revenue. This is a contradiction.\n\nThese arguments establish that in the case of orthogonal types an optimal mechanism sets $r_{j}=0, p_{j}=p$, and $E_{j}$ is a binary monotone partition of $x_{j}$ such that $\\mathbb{E}\\left[x_{j} \\mid E_{j}, s^{+}\\right]=$ $\\max \\left\\{p, \\mathbb{E}\\left[x_{j}\\right]\\right\\}$ for all $j=1, \\ldots, J$.\n\nNow, consider a general setting. Introduce auxiliary attributes as in the proof of Corollary 1. By the arguments above, $M^{S P}\\left(p^{*}\\right)$ solves a relaxed problem in which the constraints (16) and (17) are dropped. If $M^{S P}\\left(p^{*}\\right)$ is incentive compatible, then these relaxed constraints are satisfied and, thus, $M^{S P}\\left(p^{*}\\right)$ solves an original problem.\n\nProof of Corollary 2. For a profile $\\left(\\theta_{1}, \\ldots, \\theta_{J}\\right)$, denote by $p^{*}\\left(\\theta_{1}, \\ldots, \\theta_{J}\\right)$ an optimal price in a separate persuasion mechanism, by $\\alpha_{0 j}\\left(\\theta_{1}, \\ldots, \\theta_{J}\\right)$ optimal persuasion thresholds, and by $\\Pi^{S P}\\left(p, \\theta_{1}, \\ldots, \\theta_{J}\\right)$ the revenue function. As attributes are continuously distributed, $\\Pi^{S P}(\\cdot)$ is continuous. If $p^{*}\\left(\\theta_{1}, \\ldots, \\theta_{J}\\right)$ is a singleton, then, by the Maximum Theorem, $p^{*}(\\cdot)$ and $\\alpha_{0}(\\cdot)$ are continuous functions in a neighborhood of $\\left(\\theta_{1}, \\ldots, \\theta_{J}\\right)$.\n\nFix an orthogonal type profile $\\hat{\\Theta}=\\left(\\hat{\\theta}_{1}, \\ldots, \\hat{\\theta}_{J}\\right)$. If $\\mathbb{E}[\\hat{\\theta} \\cdot x] \\neq p\\left(\\hat{\\theta}_{1}, \\ldots, \\hat{\\theta}_{J}\\right)$ for all $\\hat{\\theta} \\in \\hat{\\Theta}$, then the types, when deviating, strictly prefer to not act contrary to their no-information action. As $\\hat{\\Theta} \\geq 0$, the mismatching strategies are irrelevant; moreover, for all $\\hat{\\theta}_{j}, \\hat{\\theta}_{k} \\in \\hat{\\Theta}$, $k \\neq j$ :\n\n$$\n\\begin{aligned}\n& \\mathbb{E}\\left[\\hat{\\theta}_{j} \\cdot x \\mid \\hat{\\theta}_{k} \\cdot x>\\alpha_{0 k}\\left(\\hat{\\theta}_{1}, \\ldots, \\hat{\\theta}_{J}\\right)\\right]<p^{*}\\left(\\hat{\\theta}_{1}, \\ldots, \\hat{\\theta}_{J}\\right), \\text { if } \\mathbb{E}\\left[\\hat{\\theta}_{j} \\cdot x\\right]<p, \\\\\n& \\mathbb{E}\\left[\\hat{\\theta}_{j} \\cdot x \\mid \\hat{\\theta}_{k} \\cdot x<\\alpha_{0 k}\\left(\\hat{\\theta}_{1}, \\ldots, \\hat{\\theta}_{J}\\right)\\right]>p^{*}\\left(\\hat{\\theta}_{1}, \\ldots, \\hat{\\theta}_{J}\\right), \\text { if } \\mathbb{E}\\left[\\hat{\\theta}_{j} \\cdot x\\right]>p .\n\\end{aligned}\n$$\n\nNow, replace orthogonal $\\hat{\\Theta}$ by a generic $\\Theta$. As long as all types in $\\Theta$ are positive, the mismatching strategies remain irrelevant. As attributes are continuously distributed, in some neighborhood of $\\hat{\\Theta}$ both sides of the inequalities are continuous and the constraints remain satisfied. Hence, the buyer cannot benefit from misreporting, and the result follows\nfrom Theorem 4.\n\nProof of Proposition 4. Consider any optimal menu and a type $\\theta$ with $r(\\theta)>0$. Let $r^{\\prime}(\\theta)=r(\\theta)-\\varepsilon, p^{\\prime}(\\theta)=p(\\theta)+\\varepsilon / \\mathcal{Q}(\\theta)$ for a small $\\varepsilon>0$. If incentive compatible, this modification preserves the seller's revenue and the buyer's payoff. The constraint (16) remains the same. The constraint (18) is relaxed and is strictly so when $\\mathcal{Q}(\\theta)<1$. The constraint (19) is satisfied by the individual rationality. For the constraint (17):\n\n$$\nr^{\\prime}(\\theta)+(1-\\mathcal{Q}(\\theta)) p^{\\prime}(\\theta)=r(\\theta)+(1-\\mathcal{Q}(\\theta)) p(\\theta)+\\varepsilon \\frac{1-2 \\mathcal{Q}(\\theta)}{\\mathcal{Q}(\\theta)} .\n$$\n\nFor the modification to violate incentive constraints, it must be that $1-2 \\mathcal{Q}(\\theta)<0$ and the constraint (17) binds for some type $\\theta^{\\prime}$.\n\nProof of Corollary 3. I present the arguments for the case when the higher signal is sent with probability one at the threshold. The cases with randomization are analogous.\n\ni. ) Whenever $\\alpha_{j} \\geq 0$ :\n$$\n\\mathbb{E}\\left[x_{j} \\mid \\alpha x \\geq \\alpha_{0}\\right]=\\mathbb{E}\\left[x_{j} \\mid \\alpha_{j} x_{j} \\geq \\alpha_{0}-\\sum_{k \\neq j} \\alpha_{k} x_{k}\\right] \\geq \\mathbb{E}\\left[x_{j} \\mid \\alpha_{j} x_{j}<\\alpha_{0}-\\sum_{k \\neq j} \\alpha_{k} x_{k}\\right] .\n$$\n$\\mathbb{E}\\left[v(\\theta, x) \\mid \\alpha x \\geq \\alpha_{0}\\right]=\\sum_{j=1}^{J} \\theta_{j} \\mathbb{E}\\left[x_{j} \\mid \\alpha x \\geq \\alpha_{0}\\right] \\geq \\sum_{j=1}^{J} \\theta_{j} \\mathbb{E}\\left[x_{j} \\mid \\alpha x<\\alpha_{0}\\right]=\\mathbb{E}\\left[v(\\theta, x) \\mid \\alpha x<\\alpha_{0}\\right]$.\nii. ) For any increasing function $v(\\theta, \\cdot)$, and $x_{0} \\in \\mathbb{R}, \\mathbb{E}\\left[v(\\theta, x) \\mid x \\geq x_{0}\\right] \\geq \\mathbb{E}\\left[v(\\theta, x) \\mid x<x_{0}\\right]$.","text_sha256":"f8d66602f07b58b94a1ecf3db947915dd11ae0dcf2969a5c2a10d4079dade4e4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0029","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAliprantis, C. and K. Border (2006): \"Infinite Dimensional Analysis,\" .\n\nAnderson, S. P. and R. Renault (2006): \"Advertising Content,\" American Economic Review, 96, 93-113.\n\nBar-Isaac, H., G. Caruana, and V. Cuñat (2010): \"Information Gathering and Marketing,\" Journal of Economics \\& Management Strategy, 19, 375-401.\n\nBergemann, D., A. Bonatti, and A. Smolin (2018): \"The Design and Price of Information,\" American Economic Review, 108, 1-48.\n\nBergemann, D., F. Castro, and G. Weintraub (2020): \"The Scope of Sequential Screening with Ex-Post Participation Constraints, \" Journal of Economic Theory, 105055.\n\nBergemann, D. and M. Pesendorfer (2007): \"Information Structures in Optimal Auctions,\" Journal of Economic Theory, 137, 580-609.\n\nBergemann, D., J. Shen, Y. Xu, and E. Yeh (2012): \"Multi-Dimensional Mechanism Design with Limited Information,\" in Proceedings of the 13th ACM Conference on Electronic Commerce, ACM, 162-178.\n\nBlackwell, D. (1953): \"Equivalent Comparisons of Experiments,\" Annals of Mathematical Statistics, 24, 265-272.\n\nCarville, O. (2018): \"ZipRecruiter Is Valued at \\$1.5 Billion in a Bet on AI Hiring,\" Bloomberg.com, accessed at https://www.bloomberg.com/news/articles/2018-10-04/ ziprecruiter-is-valued-at-1-5-billion-in-a-bet-on-ai-hiring.\n\nChakraborty, A. and R. Harbaugh (2010): \"Persuasion by Cheap Talk,\" American Economic Review, 100, 2361-82.\n\nCourty, P. and H. Li (2000): \"Sequential Screening,\" Review of Economic Studies, 67, 697-717.\n\nDaskalakis, C., A. Deckelbaum, and C. Tzamos (2017): \"Strong Duality for a Multiple-Good Monopolist,\" Econometrica, 85, 735-767.\n\nDoval, L. and J. C. Ely (2020): \"Sequential information design,\" Econometrica, 88, 2575-2608.\n\nDubé, J.-P. and S. Misra (2019): \"Personalized Pricing and Customer Welfare,\" Discussion Paper.\n\nDworczak, P. (2020): \"Mechanism Design with Aftermarkets: Cutoff Mechanisms,\" Discussion paper.\n\nDworczak, P. and G. Martini (2019): \"The Simple Economics of Optimal Persuasion,\" Journal of Political Economy, 127.\n\nDye, R. A. (1985): \"Disclosure of Nonproprietary Information,\" Journal of Accounting Research, 123-145.\n\nEső, P. and B. Szentes (2007): \"Optimal Information Disclosure in Auctions and the Handicap Auction,\" Review of Economic Studies, 74, 705-731.\n\n- (2017): \"Dynamic Contracting: An Irrelevance Theorem,\" Theoretical Economics, 12, 109-139.\n\nGabaix, X. and D. Laibson (2006): \"Shrouded Attributes, Consumer Myopia, and Information Suppression in Competitive Markets,\" Quarterly Journal of Economics, 121, 505-540.\n\nHartline, J. (2020): \"Mechanism Design and Approximation,\" Accessed at http:// jasonhartline.com/MDnA/.\n\nHeumann, T. (2020): \"Information Design and Sequential Screening with Ex Post Participation Constraint,\" Theoretical Economics, 15, 319-359.\n\nJohnson, J. P. and D. P. Myatt (2006): \"On the Simple Economics of Advertising, Marketing, and Product Design,\" American Economic Review, 96, 756-784.\n\nKamenica, E. and M. Gentzkow (2011): \"Bayesian Persuasion,\" American Economic Review, 101, 2590-2615.\n\nKoessler, F. and R. Renault (2012): \"When Does a Firm Disclose Product Information?\" The RAND Journal of Economics, 43, 630-649.\n\nKolotilin, A., T. Mylovanov, A. Zapechelnyuk, and M. Li (2017): \"Persuasion of a Privately Informed Receiver,\" Econometrica, 85, 1949-1964.\n\nKrähmer, D. (2020): \"Information Disclosure and Full Surplus Extraction in Mechanism Design,\" Journal of Economic Theory, 105020.\n\nKrähmer, D. and R. Strausz (2015a): \"Ex Post Information Rents in Sequential Screening,\" Games and Economic Behavior, 90, 257-273.\n\n- (2015b): \"Optimal Sales Contracts with Withdrawal Rights,\" Review of Economic Studies, 82, 762-790.\n\nLancaster, K. J. (1966): \"A New Approach to Consumer Theory,\" Journal of Political Economy, 74, 132-157.\n\nLehmann, D., L. I. Oćallaghan, and Y. Shoham (2002): \"Truth Revelation in Approximately Efficient Combinatorial Auctions,\" Journal of the ACM, 49, 577-602.\n\nLewis, T. R. and D. E. Sappington (1994): \"Supplying Information to Facilitate Price Discrimination,\" International Economic Review, 309-327.\n\nLi, H. and X. Shi (2017): \"Discriminatory Information Disclosure,\" American Economic Review, 107, 3363-85.\n\nLuenberger, D. G. (1969): Optimization by Vector Space Methods, John Wiley \\& Sons.\n\nMyerson, R. B. (1981): \"Optimal Auction Design,\" Mathematics of Operations Research, 6, 58-73.\n\n- (1982): \"Optimal Coordination Mechanisms in Generalized Principal-Agent Problems,\" Journal of Mathematical Economics, 10, 67-81.\n\nRayo, L. and I. Segal (2010): \"Optimal Information Disclosure,\" Journal of Political Economy, 118, 949-987.\n\nRiley, J. and R. Zeckhauser (1983): \"Optimal Selling Strategies: When to Haggle, When to Hold Firm,\" Quarterly Journal of Economics, 98, 267-290.\n\nRochet, J.-C. and P. Choné (1998): \"Ironing, Sweeping and Multidimensional Screening,\" Econometrica, 66, 783-826.\n\nRockafellar, R. T. (1970): Convex Analysis, Princeton University Press.\n\nSamuelson, W. (1984): \"Bargaining under Asymmetric Information,\" Econometrica, 995-1005.\n\nWei, D. and B. S. Green (2019): \"(Reverse) Price Discrimination with Information Design,\" Discussion paper.\n\n[^0]:    *Toulouse School of Economics, alexey.v.smolin@gmail.com. I thank Dirk Bergemann and Daniel Krähmer for helpful conversations as well as Heski Bar-Isaac and Levent Celik for excellent discussions. I am grateful to the participants in research seminars at numerous institutions. Finally, I thank the coeditor, Nicola Persico, and the anonymous referees for their many productive suggestions. I acknowledge funding from ANR under grant ANR-17-EURE-0010 (Investissements d'Avenir program).\n\n[^1]:    ${ }^{1}$ \"ZipRecruiter Is Valued at \\$1.5 Billion in a Bet on AI Hiring,\" (Carville (2018)).\n\n[^2]:    ${ }^{2}$ See, for example, Anderson and Renault (2006), Eső and Szentes (2007), and Li and Shi (2017).\n    ${ }^{3}$ This terminology follows Rayo and Segal (2010) and is not to be confused with the voluntary disclosure of Dye (1985), studied for example by Koessler and Renault (2012).\n\n[^3]:    ${ }^{4}$ Chakraborty and Harbaugh (2010) use linear disclosure rules to construct informative equilibria in a multidimensional cheap talk game.\n    ${ }^{5}$ This intuition leads in the right direction but does not consider discriminatory menus and information pricing. I formally complete the argument and confirm the result by building on single-dimensional mechanism-design machinery.","text_sha256":"f3b818b892618c8e48ee7ed01b9e835e68b516937c86d62f7b5fcb3625e00cb5"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0030","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"[^4]:    ${ }^{6}$ See, however, Anderson and Renault (2006), who show that optimal disclosure rule is partial if the purchase is associated with search costs and the seller cannot commit to prices. Similarly, Dworczak (2020) shows that optimal disclosure rule can be partial in the presence of aftermarkets. See also Bar-Isaac, Caruana, and Cuñat (2010).\n    ${ }^{7}$ Wei and Green (2019) show that withholding information may also be optimal if the information must be provided free of charge.\n\n[^5]:    ${ }^{8}$ Almost all of this literature studies one-dimensional settings. The works of Rayo and Segal (2010) and Dworczak and Martini (2019) provide elegant exceptions.\n    ${ }^{9}$ As I discuss in Sections 4.4 and 5.3, this formulation can be generalized.\n\n[^6]:    ${ }^{10} S$ can be any Polish space. Throughout the paper, all introduced functions are (Borel) measurable.\n    ${ }^{11}$ This setting is equivalent to one in which the seller provides information for free but can charge the buyer for opting out from the consecutive sale.\n    ${ }^{12}$ For instance, the buyer cannot claim a refund ex post. See Krähmer and Strausz (2015b), Heumann (2020) and Bergemann et al. (2020) for recent studies of ex post incentive constraints.\n\n[^7]:    ${ }^{13}$ Krähmer (2020) and Doval and Ely (2020) emphasize the usefulness of such schemes in screening problems and in general games of incomplete information, respectively.\n\n[^8]:    ${ }^{14}$ Rochet and Choné (1998), Bergemann et al. (2012) and Daskalakis et al. (2017) highlight the difficulties associated with the multidimensional screening problems.\n\n[^9]:    ${ }^{15}$ The attribute surplus should not be confused with the trade surplus which depends on the match between the attribute surplus and the buyer type.\n\n[^10]:    ${ }^{16} F$ has the monotone hazard rate property if $f(\\theta) /(1-F(\\theta))$ is monotonically increasing.\n\n[^11]:    ${ }^{17}$ The name is inspired by \"single-minded\" bidders in combinatorial auctions who value specific bundles. See, for example, Lehmann et al. (2002).\n    ${ }^{18}$ For example, see Section 8.4 of Hartline (2020).\n\n[^12]:    ${ }^{19}$ In any given setting, the incentive compatibility of this mechanism can be straightforwardly checked.\n\n[^13]:    ${ }^{20}$ Eső and Szentes (2017) generalize the latter finding to dynamic environments. Krähmer and Strausz (2015a) discuss settings in which the full-disclosure distributional assumptions are violated.\n\n[^14]:    ${ }^{21}$ This step might fail if there are infinitely many attributes, $|J|=\\infty$. If $\\mathcal{F}$ has infinite dimensions, then it might have some boundary points that cannot be supported by a hyperplane. In such cases, a sufficient condition for the existence of a supporting hyperplane is that $\\mathcal{F}$ has a nonempty interior.","text_sha256":"6cbc0c59b758cc320a793adfef2f72aea82dc9c870981cf2312341efb138827a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08:0031","work_id":"alex-smolin:disclosure-and-pricing-of-attributes","paper_id":"alex-smolin:disclosure-and-pricing-of-attributes:2022-08-08","title":"Disclosure and Pricing of Attributes","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2022-08-08","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md","source_record":"https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf","doi":"https://doi.org/10.1111/1756-2171.12451","citation":"Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Alex Smolin\n\n**Canonical citation:** Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/disclosure-and-pricing-of-attributes.md\n\n**Source record:** https://alexsmolin.com/files/disclosure-and-pricing-of-attributes-working-paper.pdf\n\n**Published record:** https://doi.org/10.1111/1756-2171.12451\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"e64d74bb062ed0d96948319bbd2dfe4a7a79fa6268f9652005b1f542e78334cd"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0001","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Alex Smolin.\n> Canonical citation: Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"1b9ec8c1e0df97192d00154bdadfe60da942570ec4043460a1c65a9afbf658a5"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0002","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Dynamic Evaluation Design","text":"# Dynamic Evaluation Design\n\n**Authors:** Alex Smolin\n\n**Manuscript date:** 2020-10-28\n\n#### Abstract\n\nA principal owns a firm, hires an agent of uncertain productivity, and designs a dynamic policy for evaluating his performance. The agent observes ongoing evaluations and decides when to quit. When not quitting, the agent is paid a wage that is linear in his perceived productivity; the principal claims the residual performance. After quitting, the players secure fixed outside options. I show that equilibrium evaluation policies are Pareto efficient. In a minimally informative equilibrium, for a broad class of performance technologies, the agent's wage deterministically grows with tenure. My analysis suggests that endogenous performance evaluation plays an important role in shaping careers in organizations.\n\nKeywords: evaluation, information design, career concerns, bandit experimentation, downward wage rigidity, up-or-out, internal labor markets\n\nJEL Codes: C72, D82, D83, M52\n\n[^0]","text_sha256":"de5ee4cc687c9078c7dcf012f00cbd89fc0370e895d8848ca8b694715502d34e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0003","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nPerformance evaluation is an important part of organizational life. Although much evaluation is informal, most organizations have formal evaluation policies designed to collect and distribute performance information to employees. ${ }^{1}$ As communication and information technologies advance, many companies find it easier to provide more evaluation. As a recent example, in August 2015 General Electric (GE) announced an ongoing shift from its legacy system of annual performance reviews to more frequent conversations between managers and employees via an online application. ${ }^{2}$ In this way, GE joined other high-profile companies such as Microsoft, Accenture, and Adobe in a move towards more frequent, exhaustive, and real-time evaluation. However, whenever adopting new evaluation policies, companies should ask: What is their effect on overall performance? Would other evaluation policies perform better? Ultimately, which evaluation policy is the best for the company?\n\nIn this paper, I develop a framework to analyze the design of evaluation policies. I consider a principal who owns a firm and hires an agent to work over time. The agent's productivity, his type, is initially uncertain to both players but affects the agent's ongoing performance via a general production technology. The agent's performance is not directly observed but can be revealed through evaluations. While at the firm, the agent's wage is linear in his expected productivity; the principal claims the residual performance. In every period, the agent evaluates his career prospects and decides whether to quit. When the agent quits, the players secure exogenous outside options. Both players are risk neutral and discount the future at the same rate.\n\nThe principal designs and adopts a dynamic evaluation policy. The policy is a sequence of statistical experiments that are informative of past performance. The experiments can vary in what and when performance is assessed. The evaluation is costless but its design should take into account the agent's incentives. On the one hand, the promise of future evaluations motivates the agent to stay at the firm and learn whether he is able to perform well. On the other hand, an evaluation may turn out negative and persuade the agent to quit.\n\nIn Section 4, I investigate equilibrium evaluation policies by developing a novel efficiency argument. First, I show that the design problem can be viewed as a dynamic persuasion problem in a bandit experimentation setting. It allows me to apply the revelation principle and characterize the set of feasible payoffs that can be possibly achieved in the relationship.\n\n[^1]Second, I study the set of implementable payoffs-the payoffs that can be achieved by some evaluation policy and the agent's best response to it. I show that this set includes all Pareto-efficient payoffs that deliver the agent at least his safe option, and I conclude that any equilibrium evaluation policy is efficient. Under a minimally informative equilibrium policy, the agent is informed about whether he would quit if he could fully observe past performance but had a lower outside option.\n\nIn Section 5, I study the effects of optimal evaluation on the agent's career and wage dynamics. I observe several qualitative properties that hold in the minimally informative equilibrium. First, the agent's wage is a deterministic function of tenure. Second, the agent's continuation value grows with tenure so that the agent becomes more optimistic about his prospects the longer he remains at the firm. Third, for a broad class of performance technologies, the agent's wage also grows with tenure. The increase reflects the ongoing positive selection and the corresponding growth of expected productivity. The shape of the wage profile depends on the performance technology. If the technology is coarse, such that performance comes as a stream of infrequent successes, then the wage increases at the revision dates, which are spaced sparsely over the agent's career. In contrast, if the technology is detailed, such that performance can always reveal the agent's incompetence, then performance is constantly monitored, and the wage gradually grows in time.\n\nIn Section 6, I study the joint design of wage contract parameters and an evaluation policy. I show that once the principal can control both the information flow and monetary incentives, she is able to extract the full surplus from the relationship: in equilibrium, the joint surplus is maximized, and the agent is left with no rents. The principal can achieve this by offering a fixed-wage contract and, in many cases, by offering a pure-bonus contract.\n\nI discuss the findings in Section 7. First, my analysis suggests that endogenous evaluation policies may be important in explaining economic dynamics commonly observed within firms. These include a lack of wage variation within same-tenure cohorts, downward wage rigidity, and up-or-out contracts. My results further speak in favor of retrospective evaluation and provide a rationale for rating compression and leniency bias as techniques to maintain workforce morale. Second, I highlight the important commitment role that human resource (HR) departments may play in implementing optimal evaluation policies. Third, I show that the equilibrium evaluation policy is robust to a multitude of performance leaks. Finally, I discuss how the framework can be extended to incorporate advisory feedback.\n\nRelated Literature My paper contributes to the literature on dynamic persuasion and information design built from the static models of Rayo and Segal (2010) and Kamenica and Gentzkow (2011). Orlov (2016) studies the joint design of performance evaluations and\nmonetary contracts when the agent exerts private effort. Renault, Solan, and Vieille (2017) and Ely (2017) study dynamic persuasion with an exogenous information flow and a myopic agent. Orlov, Skrzypacz, and Zryumov (2020) investigate a setting in which the principal lacks commitment across different periods. Importantly, my paper introduces the efficiency argument that permits the characterization of the optimal information policies by studying Pareto-efficient allocations. ${ }^{3}$","text_sha256":"e6fbe88e1c0917e734a43f4faa06c7117a1ecba6c06d3eda75ac0bc270c768c8"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0004","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"My framework highlights the interplay between career concerns and performance evaluations. It complements the turnover theory of Jovanovic (1979) by endogenizing the information flow. Relatedly, several papers investigate evaluation effects on private efforts in the framework of Holmström (1999). Hansen (2013) studies static incentives and focuses on partition evaluations. Hörner and Lambert (2020) study dynamic incentives with a focus on Gaussian policies.\n\nSimilar incentive effects are present in multistage contests and tournaments. Ederer (2010) compares the effectiveness of complete- and no-evaluation policies in two-stage tournaments. Halac, Kartik, and Liu (2016) study the optimal design of general multistage contests and similarly focus on the extreme evaluation policies within each period. Nevertheless, Goltsman and Mukherjee (2011) highlight that the optimal evaluation policies in tournaments are generally partially informative.\n\nFinally, my paper contributes to the literature on dynamic contracts without transfers. Guo (2016) studies dynamic delegation when the agent is privately informed. Hörner and Guo (2015) study dynamic resource allocation when the agent's private information evolves over time. My paper highlights that information control may complement delegation and action control as a powerful management tool.","text_sha256":"97f2852e015343f65d889aea4ae2f143dc68be2abb1a1f828346a302145dfc1a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0005","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"## 2 Model\n\nA principal owns a firm and hires an agent. The relationship takes place in consecutive periods $t=0,1,2, \\ldots$ At time 0, the agent's productivity $\\theta \\in \\Theta \\subseteq \\mathbb{R}$ is drawn according to a cumulative distribution $G_{0}$. The productivity is fixed throughout the relationship and is not directly observed by either principal or agent. The players are symmetrically informed about productivity with the prior expectation of productivity, $\\mathbb{E}[\\theta]$, being equal to $\\theta_{0}$.\n\nPerformance Productivity affects the agent's performance at firm $y_{t} \\in Y \\subseteq \\mathbb{R}$. Conditional on productivity, performance is independently and identically distributed across\n\n[^2]periods according to a cumulative distribution $F_{\\theta}$. The collection of distributions $\\left\\{F_{\\theta}\\right\\}_{\\theta \\in \\Theta}$ defines production capabilities of the firm and is called the (performance) technology. I associate productivity with its expected performance: $\\mathbb{E}\\left[y_{t} \\mid \\theta\\right]=\\theta$.\n\nIt follows that performance is informative of the agent's productivity: consistently higher performance suggests higher productivity. The overall informativeness and details of the learning process are determined by technology in place. I impose no assumptions on technology in the characterization of equilibrium payoffs in Section 4. I will impose a regularity assumption in Section 5 to establish downward wage rigidity. Performance is not directly observed by either party but can be revealed through evaluations as discussed below.\n\nStrategies The principal can publicly reveal past performance through an evaluation policy that she designs. The policy is costless and governs when and what performance information is available. The evaluations are objective; their outcomes cannot be manipulated by the principal. At the same time, I place no restrictions on which evaluations the principal can conduct. That is, she can conduct a complete evaluation, no evaluation, periodic reviews, grade evaluations, and so forth.\n\nFormally, the principal chooses an evaluation policy $m$ among all stochastic processes measurable with respect to past performance and evaluations. ${ }^{4}$ The policy can be represented by a sequence of random messages $\\left\\{m_{t}\\right\\}_{t=0}^{\\infty}$ that are sent to the agent: ${ }^{5}$\n\n$$\nm_{t}: Y^{t-1} \\times M^{t-1} \\rightarrow \\Delta(M) .\n$$\n\nThe message space $M$ is the same in all periods and can be freely chosen by the principal. The exact message labels are irrelevant because their meaning is determined solely by the law of $m$. Associate a complete-evaluation policy $\\bar{m}$ with $m_{t} \\equiv y_{t-1}$ and a no-evaluation policy $\\underline{m}$ with $m_{t} \\equiv \\emptyset$. Denote the set of all possible evaluation policies of the form (1) by $\\mathcal{M}$.\n\nThe concept of an evaluation policy is an extension of Kamenica and Gentzkow (2011)'s static persuasion policy and admits two possible interpretations. First, it can be viewed as a disclosure policy. In this interpretation, the principal constantly monitors performance but is bound to communicate according to the chosen policy. Second, it can be viewed as a sequence of public experiments. In this interpretation, the principal does not directly observe performance but commits to a sequential policy of public tests to inform both players of past\n\n[^3]performance.\nThe evaluations may be understood as being conducted by the HR department of a firm. In this case, a realization $m_{t}$ corresponds to an outcome of a particular evaluation. The evaluation policy $m$ corresponds to the operating rules of the department and specifies which and how past performance is evaluated in any given period. I discuss the implementation details in Section 7.2.\n\nFaced with the evaluation policy, the agent chooses whether and when to quit the firm. He decides based on past evaluations, which he correctly interprets according to Bayes' rule. Quitting is irreversible and ends the game.\n\nFormally, the agent chooses a quitting time $\\tau$, which is a stopping time measurable with respect to the evaluation policy $m$ :\n\n$$\n\\tau \\text { is a stopping time w.r.t. } m_{0}, m_{1}, \\ldots\n$$\n\nThe quitting time is a random variable. If $\\tau \\equiv 0$, then the agent quits at time 0 and does not generate any performance. If $\\tau \\equiv \\infty$, then the agent stays at the firm forever, irrespective of past evaluations. Denote the set of all possible quitting times by $\\mathcal{T}$.\n\nPayoffs As long as the agent stays at the firm, the principal appropriates the performance outcomes and pays the agent a wage $w_{t}$. I assume that the wage is set according to a linear contract:\n\n$$\nw_{t}\\left(m^{t}\\right)=w^{F}+\\alpha \\mathbb{E}\\left[y_{t} \\mid m^{t}\\right],\n$$\n\nwith $w^{F}>0$ being a fixed base wage and $\\alpha \\in(0,1)$ being a bonus rate. Linear contracts are widely used in practice and capture in the simplest form the reputation effects of performance evaluations (see Carroll (2015) and the references therein). The agent wants to receive positive evaluations to be perceived as more productive because, in this case, he will be paid a higher bonus. For now, I will treat $w^{F}$ and $\\alpha$ as exogenously fixed. I study endogenous contracts in Section 6.\n\nThe timing within each period $t$ is as follows. First, the worker receives an evaluation $m_{t}$. Then, he decides whether to stay at the firm. If he stays, then the output $y_{t}$ is produced and the agent is paid wage $w_{t}$ according to (3).\n\nAs soon as the agent quits the firm, he secures a total payoff of $V^{A} \\in \\mathbb{R}$, and the principal secures $V^{P} \\in \\mathbb{R}$. These payoffs are exogenous, commonly known, and fixed throughout the relationship. They can be viewed as the opportunity costs of the players.\n\nBoth players are risk neutral and discount the future with a common discount factor $\\delta$. For given evaluation policy $m \\in \\mathcal{M}$ and quitting strategy $\\tau \\in \\mathcal{T}$, the normalized expected\npayoffs of the players are:\n\n$$\n\\begin{aligned}\n& U^{P}(m, \\tau)=\\mathbb{E}_{m, \\tau}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t}\\left(y_{t}-w_{t}\\right)+\\delta^{\\tau} V^{P}\\right] \\\\\n& U^{A}(m, \\tau)=\\mathbb{E}_{m, \\tau}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t} w_{t}+\\delta^{\\tau} V^{A}\\right]\n\\end{aligned}\n$$","text_sha256":"4aa263019f4805ae5de5181ca5e4ee092468790eb3cad9ba2cb0d68494d1eae4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0006","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"Note that evaluation policy plays two roles in the payoffs. First, it shapes the agent's wage. Second, it provides the agent with information that guides his quitting decision.\n\nEquilibrium I study perfect Bayesian equilibria of this game. For a given evaluation policy, the agent chooses a quitting time to maximize his total expected payoff. The principal anticipates the agent's best response and designs the evaluation policy to maximize her expected payoffs.\n\nDefinition 1. An evaluation policy $m^{*}$ and a quitting time $\\tau^{*}$ constitute an equilibrium if they solve the problem:\n\n$$\n\\begin{array}{ll}\n& \\max _{m \\in \\mathcal{M}, \\tau \\in \\mathcal{T}} U^{P}(m, \\tau) \\\\\n\\text { s.t. } & \\tau \\in \\arg \\max _{\\tau \\in \\mathcal{T}} U^{A}(m, \\tau) .\n\\end{array}\n$$\n\nMy goal is to characterize an equilibrium evaluation policy and payoffs in this game. This characterization further allows me to study equilibrium wage dynamics. To this end, for given strategies $m$ and $\\tau$, let observed wage $W_{t}$ equal $w_{t}$ if the agent remains at the firm, $\\tau>t$, and, to complete the definition, equal to 0 otherwise. The observed wage at time $t$ is a random variable that may take many values because the agent can possibly stay at the firm under a wide range of past evaluations. Define a wage profile as a collection of observed wages at different times $W=\\left\\{W_{t}\\right\\}_{t=0}^{\\infty}$. The wage profile captures the dynamics of the agent's wage throughout his career at the firm.","text_sha256":"818df96e8193ea4ee46bab3e6b76a98394bc111f4601c110e7706792141cdec6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0007","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Binary Example","text":"## 3 Binary Example\n\nIn this section, I illustrate the workings of the setting by means of a simple example. The agent's performance is binary, low or high, $Y=\\left\\{y^{L}, y^{H}\\right\\}$. Let $y^{L}=0$ and $y^{H}=1$, and call $y^{H}$ a \"success.\" There are two possible types, $\\Theta=\\left\\{\\theta^{L}, \\theta^{H}\\right\\}$, and each type is equally likely. By the definition of productivity, a type is equal to expected productivity, which in this example coincides with the probability of success, $\\operatorname{Pr}\\left(y_{t}=y^{H} \\mid \\theta\\right)=\\theta$. Assume that\nonly the high type is productive, $\\theta^{L}=0$ and $\\theta^{H}=1 / 2$. Let the payoff structure be $w^{F}=0$, $\\alpha=1 / 2, V^{A}=1 / 5$, and $V^{P}=0$. Finally, let the discount factor be $\\delta \\simeq 0.97 .{ }^{6}$\n\nNo Evaluation First, consider the case in which the principal adopts a no-evaluation policy $\\underline{m}$. In this case, the same evaluation message is sent irrespective of past performance and thus is completely uninformative. The firm effectively provides no feedback. As a result, the wage is fixed at:\n\n$$\nw_{t} \\equiv w_{\\emptyset}=w^{F}+\\alpha \\theta_{0}=\\frac{1}{8} .\n$$\n\nWhenever staying at the firm, the agent receives a flow payoff of $w_{\\emptyset}=1 / 8$ and foregoes the opportunity flow of $V^{A}=1 / 5$. Because $w_{\\emptyset}<V^{A}$, the agent quits at time 0 .\n\nConsequently, in the absence of informative evaluations, the wage profile is nil and the firm is effectively not operating.\n\nComplete Evaluation Now, consider the case in which the principal adopts a completeevaluation policy $\\bar{m}$. In this case, each evaluation fully reveals the agent's performance in the last period, and the agent's wage depends on past evaluations. The wage starts at $w_{\\emptyset}=1 / 8$. As long as no successes occur, the wage gradually decreases according to Bayes' rule, $w_{t}^{L}=\\frac{1}{2\\left(1+2^{t}\\right)}$. If a success occurs, it indicates the the agent is of the high type; the wage jumps to $w^{H} \\equiv 1 / 4$, and remains there forever. The transition to one of these two wages ensures that the wage is a martingale, the property guaranteed by Bayes' rule.\n\nFaced with these career prospects, the agent optimally quits whenever his wage drops below a cutoff wage $\\hat{w}$. The cutoff depends on the discount factor. A higher $\\delta$ translates into lower $\\hat{w}$, because career concerns are more important. For the considered discount factor, the agent works until $\\hat{T}=2$ and continues working if and only if success occurred in the past.\n\nThe resulting wage profile is random. Viewing the agent as a representative employee, one out of many independent draws would result in two distinct features. First, there would be cross-sectional variation in employee wages in periods before $\\hat{T}$ : some employees are proven to be high types, and some still attempt to achieve a success. Second, the wage of a given employee is likely to decrease during his career in the firm (Figure 1).\n\nEquilibrium Evaluation Now, consider a partial evaluation policy that reveals whether the principal's belief falls below a certain cutoff. Given the success technology, this policy is equivalent to a revision policy, $m_{T}=y^{T-1}$ and $m_{t}=\\emptyset$ for $t \\neq T$. The firm provides no evaluations before or after the revision date $T$ at which all past performance is evaluated.\n\n[^4]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: Wage profile under complete evaluation policy (left) and equilibrium evaluation policy (right).\n\nUnder this revision policy, the agent's wage prior to revision remains at $w_{\\emptyset}=1 / 8$. At $T$, the full evaluation of past performance is conducted. If a success is revealed, the agent is proven to be of a high type, and his wage jumps up to $w_{T}^{H}=1 / 4$. If no success is observed, then the productivity expectation drops, as does the wage, to $w_{T}^{L}=\\frac{1}{2\\left(1+2^{T}\\right)}$. After the revision time, no evaluations are conducted, so the wage remains constant.\n\nGiven these career prospects, as $w_{T}^{H}>V^{A}>w_{T}^{L}$, the agent's quitting problem reduces to a binary choice: to either quit at time 0 or remain until the revision time and quit only if no successes are revealed. If the revision date equals the complete evaluation time $T=\\hat{T}$, then the agent prefers to remain until the revision because this strategy delivers him the same payoff as the best response to a complete evaluation policy. However, the principal can induce the agent to generate more surplus by postponing the revision until time $T>\\hat{T}$. There is a limit on how late the revision can be performed because the agent may prefer to quit at time 0: the maximal revision time can be calculated to be $T^{*}=10$.\n\nIn fact, the revision policy with revision time $T^{*}$ is optimal for the principal. Indeed, it delivers payoffs $U^{A *}=V^{A}=0.2$ and $U^{P *} \\simeq 0.12$. These payoffs are Pareto efficient because the agent never quits when successful. Because the agent can guarantee his safe option by quitting at time 0, the principal cannot achieve payoffs above $U^{P *}$, and the result follows. The equilibrium wage profile is illustrated in Figure 1.\n\nIn what follows, I study the general setting and demonstrate that the main features of this example are general. First, the evaluation policy affects payoffs only through its effect on the quitting time and not on the wage. Second, equilibrium payoffs are Pareto efficient. Third, if productivity is binary or technology is regular, then an equilibrium wage profile is deterministically increasing in tenure. However, under general technology, an optimal evaluation policy needs to be stated in terms of the principal's beliefs and cannot be implemented\nvia a simple revision policy.","text_sha256":"6e828189986f051aa5cf116eb42a5fd79d41ce5f0c8f32e2401e4289eb57caca"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0008","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Equilibrium Analysis","text":"## 4 Equilibrium Analysis\n\nEquilibrium characterization requires solving a dynamic information disclosure problem. Such problems are known to be difficult due to their inherent multidimensionality. The characterization is further complicated because the information that can be disclosed is generated gradually, and the agent is forward looking. Because of these features, I cannot use the existing techniques of Kremer, Mansour, and Perry (2014) and Ely (2017). Instead, I develop and use an efficiency approach. First, I characterize the set of Pareto-efficient payoffs (Sections 4.1 and 4.2). Second, I provide an upper bound on the principal's equilibrium payoffs. Finally, I demonstrate that the upper bound can be achieved with a particular information policy (Section 4.3).","text_sha256":"2076533be20fe053289bafe5c6c252f4c5958791a104d63056082fa838105c9d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0009","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.1 Payoff Transformation","text":"### 4.1 Payoff Transformation\n\nTo characterize the set of feasible and Pareto-efficient payoffs, it is useful to observe that because both players are risk neutral, the mean-preserving spread of a wage should not affect their payoffs in any period. For example, receiving a fixed wage proportional to the expected performance at the beginning of a period should be payoff equivalent to receiving a bonus proportional to performance at the end of a period.\n\nThis intuition can be formalized. Applying the law of iterated expectations and the optional stopping theorem, the player's payoffs can be written solely in terms of performance histories and payoff parameters.\n\nLemma 1. (Payoff Transformation) The players' payoffs can be written as:\n\n$$\n\\begin{aligned}\n& U^{P}(m, \\tau)=\\mathbb{E}_{m, \\tau}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t} y_{t}+\\delta^{\\tau} \\hat{V}^{P}\\right] \\times(1-\\alpha)-w^{F}, \\\\\n& U^{A}(m, \\tau)=\\mathbb{E}_{m, \\tau}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t} y_{t}+\\delta^{\\tau} \\hat{V}^{A}\\right] \\times \\alpha+w^{F},\n\\end{aligned}\n$$\n\nwith $\\hat{V}^{P}=\\left(V^{P}+w^{F}\\right) /(1-\\alpha)$ and $\\hat{V}^{A}=\\left(V^{A}-w^{F}\\right) / \\alpha$.\nLemma 1 implies that the current setting is strategically equivalent in the sense of Thompson (1952) to the setting of persuasion in bandit experimentation. ${ }^{7}$ In this alternative setting, the agent sequentially pulls the arm of a slot machine and decides when to stop. Pulling the\n\n[^5]arm generates stochastic rewards for both players. The rewards depend on the machine's type and are not observed by the agent. Stopping delivers the players their safe options. The principal designs what reward information the agent observes to maximize her own payoffs.\n\nThe payoff representation (8) and (9) shows that the conflict of interest between the players is captured by their safe options $\\hat{V}^{P}$ and $\\hat{V}^{A}$. If these options are equal, then there is no conflict of interest. In this case, the optimal evaluation policy is to provide complete evaluation because it allows the agent to make maximally informed decisions. In contrast, if these options differ, the principal may find it optimal to coarsen the evaluations to steer agent decisions towards her interests.\n\nAssumption 1. (Conflict of Interest) $\\hat{V}^{P}<\\hat{V}^{A}$.\nAssumption 1 implies that there is a conflict of interest. As the principal's safe option is lower, she prefers the agent to remain longer at the firm than the agent would ideally prefer. This conflict of interest is typical in the literature on Bayesian persuasion. For example, it holds whenever the agent needs some evaluation to remain at the firm, $V^{A}>w^{F}$, and either (i) the firm appropriates a substantial share of output such that $\\alpha$ is sufficiently small (cf. Ely and Szydlowski (2020)), or (ii) the principal's total opportunity costs, $V^{P}+w^{F}$, are sufficiently low.","text_sha256":"336404e1da66069cd6410267f7debce25a01d4ac3a12c1e83521e01859b8600b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0010","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.2 Feasible Payoffs","text":"### 4.2 Feasible Payoffs\n\nI proceed with characterizing the set of feasible payoffs. Recall from standard game-theoretic terminology that a pair of strategies $m \\in \\mathcal{M}, \\tau \\in \\mathcal{T}$ delivers payoffs $\\left(u^{A}, u^{P}\\right)$ if given the strategies, the payoff of the agent equals $u^{A}$ and the payoff of the principal equals $u^{P}$. In turn, payoffs $\\left(u^{A}, u^{P}\\right)$ are feasible if they can be delivered by some players' strategies. The payoffs are (weakly Pareto) efficient if there are no strategies that deliver strictly greater payoffs to both players. Denote the set of all feasible payoffs by $\\mathcal{F}$ :\n\n$$\n\\mathcal{F} \\triangleq\\left\\{\\left(U^{A}(m, \\tau), U^{P}(m, \\tau)\\right) \\mid m_{t}: Y^{t-1} \\times M^{t-1} \\rightarrow \\Delta(M), \\tau \\text { is a stopping time w.r.t. } m\\right\\} .\n$$\n\nIt is possible to characterize the efficient strategies by a solution to an auxiliary problem. Define the payoffs of a fictitious agent with a virtual safe option $\\hat{V}^{F}$ as:\n\n$$\nU^{F}\\left(m, \\tau, \\hat{V}^{F}\\right)=\\mathbb{E}_{m, \\tau}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t} y_{t}+\\delta^{\\tau} \\hat{V}^{F}\\right]\n$$\n\nLemma 2. (Efficient Payoffs) The set of feasible payoffs $\\mathcal{F}$ is compact and convex. Efficient payoffs are spanned by $\\left(\\bar{m}, \\tau^{\\prime}\\right)$, where $\\tau^{\\prime} \\in \\arg \\max _{\\tau} U^{F}\\left(\\bar{m}, \\tau, \\hat{V}^{F}\\right)$ and $\\hat{V}^{F} \\in\\left[\\hat{V}^{P}, \\hat{V}^{A}\\right]$.\n\nThe intuition behind the lemma is as follows. First, any feasible payoff can be obtained with a complete-evaluation policy $\\bar{m}$ because this policy provides the most opportunities to respond to performance information. Second, $\\mathcal{F}$ is compact as a continuous image of a compact set of quitting strategies; $\\mathcal{F}$ is convex because randomizing over two quitting strategies can obtain any convex combination of their corresponding payoffs. As such, by the separating hyperplane theorem, any boundary point of $\\mathcal{F}$ is achieved by a quitting strategy that maximizes a linear combination of players' payoffs. For efficient payoffs, the combination weights are positive, and the problem is equivalent to a stopping problem of a fictitious agent with a virtual safe option $\\hat{V}^{F} \\in\\left[\\hat{V}^{P}, \\hat{V}^{A}\\right]$. As the weight is shifted from the agent to the principal, $\\hat{V}^{F}$ gradually moves from $\\hat{V}^{A}$ to $\\hat{V}^{P}$.\n\nIn fact, these arguments allow us to characterize the whole feasibility set $\\mathcal{F}$, not only its efficient payoffs. To do so, it suffices to solve a collection of optimal stopping problems that maximize various linear combinations of players' payoffs. If these stopping problems can be solved, analytically or numerically, then $\\mathcal{F}$ can be reconstructed.\n\nIn particular, this procedure allows us to depict the feasibility set of the binary example in Section 2 (Figure 2). Points $A$ and $D$, as well as any payoffs on the segment $A D$, can be delivered by a quitting time that does not depend on evaluations. Point $A$ is delivered by the agent never quitting, $\\tau \\equiv \\infty$. Point $D$ is delivered by the agent quitting at time zero, $\\tau \\equiv 0$. The payoffs on segment $A D$ can be delivered by a randomization between these two quitting times.\n\nIn contrast, to obtain Pareto-efficient payoffs on the arc $A C$, the quitting strategy must use performance information. By Lemma 2, those payoffs are spanned by solutions to the virtual problems with outside options $\\hat{V}^{F} \\in\\left[\\hat{V}^{P}, \\hat{V}^{A}\\right]$. Point $A$ corresponds to a virtual safe option $\\hat{V}^{F}=\\hat{V}^{P}$, point $C$ corresponds to a virtual safe option $\\hat{V}^{F}=\\hat{V}^{A}$. A somewhat peculiar point $E$ is delivered by a strategy that minimizes the agent's payoff: it prescribes remaining at the firm at low expected productivity and quitting the firm at high expected productivity. Point $B$ maximizes the principal's payoff among all payoffs that deliver at least a safe payoff $V^{A}$ to the agent. It plays an important role in the equilibrium characterization in the next section.","text_sha256":"b549f45a3a6406f992c1d910e7beeec7fc357148914c66c39d2ba763def19a31"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0011","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.3 Equilibrium Payoffs","text":"### 4.3 Equilibrium Payoffs\n\nThe notion of feasibility ignores players' incentives. As shown in the previous section, all feasible payoffs can be delivered by a complete evaluation policy because the quitting time\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 2: Feasible payoffs in the example from Section 3. $\\Theta=\\{0,1 / 2\\}, \\theta_{0}=0.25, Y=\\{0,1\\}$, $\\operatorname{Pr}(y=1 \\mid \\theta)=\\theta, w^{F}=0, \\alpha=0.5, V^{A}=0.2, V^{P}=0, \\delta \\simeq 0.97$. Computed numerically.\n\ncan ignore any additional information. However, under complete evaluation, the agent would act in his own interests and choose a quitting time to maximize his payoff. In Figure 2, it would correspond to point $C$.\n\nTo incorporate the players' incentives, I refer to mechanism-design terminology and say that an evaluation policy $m \\in \\mathcal{M}$ implements payoffs $\\left(u^{A}, u^{P}\\right)$ if the payoffs are delivered by that policy and some agent's best response to it. In turn, payoffs $\\left(u^{A}, u^{P}\\right)$ are implementable if they can be implemented by some evaluation policy.\n\nIn these terms, a complete-evaluation policy implements the payoffs of point $C$. However, it is not the only evaluation policy that implements them. Consider a policy that sends only two messages: \"stay\" and \"quit.\" Let the policy mimic the agent's best response under a complete-evaluation policy; that is, to send a \"quit\" message only after those performance histories at which the agent himself would quit. If the agent follows the recommendations, then the joint distribution of quitting time and performance will be the same. By Lemma 1, his payoff then equals the payoffs of point $C$. Because it is his maximal feasible payoff, he cannot do better than follow the recommendations, and so this recommendation policy would implement payoffs $C$.\n\nDefinition 2. An evaluation policy is a recommendation policy if it places a positive probability on at most two messages: \"stay\" and \"quit.\" A recommendation policy is incentive compatible if following the recommendations is an agent's best response.\n\nIn fact, the agent cannot do better than follow the recommendations of an arbitrary recommendation policy that mimics his best response to some evaluation policy. This fol-\nlows from the standard argument of direct mechanisms of Myerson (1986). Consider an arbitrary evaluation policy $m$ and a recommendation policy $m^{\\prime}$ that mimics an agent's best response to $m .^{8}$ Because the agent always knows his actions, policy $m^{\\prime}$ provides weakly less information than $m$. Hence, his payoff cannot be higher than that under $m$. Following the recommendations delivers the agent the same payoff as under $m$ and hence is a best response to $m^{\\prime}$.\n\nIn other words, the recommendation policies provide minimal information for the agent to make his quitting decision. The principal does not need to provide any information beyond that. Note that Lemma 1 is crucial for this observation because it establishes that the net payoff effect on evaluation policy comes only through its effect on a quitting time and not on a wage.\n\nLemma 3. (Recommendation Principle) All implementable payoffs can be implemented by incentive-compatible recommendation policies.\n\nI proceed with a characterization of implementable efficient payoffs. The agent can secure the payoff $V^{A}$ by quitting at time 0. Hence, payoffs that yield the agent less than $V^{A}$ cannot be implemented. In Figure 2, this means that no efficient payoffs to the west of line $B D$ can be implemented. At the same time, point $C$ is implementable by a complete-evaluation policy. In fact, all efficient payoffs on segment $B C$ are also implementable.\n\nLemma 4. (Implementable Payoffs) No payoffs with $u^{A}<V^{A}$ are implementable. All efficient payoffs with $u^{A} \\geq V^{A}$ are implementable.\n\nThis lemma builds on the efficient payoff characterization of Lemma 2 and the recommendation principle of Lemma 3. By the payoff characterization, the efficient payoffs can be delivered by strategies $(m, \\tau)$ that maximize the payoff of an agent with a virtual safe option $\\hat{V}^{F} \\in\\left[\\hat{V}^{P}, \\hat{V}^{A}\\right]$. The agent, if faced with a recommendation policy that mimics $\\tau$, is willing to follow recommendations. The argument proceeds as follows. Recommendations to quit are incentive compatible because even the agent with a lower safe option $\\hat{V}^{F} \\leq \\hat{V}^{A}$ is willing to follow them-it delivers to him the first-best payoff. Recommendations to stay are incentive compatible because the agent's continuation payoff weakly increases with tenure. Indeed, the agent's continuation payoff after each recommendation to stay can be viewed as a convex combination of three terms: the current wage $w_{t}$, the next-period continuation payoff if recommended to quit $\\hat{V}^{A}$, and the next-period continuation payoff if recommended to stay. The agent's payoff must be greater than the first two terms because the agent can\n\n[^6]guarantee those payoffs by remaining at the firm forever and quitting immediately, respectively. Hence, the payoff is smaller than the third term, which is equivalent to the growth of continuation value with tenure. In other words, while staying at the firm, the agent becomes increasingly optimistic about his prospects.\n\nIn equilibrium, the principal chooses an evaluation policy to maximize her payoffs. Equivalently, the principal maximizes her payoff among all implementable payoffs. It is clear from Lemma 4 that she optimally chooses an efficient payoff that either delivers $V^{A}$ to the agent (point $B$ in Figure 2) or, if implementable, delivers the first-best payoff to the principal.\n\nTheorem 1. (Equilibrium Payoffs) Equilibrium payoffs exist and are Pareto efficient. Either the agent is left with no rents, $u^{A *}=V^{A}$, or the principal obtains her first-best payoffs, $u^{P *}=\\max _{\\left(u^{A}, u^{P}\\right) \\in \\mathcal{F}} u^{P}$.\n\nProof. By Lemma 4, the principal's problem reduces to the maximization of $u^{P}$ among all $\\left(u^{A}, u^{P}\\right) \\in \\mathcal{F}$ such that $u^{A} \\geq V$. Because $\\mathcal{F}$ is compact, the problem has a solution with the stated properties. $\\square$","text_sha256":"07129c22d2e92a6e1ea3d7858a9dcbf9cb6d8ffbb3e34fe5e48279cfec0f7ea9"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0012","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.3 Equilibrium Payoffs","text":"Theorem 1 establishes that despite the conflict of interest within the firm the equilibrium outcome is Pareto efficient. However, control of performance information is a powerful tool that allows the principal to either obtain her full control payoffs or extract all rents from the agent.","text_sha256":"52e1566350ee4ed6ec909099b37e612f585fcbac43f46480ef702fa3f3166eb3"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0013","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Wage Profile","text":"## 5 Wage Profile\n\nTheorem 1 implies that equilibrium payoffs are generically unique. However, several equilibrium evaluation policies could possibly deliver these payoffs but result in different wage profiles. In what follows, I concentrate on a particular equilibrium in which the principal uses an incentive-compatible recommendation policy. By Lemma 3, this equilibrium always exists and there are at least two reasons to concentrate on it. First, the recommendation policies are minimally informative; any other policy can be Blackwell garbled into a recommendation policy without affecting the players' payoffs. Providing minimal information may be desirable to avoid its misuse by the agent in ways not conceivable by the principal. Second, the recommendation policies minimize wage volatility, which may be desirable for a firm.\n\nDefinition 3. An equilibrium $\\left(m^{*}, \\tau^{*}\\right)$ is minimally informative if $m^{*}$ is a recommendation policy and $\\tau^{*}$ follows its recommendations.\n\nIn what follows, by equilibrium, I mean a minimally informative equilibrium. Hence, an equilibrium wage profile is deterministic. Indeed, in any period, there is a unique history of past evaluations that results in the agent remaining at the firm, namely, a sequence of recommendations to stay. Consequently, the equilibrium agent's career takes a particularly simple form. There is a deterministic wage profile $W$ and commonly known performance requirements to stay at the firm. If the agent's performance satisfies these requirements, he remains at the firm; otherwise, he quits and secures a safe option.\n\nThe equilibrium wage profile depends on the equilibrium recommendation policy, which, in turn, depends on technology details. However, by Theorem 1, the equilibrium policy is efficient. It allows me to establish wage profile properties that hold for a broad class of technologies.","text_sha256":"791a1939ee543d31c2ecf74b9e3ed5dac29f5af328279820160132f9d1d66e9b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0014","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.1 Downward Wage Rigidity","text":"### 5.1 Downward Wage Rigidity\n\nNaïve intuition suggests that average productivity and, hence, the wage should increase with tenure because the equilibrium is efficient. Indeed, efficiency is commonly associated with ongoing positive selection that eliminates poor performers and retains good performers. Such positive selection can be implemented through a sequence of history-dependent cutoffs such that the agent is recommended to quit whenever his productivity expectation drops below the corresponding cutoff. Under such a policy, the expected productivity would increase after every history, and hence, average productivity would increase with tenure.\n\nDefinition 4. A recommendation policy is a cutoff policy (in expectations) if there exists a cutoff function $q_{t}: Y^{t-1} \\rightarrow \\mathbb{R}$ such that\n\n$$\nm_{t}= \\begin{cases}\\text { \"stay,\" } & \\text { if } \\mathbb{E}\\left[\\theta \\mid y^{t-1}\\right]>q_{t-1}\\left(y^{t-2}\\right), \\\\ \\text { \"quit,\" } & \\text { if } \\mathbb{E}\\left[\\theta \\mid y^{t-1}\\right]<q_{t-1}\\left(y^{t-2}\\right) .\\end{cases}\n$$\n\nThe naïve intuition overlooks the fact that, in general, efficient selection should account for the whole profile of productivity beliefs, not just productivity expectation. Roughly, a greater productivity variance increases the chances of being highly productive and, hence, increases the option value of quitting and experimentation. An agent with a lower expected productivity but higher chances of being very productive could be worth retaining, whereas an agent with higher but certain productivity could be worth terminating. As a result, an efficient policy may not be cutoff, and the resulting equilibrium wage may decrease.\n\nNevertheless, I show that efficient policies are cutoff in many cases. By Lemmas 2 and 3, any efficient policy maximizes a payoff of an agent with some virtual safe option $\\hat{V}^{F} \\in$\n$\\left[\\hat{V}^{P}, \\hat{V}^{A}\\right]$. Such policy is Markov in the productivity beliefs, so the space of beliefs can be split into two sets: the set at which the agent stays and the set at which the agent quits. The exact characterization depends on the parameters of the problem and can be intractable. However, it can be obtained in the following cases.\n\nIf there are only two productivity types, $|\\Theta|=2$, then there is a threshold expectation $q^{c}$ such that under any efficient policy, the agent stays if the productivity expectation is above the threshold and quits otherwise. ${ }^{9}$ That is, an optimal policy is cutoff with the cutoff function being constant.\n\nIf there are more than two types, $|\\Theta|>2$, then belief is a multidimensional object, and the efficient policy can be characterized only under additional assumptions.\n\nDefinition 5. A technology is regular if it admits densities, $|\\Theta|<\\infty$, and $\\forall \\theta, \\theta^{\\prime} \\in \\Theta$, $y, y^{\\prime} \\in \\operatorname{supp} f_{\\theta} \\cap \\operatorname{supp} f_{\\theta^{\\prime}}, \\theta^{\\prime}>\\theta, y^{\\prime}>y$\n\n$$\nf_{\\theta^{\\prime}}\\left(y^{\\prime}\\right) f_{\\theta}(y) \\geq f_{\\theta^{\\prime}}(y) f_{\\theta}\\left(y^{\\prime}\\right)\n$$\n\nUnder regular technology, the conditional performance distributions satisfy the monotone likelihood property. Most technologies used in the literature satisfy this condition. The regularity assumption adds the structure necessary to analyze the multiple-type case. Banks and Sundaram (1992) use the regularity condition to establish that an optimal strategy is cutoff.\n\nTheorem 2. (Downward Wage Rigidity) If there are only two types $|\\Theta|=2$ or the technology is regular, then in any minimally informative equilibrium:\n\n1. An evaluation policy $m$ is a cutoff policy;\n2. The wage profile $W$ is deterministic and weakly increasing.\n\nThe theorem establishes sufficient conditions under which the equilibrium wage exhibits downward rigidity. The conditions are plausible in that they are satisfied in most existing models of career concerns and experimentation. Interestingly, the proof presents an even stronger statement: not only does expected productivity increase, but the whole profile of productivity beliefs also shifts upwards in an MLRP sense.","text_sha256":"957893ad26f4bc9d210a82007d237566fc92e011cae770d3176613460f8b098b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0015","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5.2 Wage Profile Shape","text":"### 5.2 Wage Profile Shape\n\nTheorem 2 establishes that under general conditions, the equilibrium wage profile is deterministic and weakly increasing. Nevertheless, the exact shape of the wage profile depends on\n\n[^7]technology details. Calculating the wage profile in closed form is intractable outside of very specific examples. To illustrate the role of technology in determining the equilibrium, in this section, I calculate wage profiles numerically and contrast them under different performance technologies.\n\nCoarse Performance In many industries, performance measures are coarse. A lawyer's performance is captured by the number of successful trials, a consultant's performance is measured by the outcomes of his past projects, and a drug laboratory's performance is assessed by the number of new drugs developed. In all these cases, a performance outcome within any period is limited to a few options that cannot fully reveal productivity. I illustrate such cases with the following example. Here, performance is binary, low or high, $Y=$ $\\left\\{y^{L}, y^{H}\\right\\}, y^{L}=0, y^{H}=1$. There are two equally likely types, $\\Theta=\\left\\{\\theta^{L}, \\theta^{H}\\right\\}$, and the technology is:\n\n| $f_{\\theta}(y)$ | $y^{L}$ | $y^{H}$ |\n| :--- | :--- | :--- |\n| $\\theta^{L}$ | $1-\\theta^{L}$ | $\\theta^{L}$, |\n| $\\theta^{H}$ | $1-\\theta^{H}$ | $\\theta^{H}$ |\n\nwith $0<\\theta^{L}<\\theta^{H}<1$, meaning that no performance realization is conclusive. Observing $y^{H}$ increases expected productivity; observing $y^{L}$ decreases it. $V^{A}>\\theta_{0}>0, V^{P}=0$ that ensures that the principal cannot obtain her first-best payoff.\n\nBecause the type is binary, as discussed in Section 5.1, an equilibrium evaluation policy is cutoff with a constant cutoff $q$. The cutoff is chosen such that the agent obtains a payoff $V^{A}$ by following the recommendations. The cutoff and the corresponding wage profile and quitting rate can be calculated by Monte Carlo simulations and are presented in Figure 3.\n\nThe agent's equilibrium career can be read off these plots. It features a clear promotion ladder in which infrequent performance revisions are followed by either quitting or obtaining a permanently higher wage.\n\nDetailed Performance In other industries, the performance measures are detailed. For example, a manager can reveal his incompetency in day-to-day interactions with his clients and employees. I illustrate such settings with the following example. Performance outcomes are rich, $Y=\\mathbb{R}$. There are two equally likely types, $\\Theta=\\left\\{\\theta^{L}, \\theta^{H}\\right\\}$. Performance is distributed according to a Gaussian distribution:\n\n$$\ny_{t}=\\theta+\\varepsilon, \\quad \\varepsilon \\sim N\\left(0, \\sigma^{2}\\right) .\n$$\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 3: Wage profile (left) and quitting rate (right) in a minimally informative equilibrium. Coarse performance. $\\Theta=\\left\\{\\theta^{L}, \\theta^{H}\\right\\}, Y=\\{0,1\\}, \\operatorname{Pr}(y=1 \\mid \\theta)=\\theta, w^{F}=0, \\alpha=0.5$, $\\theta^{L}=0.2, \\theta^{H}=0.4, V^{A}=0.6, V^{P}=0$. Computed numerically by Monte Carlo simulation.\n\nExpected productivity increases in performance. Moreover, the performance can be very conclusive-for any prior expectation below $\\theta^{H}$, the probability of interim expectations being arbitrarily close to $\\theta^{L}$ is positive. With the payoff structure of the previous example, the equilibrium cutoff, wage profile, and quitting rate are computed numerically and are presented in Figure 4.\n\nThe quitting may occur in any period, and the wage strictly increases with tenure. Nevertheless, similar to coarse technology, the wage profile exhibits a (roughly) S shape that reflects the learning pattern and the rate of selection. At the beginning of the career, the quitting rate is low because there is little time to accumulate sufficiently negative performance information. Late in the career, the quitting rate is also low because, if retained, the agent was proven to have performed well and is likely to be a high type. Most selection occurs in the middle of the career, when information sufficient for selection is likely to be accumulated, but much productivity uncertainty remains.","text_sha256":"3c654f8533c83f6082ce1c4e92bb92b82c17e676e5a6cf191b79487db5e45578"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0016","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Optimal Wage Contracts","text":"## 6 Optimal Wage Contracts\n\nIn the previous sections, I have studied the design of optimal evaluation policies while holding the wage contract exogenously fixed. In organizations, the focus on information control can be motivated when payment schemes are more rigid and difficult to change than evaluation policies. However, it is natural to ask what contract the principal would prefer if she expected to complement it with optimal information provision. To address this question, in this section, I drop the Assumption 1 and allow the principal to freely choose the parameters of\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 4: Wage profile (left) and quitting rate (right) in a minimally informative equilibrium. Rich performance. $\\Theta=\\left\\{\\theta^{L}, \\theta^{H}\\right\\}, y_{t}=\\theta+\\varepsilon_{t}, \\varepsilon_{t} \\sim N\\left(0, \\sigma^{2}\\right), \\sigma=0.6, w^{F}=0, \\alpha=0.5$, $\\theta^{L}=0, \\theta^{H}=1, V^{A}=0.6, V^{P}=0$. Computed numerically by Monte Carlo simulation.\n\nthe linear wage contract $w^{F}$ and $\\alpha$.\nNot very surprisingly, if the principal has both information and monetary control, then she can extract the full surplus from the relationship. To be precise, call a pair of strategies $m^{E}, \\tau^{E}$ surplus-efficient if they deliver the maximal total payoff to the players. As payoffs are quasilinear in payments, wage contract details are irrelevant for efficiency. As before, $m^{E}$ can be set to a complete-evaluation policy $\\bar{m}$, because the quitting time can always ignore redundant information. Lemma 1 applies, and an efficient quitting time $\\tau^{E}$ solves an optimal stopping problem with an outside option $V^{E}=V^{A}+V^{P}$. Call the corresponding delivered payoffs surplus efficient.\n\nI say that a given scheme $\\left(w^{F}, \\alpha, m\\right)$ extracts the full surplus if it implements the surplusefficient payoffs in which the agent's payoff is equal to $V^{A}$, meaning that he obtains zero rents. As the agent can guarantee $V^{A}$ by quitting at time zero, if a scheme extracts the full surplus, then this scheme is optimal for the principal among all possible schemes, with not necessarily linear contracts.\n\nAssumption 2. (Positive Outside Options) $V^{A}>0$ and $V^{P}+V^{A}>0$.\nThe first part of Assumption 2 guarantees that the agent's would not prefer to remain at firm if offered zero compensation. The second part ensures that the principal's outside option is not too negative. It implies that the agent prefers following the surplus-efficient policy if offered a contract with $w^{F}=0$ and $\\alpha=1$ to quitting at time zero.\n\nProposition 1. (Joint Scheme) By jointly controlling the parameters of the wage contract and the evaluation policy, the principal can extract the full surplus. The equilibrium evaluation policy recommends a surplus-efficient quitting policy. The wage parameters can be set to either (i) $w^{F}=V^{A}, \\alpha=0$, or, under Assumption 2, (ii) $w^{F}=0$, $\\alpha=\\alpha^{*} \\in\\left[0, V^{A} /\\left(V^{A}+V^{P}\\right)\\right]$.\n\nThe proof is constructive. As the aim of the principal is to extract the full surplus, by Lemma 3, she can focus on evaluations that recommend a surplus-efficient quitting time. The wage parameters are chosen such that the agent is willing to follow the recommendations and obtains zero rents. In the first scenario, $w^{F}=V^{A}, \\alpha=0$, the agent is effectively compensated according to his outside option. His wage is not linked to performance, and his payoff is fixed at $V^{A}$ irrespective of when he quits. The incentives are trivially satisfied, and the agent obtains zero rents. In the second scenario, $w^{F}=0, \\alpha=\\alpha^{*}$, the agent is compensated solely by a bonus. The incentive constraints are satisfied over an interval of bonuses. I show that it is possible to find a bonus that also leaves no rents to the agent.\n\nTo summarize, by using just one monetary tool, either a fixed wage or a bonus, the principal can extract the full surplus. A properly chosen fixed wage makes the agent indifferent between any quitting policy. It means that the principal can in principle provide a completeevaluation policy, expecting the agent to quit efficiently. However, the agent's indifference may decrease the appeal of this policy in practice. In contrast, a properly chosen bonus compensation links the pay to performance and provides stronger incentives throughout the relationship. However, it should be complemented by a coarse evaluation policy: if the agent were provided complete evaluation, he would quit at his personally optimal time.\n\nLet me highlight that in both scenarios the principal needs to provide informative evaluation to guide the agent's decisions as long as surplus-efficient decisions depend on performance. No evaluation is generically not optimal even though the principal controls the monetary incentives.","text_sha256":"9630309af2c1be6cc1f8429458bce8dc7bc385c77d228b1288ceb7ef4a898b2d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0017","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"7 Discussion","text":"## 7 Discussion","text_sha256":"3df2b30553e035c537216cb4c1df6810c1e1bcfe2a87f106695f5ca2d5de8ac0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0018","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"7.1 Organizational Implications","text":"### 7.1 Organizational Implications\n\nWages and Career My analysis draws attention to a role that evaluations may play in explaining internal labor market dynamics. Internal labor markets, or personnel economics, has attracted considerable attention in the economic literature. ${ }^{10}$ Some of the robust empirical findings are (i) a well-defined career ladder within a firm (Baker, Gibbs, and Holmstrom\n\n[^8](1994a,b); Seltzer and Merrett (2000); Dohmen (2004)), (ii) nominal, and to a large extent real, downward wage rigidity (Baker et al. (1994a,b); Card and Hyslop (1997)), (iii) wage growth that cannot be attributed to human capital formation (Baker et al. (1994a,b); Medoff and Abraham (1980, 1981)), and (iv) wage compression, i.e., insufficient wage variation within worker cohorts (Akerlof and Yellen (1990); Baker et al. (1994a,b)). ${ }^{11}$\n\nIt has been presumed that these patterns cannot be explained by learning models of wage formation. Indeed, if a worker is provided detailed performance information and his wage is proportional to expected productivity, then, for the duration of his stay at the firm, his wage should follow a random walk. If one examined many workers within the same cohort, this would imply that over time, the cohort wage distributions present a sequence of mean-preserving spreads.\n\nHowever, I show that it is possible to reconcile all these organizational patterns within a pure learning model with endogenous quitting, if a wage responds not to detailed performance but to optimally designed evaluations. In my model, the equilibrium wage is a deterministic function of tenure such that all workers from the same cohort receive the same salary. Moreover, the wage never decreases: poor evaluations are infrequent and result in an immediate resignation, whereas good evaluations lead to a permanent wage increase. Wage growth is fueled by positive ongoing selection and shapes an endogenous career ladder.\n\nOf course, any pure learning model would be too stylized to claim universal applicability; a hybrid model that combines several features such as that of Gibbons and Waldman (1999) would be more appropriate. However, my analysis suggests that evaluation practices should be given closer attention in applications and that learning theories of wage formation have the potential to explain evidence. The learning considerations may be particularly relevant for professional firms in which employee talents have a high impact on performance and are learned gradually over time (O'Flaherty and Siow (1995)).\n\nEvaluation Practices My analysis provides a novel perspective on evaluation practices in organizations. First, it is often observed that evaluations feature rating compression and leniency bias-too many employees are bunched at the highest grades (Murphy and Cleveland (1995)). This feature is often attributed to behavioral biases of evaluators and criticized for reducing evaluation informativeness. However, my analysis shows that these practices can in fact benefit a firm by encouraging employees to supply more effort. ${ }^{12}$ Second, there is no consensus in the literature on whether evaluations should be retrospective, that is, account\n\n[^9]for past performance. My analysis suggests that such an accounting is generally necessary. The equilibrium evaluation policy is Markov in belief based on the entire performance history; in general, this policy cannot be implemented through a sequence of grades based solely on current performance.\n\nThe equilibrium evaluation policy can also have a psychological interpretation for maintaining workforce morale. It has been confirmed both in an experimental setting (Kuhnen and Tymula (2012)) and in the field (Kolstad (2013)) that information provision is an important incentive device that interacts with workers' intrinsic motivation. At the same time, some organizational literature has informally argued that evaluation provision should balance two opposing forces-the employee's need for evaluation and its damage to his self-esteem (Murphy and Cleveland (1995)). My model can be viewed as a formalization of these forces with the agent's morale captured by his productivity belief. ${ }^{13}$ On one hand, evaluation is a valuable input for employees' future decisions. On the other hand, any evaluation may turn out negative and discourage an employee. The equilibrium policy can then be interpreted as the policy that optimally trades off these two competing effects.","text_sha256":"289ca6e9837fc2a4031ea3bdd2721b1ca55f4617aa06ac85ee7e613ffde044de"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0019","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"7.2 Value of Human Resource Departments","text":"### 7.2 Value of Human Resource Departments\n\nMy analysis relies on the ability of the firm to commit to its evaluation policy. In fact, several layers of commitment matter. First, I assume that evaluations are objective, so that the information is verifiable. If the evaluation were \"cheap talk,\" then the equilibrium recommendation policy would not be credible-the principal would always recommend that the agent stay. Second, I assume that the principal commits to the evaluation rules at the beginning of the relationship and cannot change them in the future. This assumption matters because under the equilibrium policy, as long as the agent remains at a firm, his continuation value increases, and his incentive constraints become slack. In the absence of intertemporal commitment, even if the players are symmetrically informed, the principal would be tempted to change the evaluation policy to extract more continuation surplus.\n\nFor these reasons, some degree of commitment power is required to implement the equilibrium policy. In practice, this commitment can be plausibly achieved by having an HR department. Nearly all of the large firms have such departments (Gomez-Mejia, Balkin, and Cardy (2014)). They are relatively independent and administer performance appraisal of employees, controlling what information is collected and transmitted throughout a firm. HR departments provide clear instructions regarding the frequency of evaluations, the metrics that are being assessed, and the grading scales. The rules are transparent and the\n\n[^10]performance metrics are largely objective. ${ }^{14}$ The question of evaluation design can then be interpreted as precisely what evaluation polices should be administered by an HR department to maximize a firm's objectives.\n\nRelatedly, my framework can be used to assess the value of having such an HR department: if the evaluations were subjective and given by biased parties, then the value of information control could be lost altogether.","text_sha256":"b9325c7bae0e7c92ece1041862c617159b23e03d97817fd6d9419f6ad0672928"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0020","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"7.3 Performance Leaks","text":"### 7.3 Performance Leaks\n\nThroughout the analysis, I assumed that the principal has full control over the information flow of the agent. This a typical assumption in the information design literature, and it provides an upper bound on what the principal can possibly achieve. However, it is possible that some performance information may leak; for example, the agent may have \"gut feelings\" about when he performed very well in the past. Such leaks can both limit the scope for information control and increase the agent's guaranteed payoff. At the extreme, if all performance information is leaked, then there is no scope for information design, and the agent obtains his first best payoff.\n\nHowever, some performance leaks are benign because they may be not acted upon. To illustrate, I let the productivity space be binary, $\\Theta=\\left\\{\\theta^{L}, \\theta^{H}\\right\\}$, and assume that the technology is strongly regular, meaning that it is regular, $\\operatorname{supp} f_{\\theta^{L}}=\\operatorname{supp} f_{\\theta^{H}}=\\mathbb{R}$, and $f_{\\theta^{H}}(y) / f_{\\theta^{L}}(y)$ spans $[0,+\\infty)$ as $y$ spans $\\mathbb{R}$. Under strongly regular technology, performance is inconclusive, meaning that the principal's belief $\\mu_{t} \\triangleq \\operatorname{Pr}\\left(\\theta^{H} \\mid y^{t-1}\\right)$ never equals 0 or 1. At the same time, the belief is variable, meaning that for any $\\mu_{t-1} \\in(0,1)$, performance $y_{t-1}$ can swing belief $\\mu_{t}$ to anywhere in (0, 1). Call a performance leak a partitional disclosure with a cutoff $\\kappa_{t}$ if it reveals whether the principal's belief $\\mu_{t}$ is above $\\kappa_{t}$. A partitional disclosure reveals whether the agent is a top performer. Within each period, let the leak occur after the evaluation but before the agent's action.\n\nClaim 1. (Performance Leaks) Let $|\\Theta|=2$, the technology be strongly regular, and the principal evaluate the agent according to the equilibrium recommendation policy. Then, there exist cutoffs $\\underline{\\kappa}_{t}<1$ for $t=0,1, \\ldots$ such that the leaks of partitional disclosures with cutoffs $\\kappa_{t} \\geq \\kappa_{t}$ do not affect players' equilibrium behavior or payoffs.\n\nIntuitively, the leaks in Claim 1 reveal only sufficiently top performers. Such leaks make top performers even more optimistic and keep them at the firm. Less-than-top performers do become less optimistic but not sufficiently so to find it worthwhile to quit the firm. Hence,\n\n[^11]the leaks do not interfere with equilibrium evaluations.\nMore generally, first, note that whenever the agent is recommended to quit, he may obtain full access to his performance history: the principal recommends quitting only at beliefs at which the agent himself would prefer to quit. Second, as discussed in Section 4.3, as long as the agent is recommended to stay, his continuation value increases with tenure. Hence, the incentives to quit are being progressively relaxed. Consequently, if the leaks do not drive the agent's continuation value below his outside option, he is willing to follow the recommendations. This clearly occurs if the leaks are sufficiently uninformative. However, even conclusive leaks may not interfere with equilibrium evaluations as long as they do not bring very negative news, as shown in Claim 1.","text_sha256":"87c2b405c83d9f0ff5e8c0dfe2c9ba18845b475540d22dc5fd9ee6426a95396e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0021","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"7.4 Evaluation and Advice","text":"### 7.4 Evaluation and Advice\n\nThroughout the paper, I focused on evaluation-which can be regarded as feedback informative about the agent's fit at the firm. The corresponding allocation of effort is vertical-whether to stay at the firm. I showed that this feedback should be coarse to avoid discouraging the agent too quickly and prevent him from quitting prematurely. Evaluation is to not be confused with another kind of feedback-advice-that guides the agent's actions within a firm and promotes his professional growth (Gomez-Mejia et al. (2014)). For example, a manager may guide an employee to choose the best fitting project or otherwise coach him throughout his career. The corresponding allocation of effort may be regarded as being horizontal. It is plausible that in practice, the agent's and the firm's interests are aligned on this horizontal dimension.\n\nIn some cases, advice and evaluation can be separated. For example, the efficient advice may be driven by firm-specific goals and not depend on the agent's fit within the firm. In those cases, the current analysis can be readily applied. The firm should optimally provide exhaustive advice but coarse evaluation: the former would maximize the efficiency of the agent's stay at the firm, while the latter would shield against premature quitting. However, it is possible to envision settings in which these two kinds of feedback cannot be disentangled; for example, the manager's advice to leave a trending project may inevitably be interpreted as a signal of a poor overall fit within the company. If such entanglement is severe, then additional analysis needs to be performed to understand an optimal information policy. However, in light of the current findings, I expect the policy to be coarse and distort the agent's actions towards the firm's interests.","text_sha256":"f96b0d52b404e77dfae9a1905459f571285e66a6bd36fc3196cfaa22beb60b80"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0022","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"8 Conclusion","text":"## 8 Conclusion\n\nPerformance evaluation is an important part of organizational life. Although much evaluation is informal, most firms have formal evaluation policies designed to collect and distribute performance information to employees. In this work, I showed that an equilibrium design of these policies can explain many features of internal labor markets observed in practice. Information control allows the organization to extract rents from its employees while maintaining operational efficiency.\n\nFor clarity and conciseness, I abstracted away from many realistic features of internal labor markets. Incorporating these features would make the model more applicable in practice and constitutes a plausible venue for future work. The efficiency approach developed in this paper may facilitate analysis of these richer settings. In the meantime, the proposed theory of endogenous evaluation could be viewed as complimenting existing theories of internal labor markets.","text_sha256":"7a44982b9100a45037407a301dd27935e47781fadb4ffd744633a0644a48a278"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0023","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAkerlof, G. A. and J. L. Yellen (1990): \"The Fair Wage-Effort Hypothesis and Unemployment,\" Quarterly Journal of Economics, 105, 255-283.\n\nBaker, G., M. Gibbs, and B. Holmstrom (1994a): \"The Internal Economics of the Firm: Evidence from Personnel Data,\" Quarterly Journal of Economics, 109, 881-919.\n\n- (1994b): \"The Wage Policy of a Firm,\" Quarterly Journal of Economics, 109, 921-955.\n\nBanks, J. S. and R. K. Sundaram (1992): \"Denumerable-Armed Bandits,\" Econometrica, 1071-1096.\n\nBergemann, D. and J. Välimäki (2008): \"Bandit Problems,\" in New Palgrave Dictionary of Economics, ed. by S. N. Durlauf and L. E. Blume, Palgrave Macmillan.\n\nBerry, D. A. and B. Fristedt (1985): \"Bandit Problems: Sequential Allocation of Experiments,\" London: Chapman and Hall, 5, 7-7.\n\nCard, D. and D. Hyslop (1997): \"Does Inflation \"Grease the Wheels of the Labor Market\"?\" in Reducing Inflation: Motivation and Strategy, University of Chicago Press, 71-122.\n\nCarroll, G. (2015): \"Robustness and Linear Contracts,\" American Economic Review, 105, 536-63.\n\nDohmen, T. J. (2004): \"Performance, Seniority, and Wages: Formal Salary Systems and Individual Earnings Profiles,\" Labour Economics, 11, 741-763.\n\nEderer, F. (2010): \"Feedback and Motivation in Dynamic Tournaments,\" Journal of Economics \\& Management Strategy, 19, 733-769.\n\nEly, J. C. (2017): \"Beeps,\" American Economic Review, 107, 31-53.\n\nEly, J. C. and M. Szydlowski (2020): \"Moving the Goalposts,\" Journal of Political Economy, 128, 468-506.\n\nFang, H. and G. Moscarini (2005): \"Morale Hazard,\" Journal of Monetary Economics, 52, 749-777.\n\nGibbons, R. and M. Waldman (1999): \"A Theory of Wage and Promotion Dynamics Inside Firms,\" Quarterly Journal of Economics, 114, 1321-1358.\n\nGittins, J. C. and D. M. Jones (1974): \"A Dynamic Allocation Index for the Sequential Design of Experiments,\" in Progress in Statistics, ed. by I. Vincze, J. Gani, and K. Sarkadi, Amsterdam: North-Holland Pub. Co., 241-266.\n\nGoltsman, M. and A. Mukherjee (2011): \"Interim Performance Feedback in Multistage Tournaments: The Optimality of Partial Disclosure,\" Journal of Labor Economics, 29, 229-265.\n\nGomez-Mejia, L. R., D. B. Balkin, and R. L. Cardy (2014): Managing Human Resources, Pearson, 8 ed.\n\nGuo, Y. (2016): \"Dynamic Delegation of Experimentation,\" American Economic Review, 106, 1969-2008.\n\nHalac, M., N. Kartik, and Q. Liu (2016): \"Optimal Contracts for Experimentation,\" Review of Economic Studies, 83, 1040-1091.\n\nHansen, S. E. (2013): \"Performance Feedback with Career Concerns,\" Journal of Law, Economics, and Organization, 29, 1279-1316.\n\nHolmström, B. (1999): \"Managerial Incentive Problems: A Dynamic Perspective,\" The Review of Economic Studies, 66, 169-182.\n\nHörner, J. and Y. Guo (2015): \"Dynamic Mechanisms without Money,\" Working Paper.\n\nHörner, J. and N. Lambert (2020): \"Motivational Ratings,\" Review of Economic Studies, Forthcoming.\n\nJovanovic, B. (1979): \"Job Matching and the Theory of Turnover,\" Journal of Political Economy, 87, 972-990.\n\nKamenica, E. and M. Gentzkow (2011): \"Bayesian Persuasion,\" American Economic Review, 101, 2590-2615.\n\nKaur, S. (2019): \"Nominal Wage Rigidity in Village Labor Markets,\" American Economic Review, 109, 3585-3616.\n\nKolstad, J. T. (2013): \"Information and Quality when Motivation is Intrinsic: Evidence from Surgeon Report Cards,\" American Economic Review, 103, 2875-2910.\n\nKremer, I., Y. Mansour, and M. Perry (2014): \"Implementing the 'Wisdom of the Crowd',\" Journal of Political Economy, 122, 988-1012.\n\nKuhnen, C. M. and A. Tymula (2012): \"Feedback, Self-Esteem, and Performance in Organizations,\" Management Science, 58, 94-113.\n\nMedoff, J. L. and K. G. Abraham (1980): \"Experience, Performance, and Earnings,\" Quarterly Journal of Economics, 95, 703-736.\n\n- (1981): \"Are Those Paid More Really More Productive? The Case of Experience,\" Journal of Human resources, 186-216.\n\nMurphy, K. R. and J. Cleveland (1995): Understanding Performance Appraisal: Social, Organizational, and Goal-Based Perspectives, Sage.\n\nMyerson, R. B. (1986): \"Multistage Games with Communication,\" Econometrica, 54, 323-358.\n\nO'Flaherty, B. and A. Siow (1995): \"Up-or-out Rules in the Market for Lawyers,\" Journal of Labor Economics, 13, 709-735.\n\nOrlov, D. (2016): \"Optimal Design of Internal Disclosure,\" Working Paper.\n\nOrlov, D., A. Skrzypacz, and P. Zryumov (2020): \"Persuading the Principal to Wait,\" Journal of Political Economy, 128, 2542-2578.\n\nOstrovsky, M. and M. Schwarz (2010): \"Information Disclosure and Unraveling in Matching Markets,\" American Economic Journal: Microeconomics, 2, 34-63.\n\nRayo, L. and I. Segal (2010): \"Optimal Information Disclosure,\" Journal of Political Economy, 118, 949-987.\n\nRenault, J., E. Solan, and N. Vieille (2017): \"Optimal Dynamic Information Provision,\" Games and Economic Behavior, 104, 329-349.\n\nRoss, S. (1983): Introduction to Stochastic Dynamic Programming, New York: Academic Press.\n\nSchmitt-Grohé, S. and M. Uribe (2016): \"Downward Nominal Wage Rigidity, Currency Pegs, and Involuntary Unemployment,\" Journal of Political Economy, 124, 1466-1514.\n\nSeltzer, A. and D. Merrett (2000): \"Personnel Policies at the Union Bank of Australia: Evidence from the 1888-1900 Entry Cohorts,\" Journal of Labor Economics, 18, 573-613.\n\nThompson, F. B. (1952): \"Equivalence of Games in Extensive Form,\" Technical report rm-759, RAND Corporation, Washington, D.C.\n\nWaldman, M. (2013): \"Theory and Evidence in Internal Labor Markets,\" in The Handbook of Organizational Economics, ed. by R. S. Gibbons and J. Roberts, Princeton University Press.","text_sha256":"67355b50b38e64cea27dd6b28691b454d46ef466c59de391e052f6dfce954202"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0024","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"9 Appendix","text":"## 9 Appendix\n\nProof of Lemma 1 Take any evaluation policy $m$ and consider a stochastic process $X=$ $\\left\\{X_{s}\\right\\}_{s \\geq 1}$ with:\n\n$$\nX_{s} \\triangleq(1-\\delta) \\sum_{t=0}^{s-1} \\delta^{t}\\left(y_{t}-\\mathbb{E}\\left[\\theta \\mid m^{t}\\right]\\right) .\n$$\n\nThe process $X$ is a bounded martingale with respect to the filtration generated by $m$ because:\n\n$$\n\\begin{aligned}\n\\mathbb{E}\\left[X_{s+1} \\mid m^{s}\\right] & =X_{s}+(1-\\delta) \\delta^{t} \\mathbb{E}\\left[y_{s}-\\mathbb{E}\\left[\\theta \\mid m^{s}\\right] \\mid m^{s}\\right] \\\\\n& =X_{s}+(1-\\delta) \\delta^{t}\\left(\\mathbb{E}\\left[y_{s} \\mid m^{s}\\right]-\\mathbb{E}\\left[\\theta \\mid m^{s}\\right]\\right) \\\\\n& =X_{s}+(1-\\delta) \\delta^{t}\\left(\\mathbb{E}\\left[\\theta \\mid m^{s}\\right]-\\mathbb{E}\\left[\\theta \\mid m^{s}\\right]\\right)=X_{s} .\n\\end{aligned}\n$$\n\nThe second line follows from the law of iterated expectations and the third line follows from the definition of productivity. It follows from the optional stopping theorem that for any stopping time $\\tau$ measurable with respect to $m, \\mathbb{E}\\left[X_{\\tau}\\right]=\\mathbb{E}\\left[X_{0}\\right]=0$. Equivalently, by the definition of wage (3), for any evaluation policy $m$ and stopping time $\\tau$ :\n\n$$\n\\begin{aligned}\n& U^{P}(m, \\tau)=\\mathbb{E}_{m, \\tau}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t}\\left(y_{t}-w_{t}\\right)\\right]+\\delta^{\\tau} V^{P}=\\mathbb{E}_{m, \\tau}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t}\\left((1-\\alpha) y_{t}-w^{F}\\right)+\\delta^{\\tau} V^{P}\\right] \\\\\n& =\\mathbb{E}_{m, \\tau}\\left[(1-\\alpha)(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t} y_{t}+\\delta^{\\tau}\\left(V^{P}+w^{F}\\right)\\right]-w^{F}=(1-\\alpha) \\mathbb{E}_{m, \\tau}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t} y_{t}+\\delta^{\\tau} \\hat{V}^{P}\\right]-w^{F} .\n\\end{aligned}\n$$\n\nThe agent's payoff is derived analogously. $\\square$\n\nProof of Lemma 2. Any feasible payoff can be obtained with a complete evaluation policy $\\bar{m}$. Indeed, any filtration generated by $m^{t}$ is coarser than a filtration generated by $y^{t-1}$. Hence, the set of feasible joint distributions of $y$ and $\\tau$ is maximized by setting $m^{t} \\equiv y^{t-1}$. Consequently, if payoffs $\\left(u^{A}, u^{P}\\right)$ can be generated by some $(m, \\tau)$, then they can be generated by $\\left(\\bar{m}, \\tau^{\\prime}\\right)$, where $\\bar{m}$ is a complete evaluation policy.\n\nHence, $\\mathcal{F}$ is a set of payoffs that can be generated by some quitting time $\\tau$ under complete evaluation. The quitting time can equivalently be identified with a quitting rule $\\sigma: Y^{t-1} \\rightarrow$ $\\Delta$ (\"stay\", \"quit\"); the players' expected payoffs $\\left(u^{A}(\\sigma), u^{P}(\\sigma)\\right)$ are linear as functions of $\\sigma$. Endowed with the sup norm, the set of quitting rules is a compact and convex vector space. Moreover, by the feasibility argument presented above, $\\mathcal{F}$ is a subset of $\\hat{\\mathcal{F}}$-the set of payoffs feasible if $\\theta$ were revealed at time 0 . As $\\mathbb{E}\\left[\\theta_{0}\\right]$ exists, $\\hat{\\mathcal{F}}$, and thus $\\mathcal{F}$ is bounded. Hence, $\\mathcal{F}$ is convex and compact as a bounded linear image of a convex and compact set.\n\nDenote the boundary of $\\mathcal{F}$ by $\\partial \\mathcal{F}, \\partial \\mathcal{F} \\triangleq \\mathcal{F} \\backslash \\operatorname{int}(\\mathcal{F})$. By the separating hyperplane\ntheorem, for payoffs $\\left(u^{A}, u^{P}\\right) \\in \\partial \\mathcal{F}$, there exist $\\lambda^{A}, \\lambda^{P} \\in \\mathbb{R}$ not both equal to zero such that:\n\n$$\n\\left(u^{A}, u^{P}\\right) \\in \\arg \\max _{\\left(u^{A}, u^{P \\prime}\\right) \\in \\mathcal{F}} \\lambda^{A} u^{A \\prime}+\\lambda^{P} u^{P \\prime} .\n$$\n\nConversely, for any $\\lambda^{A}, \\lambda^{P} \\in \\mathbb{R}$ not both equal to zero, a solution to (12) belongs to $\\partial \\mathcal{F}$. By Lemma 1, (12) is equivalent to maximizing:\n\n$$\n\\mathbb{E}_{\\bar{m}, \\tau}\\left[(1-\\delta) \\sum_{t=0}^{\\tau} \\delta^{t}\\left(\\lambda^{A} \\alpha+\\lambda^{P}(1-\\alpha)\\right) y_{t}+\\delta^{\\tau}\\left(\\lambda^{A} \\alpha \\hat{V}^{A}+\\lambda^{P}(1-\\alpha) \\hat{V}^{P}\\right)\\right]+w^{F}\\left(\\lambda^{A}-\\lambda^{P}\\right),\n$$\n\nover all quitting times $\\tau$. Efficient payoffs correspond to $\\lambda^{A}, \\lambda^{P} \\geq 0$, not both equal to zero. In this case, $\\lambda^{A} \\alpha+\\lambda^{P}(1-\\alpha)>0$ and maximizing the combination is equivalent to maximizing a payoff of an agent with a virtual safe option $\\hat{V}^{F}=\\frac{\\lambda^{A} \\alpha}{\\lambda^{A} \\alpha+\\lambda^{P}(1-\\alpha)} \\hat{V}^{A}+\\frac{\\lambda^{A}(1-\\alpha)}{\\lambda^{A} \\alpha+\\lambda^{P}(1-\\alpha)} \\hat{V}^{P}$. By Assumption 1, $\\hat{V}^{A}>\\hat{V}^{P}$; thus, as $\\lambda^{A}, \\lambda^{P}$ span all possible values, $\\hat{V}^{F}$ spans $\\left[\\hat{V}^{P}, \\hat{V}^{A}\\right]$. $\\square$\n\nProof of Lemma 4. Consider any efficient payoffs $\\left(u^{A}, u^{P}\\right)$. If $u^{A}<V^{A}$, then the payoffs cannot be implemented because the agent can secure a payoff $V^{A}$ by quitting at time 0 . If $u^{A} \\geq V^{A}$, then by Lemma 2 the payoffs can be delivered through strategies $(\\bar{m}, \\tau)$ that maximize the payoff of an agent with a safe option $\\hat{V}^{F} \\in\\left[\\hat{V}^{P}, \\hat{V}^{A}\\right]$. Consider the recommendation policy $m^{\\prime}$ that mimics $\\tau$. I show that $m^{\\prime}$ is incentive compatible.\n\nRecommendations to quit are incentive compatible. If the agent is recommended to quit at time $t$, his wage at the firm remains constant at some level $w_{t}^{Q}$ hereafter. Because the strategy maximizes the payoff of an agent with a safe option $\\hat{V}^{F}, w_{t}^{Q} \\leq \\hat{V}^{F}$; otherwise, remaining at firm forever onward would increase the payoff. Since $\\hat{V}^{F} \\leq \\hat{V}^{A}$, the agent with a safe option $\\hat{V}^{A}$ also agrees to quit.\n\nRecommendations to stay are also incentive compatible. A recommendation at time 0 is incentive compatible because it gives the agent a payoff of at least $u^{A}$ and $u^{A} \\geq V^{A}$. Incentive compatibility at time $t+1$ follows from incentive compatibility at time $t$. Indeed, denote by $\\tilde{U}_{t}^{A}$ the agent's optimal continuation payoff conditional on being recommended to stay at time $t$. Because the recommendation to stay is incentive compatible at time $t$ and the recommendations to quit are incentive compatible everywhere, $\\tilde{U}_{t}^{A}$ is a convex combination of $w_{t}, \\tilde{U}_{t+1}^{A}$, and $V^{A}$ :","text_sha256":"d9f1f05955d138ac2b11ba026454dc546634ac6c3787914f7f5682e9a4968c9e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0025","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"9 Appendix","text":"$$\n\\tilde{U}_{t}^{A}=(1-\\delta) w_{t}+\\delta \\operatorname{Pr}\\left(m_{t+1}=\\text { \"stay\" } \\mid m^{t} \\equiv \\text { \"stay\" }\\right) \\tilde{U}_{t+1}^{A}+\\delta \\operatorname{Pr}\\left(m_{t+1}=\\text { \"quit\" } \\mid m^{t} \\equiv \\text { \"stay\" }\\right) V^{A} .\n$$\n\nThe agent obtains a continuation payoff $V^{A}$ if he quits immediately so $\\tilde{U}_{t}^{A} \\geq V^{A}$. The agent obtains a continuation payoff $w_{t}$ if he never quits so $\\tilde{U}_{t}^{A} \\geq w_{t}$. Thus, for the equality to hold,\nit must be that $\\tilde{U}_{t+1}^{A} \\geq \\tilde{U}_{t}^{A}$. By forward induction, all recommendations to stay are incentive compatible. This completes the proof. $\\square$\n\nLemma 5. Fix $V \\in \\mathbb{R}, \\Theta=\\left\\{\\theta^{L}, \\theta^{H}\\right\\}$, and an arbitrary technology. Consider the following maximization problem:\n\n$$\n\\max _{\\tau \\in \\mathcal{T}} \\mathbb{E}_{\\bar{m}, \\tau}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t} \\mathbb{E}\\left[\\theta \\mid \\bar{m}^{t}\\right]+\\delta^{\\tau} V\\right] .\n$$\n\nThen, (i) if $V>\\theta^{H}$, then $\\tau \\equiv 0$ is uniquely optimal; (ii) if $V<\\theta^{L}$, then $\\tau \\equiv \\infty$ is uniquely optimal; and (iii) if $V \\in\\left[\\theta^{L}, \\theta^{H}\\right]$, then any optimal strategy $\\tau$ is equivalent in terms of quitting distributions to a strategy that follows the recommendations of a cutoff recommendation policy with some fixed cutoff $q \\in\\left[\\theta^{L}, \\theta^{H}\\right]$.\n\nProof. This is a standard bandit problem of an agent choosing between a risky arm with payoff $\\mathbb{E}\\left[\\theta \\mid \\bar{m}^{t}\\right]$ and a safe arm with payoff $V$. This is a Markov decision problem with an expected productivity $\\hat{\\theta}_{t} \\triangleq \\mathbb{E}\\left[\\theta \\mid y^{t-1}\\right]$ being a sufficient statistic for beliefs. An optimal quitting strategy is characterized by Gittins and Jones (1974) as an index policy. The risky arm is assigned an index $\\xi\\left(\\hat{\\theta}_{t}\\right)$. The agent quits the arm as soon as the index drops below $V$; he can arbitrarily randomize at $\\xi\\left(\\hat{\\theta}_{t}\\right)=V$.\n\nThe fact that $\\xi\\left(\\hat{\\theta}_{t}\\right)$ is increasing in $\\hat{\\theta}_{t}$ is a direct consequence of Lemma 3.2 of Banks and Sundaram (1992) that states that the index is increasing along any belief ray that passes through the \"worst\" prior. For completeness, I provide a direct argument here. By Gittins and Jones (1974), the index can be calculated as:\n\n$$\n\\begin{aligned}\n\\xi\\left(\\hat{\\theta}_{t}\\right) & \\triangleq \\sup _{\\tau} \\frac{\\mathbb{E}\\left[\\sum_{t=0}^{\\tau} \\delta^{t} \\mathbb{E}\\left[\\theta \\mid \\bar{m}^{t}\\right] \\mid \\hat{\\theta}_{t}\\right]}{\\mathbb{E}\\left[\\sum_{t=0}^{\\tau} \\delta^{t} \\mid \\hat{\\theta}_{t}\\right]} \\\\\n& =\\sup _{\\tau} \\frac{\\frac{\\hat{\\theta}_{t}-\\theta^{L}}{\\theta^{H}-\\theta^{L}} \\mathbb{E}\\left[\\sum_{t=0}^{\\tau} \\delta^{t} y_{t} \\mid \\theta=\\theta^{H}\\right]+\\frac{\\theta^{H}-\\hat{\\theta}_{t}}{\\theta^{H}-\\theta^{L}} \\mathbb{E}\\left[\\sum_{t=0}^{\\tau} \\delta^{t} y_{t} \\mid \\theta=\\theta^{L}\\right]}{\\frac{\\hat{\\theta}_{t}-\\theta^{L}}{\\theta^{H}-\\theta^{L}} \\mathbb{E}\\left[\\sum_{t=0}^{\\tau} \\delta^{t} \\mid \\theta=\\theta^{H}\\right]+\\frac{\\theta^{H}-\\hat{\\theta}_{t}}{\\theta^{H}-\\theta^{L}} \\mathbb{E}\\left[\\sum_{t=0}^{\\tau} \\delta^{t} \\mid \\theta=\\theta^{L}\\right]} \\\\\n& =\\sup _{\\tau} \\frac{\\left(\\hat{\\theta}_{t}-\\theta^{L}\\right) \\theta^{H} \\mathbb{E}\\left[1-\\delta^{\\tau+1} \\mid \\theta=\\theta^{H}\\right]+\\left(\\theta^{H}-\\hat{\\theta}_{t}\\right) \\theta^{L} \\mathbb{E}\\left[1-\\delta^{\\tau+1} \\mid \\theta=\\theta^{L}\\right]}{\\left(\\hat{\\theta}_{t}-\\theta^{L}\\right) \\mathbb{E}\\left[1-\\delta^{\\tau+1} \\mid \\theta=\\theta^{H}\\right]+\\left(\\theta^{H}-\\hat{\\theta}_{t}\\right) \\mathbb{E}\\left[1-\\delta^{\\tau+1} \\mid \\theta=\\theta^{L}\\right]} \\\\\n& =\\sup _{\\tau} \\theta^{L}+\\frac{\\left(\\hat{\\theta}_{t}-\\theta^{L}\\right)\\left(\\theta^{H}-\\theta^{L}\\right) \\mathbb{E}\\left[1-\\delta^{\\tau+1} \\mid \\theta=\\theta^{H}\\right]}{\\left(\\hat{\\theta}_{t}-\\theta^{L}\\right) \\mathbb{E}\\left[1-\\delta^{\\tau+1} \\mid \\theta=\\theta^{H}\\right]+\\left(\\theta^{H}-\\hat{\\theta}_{t}\\right) \\mathbb{E}\\left[1-\\delta^{\\tau+1} \\mid \\theta=\\theta^{L}\\right]} \\\\\n& =\\theta^{L}+\\frac{\\left(\\hat{\\theta}_{t}-\\theta^{L}\\right)\\left(\\theta^{H}-\\theta^{L}\\right)}{\\left(\\hat{\\theta}_{t}-\\theta^{L}\\right)+\\left(\\theta^{H}-\\hat{\\theta}_{t}\\right) \\kappa},\n\\end{aligned}\n$$\n\nwhere $\\kappa \\triangleq \\inf _{\\tau} \\frac{\\mathbb{E}\\left[1-\\delta^{\\tau+1} \\mid \\theta=\\theta^{L}\\right]}{\\mathbb{E}\\left[1-\\delta^{\\tau+1} \\mid \\theta=\\theta^{H}\\right]}>0$, a constant fully determined by $F_{\\theta}$, and the second and\nthird lines follow from the optional stopping theorem and the law of iterated expectations. Thus, $\\xi\\left(\\hat{\\theta}_{t}\\right)$ strictly increases with $\\hat{\\theta}_{t}$. $\\square$\n\nProof of Theorem 2. I start backwards and first show that if the evaluation policy is cutoff, then the wage profile is weakly increasing. By the law of iterated expectations:\n\n$$\n\\mathbb{E}\\left[\\theta \\mid m^{t}\\right]=\\mathbb{E}\\left[\\mathbb{E}\\left[\\theta \\mid y^{t-1}\\right] \\mid m^{t}\\right]=\\mathbb{E}\\left[\\hat{\\theta}_{t} \\mid m^{t}\\right] .\n$$\n\nIf the policy always recommends to stay at the boundary, when $\\hat{\\theta}_{t}=q_{t-1}\\left(y^{t-2}\\right)$, then\n\n$$\n\\begin{gathered}\n\\mathbb{E}\\left[\\hat{\\theta}_{t} \\mid m^{t}=(\\text { \"stay\" }, \\ldots, \\text { \"stay\" })\\right]=\\mathbb{E}\\left[\\hat{\\theta}_{t} \\mid \\hat{\\theta}_{1} \\geq q_{0}, \\ldots, \\hat{\\theta}_{t-1} \\geq q_{t-2}\\left(y^{t-2}\\right), \\hat{\\theta}_{t} \\geq q_{t-1}\\left(y^{t-1}\\right)\\right] \\geq \\\\\n\\mathbb{E}\\left[\\hat{\\theta}_{t} \\mid \\hat{\\theta}_{1} \\geq q_{0}, \\ldots, \\hat{\\theta}_{t-1} \\geq q_{t-2}\\left(y^{t-2}\\right)\\right]=\\mathbb{E}\\left[\\hat{\\theta}_{t} \\mid m^{t-1}=(\\text { \"stay\" }, \\ldots, \\text { \"stay\" })\\right],\n\\end{gathered}\n$$\n\nwhere the inequality follows by the definition of a conditional expectation and the implied positive selection. The case of randomization at the boundary is analogous. As the wage is proportional to expected ability, the wage profile is weakly increasing.\n\nI proceed with establishing that under the conditions of the theorem, the evaluation policy is indeed cutoff. If the type is binary, $|\\Theta|=2$, then by Lemma 5, an optimal policy is cutoff ( $\\tau \\equiv 0$ and $\\tau \\equiv \\infty$ correspond to cutoffs greater than $\\theta^{H}$ and smaller than $\\theta^{L}$, respectively).","text_sha256":"c0af0ea416954f9d9da9484e1882298c4d5227849c1b9b31ff64649abeb9a3ac"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0026","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"9 Appendix","text":"If the type is not binary, $|\\Theta|>2$, but the performance technology is regular, then I can appeal to the results of Ross (1983) and Banks and Sundaram (1992). For completeness, I present the proof here.\n\nTo proceed, we need some additional definitions. Associate the $k$-highest type with an index $k$ and let $K \\triangleq|\\Theta|$. For any $g_{1}, g_{2} \\in \\Delta(\\Theta)$, say that $g_{2}$ discretely (stochastically) dominates $g_{1}, g_{2} \\succeq_{d} g_{1}$, if for all $l \\in\\{1, \\ldots, K\\}, \\sum_{k=1}^{l} g_{2 k} \\geq \\sum_{k=1}^{l} g_{1 k}$. Similarly, for any $f_{1}, f_{2}$ that are probability densities over $Y$, say that $f_{2}$ continuously (stochastically) dominates $f_{1}, f_{2} \\succeq_{c} f_{1}$, if for any increasing function $h, \\int h(y) f_{g_{2}}(y) d y \\geq \\int h(y) f_{g_{1}}(y) d y$. Denote by $\\beta(g, y) \\in \\Delta(\\Theta)$ the posterior distribution calculated by Bayes' rule given a prior distribution $g$ and a one-period performance outcome $y \\in Y$. With a small abuse of notation, define $f_{g}(y) \\triangleq \\sum_{\\theta \\in \\Theta} g(\\theta) f_{\\theta}(y)$, and denote by $\\hat{Y} \\triangleq \\cup_{\\theta} \\operatorname{supp}\\left(f_{\\theta}(y)\\right)$.\n\nI provide some intuitive and standard consequences of regularity of technology driven my the MLRP property without a proof (see Ross (1983) and Banks and Sundaram (1992) for details).\n\nClaim 2. For any $g \\in \\Delta(\\Theta)$ and $y, y^{\\prime} \\in \\hat{Y}$ such that $y^{\\prime}>y, \\beta\\left(g, y^{\\prime}\\right) \\succeq_{d} \\beta(g, y)$.\nClaim 3. For any $g, g^{\\prime} \\in \\Delta(\\Theta)$ such that $g^{\\prime} \\succeq_{d} g, f_{g^{\\prime}} \\succeq_{c} f_{g}$.\n\nWe are ready to proceed to the main part of the proof. By Lemma 2 and Theorem 1, any optimal recommendation policy solves an optimal stopping problem of a fictitious agent. This is a Markov decision problem with the agent's beliefs $g_{t} \\in \\Delta(\\Theta)$ being a state variable. Denote the agent's continuation value as a function of beliefs $\\mathcal{V}(g)$ :\n\n$$\n\\mathcal{V}(g)=\\max \\left[\\hat{V}^{F}, \\mathbb{E}_{g}[\\theta]+\\delta \\int \\mathcal{V}(\\beta(g, y)) f_{g}(y) \\mathrm{d} y\\right]\n$$\n\nBy Claim 2, any beliefs following the same history $g_{t}=\\beta\\left(g_{t-1}, y\\right), g_{t}^{\\prime}=\\beta\\left(g_{t-1}, y^{\\prime}\\right)$ can be ranked according to $\\succeq_{d}$. To establish that the policy is cutoff, it then suffices to show that in any period $t$ if the agent stays at belief $g_{t 1}$, he does so for all beliefs $g_{t 2} \\succeq g_{t 1}$. As quitting results in an immediate payoff $\\hat{V}^{F}$, the policy is cutoff if for any $g_{0} \\in \\Delta(\\Theta)$ and $y_{1}, y_{2} \\in \\hat{Y}$ such that $y_{2}>y_{1}, \\mathcal{V}\\left(\\beta\\left(g_{0}, y_{2}\\right)\\right) \\geq \\mathcal{V}\\left(\\beta\\left(g_{0}, y_{1}\\right)\\right)$. This can be shown by an induction proof. Denote $g_{2}=\\beta\\left(g_{0}, y_{2}\\right)$ and $g_{1}=\\beta\\left(g_{0}, y_{1}\\right)$. By Claim 2, $g_{2} \\succeq_{d} g_{1}$. Consider the stopping problem in which the horizon is truncated to $T$ periods, $T=0,1, \\ldots$; this corresponds to altering the discount sequence to $\\left\\{1, \\delta, \\delta^{2}, \\ldots, \\delta^{T-1}, 0,0, \\ldots\\right\\}$. Denote by $\\mathcal{V}_{T}(g)$ the value function of this problem at time 0 . If $T=0, \\mathcal{V}_{0}\\left(g_{2}\\right) \\geq \\mathcal{V}_{0}\\left(g_{1}\\right)$ for any $g_{1}, g_{2}$ as both sides of the inequality are nil. As an induction hypothesis, assume that $\\mathcal{V}_{T}\\left(g_{2}\\right) \\geq \\mathcal{V}_{T}\\left(g_{1}\\right)$ for all $g_{1}, g_{2}$ such that $g_{2} \\succeq_{d} g_{1}$. We show that $\\mathcal{V}_{T+1}\\left(g_{2}\\right) \\geq \\mathcal{V}_{T+1}\\left(g_{1}\\right)$. If the optimal strategy at belief $g_{1}$ is to quit immediately, then $\\mathcal{V}_{T+1}\\left(g_{1}\\right)=\\hat{V}^{F}$ and the inequality is trivially satisfied. Otherwise,\n\n$$\n\\mathcal{V}_{T+1}\\left(g_{2}\\right)-\\mathcal{V}_{T+1}\\left(g_{1}\\right) \\geq \\mathbb{E}_{g_{2}}[\\theta]-\\mathbb{E}_{g_{1}}[\\theta]+\\delta \\int \\mathcal{V}_{T}\\left(\\beta\\left(g_{2}, y\\right)\\right) f_{g_{2}}(y)-\\mathcal{V}_{T}\\left(\\beta\\left(g_{1}, y\\right)\\right) f_{g_{1}}(y) \\mathrm{d} y .\n$$\n\nThe first term is nonnegative because $g_{2} \\succeq_{d} g_{1}$ implies $\\mathbb{E}_{g_{2}}[\\theta] \\geq \\mathbb{E}_{g_{1}}[\\theta]$. To establish the nonnegativity of the integral, rearrange it as:\n\n$$\n\\int \\mathcal{V}_{T}\\left(\\beta\\left(g_{2}, y\\right)\\right)\\left(f_{g_{2}}(y)-f_{g_{1}}(y)\\right) \\mathrm{d} y+\\int\\left(\\mathcal{V}_{T}\\left(\\beta\\left(g_{2}, y\\right)\\right)-\\mathcal{V}_{T}\\left(\\beta\\left(g_{1}, y\\right)\\right)\\right) f_{g_{1}}(y) \\mathrm{d} y .\n$$\n\nBy Claim 2 and the induction hypothesis, $\\mathcal{V}_{T}\\left(\\beta\\left(g_{2}, y\\right)\\right)$ is increasing in $y$. By Claim 3, $f_{g_{2}} \\succeq f_{g_{1}}$, so, by the definition of continuous domination, the first term is nonnegative. For the second term, note that $\\beta\\left(g_{i}, y\\right)=\\beta\\left(\\beta\\left(g_{0}, y_{i}\\right), y\\right)=\\beta\\left(\\beta\\left(g_{0}, y\\right), y_{i}\\right)$ for $i=1,2$. Hence, by Claim 2, $\\beta\\left(g_{2}, y\\right) \\succeq_{d} \\beta\\left(g_{1}, y\\right)$, so by the induction hypothesis $\\mathcal{V}_{T}\\left(\\beta\\left(g_{2}, y\\right)\\right) \\geq \\mathcal{V}_{T}\\left(\\beta\\left(g_{1}, y\\right)\\right)$, and the second term is also nonnegative. Consequently, $\\mathcal{V}_{T+1}\\left(g_{2}\\right) \\geq \\mathcal{V}_{T+1}\\left(g_{1}\\right)$.\n\nThe induction argument establishes that for any finite $T$ and $g_{2} \\succeq_{d} g_{1}, \\mathcal{V}_{T}\\left(g_{2}\\right) \\geq \\mathcal{V}_{T}\\left(g_{1}\\right)$. At the same time, by Theorem 2.5.1 of Berry and Fristedt (1985), $\\mathcal{V}_{T} \\rightarrow \\mathcal{V}$ as $T \\rightarrow \\infty$. Thus, $\\mathcal{V}\\left(g_{2}\\right) \\geq \\mathcal{V}\\left(g_{1}\\right)$. The result follows. $\\square$\n\nProof of Proposition 1. The case in which $w^{F}=V^{A}, \\alpha_{0}=0$ is presented in the main text. Consider the case in which $w^{F}=0$. By the proof of Lemma 4, if the agent's payoff from following recommendations is at least $V^{A}$ and $V^{E}=V^{A}+V^{P} \\leq \\hat{V}^{A}$, then his incentives are satisfied. Under Assumption 2, the latter condition can be written as:\n\n$$\n\\alpha \\leq \\frac{V^{A}}{V^{A}+V^{P}} .\n$$\n\nIt is left to show that there exists such an $\\alpha$ that under the surplus-efficient quitting policy $\\tau^{E}$ delivers the payoff $V^{A}$ to the agent. By Lemma 1, the agent's payoff can be written as:","text_sha256":"f31ff615e9b6d7f05d5f6a80b96951b514e58ae26a097622ce27abc4576ffcd2"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0027","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"9 Appendix","text":"$$\nU^{A}\\left(\\bar{m}, \\tau^{E}\\right)=\\mathbb{E}_{\\bar{m}, \\tau^{E}}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t} y_{t}+\\delta^{\\tau} \\hat{V}^{A}\\right] \\times \\alpha=V^{A}+\\mathbb{E}_{\\bar{m}, \\tau^{E}}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t}\\left(\\alpha y_{t}-V^{A}\\right)\\right] .\n$$\n\nAt $\\alpha=0$, the payoff is:\n\n$$\nV^{A}-\\mathbb{E}_{\\bar{m}, \\tau^{E}}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t} V^{A}\\right] \\leq V^{A},\n$$\n\nwhere the inequality holds because $V^{A}>0$. At $\\alpha=V^{A} /\\left(V^{A}+V^{P}\\right)$, the payoff is:\n\n$$\nV^{A}+\\frac{V^{A}}{V^{A}+V^{P}} \\mathbb{E}_{\\bar{m}, \\tau^{E}}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t}\\left(y_{t}-\\left(V^{A}+V^{P}\\right)\\right)\\right] \\geq V^{A},\n$$\n\nwhere the inequality holds because of Assumption 2 and the fact that a surplus-efficient policy delivers a total payoff that is at least as large as the total outside option:\n\n$$\n\\mathbb{E}_{\\bar{m}, \\tau^{E}}\\left[(1-\\delta) \\sum_{t=0}^{\\tau-1} \\delta^{t}\\left(y_{t}-\\left(V^{A}+V^{P}\\right)\\right)\\right]+V^{A}+V^{P} \\geq V^{A}+V^{P} .\n$$\n\nFurthermore, $U^{A}\\left(\\bar{m}, \\tau^{E}\\right)$ is linear in $\\alpha$. Hence, by the intermediate value theorem, there exists $\\alpha^{*} \\in\\left[0, V^{A} /\\left(V^{A}+V^{P}\\right)\\right]$ such that the agent's payoff is equal to $V^{A}$. $\\square$\n\nProof of Claim 1. By Theorem 2, the equilibrium evaluation policy is cutoff and the wage profile $\\left\\{W_{t}\\right\\}_{t=0}^{\\infty}$ is deterministic and weakly increasing. Because the technology is strongly regular, the principal's beliefs $\\mu_{t}$ have full support over $(0,1)$ at all $t$, and hence, the quitting probability is strictly positive in every period. Thus, there is strictly positive selection in every period, and $W_{t}$ increases strictly in $t$. Because the principal's belief is a sufficient statistic for the recommendation policy, it is a sufficient statistic for the agent's continuation\nvalue. Define by $\\hat{U}_{t}^{A}\\left(\\mu_{t}\\right)$ the agent's continuation value at time $t$ if the principal's belief is $\\mu_{t}$ :\n\n$$\n\\hat{U}_{t}^{A}\\left(\\mu_{t}\\right)=\\mathbb{E}_{\\tau}\\left[(1-\\delta) \\sum_{t^{\\prime}=t}^{\\tau-1} \\delta^{t^{\\prime}}\\left(W_{t^{\\prime}}-V^{A}\\right)\\right]+V^{A} .\n$$\n\nDenote the continuation distribution of quitting times by $f\\left(\\tau \\mid \\mu_{t}\\right)$. It is continuously and strictly decreasing in $\\mu_{t}$ at all subsequent periods: every continuation path of the principal's beliefs continuously and strictly increases in $\\mu_{t}$; as the technology is strongly regular, it means that there is an additional strictly positive measure of paths that stay above the optimal cutoffs. It follows that $\\hat{U}_{t}^{A}\\left(\\mu_{t}\\right)$ is continuous in $\\mu_{t}$ over [0, 1]. Moreover, the maximal value of $\\hat{U}_{t}^{A}\\left(\\mu_{t}\\right)$ is achieved at $\\mu_{t}=1$, because at this belief the agent never quits and ensures the maximal possible wage stream (which is preferred to securing an option $V^{A}$ by incentive compatibility).\n\nProceed by constructing the thresholds $\\underline{\\kappa}_{t}$. Start with the first period. Consider the agent's continuation value if he is recommended to stay and follows the recommendations:\n\n$$\n\\tilde{U}_{1}^{A}=\\mathbb{E}\\left[\\hat{U}_{1}^{A}\\left(\\mu_{1}\\right) \\mid m^{1} \\equiv \\text { \"stay\" }\\right] .\n$$\n\nAs the wage profile is strictly increasing, $\\tilde{U}_{1}^{A}>\\tilde{U}_{0}^{A} \\geq V^{A}$. As $\\hat{U}_{1}^{A}\\left(\\mu_{1}\\right)$ obtains its maximal value at $\\mu_{1}=1$, is continuous, and $\\tilde{U}_{1}^{A}>V^{A}$, it follows that $\\hat{U}_{1}^{A}\\left(\\mu_{1}\\right)>V^{A}$ for all $\\mu_{1}>x_{1}$ for some $x_{1}<1$. Moreover, as the technology is strongly regular, $\\mu_{1}$ is distributed continuously over $(0,1)$ so the probability of $\\mu_{1}$ being greater than $x_{1}$ is positive for all $x_{1}<1$ and continuously decreases to 0 as $x_{1}$ increases to 1. Hence, there exists a threshold $\\kappa_{1}<1$ such that for any $\\kappa_{1} \\geq \\underline{\\kappa}_{1}$ :\n\n$$\n\\mathbb{E}\\left[\\hat{U}_{1}^{A}\\left(\\mu_{1}\\right) \\mid \\mu_{1}>\\kappa_{1}, m^{1} \\equiv \\text { \"stay\" }\\right]>\\mathbb{E}\\left[\\hat{U}_{1}^{A}\\left(\\mu_{1}\\right) \\mid \\mu_{1}<\\kappa_{1}, m^{1} \\equiv \\text { \"stay\" }\\right]>V^{A} .\n$$\n\nTake this threshold as the partition threshold for period 1. The inequalities ensure that the agent is willing to follow the recommendation to stay. Recommendations to quit remain incentive compatible because the quitting recommendation occurs in the region of belief in which the agent himself would prefer to quit.\n\nGiven a sequence of thresholds $\\underline{\\kappa}_{t^{\\prime}}$ for $t^{\\prime}<t$, we now construct a threshold $\\underline{\\kappa}_{t}$. Consider a collection of continuation values at period $t-1, \\mathbb{E}\\left[\\hat{U}_{t-1}^{A}\\left(\\mu_{t-1}\\right) \\mid s^{t-1}, m^{t-1} \\equiv\\right.$ \"stay\" $]$, where $s^{t-1}$ are past realizations of performance leaks. By incentive compatibility, all these values are strictly greater than $V^{A}$. By the same argument as in the proof of Lemma 4, continuation values conditional on recommendations to stay increase and do so strictly because of the\nstrictly positive quitting probability:\n\n$$\n\\mathbb{E}\\left[\\hat{U}_{t}^{A}\\left(\\mu_{t}\\right) \\mid s^{t-1}, m^{t} \\equiv \\text { \"stay\" }\\right]>\\mathbb{E}\\left[\\hat{U}_{t-1}^{A}\\left(\\mu_{t-1}\\right) \\mid s^{t-1}, m^{t-1} \\equiv \\text { \"stay\" }\\right] .\n$$\n\nHence, we can find $2^{t-1}$ thresholds $\\kappa_{t}\\left(s^{t-1}\\right)$ such that the recommendation to stay whenever $\\mu_{t} \\geq \\kappa_{t}\\left(s^{t-1}\\right)$ is incentive compatible after leak history $s^{t-1}$. Setting $\\underline{\\kappa}_{t}=\\max _{s^{t-1}} \\kappa_{t}\\left(s^{t-1}\\right)$ creates an incentive-compatible policy. $\\square$","text_sha256":"7b36b148dd053ef573f9954ee6cb2c38614e26f86dca2c3be3cbc79489e4a3ee"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0028","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"9 Appendix","text":"[^0]:    *Toulouse School of Economics, 1 Esplanade de l'Université, 31080 Toulouse Cedex 06, France, alexey.v.smolin@gmail.com. This paper builds on a chapter of my Ph.D. dissertation written at Yale University that circulated as \"Optimal Feedback Design\" in November, 2015; I thank Dirk Bergemann, Johannes Hörner, and Larry Samuelson for support and guidance. I thank the coeditor, Michael Ostrovsky, and three anonymous referees for many productive suggestions. I am grateful to Florian Ederer, Sergei Izmalkov, Emir Kamenica, Anton Kolotilin, Chiara Margaria, Benny Moldovanu, Pauli Murto, Anne-Katrin Roesler, and Áron Tóbiás for insightful conversations. Finally, I am thankful to participants of research seminars at Berlin, Bocconi, Bonn, Cornell Johnson, HSE (Moscow), Indiana Kelley, MPI Bonn, and Yale, as well as at the 5th World Congress of the Game Theory Society and the 2016 North American Summer Meeting of the Econometric Society. I acknowledge funding from ANR under grant ANR-17-EURE-0010 (Investissements d'Avenir program).\n\n[^1]:    ${ }^{1}$ According to Murphy and Cleveland (1995), between 74\\% and 89\\% of business organizations had formal performance appraisal policies by 1995.\n    ${ }^{2}$ \"GE's Real-Time Performance Development,\" Harvard Business Review, August 12, 2015, https://hbr.org/2015/08/ges-real-time-performance-development.\n\n[^2]:    ${ }^{3}$ This argument was later applied by Ely and Szydlowski (2020) in a setting in which the relevant state deterministically evolves over time, resulting in a predictable reversal of incentives.\n\n[^3]:    ${ }^{4}$ Randomization over several evaluation policies can be represented by a single evaluation policy with a combined evaluation law.\n    ${ }^{5}$ I adopt the convention that for any stochastic process $x$ its time- $t$ realization is denoted by subscript $x_{t}$, and the history up to time $t,\\left\\{x_{s}\\right\\}_{s \\leq t}$, is denoted by superscript $x^{t}$. For any set $X$, a set $X^{t}$ is the $t$-fold Cartesian product of $X$.\n\n[^4]:    ${ }^{6}$ The exact value required for a clean demonstration is $2 / 1365^{1 / 10}$.\n\n[^5]:    ${ }^{7}$ Bergemann and Välimäki (2008) provide a recent overview of the bandit experimentation literature.\n\n[^6]:    ${ }^{8}$ Such policy exists since any quitting time is measurable with respect to $m$ and, hence, with respect to past performance.\n\n[^7]:    ${ }^{9}$ This standard result is also proven in the Appendix.\n\n[^8]:    ${ }^{10}$ Waldman (2013) provides an excellent overview of the related literature.\n\n[^9]:    ${ }^{11}$ In addition, downward wage rigidity has attracted considerable attention in the development economics literature (e.g., Schmitt-Grohé and Uribe (2016); Kaur (2019)).\n    ${ }^{12}$ Relatedly, Ostrovsky and Schwarz (2010) demonstrate how grade inflation can emerge as an equilibrium grading policy in school competition.\n\n[^10]:    ${ }^{13}$ Fang and Moscarini (2005) study a similar tradeoff in a static setting.\n\n[^11]:    ${ }^{14}$ Even when seeking input from employee's supervisor, appraisal systems typically present questions in a way to minimize response bias (relative scales, numerous categories, etc.).","text_sha256":"cf1505360166ff283ed04cd3002c56a654fe90705105f1edba6ffa6ec4ae1540"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-evaluation-design:2020-10-28:0029","work_id":"alex-smolin:dynamic-evaluation-design","paper_id":"alex-smolin:dynamic-evaluation-design:2020-10-28","title":"Dynamic Evaluation Design","authors":[{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2020-10-28","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md","source_record":"https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170405","citation":"Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Alex Smolin\n\n**Canonical citation:** Smolin, Alex. “Dynamic Evaluation Design.” American Economic Journal: Microeconomics 13, no. 4 (2021): 300–331.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/dynamic-evaluation-design.md\n\n**Source record:** https://alexsmolin.com/files/dynamic-evaluation-design-working-paper.pdf\n\n**Published record:** https://doi.org/10.1257/mic.20170405\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"6c902845de0da7eae8e72e49f0e6d4c64cb4061ecffbc564fe3e5ca89aad26c6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0001","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Gleb Romanyuk; Alex Smolin.\n> Canonical citation: Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"ca2092cc06cfc6ffa323e9aee002bc35340f4a99591495334b3bef046f0077f8"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0002","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Cream Skimming and Information Design in Matching Markets","text":"# Cream Skimming and Information Design in Matching Markets\n\n**Authors:** Gleb Romanyuk; Alex Smolin\n\n**Manuscript date:** 2018-05-14\n\n#### Abstract\n\nShort-lived buyers arrive to a platform over time and randomly match with sellers. The sellers stay at the platform and sequentially decide whether to accept incoming requests. The platform designs what buyer information the sellers observe before deciding to form a match. We show full information disclosure leads to a market failure because of excessive rejections by the sellers. If sellers are homogeneous, then coarse information policies are able to restore efficiency. If sellers are heterogeneous, then simple censorship policies are often constrained efficient as shown by a novel method of calculus of variations.\n\n[^0]","text_sha256":"c06eddb0a36d7e207ce6e83fe3a1b649928ebe99797b97713245e03866bb026c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0003","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nA primary objective for matching platforms, such as platforms for ride hailing (Uber, Lyft), accommodation rental (Airbnb, Craigslist), and freelance labor (Upwork, TaskRabbit) is to facilitate value-creating transactions. Providing detailed information to the market participants about the goods or trading partners is a way to ensure that they identify and pursue the most valuable matches. Matching platforms spend significant resources eliciting matchrelevant information from users and incorporating it into the platform's design. ${ }^{1}$ Sometimes, however, the relevant information is only partially revealed to the users. As an example, Uber does not show its drivers the passenger destination until after the driver has picked up the passenger. Why do such platforms choose not to fully disclose all relevant information? On what does an optimal information policy depend?\n\nWe study the information intermediation problem in the context of a dynamic matching market. Buyers and sellers try to match on a platform over time. A stream of buyers contact sellers, who then review buyer information and choose whether or not to accept them. Buyers are short lived and indifferent to with whom seller they match. Sellers are long lived and derive heterogeneous payoffs from being matched with different buyers. The utilities are nontransferable. Importantly, the market has capacity constraints-if a seller accepts a buyer, the seller becomes unavailable for a fixed time. The platform designs a stationary and non-discriminatory information policy that governs what buyer information is revealed to sellers before they accept buyer requests. It does so to maximize a weighted average of a buyer and seller steady-state surplus.\n\nWe analyze the setting in Section 3. First, we show in Proposition 1 that for an arbitrary information policy, the steady state equilibrium exists and is essentially unique. Intuitively, the more sellers available in the steady state, the lower the rate at which they are matched with the buyers. It reduces their continuation value and increases their frequency\n\n[^1]of acceptance. This results in less sellers being available and leads to unique equilibrium characterization.\n\nSecond, we show in Proposition 2 that the full disclosure outcome is inefficient: There exists a seller's strategy profile that generates higher surplus on both market sides. The market failure is driven by cream-skimming seller behavior: Each seller tries to keep his schedule open by rejecting low-value matches in order to increase his individual chances of getting high-value matches. These rejections impose negative externalities on both sides of the market. On the buyer side, the set of matches that create positive surplus for the sellers is distinct from the set of matches that create positive surplus for the buyers. When the sellers make the acceptance decision, they do not internalize buyer surplus. On the seller side, by rejecting a buyer, a seller remains available in the marketplace and competes with other sellers for incoming requests. As a result, the other sellers face fewer valuable buyers and realize lower profits.\n\nThird, we study the platform-optimal information policies that are constrained efficient. We highlight that an information policy determines not only a seller's stage payoff, but also his continuation value. An optimal information policy must balance the positive effect of information disclosure on the match quality and the negative effect on the match rate. The effect of information disclosure on the match quality is straightforward. Holding the match rate fixed, each seller benefits from more details about buyers because it allows him to make a more informed decision. However, information disclosure can exacerbate cream-skimming behavior and decrease the match rate because greater transparency increases return on search.\n\nWe find that if sellers are homogeneous, information control can fully restore efficiency-the platform can implement any Pareto efficient outcome that delivers positive surplus to sellers. Theorem 1 demonstrates that the optimal policies are partially informative and take a simple form. They send only two possible signals, which can be interpreted as recommendations to accept or reject the buyer request. The signal likelihoods are chosen to achieve a given\noutcome.\nIf sellers are heterogeneous, then it is generally impossible to fully restore efficiency. Moreover, binary recommendation policies are less effective because different sellers respond differently to the same information. We focus on a linear payoff environment and show how the form of the optimal information policies depends on details of the seller type distribution in Theorem 2. If its density is log-concave, or the density is unimodal and the platform maximizes only the matching rate, then a censorship policy is optimal. Such a policy pools more attractive buyers and fully reveals less attractive buyers. If the seller density is weakly decreasing, then, interestingly, the full disclosure policy is optimal.\n\nWe discuss how our analysis can inform the design of digital marketplaces in Section 4. We observe that most platforms actively engage in information design by disclosing only selected information categories to sellers. We highlight how such information selection can avoid inefficient cream skimming and create a balanced market. Finally, we discuss the role that strategic buyers and flexible pricing schemes can play in the optimal platform design.","text_sha256":"f46ae61b498be5f5ae4c9bfda5cfded1313374a28af17f5fb06ad5e77826ae07"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0004","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"Related literature. The role of information in matching and peer-to-peer markets is an active area of economic research. Chade (2006) highlights an adverse selection problem leading to the acceptance curse - a negative inference about the accepting party when it is privately informed. In contrast, we focus on cream skimming driven by moral hazard. Hoppe, Moldovanu, and Sela (2009) show that the option to disclose information may lead to wasteful, costly signaling that can offset the benefits of improved matching. Coles, Kushnir, and Niederle (2013) study the case of costless but restricted signalling and show that it generally improves the match rate but can decrease the receiver's welfare. Levin and Milgrom (2010) show that in online advertising markets, standardization, or \"conflation,\" helps to address at least two problems associated with excessive targeting: adverse selection and reduced competition. Ostrovsky and Schwarz (2010) study information disclosure in a static matching market and show that equilibrium information policies provide no incentives for\nearly-contracting unraveling. Lauermann (2012) highlights the cream-skimming behavior in search markets and the resulting inefficiencies caused by information provision which is also the focus of our paper. ${ }^{2}$\n\nOur work differs from the theory of centralized matching because the market participants must inspect potential matches to identify the valuable ones. Anderson and Smith (2010) show how learning and reputation concerns can preclude positive assortative matching in dynamic markets. Ashlagi, Jaillet, and Manshadi (2013) and Akbarpour, Li, and Oveis Gharan (2017) study the role of uncertainty in centralized matching markets and show that waiting for the market to thicken improves matching in kidney exchanges because individual agents do not fully incorporate the benefits of waiting. We emphasize the opposite effect: individual sellers wait too long because they do not fully incorporate the buyer loss from rejections.\n\nAt the same time, this paper contributes to the rapidly growing literature on information design in dynamic settings (e.g., Ely (2017); Che and Hörner (2018); Smolin (2017); Orlov, Skrzypacz, and Zryumov (2018)). ${ }^{3}$ In contrast to the existing studies, we investigate stationary information policies with heterogeneous and forward-looking audience. In a related paper, Kovbasyuk and Spagnolo (2017) study information policies in a dynamic market with competitive prices and myopic players. They analyze the effect of past experience disclosure and highlight that revealing negative experiences can exclude too many sellers from the market. Notably, in solving for an optimal policy in the case of homogeneous sellers, we relate to the efficiency approach of Smolin (2017). ${ }^{4}$\n\n[^2]","text_sha256":"ef1b1b99994da684b64ebf4e16d8b41e3ded4692bf5c2fbe57f1eeb289bf0114"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0005","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"## 2 Model\n\nMatching Market. Three parties are involved in the matching process: sellers, buyers, and the platform itself. Time is continuous.\n\nThe unit mass of sellers always stays on the platform; they never leave or arrive. At each moment in time, a seller is either available or busy. An available seller is sequentially presented with buyer requests and decides whether to accept or reject them in order to maximize his total payoff. Busy sellers do not receive any requests. When an available seller accepts a buyer, he becomes busy for a fixed time $\\tau$. The higher the value of $\\tau$, the fewer buyers the seller can match with during the same period of time.\n\nBuyers gradually arrive to the platform at a flow rate $\\beta$ so that within time interval $d t$ a deterministic mass $\\beta d t$ of buyers arrives. Each new buyer contacts one of the available sellers. The seller is chosen uniformly at random from the pool of available sellers. If the buyer is accepted, he stays while the match lasts, for the time $\\tau$; otherwise, he leaves the platform.\n\nAssumption 1. (Market Capacity) It is possible for sellers to accept all buyers: $\\beta \\tau<1$.\n\nThe capacity assumption simplifies the exposition. Relaxing it requires more notation to deal with either automatic rejections or queues. Figure 1 illustrates the matching process.\n\nMatch Heterogeneity. There are two dimensions of heterogeneity in the market. First, each seller has a heterogeneous match payoff across buyers. Second, different sellers have different payoff functions. Let $x$ denote a buyer type, with the interpretation that $x$ is comprised of buyer characteristics observed by the platform. ${ }^{5}$ The buyer types are distributed over a compact convex set $X \\subseteq \\mathbb{R}$ according to a distribution function $F$ with full support. Let $y$ denote a seller type, with the interpretation that $y$ is comprised of seller characteristics\n\n[^3]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: Matching process. Buyers arrive at exogenous rate $\\beta$ and contact available sellers. If rejected, a buyer leaves the platform. If accepted, the buyer forms a match which lasts for time $\\tau$. After that time elapses, the buyer leaves the platform, and the seller returns to waiting.\n\nunobserved by the platform. ${ }^{6}$ The seller types are distributed over a compact convex set $Y \\subseteq \\mathbb{R}$ according to a distribution function $G$ with full support that admits a differentiable density $g$. The seller profit for one match is $\\pi(x, y)$, the buyer utility is $u>0 .{ }^{7}$\n\nAssumption 2. (Profit Variability) Profits $\\pi(x, y)$ are continuous and for almost all $y \\in Y$ there exist $x_{1}, x_{2} \\in X$ such that $\\pi\\left(x_{1}, y\\right) \\leqslant 0$ and $\\pi\\left(x_{2}, y\\right)>0$.\n\nIn other words, all sellers can make profit or losses on a match, whereas all buyers benefit. This condition allows us to focus on sellers' screening incentives. An important special case of a linear profit function is studied in Section 3.4.\n\nEach seller seeks to maximize an average profit flow. ${ }^{8}$ Each buyer is essentially non-\n\n[^4]strategic as it does not make any decisions. We focus solely on the information design and assume the player utilities are nontransferable. We discuss the potential role of pricing schemes in Section 4.\n\nInformation Design. Before the matching process begins, the platform designs and commits to a stationary and non-discriminatory information policy that governs what buyer information is disclosed to sellers. Formally, the platform observes buyer type $x$ and sends a signal about it to the seller. Let $S=\\Delta(X)$ be the set of all posterior distributions over $X$. An information policy $\\lambda \\in \\Delta(S)$ is a probability distribution of posteriors. ${ }^{9}$ The interpretation is that $s \\in S$ is an induced posterior of a seller and $\\lambda\\left(S^{\\prime}\\right)$ is the fraction of buyers with signals $S^{\\prime} \\subset S .{ }^{10}$ The set of possible information policies is\n\n$$\n\\left\\{\\lambda \\in \\Delta(S): \\int s \\mathrm{~d} \\lambda(s) \\sim F\\right\\} .\n$$\n\nWhen a buyer of type $x$ requests an available seller, the platform sends a signal to the seller according to $\\lambda$. The seller knows the platform's choice of $\\lambda$ and forms posterior beliefs via Bayes' rule. The full disclosure policy, denoted by $\\lambda^{F D}$, perfectly reveals buyer type $x$ to the sellers. The no disclosure policy fully conceals $x$. We say an information policy $\\lambda^{\\prime}$ is coarser than $\\lambda^{\\prime \\prime}$ if $\\lambda^{\\prime}$ is a Blackwell garbling of $\\lambda^{\\prime \\prime}$.\n\nSteady State. The matching process is the dynamic system in which sellers become repeatedly busy and available. Let $\\alpha(y) \\geqslant 0$ be the acceptance rate, the fraction of buyers accepted by type- $y$ sellers. Let $\\rho(y)$ be the utilization rate, the fraction of type- $y$ sellers who are busy. Denote the average utilization rate by $\\bar{\\rho} \\triangleq \\int_{Y} \\rho(y) d G(y)$. As the total mass of sellers is $1, \\bar{\\rho}$ is also the mass of busy sellers.\n\nThe seller's strategy is a function $\\sigma(\\cdot, y): S \\rightarrow[0,1]$ that for every seller of type $y$ maps\n\n[^5]a signal to the probability of accepting the request. The seller's acceptance rate is then:\n$$\n\\alpha(y)=\\int \\sigma(s, y) \\mathrm{d} \\lambda(s) .\n$$\n\nIn a steady state, the flow of sellers who form new matches should be equal to the flow of sellers who return to waiting. The flow of new matches is equal to the product of the buyer flow that type- $y$ sellers receive $\\beta \\frac{(1-\\rho(y)) g(y)}{1-\\bar{\\rho}}$ and type- $y$ sellers' acceptance rate $\\alpha(y)$. The flow of returning sellers is equal to $g(y) \\rho(y) / \\tau$, because the mass of busy type- $y$ sellers is $g(y) \\rho(y)$ and matches last time $\\tau$. Hence, a steady state is accounted by the following equation:\n\n$$\n\\beta \\frac{g(y)(1-\\rho(y))}{1-\\bar{\\rho}} \\alpha(y)=\\frac{g(y) \\rho(y)}{\\tau}, \\quad \\forall y \\in Y .\n$$","text_sha256":"9223f7fb4a463e68c0d2af31d13dc28f1f4e9126623f8382c8458945f625603f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0006","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"In a steady state, buyers request an available seller at a Poisson rate denoted by $\\beta_{A}$. On one hand, an individual available seller faces a stochastic request process such that the probability of a new request over the time interval $d t$ is $\\beta_{A} d t+o(d t)$. On the other hand, the available sellers jointly face the deterministic arrival process of buyers, where over time interval $d t$ the mass $\\beta d t$ of buyers arrives. Uniform assignment of buyers across sellers results in a particularly simple form of the relationship between the two rates (Myerson (2000)):\n\n$$\n\\beta_{A} \\triangleq \\frac{\\beta}{1-\\bar{\\rho}} .\n$$\n\nA steady-state equilibrium is a market outcome in which the sellers take the request rate $\\beta_{A}$ as given and optimize independently, and the busy-available seller flows balance.\n\nDefinition 1. The pair $(\\sigma, \\bar{\\rho})$, where $\\sigma: S \\times Y \\rightarrow[0,1]$ is the sellers' strategies and $\\bar{\\rho} \\in[0,1]$ is the mass of busy sellers, constitutes a steady-state equilibrium if the following hold:\n\n1. (Steady state) The average utilization rate $\\bar{\\rho}$ is given by (2) and (3). The request rate $\\beta_{A}$ is given by (4).\n2. (Optimality) For all $y$ and every type- $y$ seller, $\\sigma(\\cdot, y)$ is an optimal strategy given the\n\nrequest rate $\\beta_{A}$ and an information policy $\\lambda$.\n\nThe platform's objective is to maximize a weighted average of the buyer surplus and the joint seller profits in a steady-state equilibrium. For a given steady-state equilibrium, define by $V(y)$ the seller- $y$ profits, by $V=\\int_{Y} V(y) \\mathrm{d} G(y)$ the joint seller profits, and by $U$ the buyers' utility. The platform then formally maximizes\n\n$$\n\\mathcal{J}(\\gamma)=\\gamma U+(1-\\gamma) V .\n$$\n\nThe general objective $\\mathcal{J}(\\gamma)$ includes as special cases maximization of total surplus at $\\gamma=1 / 2$, maximization of the joint seller profits at $\\gamma=0$ and maximization of buyer surplus at $\\gamma=1$. As $\\gamma$ spans the interval [0, 1] the platform-optimal allocations span constrained-Pareto-efficient allocations.\n\nAssumption Discussion. Before proceeding with the main analysis, we highlight two important features of our model that are motivated by the stylized facts about the online platforms.\n\nFirst, no two buyers simultaneously contact the same seller. Such coordination frictions in matching markets have been extensively studied in the theoretical literature (Burdett, Shi, and Wright (2001); Kircher (2009)), and digital platforms now usually have good technological means of resolving the simultaneity-driven friction. Instead, we focus on the matching friction that pertains to preference heterogeneity and screening. ${ }^{11}$\n\nSecond, the buyers make a single search attempt and pick the seller uniformly at random. The single search attempts can be attributed to rejection intolerance and capture an aspect of real-life matching markets in that rejections are costly to buyers (wasted time, search efforts, bidding costs). ${ }^{12}$ It allows us to not have an endogenous distribution of buyer types.\n\n[^6]The uniform assignment is plausible in settings in which the buyers are indifferent across the sellers or the search costs are prohibitively high. It allows us to analyze the match-quality and match-rate tradeoff in a cleaner model, in which any seller faces the same request rate, $\\beta_{A}$, and the same distribution of buyer types, $F$. Altogether, this implies that the buyers are effectively non-strategic and allows us to focus on the sellers' side of the market. We discuss possible implications of buyers' strategic behavior in Section 4.","text_sha256":"5efa3d43fb8456e67330a77dc8732b0c9b292cb2ef68f24789e203085b2a5c5e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0007","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Analysis","text":"## 3 Analysis\n\nWe proceed with studying how the platform's information policy affects the equilibrium market outcome. We show that for an arbitrary information policy, an equilibrium exists and is essentially unique. We then show that full disclosure policy aggravates cream skimming and produces an inefficient market outcome. We then characterize optimal information policies in the cases of homogeneous sellers and vertically differentiated sellers.","text_sha256":"c922c0adcb70a72be830aa80f98d62505a1f7d72e8132b6a3ad17cd30aca0726"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0008","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.1 Equilibrium Existence and Uniqueness","text":"### 3.1 Equilibrium Existence and Uniqueness\n\nFor a given information policy, each seller is facing a dynamic decision problem of maximizing an average profit flow. He is presented with a sequence of buyers arriving at a Poisson rate $\\beta_{A}$. For each buyer request, the seller observes the signal realization $s$ and chooses whether to accept the request (see Figure 2). With a small abuse of notation, let $\\pi(s, y) \\triangleq \\mathbb{E}[\\pi(x, y) \\mid s]$ be the type- $y$ seller's flow profit if he accepts a buyer with signal $s$ and let $V(y)$ be the seller's value function, an average profit flow when he acts optimally. ${ }^{13}$ An opportunity cost of accepting is equal to $\\tau V(y)$, and the seller optimization problem corresponds to the following Bellman equation\n\n[^7]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 2: Seller's dynamic optimization problem with screening and waiting. An available seller receives buyer requests at a Poisson rate $\\beta_{A}$. If a request is accepted, the seller becomes busy for time $\\tau$, during which he does not receive new requests.\n\n$$\nV(y)=\\beta_{A} \\int \\max \\{0, \\pi(s, y)-\\tau V(y)\\} \\mathrm{d} \\lambda(s) .\n$$\n\nIt is immediate that an optimal seller strategy is cutoff with him accepting all buyers with $s$ such that $\\pi(s, y)>\\hat{\\pi}$ and rejecting all buyers with $s$ such that $\\pi(s, y)<\\hat{\\pi}$. The optimal threshold $\\hat{\\pi}$ naturally depends on the information policy.\n\nProposition 1. (Existence and Uniqueness) For an arbitrary information policy $\\lambda$, a steadystate equilibrium exists and is unique up to the acceptance of marginal buyers: if $(\\sigma, \\bar{\\rho})$ and $\\left(\\sigma^{\\prime}, \\bar{\\rho}^{\\prime}\\right)$ are two steady-state equilibria, then (1) $\\bar{\\rho}=\\bar{\\rho}^{\\prime}$, (2) for any $y \\in Y, \\sigma(\\cdot, y)$ and $\\sigma^{\\prime}(\\cdot, y)$ coincide except on the set $\\{s: \\pi(s, y)=\\tau V(y)\\}$, and (3) the surpluses coincide, $V(y) \\equiv V^{\\prime}(y), U=U^{\\prime}$.\n\nTo prove this result, we show that for an arbitrary acceptance rate profile $\\alpha(y)$, there is a unique steady-state value of the average utilization rate $\\bar{\\rho}$. The equilibrium existence and uniqueness roughly follows from the continuity and monotonicity of the reaction curves of $\\alpha$ in $\\bar{\\rho}$ and $\\bar{\\rho}$ in $\\alpha$. On one hand, if the average utilization rate $\\bar{\\rho}$ increases, then the request traffic to each available seller increases. As a result, sellers become pickier, threshold $\\hat{\\pi}$ increases, and the acceptance rate $\\alpha(y)$ decreases. On the other hand, if $\\alpha(y)$ increases,\nsellers become less available, and $\\bar{\\rho}$ strictly decreases. The equilibrium values of $\\alpha$ and $\\bar{\\rho}$ then determine the sellers' strategies up to the acceptance of marginal buyers and, in turn, the sellers' surpluses. The buyer surplus is uniquely determined by the utilization rate as well.","text_sha256":"3e28d9b9fd090bc25b654531ff65bd78bc2e06dc58e806c6f17920273c303454"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0009","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.2 Market Failure Under Full Disclosure","text":"### 3.2 Market Failure Under Full Disclosure\n\nIn this subsection, we establish that full disclosure leads to a Pareto inefficient market outcome. The market failure is driven by the sellers' decentralized decision-making and their capacity constraints.\n\nWe first define Pareto efficiency in our setting. An outcome $O=(V(\\cdot), U)$ is a combination of seller profit profile and buyer surplus. We say that a market outcome is feasible if there is a seller strategy profile under full disclosure that achieves it. A feasible outcome $O$ is Pareto efficient if there is no other feasible $O^{\\prime}$ such that $V^{\\prime}(y) \\geqslant V(y)$ for all $y$ and $U^{\\prime} \\geqslant U$, and at least one seller type or buyers are strictly better off. The Pareto frontier is the set of all Pareto-efficient outcomes. An outcome $O$ is implementable if there is an information policy that induces it in a steady state equilibrium.\n\nImagine that the platform starts with full disclosure as its default information policy. Write $\\alpha^{F D}(y), V^{F D}(y), U^{F D}$ for equilibrium acceptance rates, seller profits, and consumer surplus. Correspondingly, write $\\alpha^{\\sigma}(y), V^{\\sigma}(y), U^{\\sigma}$ for these parameters when a given strategy profile $\\sigma$ is played.\n\nProposition 2. (Market Failure) There exists a strategy profile $\\sigma$ under which all market participants match more often and are strictly better off than in a full-disclosure equilibrium. That is for all $y \\in Y$ :\n\n$$\n\\begin{aligned}\n\\alpha^{\\sigma}(y) & >\\alpha^{F D}(y), \\\\\nV^{\\sigma}(y) & >V^{F D}(y), \\\\\nU^{\\sigma} & >U^{F D} .\n\\end{aligned}\n$$\n\nA full proof is given in the Appendix. The basic intuition for the coordination problem is cream skimming: A seller keeps his schedule open by rejecting low-value matches in order to increase his own chances of getting high-value matches. As a result, in equilibrium, sellers spend considerable time waiting for high-value matches. Collectively, this behavior is suboptimal because some low-value matches have to be accepted to maximize the joint seller surplus. The market feature that gives rise to the coordination failure is that collectively, the sellers are not capacity constrained, whereas individually, they are capacity constrained. Notably, unlike price competition that benefits buyers and improves market efficiency, seller competition for better buyers hurts buyers and decreases market efficiency.\n\nIt is instructive to highlight the inefficiencies on the seller and the buyer sides of the market as outlined in the introduction. Each seller rejects low-value matches to increase his chances of getting high-value matches. Such rejections directly harm the buyers because the sellers do not internalize their surplus. Such rejections hurt other sellers as well, because the seller remains available on the marketplace and competes with other sellers for incoming buyers.\n\nCan information design mitigate the market failure? To answer this question, it is important to understand competing effects of information provision. Define the match rate $m$ as the number of matches formed on the platform over a unit time interval. Higher $m$ implies that more buyers are served, and the buyer surplus is proportional to it, $U=m u$. Similarly, define the seller average match quality by $q$ so that $V=m q$. Evidently, both surpluses increase in the match rate $m$. However, the match rate and the match quality are in conflict, and the information disclosure affects them through three different channels.\n\nFirst, information provision has a positive effect on the seller match quality. From an individual seller's point of view, more information increases his set of attainable payoffs. Holding the match rate fixed, he individually benefits from more information about buyers. Second, information provision reduces the match rate by allowing the sellers to reject less valuable buyers. The platform can limit information and induce sellers to accept more\nbuyers. This effect is reminiscent of static persuasion settings. Third, information provision reduces the match rate through increasing sellers' return on search. With returns on search, the sellers reject more often, and the match rate goes down.\n\nThe match quality effect motivates the platform to provide more information, while the effects on match rate work in the opposite direction. The interaction of these effects depends on the primitives of the economic environment and shapes the preferred information policy. In the next section, we show that with homogeneous sellers, information design can fully mitigate the market failure and achieve an efficient allocation.","text_sha256":"a68123deb57cbe274a477a8bb25800157a11a44d34c7cde9823874491b7c7c3a"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0010","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.3 Optimal Policy with Homogeneous Sellers","text":"### 3.3 Optimal Policy with Homogeneous Sellers\n\nWe start with the case of homogeneous sellers so that $Y$ is a singleton or, equivalently:\n\n$$\n\\pi(x, y)=\\pi(x) .\n$$\n\nIt is instructive to think about what seller strategies are Pareto efficient in this environment. It is straightforward to see that any efficient strategy is cutoff-all sellers should accept buyers that generates profits above some threshold. Intuitively, any given acceptance rate $\\alpha$ results in the same buyer utility and a cutoff strategy maximizes the seller surplus among all strategies with the same acceptance rate. Conversely, a cutoff monotonically affects the acceptance rate and hence corresponds to some efficient outcome. We show that information control allows the platform to restore efficiency and bring the payoffs to the Pareto frontier.\n\nTheorem 1. (Homogeneous Sellers) Suppose the sellers are homogeneous. Then, for any Pareto-efficient outcome ( $U, V$ ) with $V \\geqslant 0$, there is an information policy that sends only two signals that implement $i t$.\n\nThe proof follows the efficiency approach of Smolin (2017). ${ }^{14}$ Consider an arbitrary seller\n\n[^8]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 3: A sketch of a set of feasible and implementable outcomes in the case of homogeneous sellers. $V$ denotes the seller surplus; $U$ denotes buyer surplus. Any point on the Pareto frontier (thick solid line) can be implemented by some information policy. The full-disclosure outcome is suboptimal.\n\nstrategy that under full disclosure achieves a Pareto-efficient outcome with positive seller surplus. Consider an information policy that sends two signals and effectively recommends an action according to the strategy. Under the recommendation policy, there is only one signal with a recommendation to accept, so there is only one type of acceptable buyer. Hence, all profitable buyers are the same, and there is no return to cream skimming. For any Pareto-efficient outcome that delivers a positive surplus to sellers, they are willing to obey the recommendations.\n\nTheorem 1 characterizes at once the range of optimal information policies corresponding to different platform objectives as any point on the Pareto frontier maximizes $\\gamma U+(1-\\gamma) V$ for some $\\gamma \\in[0,1]$. Figure 3 illustrates the result. Note that by Proposition 2, the fulldisclosure equilibrium outcome is outside of the Pareto frontier.\n\nNow, note that in the case of homogeneous sellers, Pareto-efficient allocations do not depend on exact parameters $\\beta$ and $\\tau$. We have an immediate corollary.\n\nCorollary 1. Suppose the sellers are homogeneous. The optimal information policy $\\lambda^{*}$ depends only on the objective weights $\\gamma$ and not on the buyer traffic intensity $\\beta$ or the task\nlength $\\tau$.\n\nThis corollary implies that the same information policy would be optimal if sellers became available immediately after accepting a buyer, $\\tau=0$. The independence from $\\beta$ and $\\tau$ is driven by the coarse structure of an optimal policy. The rate of arrival to the available sellers, $\\beta_{A}$, matters to sellers only to the extent that a higher request rate magnifies the opportunity cost of accepting. With a binary policy, this opportunity cost is nil because all profitable buyers are the same.\n\nIn the next section, we study the setting with heterogeneous sellers and show that the optimal policy is richer and does depend on $\\beta$ and $\\tau$.","text_sha256":"0681914f771d3e7ff2bc455bbc9df391783661b70c08f628e9472e9156fd04b3"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0011","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.4 Optimal Policy with Heterogeneous Sellers","text":"### 3.4 Optimal Policy with Heterogeneous Sellers\n\nWe proceed with studying the setting in which the sellers feature vertical payoff heterogeneity. We characterize the optimal information policy and show that, unlike in the case of homogeneous sellers, the Pareto frontier is generally not attainable.\n\nFormally, consider the linear payoff environment in which the seller profit is a linear function of buyer and seller types: $X=[0, \\bar{x}], Y=[0, \\bar{y}]$, and\n\n$$\n\\pi(x, y)=y-x .\n$$\n\nWe interpret $x$ as the level of difficulty of the buyer's task and interpret $y$ as the seller's skill level. In this way, buyers are vertically differentiated in the eyes of the sellers, and high-type sellers can profitably match with more buyers than can low-type sellers. Note that by the profit variability assumption, $\\bar{y} \\leqslant \\bar{x}$.\n\nSimilar to the case of a homogeneous sellers, it is easy to see that the Pareto efficient strategies are cutoff-all seller types should accept buyers below a threshold which can now be type-dependent. At the same time, choosing the cutoff profile to maximize the total seller surplus should account for the congestion externalities due to uniform assignment. In\ngeneral, this profile depends on the details of seller and buyer type distributions. However, it is clear that the efficient cutoffs are generically different from their types and vary across them.\n\nFinding an optimal information policy in this environment is a challenging problem. First, the sellers' preferences have unobserved heterogeneity and, as discussed above, Pareto efficient strategies differ across sellers. This precludes the effective use of the recommendation policies that provided a solution in the case of homogeneous sellers. Second, the sellers are forward looking, and the information policy determines not only their stage payoff but also their continuation values. As a result, a seller's decision depends not only on the posterior mean of the current buyer type but on the general shape of the information policy.\n\nTo solve the problem, we develop a variational approach in which we map information policies to a class of convex functions and then use the calculus of variation to recover a structure of optimal information policies. First, we follow Gentzkow and Kamenica (2016) and show that optimizing $\\mathcal{J}$ with respect to information policies is equivalent to optimizing it with respect to a class of convex functions. Namely, for a given information policy $\\lambda$, denote the posterior mean of $x$ conditional on signal $s$ by $z(s) \\triangleq \\int_{X} x \\mathrm{~d} s(x)$, the induced distribution of $z(s)$ by $H$, and define the spread function $\\Lambda:[0, \\infty) \\rightarrow \\mathbb{R}_{+}$as\n\n$$\n\\Lambda(z) \\triangleq \\int_{0}^{z} H(\\zeta) \\mathrm{d} \\zeta\n$$\n\nIn the linear payoff environment, $\\Lambda$ is proportional to the option value of rejecting a buyer with expected cost $z$. Let $\\bar{\\Lambda}$ be the spread function under full disclosure, $\\bar{\\Lambda}(z)=\\int_{0}^{z} F(\\zeta) \\mathrm{d} \\zeta$, and let $\\underline{\\Lambda}$ be the spread function under no disclosure, $\\underline{\\Lambda}(z)=\\max \\{0, z-\\mathbb{E}[x]\\}$. The argument of Gentzkow and Kamenica (2016) establishes a one-to-one correspondence between the set of admissible posterior mean distributions and the following set,\n\n$$\n\\mathcal{L} \\triangleq\\left\\{\\Lambda:[0, \\infty) \\rightarrow \\mathbb{R}_{+}, \\text {convex, and } \\underline{\\Lambda} \\leqslant \\Lambda \\leqslant \\bar{\\Lambda}\\right\\} .\n$$\n\nAs a result, maximization over information policies is equivalent to maximization over admissible spread functions $\\Lambda \\in \\mathcal{L}$.\n\nWe then build on the analysis of Section 3.1 and show that for any information policy, the seller's optimal strategy has a simple cutoff structure: a type- $y$ seller accepts all buyers with expected difficulty below the cutoff $\\hat{z}(y)$ and rejects all buyers above it. The cutoff profile has a simple structure-it is continuously increasing in $y$ and satisfies the following equation\n\n$$\n\\frac{y-\\hat{z}(y)}{\\tau \\beta_{A}(\\Lambda)}=\\Lambda(\\hat{z}(y)) .\n$$\n\nThe $\\beta_{A}(\\Lambda)$ is an equilibrium request rate that is linked to the seller behavior by (3) and (4).\nThe condition (11) allows easy illustration of cream skimming in Figure 4. Under cream skimming, a type- $y$ seller accepts buyers with expected difficulty $z$ not below his type $y$ but below a cutoff $\\hat{z}(y)<y$. The cutoff can be seen as a projection of the type reflected against the spread function along the trajectory with the slope inversely proportional to the request rate $\\beta_{A}$ and the task length $\\tau$. The cutoff is, hence, increasing with type-higher seller types accept more buyers, cream skimming less. ${ }^{15}$ The higher the request rate, the flatter the slope and the greater the extent of cream skimming. Similarly, the lower the task length, the steeper the slope and the lower the extent of cream skimming. In the limit case as $\\tau \\rightarrow 0$, the slope is vertical, there is no cream skimming, $\\hat{z}(y)=y$, and the problem can be seen as a static persuasion of Kolotilin, Mylovanov, Zapechelnyuk, and Li (2017).\n\nNext, to solve for an optimal spread function, we refer to the calculus of variations and consider effects of small spread-function variations on the platform's objective. Namely, we consider variations $\\varepsilon h:[0, \\infty) \\rightarrow \\mathbb{R}_{+}$such that $\\Lambda+\\varepsilon h \\in \\mathcal{L}$. We use the equilibrium conditions and calculate a variational derivative of the platform objective at $\\varepsilon=0$ :\n\n$$\n\\frac{d \\mathcal{J}}{d \\varepsilon}(0)=\\int_{0}^{\\bar{y}} h(\\hat{z}(y)) \\nu(y)\\left(-\\frac{g^{\\prime}(y)}{g(y)} \\Psi+(1-\\gamma) \\frac{\\beta}{\\bar{\\nu}}\\right) g(y) \\mathrm{d} y+h(\\hat{z}(\\bar{y})) \\nu(\\bar{y}) g(\\bar{y}) \\Psi .\n$$\n\n[^9]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 4: Admissible spread functions $\\Lambda$ and equilibrium acceptance cutoffs $\\hat{z}(y)$.\n\nHere, $\\nu(y) \\triangleq 1-\\rho(y), \\bar{\\nu} \\triangleq 1-\\bar{\\rho}$ are the equilibrium availability rates under the policy $\\Lambda$, and $\\Psi>0$ is a positive term that depends on the objective weight $\\gamma$ and the aggregate equilibrium behavior. The integral term captures the relative distribution of the platform's surplus across the seller types, whereas the stand-alone term captures its aggregate level.","text_sha256":"764881a54bee00a65dbe5124d0e287a4fc5f6f96e32a402f7934525e94156bb1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0012","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.4 Optimal Policy with Heterogeneous Sellers","text":"The behavior of the integrand expression shows the direction in which $\\mathcal{J}$ can be locally improved. If $g$ is decreasing, then the integrand is positive everywhere. Any positive admissible variation increases $\\mathcal{J}$, pushing the spread function towards full disclosure. If $g$ is log-concave, then the integrand crosses zero from below at most once. ${ }^{16}$ The same thing happens if $g$ is unimodal and the platform maximizes the matching rate, $\\gamma=1$. In these cases, an improving deviation pushes towards more disclosure in the upper tail of $x$ and less disclosure in the lower tail. We show that these deviations lead to censorship policies.\n\nDefinition. An information policy $\\lambda$ is a lower-censorship policy if for some $\\hat{x} \\in[0,1], \\lambda$ fully reveals all $x>\\hat{x}$ and pools all $x<\\hat{x}$.\n\nTheorem 2. (Heterogeneous Sellers) In the linear payoff environment:\n\n1. If the seller distribution $g$ is log-concave, then a lower-censorship policy is optimal;\n\n[^10]\n\n2. If the seller distribution $g$ is weakly decreasing, then a full disclosure policy is optimal;\n3. If the platform maximizes only the matching rate, $\\gamma=1$, and the seller distribution $g$ is unimodal then a lower-censorship policy is optimal.\n\nOptimal disclosure depends on the shape of seller type distribution because the optimal policy should satisfy incentive constraints for the sellers. Suppose there are two types of sellers, professionals and amateurs, and two types of tasks, easy and hard. Professionals accrue relatively higher value from tasks than do amateurs; but assume that under full disclosure, both professionals and amateurs accept only easy tasks. Limiting information may cause opposite reactions from them. Consider an information policy that pools easy tasks with hard tasks. A professional's average profit from the pooled task is positive, and so he starts accepting hard tasks. Conversely, an amateur's average profit from the pooled task is negative, and so he starts rejecting even easy tasks. Because the platform does not observe the seller type, the optimal information policy should strike a balance between professionals and amateurs and thus depends on their relative population sizes. If there are sufficiently many professionals, then no disclosure is optimal; if there are sufficiently many amateurs, then full disclosure is optimal.\n\nTheorem 2 extends this binary intuition to the case of many seller types. It shows that for a broad class of type distributions, lower-censorship policies are optimal. These policies pool attractive buyers with infra-marginal buyers and fully reveal the less attractive buyers. They effectively provide no useful information for low-type sellers but bring a lot of informational value to high-type sellers. Under these policies, the worst sellers are excluded from the platform, the medium sellers accept only the pooled buyers, and the best sellers screen among the less attractive buyers. In this way, the platform optimally trades off the buyer and seller surpluses.","text_sha256":"10fe7b9182e7789f696ad66436677050020dc649817f17dd7aac3a80b39fc8bc"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0013","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Applications","text":"## 4 Applications\n\nStrategic information coarsening is relevant in practice and can be found in the designs of many platforms. In this section, we discuss how our model relates to ride-hailing, accommodation, and freelance agencies, and how our analysis can guide their design.\n\n- Ride-Hailing Platforms. Sellers and buyers are, respectively, drivers and passengers. When idle, drivers receive requests from passengers. The driver type captures his home location, preference for long rides, and tolerance for congestion. The rider type captures the destination and ride history. Drivers do not like short rides or rides to remote neighborhoods. ${ }^{17}$ Passengers do not like waiting. ${ }^{18}$ The leading agency Uber applies fixed pricing per mile and per minute that can depend only on aggregate multipliers, such as surge in demand. ${ }^{19}$ The passenger destination is fully concealed and represents the platform information coarsening. ${ }^{20}$\n- Accommodation Rental Platforms. Sellers and buyers are, respectively, hosts and guests. Hosts are capacity constrained: once a room is booked, the host cannot accept a better guest. The host type captures his preference for age, race, personality, and daily schedule. The guest type captures his gender, age, socio-economic status, and lifestyle. In the leading agency Airbnb, every host sets a price that applies for all guests, but he may prefer to reject a guest whom he expects to be a bad fit. The host can adopt the \"Instant Book\" feature, with which the host commits to accepting all guests filtered by a few characteristics (e.g., pets, smoking, infants). The choice of filters acts as an implicit information coarsening.\n\n[^11]\n\n- Freelance Labor Platforms. Sellers and buyers are, respectively, service providers and clients. The service providers are constrained in the number of tasks they can do per week. The provider's type captures his skills and work ethic. The client type captures the task category and difficulty, and the client's expertise and location. In the leading agency TaskRabbit, the service providers set an hourly rate that applies to all tasks in the same category. Pulling many tasks into a single category is a form of information coarsening.\n\nOur results shed light on why limiting buyer information can improve performance of these platforms. In an example of ride-hailing platforms, if drivers have too detailed information about passenger requests, they cream skim. This results in low acceptance rates and a suboptimal demand-supply fit. The platform's policy of hiding the passenger destination from drivers can be used to force drivers to accept more rides and avoid the inefficient cream-skimming behavior.\n\nAt the same time, we show that providing some information is often beneficial. In particular, we argue that censorship policies that reveal the least valuable buyers and pool the most valuable buyers with intermediate ones can be particularly useful. In the case of ride-hailing platforms, a censorship policy can be implemented as follows. The best driver destinations are pooled with intermediate destinations. The worst destinations, like those to remote places, are flagged and accompanied with a precise description. In equilibrium, the pooled destinations are served by all drivers active on the platform. The flagged destinations are served only by the most willing drivers, depending on the description details. This policy optimally balances between the platform's objectives-it provides sufficient information and surplus to the drivers but insures a high matching rate for the riders.\n\nBefore concluding, we discuss two considerations we abstracted away from but that might be important and should be taken into account in some applications: strategic buyers and optimal pricing.\n\nStrategic Buyers. If buyers have heterogeneous match quality and search for better matches, then the platform might face the information design possibilities on both sides of the market. Disclosure to buyers can play several roles, depending on their strategic options.\n\nFirst, the platform can use the disclosure to direct buyers to the sellers with higher acceptance rates. Imagine buyers observe seller acceptance rates imperfectly. Then they request selective sellers too often compared to what would be efficient. The platform can pool selective and non-selective sellers to shift buyer flow towards non-selective sellers.\n\nSecond, the platform can use information disclosure to direct buyers to the sellers who value them the most. Imagine that the prices do not reflect the seller match payoffs perfectly. Then, the information policy that pools matches that are less valuable for buyers and more valuable for sellers with those that are, conversely, more valuable for buyers and less valuable for sellers, will improve overall efficiency. ${ }^{21}$ In particular, one consider a natural extension of our model in which the buyers have differential tastes over sellers. Our analysis can inform this setting as well. If both sides of the market are homogeneous, then the argument analogous to that of Section Section 3.3, can be used to establish that the platform can again fully restore efficiency. If the sides are heterogeneous, then the analysis is more complicated but we expect the variational approach still being useful, even if more tedious. Moreover, we conjecture that the lower-censorship policies would continue to play an important role in analysis.\n\nOptimal Pricing. Throughout the paper, we assumed the participants' payoffs are nontransferable. In practice, a platform may want to complement the information design with the pricing design. Such complementation is straightforward if the pricing is uniform and takes the form of a fixed amount a buyer should pay to a seller if he is accepted. In this case,\n\n[^12]the platform should simply perform backward induction optimization. For any given price, the platform can calculate the matching payoffs and find the optimal information policy and its corresponding payoffs by using our analysis. In the first step, the platform can then choose the price optimal.","text_sha256":"00af147870a03da5b6514841fec7b92cbbb65f71b733de36c5629c62f1cea4c9"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0014","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Applications","text":"In fact, if the sellers are homogeneous, then the uniform pricing is sufficient to implement an efficient outcome that maximizes joint surplus. Indeed, Theorem 1 establishes that any Pareto-efficient outcome can be implemented by some information policy as long as the sellers receive some rents. If the efficient outcome delivers positive rents to the sellers, then no additional price is required. Otherwise, the platform can set the price to subsidize the sellers and give them sufficient rent to participate in the platform. Then, by the same argument as in the main analysis, there exists a recommendation policy that implements the outcome.\n\nIf the sellers are heterogeneous, then Theorem 2 shows that the optimal information policy need not even achieve Pareto frontier, so that flexible pricing rules can possibly outperform the uniform pricing. However, most matching platforms adopt simple pricing. There could be several explanations. First, consumers appreciate transparent and simple pricing, both because it is easier to assess the cost of a ride and because complex pricing can feel like price gouging and prompt the risk of price-discrimination litigation. Second, flexible pricing is a non-trivial development task that requires significant resources. Relatedly, it may be impractical to condition on every variable relevant in the transportation market. There is some complexity threshold for the designer beyond which increasing the price complexity is not justified. ${ }^{22}$\n\nAs a final remark, we note that often the platforms accompany uniform pricing with a commission proportional to the price of each match. In these cases, maximizing a profit flow is equivalent to maximizing a matching rate. Our results then point out that censorship policies can be useful to bring the most revenue to the platform.\n\n[^13]","text_sha256":"95c0c4af1bd1d0967eb7256a660c61731840255635e101e851180c4ea60e37ab"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0015","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Conclusion","text":"## 5 Conclusion\n\nStrategic information disclosure is arguably less intrusive than direct control-and often more palatable than sophisticated pricing schemes. When the two sides of the market are not symmetric in their preferences for match quality and match rate, the more patient and selective side of the market tends to cream skim. In this paper, we provide a case for using information design as a way to mitigate the adverse effects of cream skimming and improve platform performance. Specifically, we highlight the use of censorship policies as a way to balance surpluses on both sides of the market.\n\nFor most of this paper, we isolated information design from other tools available to platforms. Flexible pricing, real-time auctions, recommender systems, and queue management can also help improve matching market performance. We believe that studying interaction between these schemes can complement our current insights and constitutes a promising avenue for further research.","text_sha256":"dfa47fd084105ba0d016262c0c7d0021c12d4fce3bd1bbaeb73ed48185125cb4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0016","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Appendix","text":"## 6 Appendix\n\nLemma 1. For any information policy $\\lambda$ and an equilibrium acceptance rate profile $\\alpha(y)$ :\n\n- the average utilization rate $\\bar{\\rho} \\in[0,1]$ is a solution to\n$$\n1=\\int \\frac{1}{1-\\bar{\\rho}+\\beta \\tau \\alpha(y)} \\mathrm{d} G(y) ;\n$$\n- the solution $\\bar{\\rho}$ exists and is unique; it increases in $\\beta, \\tau$, and whenever $\\alpha(y)$ is increased on a set of strictly positive measure with respect to $G$;\n- the utilization rate $\\rho(y)$ increases whenever $\\alpha\\left(y^{\\prime}\\right)$ is increased on a set of strictly positive measure with respect to $G$.\n\nProof. By (3):\n\n$$\n1-\\rho(y)=\\frac{1}{1+\\tau \\beta \\alpha(y) /(1-\\bar{\\rho})}\n$$\n\nIntegrate both sides with respect to $G(y)$ and rearrange to obtain (13). Its right-hand side is continuously and strictly increasing in $\\bar{\\rho}$. Evaluated at $\\bar{\\rho}=0$, it equals $\\int \\frac{d G(y)}{1+\\beta \\tau \\alpha(y)} \\leqslant 1$; evaluated at $\\bar{\\rho}=1$, it equals $\\int \\frac{d G(y)}{\\beta \\tau \\alpha(y)} \\geqslant 1$, by the aggregate capacity assumption. Therefore, a solution exists and is unique. The monotonicity of $\\bar{\\rho}$ follows. Finally, consider a perturbation of increasing $\\alpha$ on some positive measure of $y^{\\prime} \\neq y$. By the argument above, $\\bar{\\rho}$ increases so, by (14), $\\rho(y)$ increases. $\\square$\n\nLemma 2. For any given information policy $\\lambda$ and utilization rate $\\bar{\\rho}$, the sellers optimal strategies are cutoff:\n\n$$\n\\sigma(s, y)= \\begin{cases}1, & \\pi(s, y)>\\hat{\\pi}(y) \\\\ 0, & \\pi(s, y)<\\hat{\\pi}(y)\\end{cases}\n$$\n\nThe cutoff function $\\hat{\\pi}(y)$ is unique and for all types $\\hat{\\pi}(y)>0$ and $\\alpha(y)<1$.\nProof. Fix $y \\in Y$. By (6), the optimal seller strategy is such that all signals from $S_{a}(y) \\triangleq$ $\\{s: \\pi(s, y)>\\tau V(y)\\}$ are accepted and all signals from $S_{r}(y) \\triangleq\\{s: \\pi(s, y)<\\tau V(y)\\}$ are rejected. The seller is indifferent between accepting and rejecting signals from $S_{m}(y) \\triangleq$ $\\{s: \\pi(s, y)=\\tau V(y)\\}$. The cutoff for a type- $y$ seller is $\\hat{\\pi}(y)=\\tau V(y)$. Using (6), the cutoff of the optimal strategy is a solution to\n\n$$\n\\hat{\\pi}(y)=\\tau \\beta_{A} \\int \\max \\{0, \\pi(s, y)-\\hat{\\pi}(y)\\} \\lambda(\\mathrm{d} s) .\n$$\n\nThe solution is unique because the left-hand side of (15) is strictly increasing in $\\hat{\\pi}(y)$ while the right-hand side is decreasing in $\\hat{\\pi}(y)$. By the variability assumption, $\\hat{\\pi}(y)>0$. Finally, under the full disclosure policy, $\\pi$ is continuously distributed, so $\\alpha(y)<1$. $\\square$\n\nProof of Proposition 1. Step 1. Existence. Consider the correspondence $\\psi:[0,1] \\rightrightarrows[0,1]$ which maps $\\bar{\\rho}$ to a set of \"reaction\" $\\bar{\\rho}$ 's by the following procedure. First, find the unique\ncutoff function $\\hat{\\pi}$ from (15) with $\\beta_{A}=\\beta /(1-\\bar{\\rho})$. The cutoff $\\hat{\\pi}$ does not pin down the acceptance rate profile $\\alpha(y)$ uniquely because marginal signals can have positive probability under $\\lambda$. The acceptance rates consistent with $\\hat{\\pi}(y)$ are integrable functions such that\n\n$$\n\\lambda\\left(S_{a}(y)\\right) \\leqslant \\alpha(y) \\leqslant \\lambda\\left(S_{a}(y)\\right)+\\lambda\\left(S_{m}(y)\\right), \\quad \\forall y \\in Y .\n$$\n\nDenote by $\\mathcal{A}$ the set of $\\alpha$ consistent with $\\hat{\\pi}$. For any $\\alpha \\in \\mathcal{A}$, find $\\bar{\\rho}$ as shown in Lemma 1. Going over all $\\mathcal{A}$ will produce the set $\\psi(\\bar{\\rho})$.\n\nBy construction, $\\mathcal{A}$ is convex. By Lemma 1, $\\bar{\\rho}$ is continuous in $\\alpha$, so $\\psi(\\bar{\\rho})$ is a closed interval in [0, 1]. Moreover, $\\psi$ is upper hemicontinuous because $\\hat{\\pi}$ is continuous in $\\beta_{A}$ according to (15). By Kakutani's theorem, $\\psi$ has a fixed point, which is a solution.\n\nStep 2. Uniqueness. Towards a contradiction, suppose $\\bar{\\rho}$ and $\\bar{\\rho}^{\\prime}, \\bar{\\rho}>\\bar{\\rho}^{\\prime}$ are two distinct fixed points of $\\psi$. By uniformity of assignment, $\\beta_{A}>\\beta_{A}^{\\prime}$. By (15) and the variability assumption, $\\hat{\\pi}(y)>\\hat{\\pi}^{\\prime}(y)$ for any $y \\in Y$. Thus, whenever $\\pi(s, y) \\geqslant \\hat{\\pi}(y)$ we also have $\\pi(s, y)>\\hat{\\pi}^{\\prime}(y)$. But this means that $S_{a}^{\\prime}(y) \\supseteq S_{a}(y) \\cup S_{m}(y)$. Therefore, $\\alpha^{\\prime}(y) \\geqslant \\lambda\\left(S_{a}^{\\prime}\\right) \\geqslant$ $\\lambda\\left(S_{a} \\cup S_{m}\\right) \\geqslant \\alpha(y)$ for all $y$. By Lemma 1, this implies $\\bar{\\rho}^{\\prime} \\geqslant \\bar{\\rho}$, a contradiction. Moreover, since $\\bar{\\rho}^{\\prime}=\\bar{\\rho}, U=U^{\\prime}$, and by Lemma 2, the strategy cutoffs coincide $\\hat{\\pi}^{\\prime}(\\cdot)=\\hat{\\pi}(\\cdot)$. Hence, the seller surpluses coincide as well $V(\\cdot)=V^{\\prime}(\\cdot)$.\n\nProof of Proposition 2. Rearranging (6),\n\n$$\n\\begin{aligned}\nV(y) & =\\frac{\\beta_{A}}{1+\\tau \\beta_{A} \\alpha(y)} \\int \\pi(s, y) \\sigma(s, y) \\mathrm{d} \\lambda(s)= \\\\\n& =\\frac{\\beta}{1-\\bar{\\rho}+\\beta \\tau \\alpha(y)} \\int \\pi(s, y) \\sigma(s, y) \\mathrm{d} \\lambda(s)\n\\end{aligned}\n$$\n\nBy Lemma 2, strategy $\\sigma^{F D}$ prescribes jobs in $\\{x: \\pi(x, y)>\\tau V(y)\\}$ to be accepted with probability 1 and induces acceptance rates $\\alpha^{F D}(y)<1$. Consider a deviation $\\sigma$ that induces uniformly higher acceptance rates $\\alpha(y)=\\alpha^{F D}(y)+\\Delta$ for some small $\\Delta>0$ for all $y \\in Y$. As $Y$ is compact, such deviation exists and strictly increases the buyer surplus, $U>U^{F D}$.\n\nMoreover, by (13), $1-\\bar{\\rho}+\\beta \\tau \\alpha(y)=1-\\bar{\\rho}^{F D}+\\beta \\tau \\alpha^{F D}(y)$ for all $y$. Yet, the integral strictly increases, because $\\hat{\\pi}(y)>0$, and so $V(y)>V^{F D}(y)$.\n\nProof of Theorem 1. Take any Pareto-efficient pair $O=(V, U)$. Since $O$ is feasible, there is a seller strategy $\\sigma$ that induces $O$. Consider an information policy $\\hat{\\lambda}$ that recommends this strategy with $s_{a}$ being the recommendation to accept and $s_{r}$ the recommendation to reject. We need to check that the sellers would follow the recommendations.\n\nFrom (6) we have $v\\left(s_{a}\\right)=\\pi\\left(s_{a}\\right)-\\tau V, v\\left(s_{r}\\right)=0$, and $V=\\beta_{A}\\left(\\pi\\left(s_{a}\\right)-\\tau V\\right) \\hat{\\lambda}\\left(s_{a}\\right)$. The incentive constraints require that $\\pi\\left(s_{a}\\right) \\geqslant \\tau V$ and $\\pi\\left(s_{r}\\right) \\leqslant \\tau V$. For the former,","text_sha256":"94a6a75637b236cf62e6da8c8eddccec05492073ffe007ee03586f04cc90fd0e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0017","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Appendix","text":"$$\n\\tau V=\\frac{\\tau \\beta_{A} \\hat{\\lambda}\\left(s_{a}\\right)}{1+\\tau \\beta_{A} \\hat{\\lambda}\\left(s_{a}\\right)} \\pi\\left(s_{a}\\right) \\leqslant \\pi\\left(s_{a}\\right) .\n$$\n\nFor the latter, recall that $O$ is Pareto efficient, hence $\\sigma$ accepts all profitable jobs. This implies that\n\n$$\n\\pi\\left(s_{r}\\right) \\leqslant 0 \\leqslant \\tau V .\n$$\n\nProof of Theorem 2. The sequence of lemmas that follow support and formalize the argument outlined in the main text.\n\nLemma 3. $\\ell \\in \\mathcal{L}$ if and only if there is $\\lambda \\in \\Delta(S)$ such that $\\Lambda(\\cdot, \\lambda)=\\ell$.\n\nProof. The proof is standard and omitted. Kolotilin et al. (2017) and Gentzkow and Kamenica (2016) describe it in details. $\\square$\n\nLemma 4. In the linear payoff environment, for any information policy $\\lambda$, the seller's optimal strategy has a cutoff form such that a type-y seller accepts all buyers with expected difficulty $z<\\hat{z}(y)$ and rejects all buyers with $z>\\hat{z}(y)$. For a given $\\beta_{A}$, the seller payoff $V(y)$ and the cutoff $\\hat{z}(y)$ satisfy:\n\n$$\nV(y)=\\frac{y-\\hat{z}(y)}{\\tau}=\\beta_{A} \\Lambda(\\hat{z}(y)) .\n$$\n\nAs a result, $\\hat{z}(y)$ is increasing in $y$ and decreasing in $\\tau \\beta_{A}$.\nProof. The first steps of the proof of Proposition 1 show that in the case of arbitrary $X$ and $Y$, the seller optimal strategy has a cutoff form. They also show that the minimal acceptable profit is $\\hat{\\pi}(y)=\\tau V(y)$ and, moreover, (15) holds. In the linear payoff environment, $\\pi(s, y)=y-z(s)$ and $\\hat{\\pi}(y)=y-\\hat{z}(y)$ for some $\\hat{z}$. Plug it back into (15) and obtain\n\n$$\ny-\\hat{z}(y)=\\tau \\beta_{A} \\int_{0}^{\\hat{z}(y)}(\\hat{z}(y)-z) \\mathrm{d} H(z)=\\tau \\beta_{A} \\int_{0}^{\\hat{z}(y)} H(z) d z=\\tau \\beta_{A} \\Lambda(\\hat{z}(y))\n$$\n\nwhere the second equality follows by the integration by parts. $\\square$\n\nLemma 5. In the linear payoff environment, for any information policy $\\lambda$ and for almost all $y$,\n\n$$\n\\alpha(y)=H(\\hat{z}(y))=\\Lambda^{\\prime}(\\hat{z}(y)) .\n$$\n\nThe acceptance rate $\\alpha(y)$ is increasing in $y$ and decreasing in $\\tau \\beta_{A}$ for any $y \\in Y$. Moreover,\n\n$$\n\\hat{z}^{\\prime}(y)=\\frac{1}{1+\\tau \\beta_{A} \\alpha(y)}=\\nu(y) .\n$$\n\nProof. By Lemma 4, the accepted jobs are those with $z<\\hat{z}(y)$. Using the definition of $\\Lambda$ from (9), the probability of accepting $\\alpha(y)$ must lie between $H(\\hat{z}(y)-)$ and $H(\\hat{z}(y)+)$. The points of discontinuity of $H$ constitute a set of measure zero and seller types have full support on $[0,1]$. It follows that $\\alpha(y)=\\Lambda^{\\prime}(\\hat{z}(y))$ for almost all types $y$.\n\nBy (17), $\\hat{z}(y)$ is increasing in $y$. Since $\\Lambda$ is convex, $\\alpha(y)$ is increasing in $y$. Same argument reveals that $\\alpha(y)$ is decreasing in $\\tau \\beta_{A}$.\n\nFinally, using Lemma 4, we have $y-\\hat{z}(y)=\\tau \\beta_{A} \\Lambda(\\hat{z}(y))$. Differentiating that expression on both sides with respect to $y$ and substituting $\\Lambda^{\\prime}(\\hat{z}(y))$ with $\\alpha(y)$, the result follows. $\\square$\n\nConsider an initial information policy $\\Lambda_{0} \\in \\mathcal{L}$, and a deviation $h:[0,1] \\rightarrow[0,1]$. A deviation is admissible if $\\Lambda_{0}+h \\in \\mathcal{L}$. Since $\\mathcal{L}$ is convex, $\\Lambda_{0}+\\varepsilon h \\in \\mathcal{L}$ for all $\\varepsilon \\in[0,1]$. For a functional $\\mathcal{I}: \\mathcal{L} \\rightarrow \\mathbb{R}$ we will denote the value of $\\mathcal{I}$ under information policy $\\Lambda_{0}+\\varepsilon h$ by\n$\\mathcal{I}(\\varepsilon)$. The main step in the analysis is to find the variational derivative with respect to $\\varepsilon$ at zero:\n\n$$\n\\mathcal{J}^{\\prime}(0)=\\gamma U^{\\prime}(0)+(1-\\gamma) V^{\\prime}(0) .\n$$\n\nLemma 6. In the linear payoff environment, for any admissible deviation $h$ :\n\n$$\n\\begin{aligned}\n& U^{\\prime}(0)=-u \\beta \\Psi_{1}\\left(\\int_{0}^{\\bar{y}} h(\\hat{z}(y)) \\nu(y) g^{\\prime}(y) \\mathrm{d} y-h(\\hat{z}(\\bar{y})) \\nu(\\bar{y}) g(\\bar{y})\\right), \\\\\n& V^{\\prime}(0)=\\Psi_{2} U^{\\prime}(0)+\\frac{\\beta}{\\bar{\\nu}} \\int h(\\hat{z}(y)) \\nu(y) g(y) \\mathrm{d} y,\n\\end{aligned}\n$$\n\nwhere $\\Psi_{1}=1 / \\int_{0}^{\\bar{y}}\\left(\\nu^{2}(y)-\\tau \\nu^{\\prime}(y) V(y)\\right) g(y) \\mathrm{d} y>0$ and $\\Psi_{2}=\\frac{\\tau}{u \\bar{\\nu}} \\int \\nu(y) V(y) \\mathrm{d} G(y)>0$.\n\nProof. Consider a variation $\\varepsilon h(z)$. The resulting matching rate is by definition and Lemma 5:\n\n$$\nm(\\varepsilon)=\\frac{1}{\\tau} \\int_{0}^{1} 1-\\nu(y, \\varepsilon) \\mathrm{d} G(y)=\\frac{1}{\\tau}\\left(1-\\int \\hat{z}^{\\prime}(y, \\varepsilon) d G(y)\\right)=\\frac{1}{\\tau}\\left(1-\\hat{z}(\\bar{y}, \\varepsilon) g(\\bar{y})+\\int g^{\\prime}(y) \\hat{z}(y, \\varepsilon) \\mathrm{d} y\\right),\n$$\n\nso\n\n$$\nm^{\\prime}(\\varepsilon)=\\frac{1}{\\tau} \\int_{0}^{\\bar{y}} g^{\\prime}(y) \\hat{z}_{\\varepsilon}^{\\prime}(y, \\varepsilon) \\mathrm{d} y-g(\\bar{y}) \\hat{z}_{\\varepsilon}^{\\prime}(\\bar{y}, \\varepsilon) .\n$$\n\nBy Lemma 4,\n\n$$\n\\tau V(y ; \\varepsilon)=y-\\hat{z}(y, \\varepsilon)=\\tau \\beta_{A}(\\varepsilon)\\left(\\Lambda_{0}+\\varepsilon h\\right)(\\hat{z}(y, \\varepsilon)) .\n$$\n\nFinding $U^{\\prime}(0)$ : The surplus derivative is $U^{\\prime}(0)=u m^{\\prime}(0)=u \\bar{\\rho}_{\\varepsilon}^{\\prime}(0) / \\tau$. Differentiate the second equation in (19) with respect to $\\varepsilon$, evaluate at 0 and collect terms:\n\n$$\n\\begin{aligned}\n\\hat{z}_{\\varepsilon}^{\\prime}(y, 0) & =-\\left(1+\\tau \\beta_{A} \\alpha(y)\\right)^{-1}\\left(\\tau \\beta_{A}^{\\prime}(0) \\Lambda_{0}(\\hat{z}(y, 0))+\\tau \\beta_{A}(0) h(\\hat{z}(y, 0))\\right) \\\\\n& =-\\nu(y, 0)\\left(\\tau \\beta_{A}^{\\prime}(0) \\Lambda_{0}(\\hat{z}(y, 0))+\\tau \\beta_{A}(0) h(\\hat{z}(y, 0))\\right) \\\\\n& =-\\nu(y, 0) \\tau \\beta\\left(\\frac{\\bar{\\rho}_{\\varepsilon}^{\\prime}(0) \\Lambda_{0}(\\hat{z}(y, 0))}{\\bar{\\nu}^{2}(0)}+\\frac{h(\\hat{z}(y, 0))}{\\bar{\\nu}(0)}\\right) .\n\\end{aligned}\n$$\n\nwhere the second line follows by Lemma 5 and the last line follows as $\\beta_{A}(\\varepsilon)=\\beta / \\bar{\\nu}(\\varepsilon)$.\n\nSubstituting (20) into (18) and omitting the $\\varepsilon=0$ argument in the function we obtain,\n\n$$\n\\bar{\\rho}_{\\varepsilon}^{\\prime}=-\\int \\nu(y) \\tau \\beta\\left(\\frac{\\bar{\\rho}_{\\varepsilon}^{\\prime} \\Lambda_{0}(\\hat{z}(y))}{\\bar{\\nu}^{2}}+\\frac{h(\\hat{z}(y))}{\\bar{\\nu}}\\right) g^{\\prime}(y) \\mathrm{d} y+\\nu(\\bar{y}) \\tau \\beta\\left(\\frac{\\bar{\\rho}_{\\varepsilon}^{\\prime} \\Lambda_{0}(\\hat{z}(y))}{\\bar{\\nu}^{2}}+\\frac{h(\\hat{z}(y))}{\\bar{\\nu}}\\right) g(\\bar{y}) .\n$$\n\nRearranging,","text_sha256":"99d95bffbc2fac2718ed1664d2403d363e29cf56aae309ca996bb6c1affd8939"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0018","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Appendix","text":"$$\n\\bar{\\rho}_{\\varepsilon}^{\\prime}=-\\tau \\beta \\Psi_{1}\\left(\\int_{0}^{\\bar{y}} h(\\hat{z}(y)) \\nu(y) g^{\\prime}(y) \\mathrm{d} y-h(\\hat{z}(\\bar{y})) \\nu(\\bar{y}) g(\\bar{y})\\right)\n$$\n\nwhere\n\n$$\n\\begin{aligned}\n\\Psi_{1} & \\triangleq\\left(\\bar{\\nu}+\\tau \\beta_{A} \\int_{0}^{\\bar{y}} \\nu(y) \\Lambda(\\hat{z}(y)) g^{\\prime}(y) \\mathrm{d} y-\\tau \\beta_{A} \\nu(\\bar{y}) \\Lambda(\\hat{z}(\\bar{y})) g(\\bar{y})\\right)^{-1} \\\\\n& =\\left(\\bar{\\nu}+\\tau \\int_{0}^{\\bar{y}} \\nu(y) V(y) g^{\\prime}(y) \\mathrm{d} y-\\tau \\nu(\\bar{y}) V(\\bar{y}) g(\\bar{y})\\right)^{-1} \\\\\n& =\\left(\\int_{0}^{\\bar{y}} \\nu(y) g(y)+\\tau \\nu(y) V(y) g^{\\prime}(y) \\mathrm{d} y-\\tau \\nu(\\bar{y}) V(\\bar{y}) g(\\bar{y})\\right)^{-1} \\\\\n& =\\left(\\int_{0}^{\\bar{y}}\\left(\\nu(y)-\\tau \\nu^{\\prime}(y) V(y)-\\tau V^{\\prime}(y) \\nu(y)\\right) g(y) \\mathrm{d} y\\right)^{-1} \\\\\n& =\\left(\\int_{0}^{\\bar{y}}\\left(\\nu^{2}(y)-\\tau \\nu^{\\prime}(y) V(y)\\right) g(y) \\mathrm{d} y\\right)^{-1}\n\\end{aligned}\n$$\n\nwith the last line following from differentiating the first equation in (19) with respect to $y$ : $\\tau V^{\\prime}(y)=1-\\nu(y)$. The term $\\Psi_{1}$ is positive because $\\nu^{\\prime}(y)<0$.\n\nFinding $V^{\\prime}[0]$ : Differentiate the first equation in (19) with respect to $\\varepsilon$ and evaluate at zero. Using (20), we have:\n\n$$\n\\tau V_{\\varepsilon}^{\\prime}(y, 0)=-\\hat{z}_{\\varepsilon}^{\\prime}(y, 0)=\\nu(y, 0) \\tau\\left(\\beta_{A}^{\\prime}[0] \\Lambda_{0}(\\hat{z}(y))+\\beta_{A}[0] h(\\hat{z}(y))\\right) .\n$$\n\nIntegrating over seller types:\n\n$$\n\\begin{aligned}\nV^{\\prime}(0) & =\\int V_{\\varepsilon}^{\\prime}(y, 0) \\mathrm{d} G(y) \\\\\n& =\\beta_{A}^{\\prime}(0) \\int \\nu(y) \\Lambda_{0}(\\hat{z}(y)) \\mathrm{d} G(y)+\\beta_{A} \\int \\nu(y) h(\\hat{z}(y)) \\mathrm{d} G(y) \\\\\n& =m^{\\prime}(0) \\frac{\\tau \\beta_{A}}{\\bar{\\nu}} \\int \\nu(y) \\Lambda_{0}(\\hat{z}(y)) \\mathrm{d} G(y)+\\beta_{A} \\int \\nu(y) h(\\hat{z}(y)) \\mathrm{d} G(y) \\\\\n& =m^{\\prime}(0) \\frac{\\tau}{\\bar{\\nu}} \\int \\nu(y) V(y) \\mathrm{d} G(y)+\\frac{\\beta}{\\bar{\\nu}} \\int \\nu(y) h(\\hat{z}(y)) \\mathrm{d} G(y)\n\\end{aligned}\n$$\n\nThe second part of the result follows. $\\square$\n\nLemma 7. $\\lambda \\in \\Delta(S)$ is $x^{*}$-lower-censorship if and only if the corresponding spread function $\\Lambda$ has the following form:\n\n$$\n\\Lambda(z)= \\begin{cases}\\underline{\\Lambda}(z), & z \\in\\left[0, \\mathbb{E}\\left[x \\mid x<x^{*}\\right]\\right] \\\\ \\bar{\\Lambda}\\left(x^{*}\\right)+F\\left(x^{*}\\right)\\left(z-x^{*}\\right), & z \\in\\left(\\mathbb{E}\\left[x \\mid x<x^{*}\\right], x^{*}\\right) \\\\ \\bar{\\Lambda}(z), & z \\in\\left[x^{*}, \\bar{x}\\right]\\end{cases}\n$$\n\nProof. Denote by $z^{*}$ the expected value of $x$ on the pooled part of $X, z^{*}=\\mathbb{E}\\left[x \\mid x>x^{*}\\right]$. If $\\lambda$ is the $x^{*}$-lower-censorship, then a straightforward application of the spread-function definition obtains (21).\n\nConversely, suppose $\\Lambda$ has a form given in (21). Let $H(z)=\\Lambda^{\\prime}(z)$, where $\\Lambda^{\\prime}$ is the right derivative. Function $H(z)$ then equals to $F(z)$ for $z \\leqslant x^{*}$, to $F\\left(x^{*}\\right)$ for $x^{*}<z<z^{*}$, and to 1 for $z \\geqslant z^{*}$. Such distribution corresponds to a $x^{*}$-lower-censorship. $\\square$\n\nNow let's return to the proof of Theorem 2. Use Lemma 6 to obtain the expression for\nthe variational derivative of the objective function with respect to $\\varepsilon$ :\n\n$$\n\\begin{aligned}\n\\mathcal{J}^{\\prime}(0) & =\\gamma U^{\\prime}(0)+(1-\\gamma) V^{\\prime}(0) \\\\\n& =\\int_{0}^{\\bar{y}} h(\\hat{z}(y)) \\nu(y)\\left(-g^{\\prime}(y) \\Psi+(1-\\gamma) \\frac{\\beta}{\\bar{\\nu}} g(y)\\right) \\mathrm{d} y+h(\\hat{z}(\\bar{y})) \\nu(\\bar{y}) g(\\bar{y}) \\Psi \\\\\n& =\\int_{0}^{\\bar{y}} h(\\hat{z}(y)) \\nu(y)\\left(-\\frac{g^{\\prime}(y)}{g(y)} \\Psi+(1-\\gamma) \\frac{\\beta}{\\bar{\\nu}}\\right) g(y) \\mathrm{d} y+h(\\hat{z}(\\bar{y})) \\nu(\\bar{y}) g(\\bar{y}) \\Psi\n\\end{aligned}\n$$\n\nwhere $\\Psi \\triangleq\\left(\\gamma+\\Psi_{2}(1-\\gamma)\\right) u \\beta \\Psi_{1}>0$.\nIf $g$ is weakly decreasing, then $g^{\\prime} / g \\leqslant 0$, and so the integrand in (22) is positive for all $y \\in Y$. Therefore, for any $\\Lambda_{0}$ and any $h \\geqslant 0, \\mathcal{J}^{\\prime}(0)>0$. The only spread function that cannot be improved like that corresponds to full disclosure so full disclosure is optimal.\n\nIf $g$ is log-concave, then $g^{\\prime}(y) / g(y)$ is decreasing. Then for any $\\Lambda_{0}$, the integrand in (22) crosses zero from below at most once. If the integrand is always positive, then, again, full disclosure is optimal. If the integrand does cross zero once, then we will show that a lower censorship is optimal. Towards the contradiction, suppose that $\\Lambda_{0}$ is not a lower censorship. We will construct a platform-profitable deviation $h$ as follows. Denote by $y^{*}$ the zero of the integrand in (22), and denote by $z^{*}=\\hat{z}\\left(y^{*}, \\Lambda_{0}\\right)$ the optimal cutoff of the $y^{*}$-type seller. Let $\\tilde{\\Lambda} \\in \\mathcal{L}$ be the lower censorship that passes through the point $\\left(z^{*}, \\Lambda_{0}\\left(z^{*}\\right)\\right)$, as in (21). Essentially, $\\tilde{\\Lambda}$ is the policy that discloses as little as possible on the left tail and as much as possible on the right tail but still passes through $\\left(z^{*}, \\Lambda_{0}\\left(z^{*}\\right)\\right)$. This lower censorship is unique because there is a unique line tangent to $\\bar{\\Lambda}$ and passing through $\\left(z^{*}, \\Lambda_{0}\\left(z^{*}\\right)\\right)$. We have $\\tilde{\\Lambda}(z) \\leqslant \\Lambda_{0}(z)$ for $z<z^{*}$ and $\\tilde{\\Lambda}(z) \\geqslant \\Lambda_{0}(z)$ for $z>z^{*}$; as $\\Lambda_{0}$ is not a lower censorship,\nfor some $z$ one of these inequalities is strict. Consider a deviation $h=\\tilde{\\Lambda}-\\Lambda_{0}$. We have\n\n$$\n\\begin{aligned}\n\\mathcal{J}^{\\prime}(0)= & \\int_{0}^{\\bar{y}} h(\\hat{z}(y)) \\nu(y)\\left(-\\frac{g^{\\prime}(y)}{g(y)} \\Psi+(1-\\gamma) \\frac{\\beta}{\\bar{\\nu}}\\right) g(y) \\mathrm{d} y+h(\\hat{z}(\\bar{y})) \\nu(\\bar{y}) g(\\bar{y}) \\Psi \\\\\n= & \\int_{0}^{y^{*}} \\underbrace{h(\\hat{z}(y))}_{\\leqslant 0} \\nu(y) \\underbrace{\\left(-\\frac{g^{\\prime}(y)}{g(y)} \\Psi+(1-\\gamma) \\frac{\\beta}{\\bar{\\nu}}\\right)}_{\\leqslant 0} g(y) \\mathrm{d} y \\\\\n& +\\int_{y^{*}}^{\\bar{y}} \\underbrace{h(\\hat{z}(y))}_{\\geqslant 0} \\nu(y) \\underbrace{\\left(-\\frac{g^{\\prime}(y)}{g(y)} \\Psi+(1-\\gamma) \\frac{\\beta}{\\bar{\\nu}}\\right)}_{\\geqslant 0} g(y) \\mathrm{d} y+\\underbrace{h(\\hat{z}(\\bar{y}))}_{\\geqslant 0} \\nu(\\bar{y}) g(\\bar{y}) \\Psi .\n\\end{aligned}\n$$","text_sha256":"d7d0385bf6f4a246015a68ab2117fe643d9ae7bfdae4c5e964c81c5bd866a7b7"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0019","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Appendix","text":"By the argument above, one of the inequalities is strict, and so $\\mathcal{J}^{\\prime}(0)>0$, contradicting the optimality of $\\Lambda_{0}$.\n\nThe same logic applies if the platform maximizes only the matching rate, $\\gamma=1$, and the seller distribution $g$ is unimodal.","text_sha256":"9913e78200ac8d6f685a31b0149cdd6a7045608d1949fc30140a14e9656e824d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0020","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAkbarpour, M., S. 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Sela (2009): \"The Theory of Assortative Matching Based on Costly Signals,\" Review of Economic Studies, 76, 253-281.\n\nKamenica, E. and M. Gentzkow (2011): \"Bayesian Persuasion,\" American Economic Review, 101, 2590-2615.\n\nKircher, P. (2009): \"Efficiency of Simultaneous Search,\" Journal of Political Economy, 117, 861-913.\n\nKolotilin, A., T. Mylovanov, A. Zapechelnyuk, and M. Li (2017): \"Persuasion of a Privately Informed Receiver,\" Econometrica, 85, 1949-1964.\n\nKovbasyuk, S. and G. Spagnolo (2017): \"Memory and Markets,\" Working Paper.\n\nLauermann, S. (2012): \"Asymmetric Information in Bilateral Trade and in Markets: An Inversion Result,\" Journal of Economic Theory, 147, 1969-1997.\n\nLevin, J. and P. Milgrom (2010): \"Online Advertising: Heterogeneity and Conflation in Market Design,\" American Economic Review, 100, 603-607.\n\nMyerson, R. B. (2000): \"Large Poisson Games,\" Journal of Economic Theory, 94, 7-45.\n\nOrlov, D., A. Skrzypacz, and P. Zryumov (2018): \"Persuading the Principal to Wait,\" Working Paper.\n\nOstrovsky, M. and M. Schwarz (2010): \"Information Disclosure and Unraveling in Matching Markets,\" American Economic Journal: Microeconomics, 2, 34-63.\n\nRayo, L. and I. Segal (2010): \"Optimal Information Disclosure,\" Journal of Political Economy, 118, 949-987.\n\nSegal, I. (2007): \"The Communication Requirements of Social Choice Rules and Supporting Budget Sets,\" Journal of Economic Theory, 136, 341-378.\n\nSmolin, A. (2017): \"Dynamic Evaluation Design,\" Working Paper.\n\nTadelis, S. and F. Zettelmeyer (2015): \"Information Disclosure as a Matching Mechanism: Theory and Evidence from a Field Experiment,\" American Economic Review, 105, 886-905.\n\n[^0]:    *Romanyuk: Harvard University, 1805 Cambridge St, Cambridge, MA 02138, USA (email: gleb.romanyuk@gmail.com); Smolin: University of Bonn, Lennestraße 37, 53113, Bonn, Germany (email: alexey.v.smolin@gmail.com). We thank John Asker, Coeditor, and three anonymous referees for their productive suggestions. This paper builds on Romanyuk's dissertation chapter written at Harvard University, Department of Economics; he is indebted to Susan Athey, Drew Fudenberg, Greg Lewis, Tomasz Strzalecki, as well as to Chiara Farronato and Andrei Hagiu for their guidance and support. We are grateful to Andrey Fradkin, Ben Golub, Divya Kirti, Sergei Kovbasyuk, Jeffrey Picel, as well as seminar participants at Harvard University and MIT for helpful discussions. The views expressed in this article are those of the authors and do not necessarily reflect those of the employing organizations.\n\n[^1]:    ${ }^{1}$ Tadelis and Zettelmeyer (2015) show that in wholesale automobile auctions, information disclosure helps match heterogeneous buyers to cars of varying quality. Einav, Farronato, and Levin (2016) name eliciting and aggregating the dispersed user information as a key objective of peer-to-peer platforms.\n\n[^2]:    ${ }^{2}$ In comparison to standard search models, our market features the congestion externality but not the thick market externality.\n    ${ }^{3}$ Kamenica and Gentzkow (2011) and Bergemann and Morris (2016) outline the information design paradigm in static settings.\n    ${ }^{4}$ This approach has proven useful in other information design problems as well (e.g., Ely and Szydlowski (2017)).\n\n[^3]:    ${ }^{5}$ Buyer type $x$ captures the payoff-relevant information the platform elicits from the buyer, whether passively from the buyer's cookies and queries or actively by asking questions. For example, on Uber, $x$ would incorporate the rider's destination; on Airbnb, $x$ would incorporate the guest's race, age, and gender.\n\n[^4]:    ${ }^{6}$ Seller type $y$ captures payoff-relevant information that, for whatever reason-costly, unethical, etc.-the platform cannot elicit from the sellers. For example, on Uber, $y$ would incorporate the driver's preference for long rides and traffic; on Airbnb, $y$ would incorporate the host's preference for his guest's age, gender, socio-economic status, race, etc.\n    ${ }^{7}$ Many results generalize to the case of heterogeneous buyer utility.\n    ${ }^{8}$ The results immediately generalize to the case where the sellers have discount rate $r$ by replacing $\\tau$ with\n\n[^5]:    $\\tau_{r}=\\left(1-e^{-r \\tau}\\right) / r$.\n    ${ }^{9} \\Delta(S)$ is the set of Borel probability distributions with the weak-* topology on $\\Delta(X)$.\n    ${ }^{10}$ Equivalently, we could model an information policy as an arbitrary statistical experiment informative about $x$.","text_sha256":"6dc8d7f1453b0e22ccb16ff8c04ea2638b634ff4a57e49d2c2c1c34f148c6109"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0021","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"[^6]:    ${ }^{11}$ Fradkin (2015) finds that on Airbnb, seller screening causes around 41\\% of all failed matches, whereas simultaneity only around 12\\% (the rest is attributed to unavailable vacancies).\n    ${ }^{12}$ Fradkin (2015) reports that on Airbnb, an initial rejection decreases by 51\\% the probability that the guest eventually books any listing.\n\n[^7]:    ${ }^{13}$ For example, if a seller earns \\$1 on each buyer, and the time interval between accepting a pair of consequent buyers is 2 , then $V(y)=1 / 2$.\n\n[^8]:    ${ }^{14}$ The efficiency result also relates to that of Bergemann, Brooks, and Morris (2015). They show that segmentation of a monopolistic market can achieve every feasible combination of consumer and producer surplus\n\n[^9]:    ${ }^{15}$ Thus, the equilibrium exhibits a kind of negative stochastic assorting in terms of Chade (2006).\n\n[^10]:    ${ }^{16}$ The class of log-concave distributions includes normal, logistic, exponential, and uniform distributions, as well as their truncations.\n\n[^11]:    ${ }^{17}$ http://www.forbes.com/sites/harrycampbell/2015/03/24/just-how-far-is-your-uber-driver-willing-totake-you.\n    ${ }^{18}$ Cohen, Hahn, Hall, Levitt, and Metcalfe (2016) estimate that an increase in wait time of 1 minute decreases the probability of requesting a ride by 1.7 percent.\n    ${ }^{19}$ The surge price coefficient depends on the numbers of riders and nearby drivers who have their Uber app open at that moment: https://newsroom.uber.com/upfront-fares-no-math-and-no-surprises/.\n    ${ }^{20}$ Uber drivers can use the destination filter twice a day. If used, the platform filters passenger requests and offers only those aligned with the driver's preferred destination: http://www.ridesharingdriver.com/uberdestination-filter-get-passengers-heading-your-direction/.\n\n[^12]:    ${ }^{21}$ Rayo and Segal (2010) study the last situation in a static match model when both buyers and sellers derive heterogeneous payoffs from matches. They find that in the case of the uniform distribution of receiver reservation values, the optimal information policy pools \"non-ordered prospects,\" where two prospects are non-ordered in the sense described above.\n\n[^13]:    ${ }^{22}$ See Segal (2007) for the study of communication requirements of social choice rules. It demonstrates that in some social choice problems, the space of prices that must be discovered is prohibitively large.","text_sha256":"d4734c0b706a5278f3a2c45df9b114165b21eb60bbc599dd9fce8ce1d52cc910"}
{"schema_version":"1.0","chunk_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14:0022","work_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets","paper_id":"alex-smolin:cream-skimming-and-information-design-in-matching-markets:2018-05-14","title":"Cream Skimming and Information Design in Matching Markets","authors":[{"name":"Gleb Romanyuk"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2018-05-14","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md","source_record":"https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf","doi":"https://doi.org/10.1257/mic.20170154","citation":"Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Gleb Romanyuk; Alex Smolin\n\n**Canonical citation:** Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md\n\n**Source record:** https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf\n\n**Published record:** https://doi.org/10.1257/mic.20170154\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"1ffbec2b10830e8c369e0ac53e0bb6a3aaa7d7d3e6de4d74b7be2cc6369755e6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0001","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Chiara Margaria; Alex Smolin.\n> Canonical citation: Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"c73771935636299b57f35510c079012215b859f4e7172b41dbe63885125eacd8"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0002","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Dynamic Communication with Biased Senders","text":"# Dynamic Communication with Biased Senders\n\n**Authors:** Chiara Margaria; Alex Smolin\n\n**Manuscript date:** 2017-10-21\n\n#### Abstract\n\nWe study dynamic games in which senders with state-independent payoffs communicate to a single receiver. Senders' private information evolves according to an aperiodic and irreducible Markov chain. We prove an analog of a folk theorem-that any feasible and individually rational payoff can be approximated in a perfect Bayesian equilibrium if players are sufficiently patient. In particular, there are equilibria in which the receiver makes perfectly informed decisions in almost every period, even if no informative communication can be sustained in the stage game. We conclude that repeated interaction can overcome strategic limits of communication.\n\nKeywords: Bayesian games, repeated games, communication, folk theorem.\nJEL Codes: C72, C73, D82, D83.","text_sha256":"1fc38d4b1cfdd2047d10aa30cac879c12390bdaf276f5478907bb235a01740e2"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0003","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction\n\nCrawford and Sobel (1982) introduced cheap-talk games as a basis for the analysis of strategic transmission of unverifiable information. They considered a one-shot game between a sender who has private information and a receiver who takes an action. In equilibrium, informative communication can be sustained, but misalignment of players' interests limits the amount of information that can be transmitted. In particular, if players' preferences are misaligned, truthful communication of private information cannot be sustained; otherwise, the sender\n\n[^0]would bias reports to induce her preferred outcomes. The strategic considerations restrain communication and translate into inefficiency as both players could benefit from a betterinformed action.\n\nThe strategic limits of communication highlighted in a one-shot game provide insights into many economic situations. A buyer relies more on the recommendation of a friend than that of a sales representative; an antitrust legislator takes arguments of firms opposing a newly proposed regulation with a grain of salt; a voter feels skeptical about the promises of a politician running for office. Nonetheless, in many settings in which interests are seemingly misaligned, informative communication is sustained: companies raise funds from investors, government agencies successfully split the state budget, conglomerates allocate resources among different divisions, and so on. In many of these cases, the informed parties have strong biases so that their payoffs are determined solely by the resulting decisions. ${ }^{1}$ Importantly, all of these interactions happen repeatedly over time, with the parties trading off immediate opportunistic gains for the prospect of an ongoing relationship.\n\nTo investigate informative communication in these settings, we analyze a dynamic version of an information transmission game in which private information (states) evolve stochastically over time. We allow for many senders but focus on the case in which the senders' payoffs are state independent. In every period, each sender sends a message to the receiver, who then takes a publicly observable action. The \"cheap-talk\" nature of messages is preserved: no hard evidence can be presented, the sender cannot commit to a communication strategy, and the receiver never observes extraneous information to test the validity of the past messages. No contracts can be written between the players, so at any point in time it must be in players' interests to follow the equilibrium play.\n\nWe obtain an analog of a folk theorem-that any feasible and individually rational payoffs can be approximated in a perfect Bayesian equilibrium as the players become patient. ${ }^{2}$ This payoff set, and hence the set of equilibrium payoffs, admits a simple characterization and includes all Pareto efficient payoffs that satisfy the receiver's individual rationality. Specifically, it includes the receiver's largest feasible \"complete information\" payoff. In equilibrium, the fraction of periods in which the receiver makes perfectly informed decisions can be arbitrarily close to one, even if no informative communication can be sustained in the stage game.\n\nThese results contrast with the conventional wisdom that state independence makes it harder to maintain informative communication. Indeed, in this case, each sender has an unambiguous ranking over actions and is willing to report truthfully only if indifferent among the messages she sends. However, what comes as a curse in a one-shot game turns into a blessing when the game is dynamic. If senders' payoffs depend only on actions and not on states, then the receiver fully observes and controls the payoffs. Our equilibrium\n\n[^1]construction actively uses this feature-it targets the senders' total payoffs and ensures they do not depend on the senders' messages. To achieve it, the equilibrium play alternates between communication and adjustment phases. In communication phases, the receiver makes informed decisions relying on the senders' messages. In adjustment phases, the receiver ignores all messages and plays according to a strategy that pulls the senders' payoffs towards the target. Specification of the phases and the transitions between them is tailored to guarantee that the senders' payoffs do not depend on their messages, and thus sustains truth-telling. By the law of large numbers, as players become more patient, the play occurs in the communication phase most of the time and any individually rational payoffs can be achieved by changing the strategy the receiver plays in there.\n\nRelated literature The general idea that an ongoing relationship can overcome strategic limits of interaction is the cornerstone of the literature on repeated games as discussed in depth by Mailath and Samuelson (2006). Further, the idea of using players' payoffs as a determinant of equilibrium construction is reminiscent of Abreu et al. (1990)'s recursive technique of equilibrium payoff decomposition. In fact, if individual states are independently and identically distributed, our dynamic game can be viewed as an infinitely repeated game and the Fudenberg et al. (1994)'s standard method can be used to provide an alternative proof of our folk theorem result. However, the standard method cannot be applied to general stochastic sender-receiver games.\n\nAt the same time, the idea of linking decisions motivated a strand of mechanism design literature. Jackson and Sonnenschein (2007) showed that Pareto efficient outcomes can be achieved by linking many identical copies of a collective choice problem with private values into a single mechanism. Frankel (2016) extended these ideas into a dynamic setting where the sender has persistent private information. He introduced discounted quota contracts similar to our equilibrium construction and showed their optimality in many environments. The mechanism design setting, however, differs from ours in that it endows the receiver with commitment power and monetary transfers.","text_sha256":"e18417887fd3473a842346cf9d8397e0e07a783a9d1b294ea755678ff21e8860"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0004","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"Finally, the closest paper to ours is by Renault, Solan and Vieille (2013), who analyzed a dynamic information transmission game between a single sender and a receiver. They considered a more general payoff structure that, however, did not allow the sender's payoff to be state independent. In their setting, they showed that our analog of a folk theorem does not generally hold. In particular, the players' equilibrium payoffs do not necessarily approach the Pareto efficiency frontier as the players become patient.\n\nNeither Renault et al. (2013)'s nor our proof can be directly extended to cover both the cases of state-dependent and of state-independent payoffs. On one hand, their construction is based on the idea of statistical tests that require the sender to match the message distribution with the state distribution. However, in our setting, because payoffs are state independent, statistical tests cannot be used effectively; faced with these tests, the sender would induce\nfavorable actions earlier in time independently of the realized states, rendering her messages uninformative. On the other hand, our construction is based on the ability of the receiver to adjust the senders' continuation payoffs exactly to the target without knowing the state. ${ }^{3}$ When the payoffs are state dependent, such adjustment must depend on the state and hence rely on the sender's messages running into the same incentive compatibility problem. ${ }^{4}$\n\nThe remainder of the paper proceeds as follows: Section 2 introduces the model, Section 3 characterizes the equilibrium payoffs and discusses, and Section 4 concludes. Appendix contains a detailed proof of the main theorem.","text_sha256":"1f420ec7b3b15faf385e7d2d1345f1937c8d39d91e5b9970dd7df447b3e4183f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0005","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"## 2 Model\n\nA receiver (he) repeatedly communicates with $n$ senders (she) indexed by $i \\in N=\\{1, \\ldots, n\\}$, $n \\geq 1$. The stage game is described by the set of states $\\Omega \\triangleq \\times_{i \\in N} \\Omega_{i}$, their prior distribution $p \\in \\Delta(\\Omega)$, the set of senders' messages $M \\triangleq \\times_{i \\in N} M_{i}$, the set of actions of the receiver $A$, and the stage payoffs of the senders, $v: A \\rightarrow \\mathbb{R}^{n}$, and of the receiver $u: A \\times \\Omega \\rightarrow \\mathbb{R},^{5}$ Denote the stage game by $\\Gamma \\triangleq(N, \\Omega, p, M, A, v, u)$. We assume that the sets $\\Omega, M$, and $A$ are all finite. In the stage game, all senders first privately observe their individual states $\\omega_{i} \\in \\Omega_{i}$ and then simultaneously send public messages $m_{i} \\in M_{i}$ to the receiver who takes a publicly observable action $a \\in A,{ }^{6}$\n\nThe stage game is infinitely repeated at times $t=1,2,3, \\ldots$ with a common discount factor $\\delta$. The state profiles $\\omega_{t} \\triangleq\\left\\{\\omega_{t 1}, \\ldots, \\omega_{t n}\\right\\}$ evolve according to an irreducible and aperiodic Markov chain with a transition kernel $k\\left(\\omega_{t+1} \\mid \\omega_{t}\\right)$. Hence, the individual states can be arbitrarily correlated across senders but their inter-temporal correlation vanishes as the distance between time periods grows large. Consequently, there exists a unique stationary distribution $p \\in \\Delta(\\Omega)$ that has full support. We assume that the initial state is drawn according to this stationary distribution.\n\nWe assume that the validity of senders' messages can never be verified. First, no hard evidence is allowed-the sets of possible messages $M_{i}$ do not depend on the states. Second, the senders cannot commit to communication strategies, in contrast to the Bayesian per-\n\n[^2]suasion literature. Third, the receiver does not observe any additional information besides the senders' messages; for example, he does not observe any additional signals about past states. ${ }^{7}$ We allow the players to perfectly transmit their private information, $M_{i}=\\Omega_{i}$, thus concentrating on strategic rather than technological limits of communication. ${ }^{8}$ Denote the message profile by $m_{t} \\in M \\triangleq \\times_{i \\in N} M_{i}$. Lastly, we assume that there is a public randomization device that produces a uniformly distributed output $y_{t} \\in Y \\triangleq[0,1]$ at the beginning of each period independently of states, and all players publicly observe it. We do not require any public randomization at the interim stage, after the senders send their messages but before the receiver takes an action.\n\nOverall timing within each period of the dynamic game is as follows: the public randomization device produces an output $y_{t}$; the state profile $\\omega_{t}$ realizes and each sender $i$ privately observes her individual state $\\omega_{i t}$; the senders simultaneously send messages $m_{t i}$ to the receiver; the messages are publicly observed and the receiver takes an action $a_{t}$; the action is publicly observed and the game proceeds to the next period. Denote the resulting dynamic game by $\\Gamma^{\\infty}(\\delta)$.\n\nStrategies The timing and monitoring structure of the game $\\Gamma^{\\infty}(\\delta)$ outlined above result in the following definitions of histories and strategies. A public history $h^{t}$ at time $t$ consists of past actions, messages and the realizations of the randomization device. ${ }^{9}$ Denote the set of all public histories at time $t$ by $H^{t}$. A behavioral strategy of the receiver, $\\alpha$, maps past public history, current output of the randomization device and current messages into an action,\n\n$$\n\\begin{aligned}\nh^{t} & \\triangleq\\left\\{y_{s}, m_{s}, a_{s}\\right\\}_{s=1}^{t-1}, t>1, h^{1} \\triangleq \\emptyset \\\\\n& \\alpha:\\left(\\bigcup_{t \\geq 1} H^{t}\\right) \\times Y \\times M \\rightarrow A\n\\end{aligned}\n$$\n\nSenders' private histories $h_{i}^{t}$ contain public histories as well as the individual states they observe. Denote the set of private histories of a sender $i$ by $H_{i}^{t}$. A behavioral strategy of a sender $i, \\mu_{i}$, maps her private histories and current output of the randomization device into\n\n[^3]a message\n$$\n\\begin{gathered}\nh_{i}^{t} \\triangleq h^{t} \\cup\\left\\{\\omega_{s i}\\right\\}_{s=1}^{t} \\\\\n\\mu_{i}:\\left(\\bigcup_{t \\geq 1} H_{i}^{t}\\right) \\times Y \\rightarrow M_{i}\n\\end{gathered}\n$$\n\nPayoffs Players' interests are misaligned. The receiver's stage payoff is state dependent, $u: A \\times \\Omega \\rightarrow \\mathbb{R}$, so that he generally prefers to take different actions in different states. The senders' stage payoffs, in contrast, depend only on the receiver's actions and not on the states, $v_{i}: A \\rightarrow \\mathbb{R}$. That is each sender had an optimal action that she prefers to be taken in every state. All players discount the future at a common rate $\\delta<1$. Given strategies $\\mu \\triangleq\\left\\{\\mu_{i}\\right\\}_{i \\in N}$ and $\\alpha$, senders' and receiver's expected normalized payoffs (or simply payoffs) can be written as\n\n$$\n\\begin{aligned}\n& V_{i}(\\mu, \\alpha)=(1-\\delta) \\mathbb{E}_{\\mu, \\alpha}\\left[\\sum_{t=1}^{\\infty} \\delta^{t-1} v_{i}\\left(a_{t}\\right)\\right], \\quad i \\in N, \\\\\n& U(\\mu, \\alpha)=(1-\\delta) \\mathbb{E}_{\\mu, \\alpha}\\left[\\sum_{t=1}^{\\infty} \\delta^{t-1} u\\left(a_{t}, \\omega_{t}\\right)\\right],\n\\end{aligned}\n$$\n\nwhere the conditional expectations take into account the probability law induced by the strategies of the players, evolution of states, and the randomization device.\n\nThe set of feasible payoffs in the dynamic game $\\Gamma^{\\infty}(\\delta), \\mathscr{F} \\subseteq \\mathbb{R}^{n+1}$, with a typical element $(V, U)$ consists of all players' payoffs that can be achieved by some strategies $\\mu, \\alpha$. Because the first state is drawn from the stationary distribution, the states in all periods are ex ante identically distributed, and the set of feasible payoffs in the repeated game coincides with the set of feasible payoffs in the stage game. This set is a polytope with finitely many vertices and admits a simple characterization: it is equal to the convex hull of the payoffs in the stage game resulting from all pure mappings from states into actions. Moreover, any payoff $(U, V) \\in \\mathscr{F}$ can be supported by a stage-game strategy a: $\\Omega \\rightarrow \\Delta(A)$ played in every period.","text_sha256":"a9acfb97449dcce782eae4a9ec88c2460458c8c1157ab62b1488187bc4d1d346"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0006","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"$$\n\\mathscr{F}=\\operatorname{co}\\left\\{\\left(\\mathbb{E}_{p} v(\\mathrm{a}(\\omega)), \\mathbb{E}_{p} u(\\mathrm{a}(\\omega), \\omega)\\right) \\mid \\mathrm{a}: \\Omega \\rightarrow A\\right\\} .\n$$\n\nThe set of individually rational payoffs, $\\mathscr{F}^{*} \\subseteq \\mathscr{F}$, in a dynamic game $\\Gamma^{\\infty}(\\delta)$ consists of all feasible payoffs such that all players get at least their minmax payoffs. The senders' minmax payoffs are defined as $\\min _{\\alpha, \\mu_{-i}} \\max _{\\mu_{i}} V_{i}(\\alpha, \\mu)$ and the receivers' minmax payoff is defined as $\\min _{\\mu} \\max _{\\alpha} U(\\alpha, \\mu)$. Our payoff structure allows for a particularly tractable characterization of $\\mathscr{F}^{*}$. As the receiver can always ignore the senders' messages his minmax payoff is $\\underline{U} \\triangleq \\max _{a \\in A} \\mathbb{E}_{p}[u(a, \\omega)]$. At the same time, because the receiver fully controls the senders' payoffs, their individual rationality is innocuous. As a result,\n\n$$\n\\mathscr{F}^{*}=\\{(V, U) \\in \\mathscr{F} \\mid U \\geq \\underline{U}\\} .\n$$\n\nOur equilibrium concept is a perfect Bayesian equilibrium as described by Fudenberg and Tirole (1991). A strategy profile $(\\mu, \\alpha)$, together with a system of beliefs, is a perfect Bayesian equilibrium if $\\mu$ is sequentially rational and Bayes' rule is used to update beliefs whenever possible. ${ }^{10}$ Denote by $\\mathscr{E}(\\delta) \\subseteq \\mathbb{R}^{n+1}$ the set of equilibrium payoffs.","text_sha256":"be5a60ae5502d8177f30f18b082fe85e940653811603153226752c4500c5b673"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0007","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Folk Theorem","text":"## 3 Folk Theorem\n\nOur main result is a characterization of the limit set of equilibrium payoffs as players become arbitrarily patient, $\\lim _{\\delta \\rightarrow 1} \\mathscr{E}(\\delta)$. We obtain an analog of a folk theorem-any feasible, individually rational payoff can be approximated in an equilibrium if players are sufficiently patient.\n\nA few observations are immediate. First, no payoff outside of $\\mathscr{F}^{*}$ can be supported in equilibrium-the receiver can always guarantee at least his minmax payoff. Second, there are fully uninformative equilibria in the dynamic game $\\Gamma^{\\infty}$ in which all senders babble; that is, they send messages independently of their states, and the receiver plays a myopic best-response. The set of babbling payoffs belongs to the set of equilibrium payoffs for all $\\delta$.\n\nAssumption 1. (Valuable Communication) There exists a vector $(V, U) \\in \\mathscr{F}^{*}$ such that $U>\\underline{U}$.\n\nAssumption 1 states that communication is valuable-the receiver can strictly benefit from knowing the individual states. The assumption holds in most relevant economic environments. It is weaker than non-empty interior requirements in existing folk theorems, as it applies to the receiver only. This assumption allows to provide incentives for the receiver to follow the expected equilibrium play by threatening to cease the communication and permanently switch to babbling.\n\nTheorem 1. (Folk Theorem) For any payoffs $(V, U) \\in \\operatorname{relint}\\left(\\mathscr{F}^{*}\\right)$ there exists $\\underline{\\delta}<1$ such that for all $\\delta>\\underline{\\delta}$ there is an equilibrium with payoffs $(V, U) .{ }^{11}$ Consequently,\n\n$$\n\\lim _{\\delta \\rightarrow 1} \\mathscr{E}(\\delta)=\\mathscr{F}^{*} .\n$$\n\nThe detailed proof is relegated to the Appendix. Here we briefly outline its main ideas. For any target payoff in $\\mathscr{F}^{*}$ we pick the receiver's strategy a : $\\Omega \\rightarrow \\Delta(A)$ that supports it in the stage game and construct an equilibrium in which senders report truthfully and the receiver plays according to a most of the time.\n\n[^4]In particular, the equilibrium play switches between communication and adjustment phases. Transition between phases is determined by the senders' accumulated payoffs, which depend only on past actions and thus are publicly observed. In the communication phase, each sender reports her individual state truthfully, and the receiver plays according to the stage-game strategy a. In the adjustment phase, the receiver plays according to a strategy that brings all senders' discounted payoffs back to the target. The length and strategy of the adjustment phase depend on the senders' payoffs at the end of preceding communication phase and can always be chosen to hit the target as long as the players are sufficiently patient. Any receiver's randomization is done via the public randomization device so his deviations are immediately observable and trigger permanent babbling play.\n\nThe exact specification of the phases and their transition is tailored to ensure the players' obedience in following the equilibrium play. The senders are willing to report truthfully because they are guaranteed to obtain the target payoffs irrespectively of their messages. The receiver is effectively deferred from deviating by the assumption of valuable communication if sufficiently patient.\n\nAs players become more patient, the length of the communication phase can be increased and we can appeal to the law of large numbers for Markov chains. Hence, on average, there is less adjustment to be made, so the play occurs in the communication phase most of the time. As a result, the equilibrium payoffs approach the target payoffs.\n\nExample 1. (Resource Allocation) We illustrate the setting and the results in a resource allocation example. The receiver is a social planner who decides every period how to allocate an indivisible resource between two ex-ante symmetric regions. The senders are local representatives who privately observe the social values of allocating the resource to their region. They report the values to the planner at a regular meeting. The planner wants to put the resource to the best use and the representatives simply prefer having resource in their region. The social values are independent across regions and positively but imperfectly correlated across periods.\n\nThe example fits into our model by setting $n=2, A=\\{1,2\\}$; for $i \\in\\{1,2\\}, \\Omega_{i}=$ $\\left\\{\\omega^{0}, \\omega^{1}\\right\\}, v_{i}(a)=\\mathbf{1}(a=i), k\\left(\\omega_{i}, \\omega_{j} \\mid \\omega_{i}, \\omega_{j}\\right)=\\rho^{2}$, and $k\\left(\\omega_{i}, \\omega_{j}^{\\prime} \\mid \\omega_{i}, \\omega_{j}\\right)=\\rho(1-\\rho)$, for $\\omega_{j}^{\\prime} \\neq$ $\\omega_{j}$. Normalizing $\\omega_{i}^{0}=0$ and $\\omega_{i}^{1}=1, u(a, \\omega)=\\omega_{a}$. Positive but imperfect correlation implies $\\rho \\in(1 / 2,1)$ and corresponds to a uniform stationary distribution. The set of feasible and individually rational payoffs can be calculated by (1) and (2), and is presented in Figure 1. The set does not depend on the degree of correlation $\\rho$ and has an empty interior since the feasibility of allocation requires $V_{1}+V_{2}=1$.\n\nThe assumption of valuable communication is satisfied. In the absence of additional information the planner is indifferent between allocating the resource to one region or another and obtains a payoff $\\underline{U}=1 / 2$. However, the resource allocation matters to representatives. The \"babbling\" payoffs are generically Pareto inefficient and constitute a set\n\n$$\n\\mathscr{B}=\\left\\{(V, U) \\mid V_{1}+V_{2}=1, U=1 / 2\\right\\} .\n$$\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: The set of feasible payoffs, $\\mathscr{F}$, with a typical element $\\left(V_{1}, V_{2}, U\\right)$. Feasibility implies that the set $\\mathscr{F}$ lies on the plane $V_{1}+V_{2}=1$. The shaded area indicates individually rational payoffs $\\mathscr{F}^{*}$.\n\nIf the stage game were played only once, then strategic considerations would restrain communication and the babbling payoffs would be a unique equilibrium payoffs. The argument is standard. If the planner were to rely on a representative's report then the representative would communicate the report that maximizes a probability of getting the resource irrespectively of its social value. This would in turn make the report uninformative and preclude payoffs outside of $\\mathscr{B}$.","text_sha256":"b5d98b51f2c9fbc16aed413aa68e23f19656510bf1b1fd991d9bd7789a4cd99b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0008","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Folk Theorem","text":"However, in a dynamic game any payoffs in $\\mathscr{F}^{*}$ can be approximated in an equilibrium if the players are sufficiently patient as shown in Theorem 1. According to our equilibrium construction the planner should simply ensure the representatives of the total discounted allocation to their regions irrespectively of their reports. It can be achieved by infrequently shutting down the meetings and bringing the total discounted allocation back to the targets. In this way, the resource is allocated efficiently in a fraction of periods arbitrarily close to one. $\\square$\n\nTo recapitulate, in the absence of any contract enforcement or message verification, almost fully informed decision-making can be sustained in a dynamic game even if no informative communication can be sustained in the stage game. Moreover, the result does not require the payoff set to have a non-empty interior in contrast to most of existing folk theorems.","text_sha256":"ea20890c1c068aa0520f14760284e852a0bced6904dd65f383c1bc0bc94c04d9"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0009","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Discussion","text":"## 4 Discussion\n\nIn this section we discuss two important features of our model that are necessary for our equilibrium construction: ergodicity of state transitions and state-independence of senders' payoffs.\n\nErgodicity It is crucial for our equilibrium construction that the Markov chain according to which states evolve is ergodic, that is irreducible and aperiodic. Both of these properties are important and ensure that even though the states can be correlated across periods, this correlation vanishes as the distance between them grows. It gives the game its recurrent structure and allows to split the equilibrium play in \"almost independent\" blocks. The following example shows that the folk theorem result can fail if the chain is not irreducible.\n\nExample 2. (Reducible Chain) Consider a game with a single sender, two states $\\Omega=$ $\\left\\{\\omega^{0}, \\omega^{1}\\right\\}$, and two actions $A=\\{0,1\\}$. The payoff functions are $v(a)=a$, and $u(a, \\omega)=$ $\\mathbf{1}\\left(\\omega=\\omega^{a}\\right)$ : the sender always prefers action 1 whereas the receiver wants to match the state. In contrast to the previous analysis, the state is perfectly persistent, that is $k(\\omega \\mid \\omega)=1$. The initial state $\\omega_{1}$ is drawn according to the distribution ( $1-p_{0}, p_{0}$ ), with $p_{0}<1 / 2$.\n\nWe argue that the babbling payoff is the unique equilibrium payoff of the dynamic game for any $\\delta<1$. In fact, we show that it is the unique equilibrium payoff even if the receiver could commit to his strategy. If the receiver can commit to his strategy, by the revelation principle of Myerson (1986) any equilibrium payoff can be implemented by a direct mechanism. In the mechanism, the sender truthfully announces the initial state and the receiver follows a pre-committed dynamic strategy $a: \\Omega \\rightarrow A^{\\infty}$. For a given (incentive compatible) mechanism, the players' expected payoff are\n\n$$\n\\begin{aligned}\n& V=\\left(1-p_{0}\\right)(1-\\delta) \\sum_{t=0}^{\\infty} \\delta^{t} \\operatorname{Pr}\\left[a_{t}\\left(\\omega^{0}\\right)=1\\right]+p_{0}(1-\\delta) \\sum_{t=0}^{\\infty} \\delta^{t} \\operatorname{Pr}\\left[a_{t}\\left(\\omega^{1}\\right)=1\\right], \\\\\n& U=\\left(1-p_{0}\\right)(1-\\delta) \\sum_{t=0}^{\\infty} \\delta^{t} \\operatorname{Pr}\\left[a_{t}\\left(\\omega^{0}\\right)=0\\right]+p_{0}(1-\\delta) \\sum_{t=0}^{\\infty} \\delta^{t} \\operatorname{Pr}\\left[a_{t}\\left(\\omega^{1}\\right)=1\\right]\n\\end{aligned}\n$$\n\nSince the sender's payoffs are state-independent, incentive compatibility implies that she must be indifferent between reporting states $\\omega_{0}$ and $\\omega_{1}$. Substituting into to the formula for the receiver's payoff we obtain that the mechanism is incentive compatible only if\n\n$$\nU=1-p_{0}-V\\left(1-2 p_{0}\\right) .\n$$\n\nThe unique feasible individually rational payoff that satisfies this equation is the babbling payoff $\\left(0,1-p_{0}\\right)$. It follows the babbling payoff is also the unique equilibrium payoff of the dynamic game in which the receiver cannot commit. $\\square$\n\nThis example illustrates that when the state chain is aperiodic but reducible, the sender's incentive compatibility and the receiver's individual rationality alone can preclude equilibrium communication. We conjecture that when the chain is irreducible but periodic, the folk theorem result still fails with the receiver's incentives playing an important role. For this reason, constructing a specific counterexample in that case seems to be more complicated.\n\nState-independent payoffs Another important assumption of our model is that the senders' payoffs do not depend on the state. One might think that our equilibrium construction still work if the payoffs are \"almost\" state-independent. Unfortunately, this is not the case. In our equilibrium, the senders are indifferent between any history of messages and report truthfully in equilibrium. If the payoffs are even slightly state-dependent then the senders' best-response can be far from truthtelling.\n\nIn fact, for a generic payoff perturbation one can apply the results of Renault et al. (2013) to characterize the limit set of equilibrium payoffs, at least for the case of a single sender and Markov chain satisfying their Assumption A. They show that the equilibrium payoffs must satisfy senders' \"incentive compatibility\"-she shouldn't be able to benefit from permuting her reports. This constraint is ordinal and even small payoff perturbations can drastically restrict the equilibrium payoff set.","text_sha256":"dc5a410c619c1826edac723cbb9afe4137f1b5d97c3cf78613cbd0495147be0b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0010","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Conclusion","text":"## 5 Conclusion\n\nWe analyzed dynamic information transmission when senders' payoffs are state independent. We show that the strategic limits of communication prevalent in a stage-game disappear when the interaction is repeated. Any individually rational payoffs can be approximated in equilibrium if the players are sufficiently patient. This result complements the existing results for state-dependent payoffs and provides a rationale for informative communication to be sustained in a variety of dynamic economic settings between players with seemingly misaligned interests. In fact, our equilibrium construction delivers a clear message how to do so-to induce truthtelling the receiver should track the senders' payoff and adjust it whenever they are doing too good or too bad. This ensures the senders of their payoffs irrespective of their reports and eliminates incentives to lie.\n\nThere are many opportunities for further research. First, the limit analysis for statedependent payoffs not captured by the previous literature should be completed. Second, one can analyze the joint limit of state independency and patience. Finally, an important open question is the general analysis of equilibrium payoffs when players are impatient. We suspect that in this case finer details of Markov transition and not just its stationary distribution will come into play.","text_sha256":"d7b660f2e39347d994871b388159dbd7fba4cb0c17bef204c149afdbcddec9e1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0011","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Appendix","text":"## Appendix","text_sha256":"358db74c404180f165344e5a7796848242432a605b39156b0379338c908ec1e9"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0012","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proof of Theorem 1","text":"## Proof of Theorem 1\n\nWithout loss of generality, we normalize players' payoffs so that $\\min _{A} v_{i}(a)=0, \\forall i \\in N$, and $\\min _{(a, \\omega) \\in A \\times \\Omega} u(a, \\omega)=0$. Let $\\bar{v}_{i} \\triangleq \\max _{a \\in A} v_{i}(a)$, and $\\bar{u} \\triangleq \\max _{(a, \\omega) \\in A \\times \\Omega} u(a, \\omega)$. Recall that\n\n$$\n\\mathscr{F}^{*}=\\{(V, U) \\in \\mathscr{F} \\mid U \\geq \\underline{U}\\} .\n$$\n\nwhere $\\underline{U}=\\max _{a \\in A} \\mathbb{E}_{p}[u(a, \\omega)]$. Given any feasible and individually rational payoffs, we construct an equilibrium that achieves payoffs arbitrarily close to it provided that $\\delta$ is sufficiently high. In particular, define\n\n$$\n\\mathscr{F}^{* *} \\triangleq\\left\\{(V, U) \\in \\operatorname{relint}\\left(\\mathscr{F}^{*}\\right)\\right\\} .\n$$\n\nand let $|\\cdot|$ denote the Euclidean norm. For any $(V, U) \\in \\mathscr{F}^{* *}$ there exists $\\eta>0$ such that $\\forall V^{\\prime}:\\left|V^{\\prime}-V\\right| \\leq \\eta, V^{\\prime} \\in \\mathscr{F}^{* *}$. We will show that for any $\\varepsilon>0$ and any $(V, U) \\in \\mathscr{F}^{* *}$, there exists $\\underline{\\delta}<1$ such that for all $\\delta>\\underline{\\delta}$ we can construct an equilibrium $\\left(\\mu^{*}, \\alpha^{*}\\right)$ of $\\Gamma^{\\infty}(\\delta)$ with payoffs $\\left(V^{*}, U^{*}\\right)$, where $V^{*}=V$ and $\\left|U^{*}-U\\right|<\\varepsilon$. Since we allow for a public randomization device, the result of the theorem will follow.\n\nIn what follows, fix $\\delta<1, \\eta>0$, the target payoff vector $(V, U) \\in \\mathscr{F}^{* *}$, and the corresponding supporting strategy a: $\\Omega \\rightarrow \\Delta(A)$.\n\nStrategies We first describe the equilibrium strategies ( $\\mu^{*}, \\alpha^{*}$ ). Fix some $T_{c} \\in \\mathbb{N}$ as a function of $\\delta$ such that $\\lim _{\\delta \\rightarrow 1} \\delta^{T_{c}(\\delta)}=1, \\lim _{\\delta \\rightarrow 1} T_{c}(\\delta)=\\infty .{ }^{12}$ The equilibrium play is divided into consecutive blocks, each starting with a communication phase of length $T_{c}$ followed by an adjustment phase of (random) length $T_{a}$.\n\nOn the equilibrium path, the behavior within each block is described by strategies $\\left(\\mu_{1}, \\alpha_{1}\\right)$. According to $\\mu_{1}$, the senders always report truthfully. According to $\\alpha_{1}$, the receiver's behavior depends on the current phase within a block. In the communication phase, he plays according to the strategy a. In the adjustment phase, he ignores the senders' reports and plays according to an adjustment strategy that depends on the profile of senders' normalized discounted payoffs at the end of the communication phase,\n\n$$\n\\bar{v}_{c} \\triangleq \\frac{1-\\delta}{1-\\delta^{T_{c}}} \\sum_{t=1}^{T_{c}} \\delta^{t-1} v\\left(a_{t}\\right),\n$$\n\nand the target payoff vector $V$. In particular, define\n\n$$\n\\lambda\\left(\\bar{v}_{c}\\right) \\triangleq \\frac{\\bar{v}_{c}-V}{\\left|\\bar{v}_{c}-V\\right|}, \\quad V_{a}\\left(\\bar{v}_{c}\\right) \\triangleq V-\\lambda\\left(\\bar{v}_{c}\\right) \\eta .\n$$\n\n[^5]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 2: The choice of adjustment payoffs $V_{a}$ given the realized senders' payoff at the end of the communication phase $\\bar{v}_{c}$ and the target equilibrium payoffs $V$. Schematic illustration for the case of two senders, $n=2$.\n\nLet $\\hat{T}_{a} \\in \\mathbb{R}_{+}$satisfy\n\n$$\n\\delta^{T_{c}}\\left(1-\\delta^{\\hat{T}_{a}}\\right) \\eta=\\left(1-\\delta^{T_{c}}\\right)\\left|\\bar{v}_{c}-V\\right| .\n$$\n\nNotice that $\\hat{T}_{a}$ is well defined as long as $\\eta>\\left(1-\\delta^{T_{c}}\\right) \\delta^{-T_{c}} \\sum_{n=1}^{N} \\bar{v}_{i}$, which, in light of our choice of $\\delta$ and $T_{c}(\\delta)$, is verified for $\\delta$ high enough. Define $T_{a} \\triangleq\\left\\lfloor\\hat{T}_{a}\\right\\rfloor+1$, and let $0 \\leq r\\left(\\bar{v}_{c}\\right)<\\eta$ solve\n\n$$\n\\frac{1-\\delta}{1-\\delta^{T_{c}+T_{a}}}\\left(\\sum_{t=1}^{T_{c}} \\delta^{t-1} v\\left(a_{t}\\right)+\\sum_{t=T_{c}+1}^{T_{c}+T_{a}-1} \\delta^{t-1} V_{a}\\left(\\bar{v}_{c}\\right)+\\delta^{T_{c}+T_{a}-1}\\left(V-\\lambda\\left(\\bar{v}_{c}\\right) r\\left(\\bar{v}_{c}\\right)\\right)\\right)=V,\n$$\n\nThe adjustment phase lasts $T_{a}$ periods. In the first $T_{a}-1$ periods, the possibly random action $\\mathrm{a}_{a} \\in \\Delta A$ is played with $V_{a}\\left(\\bar{v}_{c}\\right)=\\mathbb{E} v\\left(\\mathrm{a}_{a}\\right)$. In the last period, the possibly random action $\\mathrm{a}_{r} \\in \\Delta A$ is played with $V-\\lambda\\left(\\bar{v}_{c}\\right) r\\left(\\bar{v}_{c}\\right)=\\mathbb{E} v\\left(\\mathrm{a}_{r}\\right)$.\n\nIf the receiver ever deviated, then $\\left(\\mu^{*}, \\alpha^{*}\\right)$ prescribes the senders to babble; that is, to send messages irrespectively of their individual states, and the receiver to play an action that is optimal given no information.\n\nAs for the deviation of each sender, because individual states are correlated, some deviations during the communication phase are detectable as they lead to an inconsistent sequence\nof messages. During the communication phase, whenever the current messages $m_{t}$ are inconsistent with the messages from the previous period $m_{t-1}$ the receiver plays a after replacing the reports with an artificially generated state $\\hat{\\omega} \\in \\Omega$ consistent with the previous messages and the underlying Markov chain. The strategy $\\mu^{*}$ prescribes each sender to keep playing according to $\\mu_{1}$ after any detected deviation of another sender, and if she ever privately deviated.\n\nFinally, transitions between and within blocks are as follows. Each block starts with a communication phase that lasts for $T_{c}$ periods and is followed by the adjustment phase, which lasts for $T_{a}$ periods according to (3). At period $T_{a}+1$, a new block starts and the timer is reset to $t=1$.\n\nPayoffs By construction, the equilibrium strategies deliver expected continuation payoffs $V$ to the senders at the beginning of each block and, in particular, at the beginning of the game. We show that for sufficiently high $\\delta$, the equilibrium strategies deliver the expected continuation payoff within $\\varepsilon$ of the target payoff $U$ to the receiver after any public history.\n\nFirst, we show that the expected continuation payoff of the receiver is within $\\varepsilon$ of the target payoff $U$ at the beginning of each block. Consider the average realized payoffs within each block at the end of communication and adjustments phases:","text_sha256":"ae5639c61ed76cc29568d9def2a9d55c763a931f22f042812b250d4565b1cc02"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0013","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proof of Theorem 1","text":"$$\n\\begin{aligned}\n& \\bar{u}_{c} \\triangleq \\frac{1-\\delta}{1-\\delta^{T_{c}}} \\sum_{t=1}^{T_{c}} \\delta^{t-1} u\\left(a_{t}, \\omega_{t}\\right) \\\\\n& \\bar{u}_{a} \\triangleq \\frac{1-\\delta}{1-\\delta^{T_{c}+T_{a}}} \\sum_{t=1}^{T_{c}+T_{a}} \\delta^{t-1} u\\left(a_{t}, \\omega_{t}\\right)\n\\end{aligned}\n$$\n\nso that $\\bar{u}_{c}, \\bar{u}_{a} \\in \\operatorname{proj}_{n+1} \\mathscr{F}^{*}$ and $U=\\mathbb{E}_{p}\\left[\\bar{u}_{c}\\right]$.\nWe can bound the difference between $\\bar{u}_{a}$ and $\\bar{u}_{c}$ as follows:\n\n$$\n\\begin{aligned}\n\\bar{u}_{a}-\\bar{u}_{c} & =\\frac{\\delta^{T_{c}}\\left(1-\\delta^{T_{a}}\\right)}{1-\\delta^{T_{c}+T_{a}}}\\left(\\frac{1-\\delta}{1-\\delta^{T_{a}}} \\sum_{t=T_{c}+1}^{T_{c}+T_{a}} \\delta^{t-T_{c}-1} u\\left(a_{t}, \\omega_{t}\\right)-\\bar{u}_{c}\\right), \\\\\n\\left|\\bar{u}_{a}-\\bar{u}_{c}\\right| & \\leq \\frac{\\delta^{T_{c}}\\left(1-\\delta^{T_{a}}\\right)}{1-\\delta^{T_{c}+T_{a}}} \\bar{u} \\leq \\frac{\\delta^{T_{c}}\\left(1-\\delta^{\\hat{T}_{a}+1}\\right)}{1-\\delta^{T_{c}+\\hat{T}_{a}+1}} \\bar{u}=\\frac{\\delta\\left|\\bar{v}_{c}-V\\right|+o(1) \\eta}{\\delta\\left|\\bar{v}_{c}-V\\right|+\\eta+o(1) \\eta} \\bar{u}\n\\end{aligned}\n$$\n\nwhere $o(1)$ is a function converging to 0 as $\\delta \\rightarrow 1$, and the second inequality follows by (3) and the fact that $\\delta^{T_{c}} \\rightarrow 1$ as $\\delta \\rightarrow 1$.\n\nBy construction $\\mathbb{E}_{p}\\left[\\bar{v}_{c}\\right]=V^{*}=V$. We now argue that as $\\delta$ goes to 1, the difference $\\left|\\bar{v}_{c}-V\\right|$ converges in probability to 0 independently of the starting distribution. Fix a\nsender $i \\in N$. First, by the ergodic theorem, for any starting distribution $\\pi \\in \\Delta(\\Omega)$,\n\n$$\n\\operatorname{Pr}_{\\pi}\\left[\\lim _{T \\rightarrow \\infty} \\frac{1}{T} \\sum_{t=1}^{T} v_{i}\\left(\\mathrm{a}\\left(\\omega_{t}\\right)\\right)=\\sum_{\\omega \\in \\Omega} p(\\omega) v_{i}(\\mathrm{a}(\\omega))\\right]=1,\n$$\n\nSecond, for any sequence of states $\\left\\{\\omega_{t}\\right\\}_{t=0}^{T-1}$\n\n$$\n\\left|\\frac{1-\\delta}{1-\\delta^{T}} \\sum_{t=1}^{T} \\delta^{t-1} v_{i}\\left(\\mathrm{a}\\left(\\omega_{t}\\right)\\right)-\\frac{1}{T} \\sum_{t=1}^{T} v_{i}\\left(\\mathrm{a}\\left(\\omega_{t}\\right)\\right)\\right|<\\max \\left\\{\\left|T \\frac{1-\\delta}{1-\\delta^{T}}-1\\right|,\\left|T \\frac{1-\\delta}{1-\\delta^{T}} \\delta^{T-1}-1\\right|\\right\\} \\bar{v}_{i} .\n$$\n\nUsing the fact $\\delta^{T_{c}(\\delta)} \\rightarrow 1$ and $T_{c}(\\delta) \\rightarrow \\infty$ as $\\delta \\rightarrow 1$, one can show that the right-hand side of (5) converges to 0 as $\\delta \\rightarrow 1$. Combining these two observations, by (4) it follows that for any $\\varepsilon$, for $\\delta$ sufficiently large, $\\operatorname{Pr}_{\\pi}\\left[\\left|\\bar{u}_{c}-\\bar{u}_{a}\\right|>\\varepsilon\\right]<1-\\varepsilon$.\n\nConsequently, because the receiver's stage payoffs are bounded, and $T_{c}(\\delta) \\rightarrow \\infty$ as $\\delta \\rightarrow 1, \\mathbb{E}_{\\pi}\\left[\\left|\\bar{u}_{a}-\\bar{u}_{c}\\right|\\right] \\rightarrow 0$ as $\\delta \\rightarrow 1$, where $\\pi \\in \\Delta(\\Omega)$ is the belief held by the receiver at the beginning of the block. On the other hand, by the ergodic theorem $\\mathbb{E}_{\\pi}\\left[\\bar{u}_{c}\\right] \\rightarrow U$.\n\nWe conclude the section by showing the receiver's expected continuation payoff after any public history is close to $U$. Since the equilibrium play is reset in every block, it suffices to show that (i) the contribution of a single block to the total payoff is negligible and (ii) the impact of current beliefs on payoffs in future blocks is negligible.\n\nFor (i), observe that by (3)\n\n$$\n\\left(1-\\delta^{T_{a}-1}\\right) \\eta \\leq \\frac{1-\\delta^{T_{c}}}{\\delta^{T_{c}}}\\left|\\bar{v}_{c}-V\\right| \\leq \\frac{1-\\delta^{T_{c}}}{\\delta^{T_{c}}} N \\max _{i \\in N} \\bar{v}_{i} .\n$$\n\nConsequently, $\\delta^{T_{a}-1} \\rightarrow 1$ as $\\delta \\rightarrow 1$ and $\\delta^{T_{c}} \\rightarrow 1$, so the contribution of a single block, weighted by $1-\\delta^{T_{c}+T_{a}+1}$, goes to 0 as $\\delta \\rightarrow 1$.\n\nFor (ii), observe that from before, as $\\delta \\rightarrow 1, \\mathbb{E}_{\\pi}\\left[\\bar{u}_{a}\\right]$ converges in expectation to $U$ for any starting belief $\\pi$.\n\nIncentives We now check players' incentives to follow the suggested strategies $\\left(\\pi^{*}, \\alpha^{*}\\right)$; that is, that they are sequentially rational. Consider senders' incentives. Because babbling is an equilibrium of a stage game, the off-path behavior $\\left(\\pi_{\\varnothing}, \\alpha_{\\varnothing}\\right)$ is sequentially rational. Also, since the senders' expected continuation payoffs are independent of their messages, they have no incentives to deviate. Thus, $\\pi_{1}$ is sequentially rational.\n\nWe consider now the receiver's incentives. Let $a: \\Delta(\\Omega) \\rightarrow A$ be the action that maximizes the receiver's expected payoff in the stage game for any given distribution over $\\Omega$, i.e.,\n\n$$\na(\\pi) \\in \\arg \\max _{a \\in A} \\sum_{\\omega \\in \\Omega} \\pi(\\omega) u(a, \\omega) .\n$$\n\nDefine the map $\\phi: \\Delta(\\Omega) \\rightarrow \\Delta(\\Omega)$ as $\\phi(\\pi)(\\omega):=\\sum_{\\omega^{\\prime} \\in \\Omega} \\pi\\left(\\omega^{\\prime}\\right) k\\left(\\omega \\mid \\omega^{\\prime}\\right)$. The map describes how the belief of the receiver evolves in the absence of information.\n\nThe receiver's continuation payoff following a deviation at time $t$ equals\n\n$$\n\\mathbb{E}_{\\pi_{t}}\\left[(1-\\delta) \\sum_{\\tau=1}^{\\infty} \\delta^{\\tau} u\\left(a\\left(\\phi^{(\\tau)}\\left(\\pi_{t}\\right)\\right), \\omega_{t+\\tau}\\right)\\right]\n$$\n\nwhere $\\pi_{t}$ is the receiver's belief at the end of period $t$.\nSince the Markov chain is aperiodic, for any $\\pi_{t}, \\lim _{\\tau \\rightarrow \\infty} \\phi^{(\\tau)}\\left(\\pi_{t}\\right)=p$. Hence, for any $\\varepsilon$ and any $\\pi_{t} \\in \\Delta(\\Omega)$\n\n$$\n\\left|\\mathbb{E}_{\\pi_{t}}\\left[(1-\\delta) \\sum_{\\tau=1}^{\\infty} \\delta^{\\tau} u\\left(a\\left(\\phi^{(\\tau)}\\left(\\pi_{t}\\right)\\right), \\omega_{t+\\tau}\\right)\\right]-\\underline{U}\\right|<\\varepsilon\n$$\n\nprovided $\\delta$ is high enough.\nIn words, the receiver can benefit from the current information only in the short run. In particular, the receiver's continuation payoff following a deviation can be taken to be arbitrarily close to $\\underline{U}$ as $\\delta \\rightarrow 1$.","text_sha256":"5964eed4594690e1662715f0a0504597364dbe867b457ba174c3b2688be6d2aa"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0014","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proof of Theorem 1","text":"As we showed before, the equilibrium expected continuation payoff after any history is in the neighborhood of $U$. Since $U \\in \\mathscr{F}^{* *}, U>\\underline{U}$. Hence, the receiver is willing to obey if sufficiently patient facing the threat of switching to babbling. It follows that $\\alpha_{1}$ is sequentially rational.\n\nTo complete the proof, we specify the system of beliefs. Bayes rule uniquely pins down players' belief, unless a sequence of inconsistent messages are reported. Recall that when the current messages $m_{t}$ are inconsistent with the messages from the previous period $m_{t-1}$, the receiver takes action a $(\\hat{\\omega})$, where $\\hat{\\omega}$ is an artificially generated state. At these histories, the receiver updates his belief according to Bayes rule as if the reported state was $\\hat{\\omega}$. The belief of the deviator is computed by Bayes rule. The belief of any other sender $i$ equals the posterior belief based only on her private information $\\omega_{i}^{t}$.","text_sha256":"ade7e776682add630cf684a6fc1dfa06a46a47f71f0e7c882be500d2c0b251d8"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0015","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAbreu, Dilip, David Pearce, and Ennio Stacchetti (1990) \"Toward a Theory of Discounted Repeated Games with Imperfect Monitoring,\" Econometrica, 58, 1041-1063.\n\nChakraborty, Archishman and Rick Harbaugh (2010) \"Persuasion by Cheap Talk,\" American Economic Review, 100, 2361-2382.\n\nCrawford, Vincent P. and Joel Sobel (1982) \"Strategic Information Transmission,\" Econometrica, 50, 1431-1451.\n\nEly, Jeffrey C., Johannes Hörner, and Wojciech Olszewski (2005) \"Belief-Free Equilibria in Repeated Games,\" Econometrica, 73, 377-415.\n\nEscobar, Juan F. and Juuso Toikka (2013) \"Efficiency in games with Markovian private information,\" Econometrica, 81, 1887-1934.\n\nFrankel, Alexander (2016) \"Discounted quotas,\" Journal of Economic Theory, 166, 396-444.\nFudenberg, Drew, David Levine, and Eric Maskin (1994) \"The Folk Theorem with Imperfect Public Information,\" Econometrica, 62, 997-1039.\n\nFudenberg, Drew and Jean Tirole (1991) Game Theory, Cambridge, MA: MIT Press.\nJackson, Matthew O. and Hugo F. Sonnenschein (2007) \"Overcoming Incentive Constraints by Linking Decisions,\" Econometrica, 75, 241-257.\n\nMailath, George J. and Larry Samuelson (2006) Repeated Games and Reputations: Long-Run Relationships, New York: Oxford University Press.\n\nMyerson, Roger B. (1986) \"Multistage Games with Communication,\" Econometrica, 54, 323-358.\n\nRenault, Jérôme, Eilon Solan, and Nicolas Vieille (2013) \"Dynamic sender-receiver games,\" Journal of Economic Theory, 148, 502-534.\n\n[^0]:    *We are grateful to Larry Samuelson for support and encouragement throughout this project. We also thank Dirk Bergemann, Johannes Hörner and seminar participants at Yale and the 2015 World Congress of the Econometric Society for helpful discussions and suggestions. Finally, we wish to thank the editor and the anonymous referees for many valuable comments.\n    ${ }^{\\dagger}$ Department of Economics, Boston University, margaria@bu.edu\n    ${ }^{\\ddagger}$ Institute for Microeconomics, University of Bonn, alexey.v.smolin@gmail.com\n\n[^1]:    ${ }^{1}$ Chakraborty and Harbaugh (2010) provide many more examples of such strong biases.\n    ${ }^{2}$ There is no standard formulation of a \"folk theorem\" in repeated games with incomplete information.\n\n[^2]:    ${ }^{3}$ In the equilibria we construct, after every history each sender is indifferent between sending any message, independently of the private histories observed by the other senders. Nevertheless, these equilibria are not \"belief-free\" in the sense of Ely et al. (2005) as the same property does not hold for the receiver.\n    ${ }^{4}$ Escobar and Toikka (2013) also studied limit payoffs with dynamic communication but their setting and techniques are significantly different from ours. They focus on the case of independent private values, so the players know their own payoffs. Their proof relies on statistical tests similar to those used by Renault et al. (2013) which are of limited use in our setting.\n    ${ }^{5}$ The single action of the receiver affects all senders' payoffs at once. Hence, the problem with many senders is not simply a collection of separate single-sender problems.\n    ${ }^{6}$ We can dispense with the assumption that the messages are public if we allow the receiver to announce the messages at the end of each period.\n\n[^3]:    ${ }^{7}$ This implies that the receiver does not observe his payoffs. This assumption is standard in repeated games with incomplete information and can be justified in at least two ways. First, it can approximate the situation when payoffs are observed far into the future. Second, the receiver may want to make informed decisions even if she cannot ever confirm their accuracy just like a judge wants to acquit innocents.\n    ${ }^{8}$ This assumption is still a priori restrictive because the players cannot commit to their strategies and the revelation principle does not apply. However, it suffices to obtain a folk theorem.\n    ${ }^{9}$ We adopt a convention that for any stochastic process $x$ its time- $t$ realization is denoted by subscript $x_{t}$ and the history up to time $t$ is denoted by superscript $x^{t} \\triangleq\\left\\{x_{s}\\right\\}_{s=1}^{t}$.\n\n[^4]:    ${ }^{10}$ We imposed the standard restriction (B) in Fudenberg and Tirole (1991); because types are correlated, condition (B) needs to be adjusted (see Fudenberg and Tirole, 1991, pp. 349-350).\n    ${ }^{11}$ The relative interior of a set $\\mathscr{F}^{*}$, relint $\\left(\\mathscr{F}^{*}\\right)$, is its interior under the topology induced on the affine hull of $\\mathscr{F}^{*}$. Hence the statement is meaningful even if $\\mathscr{F}^{*}$ has a dimension lower than the number of players.\n\n[^5]:    ${ }^{12}$ For example, $T_{c}(\\delta)=\\left\\lceil(1-\\delta)^{-1 / 2}\\right\\rceil$. In what follows we will often omit dependence of $T_{c}$ on $\\delta$.","text_sha256":"6bba230abf555a6c25467661f1aec918363b4741e4993c0e3d9a9b5a4d8310c1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21:0016","work_id":"alex-smolin:dynamic-communication-with-biased-senders","paper_id":"alex-smolin:dynamic-communication-with-biased-senders:2017-10-21","title":"Dynamic Communication with Biased Senders","authors":[{"name":"Chiara Margaria","url":"https://www.chiaramargaria.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-10-21","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md","source_record":"https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf","doi":"https://doi.org/10.1016/j.geb.2017.10.017","citation":"Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Chiara Margaria; Alex Smolin\n\n**Canonical citation:** Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md\n\n**Source record:** https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf\n\n**Published record:** https://doi.org/10.1016/j.geb.2017.10.017\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"946851314381fac43e18413dd19b48191ab4c4cdad040bc2f85092422706fe15"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0001","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Dirk Bergemann; Alessandro Bonatti; Alex Smolin.\n> Canonical citation: Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"848f5b162d127af4ff6cfbc27a2dd54f6f819cc63ec7f11515d7a8150d2c92b9"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0002","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"The Design and Price of Information","text":"# The Design and Price of Information\n\n**Authors:** Dirk Bergemann; Alessandro Bonatti; Alex Smolin\n\n**Manuscript date:** 2017-06-03\n\n#### Abstract\n\nA data buyer faces a decision problem under uncertainty. He can augment his initial private information with supplemental data from a data seller. His willingness to pay for supplemental data is determined by the quality of his initial private information. The data seller optimally offers a menu of statistical experiments. We establish the properties that any revenue-maximizing menu of experiments must satisfy. Every experiment is a non-dispersed stochastic matrix, and every menu contains a fully informative experiment. In the cases of binary states and actions, or binary types, we provide an explicit construction of the optimal menu of experiments.\n\nKeywords: information design, price of information, statistical experiments, mechanism design, price discrimination, hypothesis testing.\n\nJEL Codes: D42, D82, D83.\n\n[^0]","text_sha256":"e40aa9c8d6df407d235e4c6e9ea515077b504c629fe19ccfa23761a9b4d74df4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0003","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction","text_sha256":"18f8b993fceb1a7535f9d0a0f4b58ea721117844a80f48a2844eba5cc8c9a8ef"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0004","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1.1 Motivation","text":"### 1.1 Motivation\n\nThe mechanisms by which information is traded can shape the creation and the distribution of surplus in many important markets. Information about individual borrowers guides banks' lending decisions, information about consumers' characteristics facilitates targeted online advertising, and information about a patient's genome enhances health care delivery. In all these settings, information buyers (i.e., lenders, advertisers, and health care providers) have private knowledge relevant to their decision problem at the time of contracting (e.g., independent knowledge of a borrower, prior interactions with specific consumers, access to a patient's family history). Thus, potential data buyers seek to acquire supplemental information to improve the quality of their decision making.\n\nIn this paper, we develop a framework to analyze the sale of supplemental information. We consider a data buyer who faces a decision problem under uncertainty. A monopolist data seller owns a database containing information about a \"state\" variable that is relevant to the buyer's decision. Initially, the data buyer has only partial information about the state. This information is private to the data buyer and unknown to the data seller. The precision of the buyer's private information determines his willingness to pay for any supplemental information. Thus, from the perspective of the data seller, there are many possible types of the data buyer. We investigate the revenue-maximizing information policy, i.e., how much information the data seller should provide and how she should price access to the data.\n\nIn order to screen the heterogeneous data buyer types, the seller offers a menu of information products. In our context, these products are statistical experiments-signals that reveal information about the payoff-relevant state. Only the information product itself is assumed to be contractible. By contrast, payments cannot be made contingent on either the buyer's action or the realized state and signal. Consequently, the value of an experiment to a buyer is determined by the buyer's private belief and can be computed independently of the price of the experiment. We can recast the resulting screening problem as a nonlinear pricing framework wherein the buyer's type is given by his prior belief. In other words, the seller's problem is to design and price different versions of experiments, that is, different information products from the same underlying database. Because the design of information can be rephrased in terms of hypothesis testing, the present analysis can also be interpreted as a pricing model for statistical tests.\n\nA large body of literature studies the problem of versioning information goods, emphasizing that digital production allows sellers to easily customize (or degrade) the attributes of such products (Shapiro and Varian, 1999). This argument applies even more forcefully\nto information products (i.e., statistical experiments). In a nutshell, the data seller's problem consists of degrading the quality of the information sold to some buyers in order to charge higher prices to higher-value buyers. We show that the very nature of information products enriches the scope of price discrimination. Because information is valuable to the extent to which it affects decision making, buyers with different beliefs do not simply value experiments differently: they may even disagree on their ranking. In this sense, the value of information naturally has both a vertical element (the quality of the information), and a horizontal element (the position of the information).\n\nWe show that the optimal menu contains, in general, both the fully informative experiment and partially informative, \"distorted\" experiments. The distorted information products are not simply noisy versions of the same data. Instead, optimality imposes considerable structure on the distortions in the information provided. In particular, every experiment offered as part of the optimal menu is non-dispersed, i.e., it contains a signal realization that rules out one of the states. Moreover, if the buyer's decision problem is to match his action with a state, every experiment is concentrated, i.e., it induces the buyer to take the correct action with probability one, conditional on at least one realized state.\n\nWe provide a full characterization of the optimal menu in the case of binary states and actions. This setting yields sharp insights into the profitability of discriminatory pricing for selling information. In the binary-state environment, the buyer's types are one dimensional and the utilities are piecewise linear with a kink at the belief at which the buyer would switch his optimal action. If all buyer types are congruent, i.e., they take identical actions without additional information, an intuition analogous to the \"no-haggling\" result for monopoly pricing (Myerson, 1981; Riley and Zeckhauser, 1983) applies: the seller simply offers the fully informative experiment at a fixed price. In general, however, the seller's problem consists of screening types both within and across classes of congruent types.\n\nThe beneficial use of partial information can be seen with two types that are ranked according to their valuation of the fully informative experiment. The \"high\" type is ex ante less informed, while the \"low\" type is ex ante more informed. Suppose that types would pursue distinct actions in the absence of additional information. A feasible policy for the seller is to offer the high type the fully informative experiment and the low type a partially informative experiment that generates one of two signals: with small but positive probability, the signal informs the low type without noise about the state that he considers less likely ex ante; with the remaining probability, it sends a second noisy signal. This allows the low type to improve the quality of his decision making; thus, he would be willing to pay a positive amount for the experiment. By contrast, the high type would not attach a positive value to this partial information. After all, he would have chosen the action suggested by the\nnoiseless signal under his prior anyway, and given his prior, the noisy signal is too weak to modify his action.","text_sha256":"613d8965e1b6b3f982c2ae974ff7c4a54d587e5e9f932f342ab413b0438757de"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0005","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1.1 Motivation","text":"The optimal menu for two types exploits the horizontal element of information to extract value from the low type without conceding any rents to the high type. Such profitable screening by providing partial information is the novel element that distinguishes the pricing of information from other monopoly problems, such as designing insurance or goods of differentiated quality. With a continuum of types, the optimal menu still contains at most two experiments: one is fully informative, and the other contains two signals, one of which perfectly reveals the true state. In particular, the optimal menu involves discriminatory pricing (i.e., two different experiments are offered) only if \"ironing\" is required (Myerson, 1981). Intuitively, the second experiment intends to serve buyers in one group, while charging higher prices to the other group.\n\nOur findings have concrete implications for the sale of information. In Section 5, we illustrate our results in the context of the information being sold by online data brokers, focusing on a broad class of products (\"data appends\") that are used for marketing and risk-mitigation purposes. We use the language of hypothesis testing and statistical errors to demonstrate how the design of data products can be informed by the structural properties of the optimal experiments we identified in our analysis. In particular, we argue that no experiment in an optimal menu should add unbiased noise to the seller's information, and we discuss whether enabling the buyer to access only a portion of the seller's data is equivalent to introducing noise.\n\nIn Section 6, we take the first step toward more general results. We fully characterize the menu with two types that face a matching decision problem. In an optimal menu, the high type purchases the fully informative experiment, while the low type purchases an experiment that is at least partially informative. This can occur even if the types are congruent. The latter experiment provides \"directional\" information about the states the low type perceives to be relatively more likely and induces him to take the corresponding actions more often. In this way, the seller optimally reduces the information rents of the less informed type, possibly to zero, without needing to exclude the more informed type.","text_sha256":"f239e62fe84c178f0f67a3a8db8ae22b713d4b8500faa3dc8ed970d65f750457"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0006","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1.2 Related Literature","text":"### 1.2 Related Literature\n\nOur paper is part of the body of literature on selling information to imperfectly informed decision makers. In seminal papers, Admati and Pfleiderer $(1986,1990)$ analyze the sale of information to a continuum of ex ante homogeneous agents, all with the same prior information. After the purchase of supplemental information, the agents trade an asset\nwith a common value. They show that it is optimal to provide noisy, idiosyncratic and, hence, heterogeneous information. This idiosyncratic information guarantees the traders a local monopoly, which preserves the value of acquiring information even in an informative, rational-expectations equilibrium. Thus, Admati and Pfleiderer $(1986,1990)$ explicitly consider interactions among data buyers that we do not pursue here. By contrast, we focus on ex ante heterogeneous types of a single buyer who value information differently due to their different prior beliefs. The data seller in our setting offers noisy versions of the data to screen the buyer's initial information and to extract more surplus, leading to profound differences in the optimal experiments. A second contribution of this paper relative to Admati and Pfleiderer $(1986,1990)$ is that we consider all feasible statistical experiments, whereas they restrict their attention to normally distributed priors and signals. We shall see that the optimal experiment is outside of the normal class, even if the priors are normally distributed.\n\nIn recent work, Babaioff, Kleinberg, and Paes Leme (2012) also analyze the optimal mechanisms for selling information. While we consider the same general question, the details of the model, the contracting environment, and the nature of the results differ substantially. In their model, the ex post payoff function of the data buyer depends on two state variables. The seller has private information about one state variable, and the buyer has private information about the other. Their contracting environment differs from ours in that the seller is allowed to make the information disclosure and the price dependent on his privately observed signal. By contrast, we ask the data seller to commit to a selling mechanism before the realization of any state variable. The central results of their paper are statements of the revelation principle and algorithms for the optimal mechanism using surplus extraction arguments, as in Cremer and McLean (1988).\n\nWithin the mechanism design literature, our approach is related to, yet conceptually distinct from, models of discriminatory information disclosure in which the seller of a good discloses match-value information and sets a price. Several papers, including Lizzeri (1999), Ottaviani and Prat (2001), Johnson and Myatt (2006), Bergemann and Pesendorfer (2007), Eső and Szentes (2007a), Krähmer and Strausz (2015), and Li and Shi (2015), analyze this problem from an ex ante perspective, where the seller commits (simultaneously or sequentially) to a disclosure rule and a pricing policy. ${ }^{1}$ Eső and Szentes (2007b) consider a related model of selling advice. Their model is distinct from our analysis in two dimensions. First, the private information of the agent is the expected value difference between two possible actions. Thus, the private information is one dimensional rather than multidimensional.\n\n[^1]Second, the seller can make the payment contingent on both the statistical experiment and the buyer's action. By contrast, our seller can price the information but not the action itself. Commitment to a disclosure policy is present in the literature on Bayesian persuasion, e.g., Rayo and Segal (2010), Kamenica and Gentzkow (2011), and Kolotilin, Li, Mylovanov, and Zapechelnyuk (2015). In contrast to this line of work, our model admits monetary transfers and rules out any direct effect of the buyer's ex post action on the seller's utility.\n\nOur previous work (Bergemann and Bonatti, 2015) considered the information-acquisition policy of a data buyer who then decided on the placement of display advertising. This earlier model was simpler in many respects. First, the price of information was given or determined by a competitive market. Second, the data buyer did not have any private information. Third, despite allowing for a continuum of matching values (states) and advertising levels (actions), the available information structures were restricted to simple queries that perfectly revealed individual state realizations. The analysis focused on the nature of the buyer's optimal queries given the distribution of match values and the cost of advertising.\n\nHörner and Skrzypacz (2016) share a similar title but consider a very different setting. They consider a dynamic hold-up game, except that information rather than a physical object is sold. At the beginning of the game, the buyer has no private information and wants to hire a competent data seller. The data seller knows whether she is competent and can prove her competence by sequentially undertaking tests within a fixed subclass of statistical experiments. Hörner and Skrzypacz (2016) allow for sequential monetary transfers and characterize an equilibrium that is most favorable to the competent seller.\n\nFinally, our seller's problem bears some resemblance to a bundling problem. With more than two states, the buyer types are multidimensional and it is well-known-see, for example, Pavlov (2011b)-that the single-price result of Myerson (1981) and Riley and Zeckhauser (1983) does not hold. Indeed, the optimal menu involves stochastic bundling quite generally, and the structure of the bundles offered can be quite rich. ${ }^{2}$ Stochastic bundles are analogous to the partially informative experiments in our model. To further distinguish from these classic multidimensional problems, stochastic bundling can arise in our setting even when buyer types are one dimensional.\n\n[^2]","text_sha256":"5b5da71e813d01d7419c0736a03f5a90f48faa2d000a4d9fcdfb749718b9b00b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0007","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"## 2 Model\n\nA single decision maker, the data buyer, faces a decision problem under uncertainty. The state of nature $\\omega$ is drawn from a finite set $\\Omega=\\left\\{\\omega_{1}, \\ldots, \\omega_{i}, \\ldots, \\omega_{I}\\right\\}$. The data buyer chooses an action $a$ from a finite set $A=\\left\\{a_{1}, \\ldots, a_{j}, \\ldots, a_{J}\\right\\}$. The ex post utility is denoted by\n\n$$\nu\\left(\\omega_{i}, a_{j}\\right) \\triangleq u_{i j} \\in \\mathbb{R}_{+} .\n$$\n\nThe ex post payoffs can thus be represented by an $I \\times J$ matrix:\n\n| $u$ | $a_{1}$ | ⋯ | $a_{J}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | $u_{11}$ | ⋯ | $u_{1 J}$ |\n| ⋮ | ⋮ |  | ⋮ |\n| $\\omega_{I}$ | $u_{I 1}$ | ... | $u_{I J}$ |\n\nWe impose the following weak assumptions on the matrix: (i) $I \\leq J$, and (ii) $u_{i i}>u_{i j}$ for all $j \\neq i$. These two assumptions capture the idea that the action space is at least as rich as the state space and that for every state $\\omega_{i}$, there is a unique action (labeled $a_{i}$ ) that maximizes the decision maker's utility in that state.\n\nMatching Utility A useful special case is one in which the data buyer faces binary ex post payoffs in each state, i.e., he seeks to match the state and the action. In that case, we frequently drop the second subscript for the ex post utility on the diagonal. The utility function $u\\left(\\omega_{i}, a_{j}\\right)$ is then given by\n\n$$\nu\\left(\\omega_{i}, a_{j}\\right) \\triangleq \\mathbb{I}_{[i=j]} \\cdot u_{i}, \\text { and } u_{i} \\triangleq u_{i i} ;\n$$\n\nor in matrix form,\n\n| $u$ | $a_{1}$ | ... | $a_{I}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | $u_{1}$ |  | 0 |\n| ⋮ |  | ⋱ |  |\n| $\\omega_{I}$ | 0 |  | $u_{I}$ |\n\nThis formulation assumes that, in each state, the data buyer assigns the same value (normalized to zero) to each wrong action. This assumption has no bite across states because adding a state-dependent translation to the utility function does not affect preferences over actions. Under this assumption, it is without loss of generality to assume that the sets of actions and states have the same cardinality: $|A|=|\\Omega|=I=J$.\n\nPrior Information The interim belief $\\theta$ about the state is the type of the data buyer\n\n$$\n\\theta \\in \\Theta \\triangleq \\Delta \\Omega,\n$$\n\nwhere $\\theta_{i}$ denotes the interim probability that type $\\theta$ assigns to state $\\omega_{i}$, with $i=1, \\ldots, I$. The interim beliefs of the data buyer are his private information. From the perspective of the data seller, these beliefs are distributed according to a distribution\n\n$$\nF \\in \\Delta \\Theta,\n$$\n\nwhich we take as a primitive of our model. ${ }^{3}$\nIn line with the interpretation of selling supplemental information, however, we note that the beliefs $\\theta \\in \\Theta$ can be generated from a common prior and privately observed signals. Thus, suppose there is a common prior $\\mu \\in \\Delta \\Omega$. The decision maker privately observes a signal $r \\in R$ according to a commonly known experiment\n\n$$\n\\lambda: \\Omega \\rightarrow \\Delta R .\n$$\n\nThe decision maker then forms his interim belief via Bayes' rule\n\n$$\n\\theta(\\omega \\mid r) \\triangleq \\frac{\\lambda(r \\mid \\omega) \\mu(\\omega)}{\\sum_{\\omega^{\\prime} \\in \\Omega} \\lambda\\left(r \\mid \\omega^{\\prime}\\right) \\mu\\left(\\omega^{\\prime}\\right)} .\n$$\n\nThe interim beliefs $\\theta(\\omega \\mid r)$, simply denoted by $\\theta$, are thus the private information of the data buyer. From the data seller's perspective, the common prior $\\mu \\in \\Delta \\Omega$ and the distribution of signals $\\lambda: \\Omega \\rightarrow \\Delta R$ induce a distribution $F \\in \\Delta \\Theta$ of interim beliefs.\n\nSupplemental Information The data buyer seeks to augment his initial private information by obtaining additional information from the data seller in order to improve the quality of his decision making. A statistical experiment (equivalently, an information structure) $E=(S, \\pi)$ consists of a set $S$ of signals $s$ and a likelihood function:\n\n$$\n\\pi: \\Omega \\rightarrow \\Delta S .\n$$\n\nWe assume throughout that the realization of the buyer's private signal $r \\in R$ and that of the signal $s \\in S$ from any experiment $E$ are independent, conditional on the state $\\omega$. In other words, the buyer and the seller draw their information from independent sources.\n\nFor a given experiment $E=(S, \\pi)$, let $S$ denote the (finite) set of signals that are in the\n\n[^3]support of the experiment and $\\pi_{i k}$ the conditional probability of signal $s_{k} \\in S$ in state $\\omega_{i}$. Letting $K \\triangleq|S|$, we have\n$$\n\\pi_{i k} \\triangleq \\operatorname{Pr}\\left(s_{k} \\mid \\omega_{i}\\right),\n$$\nwhere $\\pi_{i k} \\geq 0$ and $\\sum_{k=1}^{K} \\pi_{i k}=1$ for all $i$. We then obtain the stochastic matrix\n\n| $E$ | $s_{1}$ | ⋯ | $s_{k}$ | ⋯ | $s_{K}$ |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | $\\pi_{11}$ | ⋯ | $\\pi_{1 k}$ | ⋯ | $\\pi_{1 K}$ |\n| ⋮ | ⋮ |  |  |  | ⋮ |\n| $\\omega_{i}$ | $\\pi_{i 1}$ |  | $\\pi_{i k}$ |  | $\\pi_{i K}$ |\n| ⋮ | ⋮ |  |  |  | ⋮ |\n| $\\omega_{I}$ | $\\pi_{I 1}$ | ... | $\\pi_{I k}$ | ⋯ | $\\pi_{I K}$ |\n\nThe following experiments are of particular interest:\n\n1. a non-dispersed experiment that contains at least one nil entry $\\pi_{i k}=0$ for some $i, k$;\n2. a concentrated experiment that contains a standard basis vector $\\pi_{i i}=1$ for some $i$;\n3. the fully informative experiment $\\bar{E}$, with $\\pi_{i i}=1$ for all $i$.\n\nIn non-dispersed experiment, one signal $s_{k}$ allows the decision maker to rule out some state $\\omega_{i}$. In a concentrated experiment, there exists a state $\\omega_{i}$ that is ruled out by all signals $s_{k} \\neq s_{i}$. The fully informative experiment perfectly reveals the true state.\n\nA menu of experiments $\\mathcal{M}=\\{\\mathcal{E}, t\\}$ (or an information policy) consists of a collection $\\mathcal{E}$ of experiments $E$ and an associated tariff function\n\n$$\nt: \\mathcal{E} \\rightarrow \\mathbb{R}_{+} .\n$$\n\nOur goal is to characterize the revenue-maximizing menu for the seller. The timing of the game is as follows:\n\n1. the seller posts a menu $\\mathcal{M}$;\n2. the true state $\\omega$ and the buyer's type $\\theta$ are realized;\n3. the buyer chooses an experiment $E \\in \\mathcal{E}$ and pays the corresponding price $t(E)$;\n4. the buyer observes a signal $s$ from experiment $E$ and chooses an action $a$.","text_sha256":"61ae51c05c747702407359a4657a842a9795915986fdd7aa694dfb29dd1f87d0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0008","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"We emphasize that the data seller is unrestricted in her choice of statistical experiment (i.e., the seller can improve upon the buyer's original information with arbitrarily accurate\nsignals), and that the marginal cost of providing the information is nil. These assumptions capture settings in which sellers hold very precise data and distribution costs are negligible.\n\nWe deliberately focus on the pure problem of the design and pricing of statistical experiments. We thus assume that the seller commits to a menu of experiments before the realization of the state $\\omega$ and the type $\\theta$. We further assume that none of the following are contractible: the buyer's action $a$, the realized state $\\omega$, or the signal $s$. Thus, scoring rules and other belief-elicitation schemes that compare the elicited beliefs with the realization of some random variable are not available to the seller. Finally, we consider only static mechanisms and do not investigate sequential protocols. We expect that the sequential sale of experiments would allow the data seller to extract additional surplus relative to static mechanisms, as this would allow the seller to correlate individual payments with the realized states. However, the nature of the incentive constraints would not be affected (at least in the last round of communication), and we expect the qualitative results to remain unchanged.","text_sha256":"83ca496b9fdf0edb0326f8c34993256edb238fc673e2a903602a0f883f42e115"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0009","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Information Design","text":"## 3 Information Design","text_sha256":"83415d0cc8c94cb8502652ff8d825cf756ab978d459852b6de1ce9d305658dbf"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0010","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.1 Value of Information","text":"### 3.1 Value of Information\n\nWe first describe the value of the buyer's initial information and then determine the incremental value of an experiment $E=(S, \\pi)$. The value of the data buyer's problem under prior information only is given by choosing the action $a_{j}$ that maximizes the expected utility given the interim belief $\\theta$ :\n\n$$\na(\\theta) \\triangleq \\underset{a_{j} \\in A}{\\arg \\max }\\left\\{\\sum_{i=1}^{I} \\theta_{i} u_{i j}\\right\\} .\n$$\n\nThe expected utility of type $\\theta$ is therefore given by\n\n$$\nu(\\theta) \\triangleq \\max _{a_{j} \\in A}\\left\\{\\sum_{i=1}^{I} \\theta_{i} u_{i j}\\right\\} .\n$$\n\nBy contrast, if the data buyer has access to an experiment $E=(S, \\pi)$, he first observes the signal realization $s_{k} \\in S$, updates his beliefs and then chooses an appropriate action. The marginal distribution of signals $s_{k}$ from the perspective of type $\\theta$ is given by\n\n$$\n\\operatorname{Pr}\\left[s_{k} \\mid \\theta\\right]=\\sum_{i=1}^{I} \\theta_{i} \\pi_{i k} .\n$$\n\nConsequently, for any signal $s_{k}$ that occurs with strictly positive probability $\\operatorname{Pr}\\left[s_{k} \\mid \\theta\\right]$, the\naction that maximizes the expected utility of type $\\theta$ is given by\n\n$$\na\\left(s_{k} \\mid \\theta\\right) \\triangleq \\underset{a_{j} \\in A}{\\arg \\max }\\left\\{\\sum_{i=1}^{I}\\left(\\frac{\\theta_{i} \\pi_{i k}}{\\sum_{i^{\\prime}=1}^{I} \\theta_{i^{\\prime}} \\pi_{i^{\\prime} k}}\\right) u_{i j}\\right\\},\n$$\n\nwhich leads to the following conditional expected utility:\n\n$$\nu\\left(s_{k} \\mid \\theta\\right) \\triangleq \\max _{a_{j} \\in A}\\left\\{\\sum_{i=1}^{I}\\left(\\frac{\\theta_{i} \\pi_{i k}}{\\sum_{i^{\\prime}=1}^{I} \\theta_{i^{\\prime}} \\pi_{i^{\\prime} k}}\\right) u_{i j}\\right\\} .\n$$\n\nIntegrating over all signal realizations $s_{k}$ and subtracting the value of prior information, the (net) value of an experiment $E$ for type $\\theta$ is given by\n\n$$\nV(E, \\theta) \\triangleq \\mathbb{E}[u(s \\mid \\theta)]-u(\\theta)=\\sum_{k=1}^{K} \\max _{j}\\left\\{\\sum_{i=1}^{I} \\theta_{i} \\pi_{i k} u_{i j}\\right\\}-u(\\theta) .\n$$\n\nIn the case of the previously defined matching utility (2), the value of information takes the simpler form\n\n$$\nV(E, \\theta)=\\sum_{k=1}^{K} \\max _{i}\\left\\{\\theta_{i} \\pi_{i k} u_{i}\\right\\}-\\max _{i}\\left\\{\\theta_{i} u_{i}\\right\\} .\n$$\n\nThe value of the prior information, given by $\\max _{i} \\theta_{i} u_{i}$, is generated by the action that has the highest value-weighted probability of matching the state. The value of an experiment $E$ is generated by choosing an action on the basis of the posterior belief $\\theta_{i} \\pi_{i k}$ induced by each signal $s_{k}$. Under matching utility, the value of information is given by the (value-weighted) incremental probability of choosing the correct action. ${ }^{4}$\n\nIn Figure 1, we illustrate the value of an experiment in a model with three actions and three states $\\omega_{i} \\in\\left\\{\\omega_{1}, \\omega_{2}, \\omega_{3}\\right\\}$. The prior belief of each agent is therefore an element of the two-dimensional simplex, $\\theta=\\left(\\theta_{1}, \\theta_{2}, 1-\\theta_{1}-\\theta_{2}\\right)$. The utility function is given by state-action matching with uniform weights, i.e., $u_{i} \\triangleq 1$. We display the value of the fully informative experiment $\\bar{E}$ and of a partially informative experiment $E$ as a function of the buyer's prior. ${ }^{5}$\n\n[^4]| $E$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | 1/2 | 1/4 | 1/4 |\n| $\\omega_{2}$ | 0 | 3/4 | 1/4 |\n| $\\omega_{3}$ | 0 | 1/4 | 3/4 |\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 1: Value of Full and Partial Information, $I=J=3$\n\nViewed as a function of the types, the value of an experiment $V(E, \\theta)$ is piecewise linear in $\\theta$ with a finite number of linear components. The linearity of the value function is a consequence of the Bayesian nature of our problem, where types are prior probabilities of states. The downward kinks are due to the max operator in the buyer's reservation utility $u(\\theta)$. They correspond to changes in the buyer's action without supplemental information. The upward kinks are generated by the max operator in (5) and reflect changes in the buyer's preferred action upon observing a signal. Finally, the experiment $E$ is only valuable if at least two signals lead to different actions. If the buyer chooses a constant action following every signal $s_{k}$, then $V(E, \\theta)=0$, as can be seen in formulations (6) and (7), as well as in Figure 1 (right).","text_sha256":"2a6d5ce55d46f357a9344f80788e33d3f4c96084aa1b48fa498b2e62ccd3d24e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0011","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.2 The Seller's Problem","text":"### 3.2 The Seller's Problem\n\nThe seller's choice of a profit-maximizing menu of experiments may involve, in principle, designing one experiment per buyer type. In turn, each experiment entails a type-dependent mapping from signals into actions. The seller's problem can, however, be simplified by reducing the set of menus of experiments to a smaller and very tractable class.\n\nFirst, by the revelation principle, we can restrict our attention to direct mechanisms $\\mathcal{M}=\\{E(\\theta), t(\\theta)\\}$ that assign an experiment $E(\\theta)=(S(\\theta), \\pi(\\theta))$ and a price $t(\\theta)$ to each type $\\theta$ of data buyer. We denote the indirect (net) utility for the truth-telling agent by\n\n$$\nV(\\theta) \\triangleq V(E(\\theta), \\theta)-t(\\theta) .\n$$\n\nThe seller's problem consists of maximizing the expected transfers\n\n$$\n\\max _{\\{E(\\theta), t(\\theta)\\}} \\int_{\\theta \\in \\Theta} t(\\theta) \\mathrm{d} F(\\theta)\n$$\n\nsubject to incentive-compatibility constraints\n\n$$\nV(\\theta) \\geq V\\left(E\\left(\\theta^{\\prime}\\right), \\theta\\right)-t\\left(\\theta^{\\prime}\\right), \\quad \\forall \\theta, \\theta^{\\prime} \\in \\Theta,\n$$\n\nand individual-rationality constraints\n\n$$\nV(\\theta) \\geq 0, \\quad \\forall \\theta \\in \\Theta .\n$$\n\nSecond, given any direct mechanism $\\mathcal{M}=\\{E(\\theta), t(\\theta)\\}$, we say that experiment $E(\\theta)$ is responsive if every signal $s \\in S(\\theta)$ leads type $\\theta$ to a different optimal choice of action and, in particular,\n\n$$\na\\left(s_{k} \\mid \\theta\\right)=a_{k} \\text { for all } s_{k} \\in S(\\theta) \\text {. }\n$$\n\nImportantly, condition (10) is only required for every experiment $E(\\theta)$ and for the corresponding type $\\theta$. In other words, we do not require this condition to hold if signals $s_{k} \\in S(\\theta)$ are evaluated by a different type $\\theta^{\\prime} \\neq \\theta$. Finally, we define an outcome of a menu as the joint distribution of states, actions, and monetary transfers resulting from every type's optimal choice of experiment and subsequent choice of action.\n\nProposition 1 establishes that it is without loss of generality to restrict attention to responsive menus-direct revelation mechanisms in which every experiment $E(\\theta)$ is responsive.","text_sha256":"18d15a411cd19a6fea169404adcd634198719dcd38a911c2f5170bb125afd511"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0012","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proposition 1 (Responsive Menus)","text":"## Proposition 1 (Responsive Menus)\n\nThe outcome of every menu $\\mathcal{M}$ can be attained by a responsive menu.\nOur proof closely follows the argument of the revelation principle for Bayesian games of communication established by Myerson (1982). We show that we can always reduce the size of the signal space to the size of the action space recommended in equilibrium. The intuition is straightforward. Consider an incentive-compatible menu that contains an experiment $E(\\theta)$ with more signals than actions. We can combine all signals in $E(\\theta)$ that lead to the same choice of action for type $\\theta$. The value of this experiment remains constant for type $\\theta$, who does not modify his behavior. However, because the new experiment is (weakly) less informative than the experiment we started with, $V\\left(E(\\theta), \\theta^{\\prime}\\right)$ decreases (weakly) for all $\\theta^{\\prime} \\neq \\theta$, relaxing the incentive constraints. Finally, because every signal sent with positive probability under experiment $E(\\theta)$ leads type $\\theta$ to a different action, we can order the signals such that each $s_{k} \\in S(\\theta)$ recommends the corresponding action $a_{k}$.\n\nThe language of the Bayesian games of communication, as suggested by Myerson (1982), is helpful for understanding the nature of the seller's problem more generally. The solution to the data seller's problem has to satisfy two different constraints, the truth-telling (or honesty) constraint given by (9) and the obedience constraint given by (4). Thus, the buyer must be jointly honest and obedient. In particular, double deviations (lying and disobeying) must not be profitable for the buyer.\n\nAn immediate implication of Proposition 1 is that, without loss of generality, we can restrict our attention to experiments in which the signal space has the cardinality of the action space, i.e., $K=J$. This insight allows us to write the likelihood function of every experiment (3) as a matrix with the same dimensions as the payoff matrix, i.e.,\n\n| $E(\\theta)$ | $s_{1}\\left(=a_{1}\\right)$ | ... | $s_{i}\\left(=a_{j}\\right)$ | ⋯ | $s_{J}\\left(=a_{J}\\right)$ |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | $\\pi_{11}$ | ⋯ | $\\pi_{1 j}$ | ⋯ | $\\pi_{1 J}$ |\n| ⋮ | ⋮ |  |  |  | ⋮ |\n| $\\omega_{i}$ | $\\pi_{i 1}$ |  | $\\pi_{i j}$ |  | $\\pi_{i J}$ |\n| ⋮ | ⋮ |  |  |  | ⋮ |\n| $\\omega_{I}$ | $\\pi_{I 1}$ | ⋯ | $\\pi_{I j}$ | ⋯ | $\\pi_{I J}$ |\n\nThus, we can replace the signal $s_{j}$ with the action recommendation $a_{j}$. This property does not require that every action $a_{j}$ is recommended with strictly positive probability. For instance, some signals $s_{j}$ may never be sent, corresponding to a column vector of zeros at the $j$-th position. The resulting value of experiment $E(\\theta)$ for type $\\theta$ can be written as\n\n$$\nV(E(\\theta), \\theta)=\\sum_{j=1}^{J} \\sum_{i=1}^{I} \\theta_{i} \\pi_{i j} u_{i j}-\\max _{j}\\left\\{\\sum_{i=1}^{I} \\theta_{i} u_{i j}\\right\\} .\n$$\n\nUnder the restriction to responsive experiments, formulation (11) removes the first max operator from the value of experiment $E(\\theta)$ for the truth-telling type $\\theta$. Because a misreporting type need not always take the recommended action, we must still use the original formulation (6) when computing the value of experiment $E(\\theta)$ for type $\\theta^{\\prime}$.\n\nWhen payoffs are given by state-action matching as in (2), the value of experiment $E(\\theta)$ for the truth-telling type $\\theta$ simplifies to\n\n$$\nV(E(\\theta), \\theta)=\\sum_{i=1}^{I} \\theta_{i} \\pi_{i i} u_{i}-\\max _{i}\\left\\{\\theta_{i} u_{i}\\right\\} .\n$$\n\nIn formulation (12), the value of experiment $E(\\theta)$ for type $\\theta$ is fully determined by its diagonal entries $\\pi_{i i}$. By contrast, the off-diagonal entries $\\pi_{i j}$ may enter the value of experiment $E(\\theta)$ for different types $\\theta^{\\prime}$, as in (7).","text_sha256":"34e334659258203a508771bc4483c9a5b05e2247f7d62f05ba4205586663f0c9"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0013","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.3 Structural Properties","text":"### 3.3 Structural Properties\n\nWe now leverage the nature of the value of information to impose additional structure on the experiments that are part of an optimal menu.","text_sha256":"ec1c4470a5db804069cc5344fad7facffbb1a437d7a5cc25e40a70a2a8f78e94"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0014","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proposition 2 (Optimal Experiments)","text":"## Proposition 2 (Optimal Experiments)\n\n1. The fully informative experiment $\\bar{E}$ is part of an optimal menu.\n2. Every experiment in any optimal menu is non-dispersed, i.e., $\\pi_{i j}=0$ for some $i \\neq j$.\n3. In the matching case, every experiment in any optimal menu is concentrated, i.e., $\\pi_{i i}=1$ for some $i$.\n\nThe first part of this result can be established via contradiction. Every type $\\theta$ values the fully informative experiment $\\bar{E}$ the most among all the experiments. Suppose, then, that $\\bar{E}$ is not part of the menu, and denote the most expensive item currently on the menu by $E^{\\prime}$. The seller can replace experiment $E^{\\prime}$ with the complete experiment $\\bar{E}$, keeping all other prices constant and charging a higher price for $\\bar{E}$ than for $E^{\\prime}$. The new menu weakly increases the seller's revenue without lowering the net utility of any buyer type. ${ }^{6}$\n\nThe second and third parts imply that every optimal experiment eliminates the buyer's uncertainty along some dimension. They are also established by an improvement argument. Fix an experiment $E$ and suppose all entries $\\pi_{i j}$ are strictly positive. In each state (row) $i$, the seller increases the probability $\\pi_{i i}$ of the signal $s_{i}$ that yields the highest payoff $u_{i i}$ for an obedient type in that state. Concurrently, the seller reduces the probability $\\pi_{i j}$ of the signal $s_{j}$ that yields the lowest payoff $\\min _{j} u_{i j}$ for an obedient type. The seller shifts a probability mass inversely proportional to the difference in payoffs $u_{i i}-u_{i j}$, and hence, the resulting payoff gain is uniform across states $\\omega_{i}$. This procedure is applied until the first entry $\\pi_{i j}$ reaches 0. Because the beliefs of each type $\\theta$ sum to one, this shift is valued uniformly by all obedient types and weakly less by any other type. A commensurate increase in the price of the experiment offsets the value of the additional information provided and, hence, strictly increases profits while (weakly) relaxing the truth-telling and participation constraints.\n\nThus, with arbitrary payoffs, every experiment $E$ contains a signal $s_{j}$ that allows the buyer to rule out some state $\\omega_{i}$. The limits of this result are related to the possibility of double deviations evoked earlier. A misreporting type $\\theta^{\\prime}$ may not necessarily choose the action recommended by every signal in the experiment $E(\\theta)$ intended for type $\\theta$. For\n\n[^5]the above improvement argument to discourage double deviations, it is critical that the probability mass is shifted away from the worst action in each state $\\omega_{i}$, thus yielding the largest possible marginal benefit $\\left(\\max _{j} u_{i j}-\\min _{j} u_{i j}\\right)$ for every obedient type in state $\\omega_{i}$. Any other shift may yield a strictly higher benefit to type $\\theta^{\\prime}$ than to type $\\theta$ and lead to the violation of the truth-telling constraints.\n\nIn the case of matching utility, the seller can shift the probability mass to the diagonal from any off-diagonal entry until the first entry $\\pi_{i i}$ reaches 1. Indeed, when the probability is shifted to the diagonal, any type that does not follow the signal's recommendation obtains a strictly lower benefit relative to an obedient type. As a result, any optimal experiment is concentrated, i.e., there exists at least one state $\\omega_{i}$ under which signal $s_{i}$ is sent with probability one, and the buyer takes the correct action $a_{i}$. Conversely, the buyer is able to rule out (at least) one state $\\omega_{i}$ after observing any signal $s_{k} \\neq s_{i}$.\n\nIn Sections 4 and 6, we show that with binary states and actions or binary types, respectively, it is sufficient to consider truth-telling and obedience separately. By contrast, Example 3 in the Supplemental Appendix demonstrates that with three (or more) actions and types, double deviations typically impose additional restrictions on the optimal menu.","text_sha256":"1a55cc2f1ca62b537b57209476f9677710bce09526279c82e82cbdd9e5689032"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0015","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Optimal Menu with Binary Actions","text":"## 4 Optimal Menu with Binary Actions\n\nIn this section, we consider an environment with two actions $a \\in\\left\\{a_{1}, a_{2}\\right\\}$ and two states $\\omega \\in\\left\\{\\omega_{1}, \\omega_{2}\\right\\}$. In this setting, the restriction to matching utility functions entails no loss of generality relative to a general payoff matrix. To wit, for every state $\\omega_{i}$, we can always subtract the (state-dependent) constant $u_{i j}$ with $i \\neq j$. This linear transformation normalizes the payoffs of the data buyer by setting $u_{12}=u_{21}=0$ without affecting the optimality conditions of the data buyer's decision problem.\n\nWe thus obtain a diagonal payoff matrix as in (2) with positive entries given by\n\n$$\nu_{1} \\triangleq u_{11}, u_{2} \\triangleq u_{22} .\n$$\n\nWith binary states, the interim belief of the data buyer (his type) is one dimensional. We identify each type with the interim probability of state $\\omega_{1}$ :\n\n$$\n\\theta \\triangleq \\operatorname{Pr}\\left[\\omega=\\omega_{1}\\right] \\in[0,1] .\n$$\n\nWe denote the interim belief type $\\theta$ that is indifferent between action $a_{1}$ and $a_{2}$ by $\\theta^{*}$ as\n\n$$\n\\theta^{*} u_{1}=\\left(1-\\theta^{*}\\right) u_{2} \\Leftrightarrow \\theta^{*}=\\frac{u_{2}}{u_{1}+u_{2}} \\text {. }\n$$","text_sha256":"50cc9c79b775303acb1d688eb446545654d5eced6572571ed7ec6c0b31b52633"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0016","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.1 Binary Experiments","text":"### 4.1 Binary Experiments\n\nWith binary actions, Proposition 1 implies that it is sufficient to consider for every type $\\theta$ experiments $E(\\theta)$ that generate (at most) two signals:\n\n| $E(\\theta)$ | $s_{1}$ | $s_{2}$ |\n| :--- | :--- | :--- |\n| $\\omega_{1}$ | $\\pi_{11}(\\theta)$ | $1-\\pi_{11}(\\theta)$. |\n| $\\omega_{2}$ | $1-\\pi_{22}(\\theta)$ | $\\pi_{22}(\\theta)$ |\n\nWith binary signals, we can simplify the notation by dropping the second subscript for diagonal entries, as we did for the payoff function:\n\n$$\n\\pi_{1}(\\theta) \\triangleq \\pi_{11}(\\theta), \\pi_{2}(\\theta) \\triangleq \\pi_{22}(\\theta) .\n$$\n\nWithout loss of generality, we assume that $\\pi_{1}(\\theta)+\\pi_{2}(\\theta) \\geq 1$. In other words, signal $s_{1}$ is relatively more likely to occur than signal $s_{2}$ under state $\\omega_{1}$ than under state $\\omega_{2}$, or\n\n$$\n\\frac{\\pi_{1}(\\theta)}{1-\\pi_{1}(\\theta)} \\geq \\frac{1-\\pi_{2}(\\theta)}{\\pi_{2}(\\theta)} .\n$$\n\nWith binary states and actions, the general payoff environment conforms to the matching utility environment. We can therefore write the value for an arbitrary experiment $E$ as follows by modifying expression (12):\n\n$$\nV(E, \\theta)=\\max \\left\\{\\theta \\pi_{1} u_{1}+(1-\\theta) \\pi_{2} u_{2}-\\max \\left\\{\\theta u_{1},(1-\\theta) u_{2}\\right\\}, 0\\right\\} .\n$$\n\nAs in the earlier formulation (12), the diagonal entries of the matrix $\\pi(\\theta)$ generate the probability that experiment $E(\\theta)$ allows type $\\theta$ to match the realized state with his action. Conversely, the first max operator accounts for the possibility that a misreporting type $\\theta^{\\prime}$ does not follow the recommendation implicit in one of the signals of $E(\\theta)$ and, hence, derives no value from the supplemental information.\n\nIn Figure 2, we illustrate how the value of information changes as a function of the type $\\theta$ for the case $u_{1}=u_{2}=1$. We compare two experiments with binary signals, namely, the fully informative experiment $E^{\\prime}=\\left(\\pi_{1}^{\\prime}, \\pi_{2}^{\\prime}\\right)=(1,1)$ and a partially informative experiment $E^{\\prime \\prime}=\\left(\\pi_{1}^{\\prime \\prime}, \\pi_{2}^{\\prime \\prime}\\right)=(1 / 2,1)$.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 2: Value of Full and Partial Information $\\left(I=J=2, \\quad u_{1}=u_{2}=1\\right)$\n\nThe value of information as a function of the buyer's interim beliefs $\\theta$ reflects many intuitive properties that we formally establish in the next subsection:\n\n1. The most valuable type for the seller is the ex ante least informed. In the examples in Figure 2, this is type $\\theta^{*}=1 / 2$. Conversely, the most informed types $\\theta \\in\\{0,1\\}$ have zero value of information. The linear decline in each direction away from $\\theta^{*}$ follows from the linearity of the value of information in the interim probability.\n2. When we consider any asymmetric experiment, such as the one displayed in the right panel of Figure 2, the distance from the least informed type $\\left|\\theta-\\theta^{*}\\right|$ is not a sufficient statistic for the value of information. The different slopes on each side of $\\theta^{*}$ indicate different marginal benefits for matching state $\\omega_{1}$ versus state $\\omega_{2}$ on the basis of differences in the interim beliefs $\\theta$.\n\nThus, even in an environment where types are clearly one dimensional, information products are inherently multidimensional (in this case, two dimensional). In particular, information always has both vertical (quality) and horizontal (positioning) dimensions. Unlike in models of nonlinear pricing (with respect to either quantity or quality) where all the types agree on the relative ranking of all the products, in the current environment, the types disagree on the very ranking of all partially informative experiments.","text_sha256":"aa7c96a5c758d7e37bb7b01b5dec9f0711ea08af536a40b63dd5c1c60e99bf43"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0017","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.2 Binary Types","text":"### 4.2 Binary Types\n\nWe derive the optimal menu with two data buyer types, $\\theta \\in \\Theta=\\left\\{\\theta^{L}, \\theta^{H}\\right\\}$. The high (value) type $\\theta^{H}$ assigns a higher value to receiving the fully informative experiment, i.e.,\n\n$$\nV\\left(\\bar{E}, \\theta^{H}\\right)>V\\left(\\bar{E}, \\theta^{L}\\right) .\n$$\n\nWith uniform weights $u_{1}=u_{2}>0$, this simply means the high (value) type is less wellinformed ex ante, i.e.,\n\n$$\n\\left|\\theta^{H}-1 / 2\\right| \\leq\\left|\\theta^{L}-1 / 2\\right| .\n$$\n\nWe refer to the distance between the interim belief $\\theta$ and the indifference belief $\\theta^{*}$ as the precision of the type $\\theta$. We denote the frequency of a high type as\n\n$$\n\\gamma \\triangleq \\operatorname{Pr}\\left(\\theta=\\theta^{H}\\right) .\n$$\n\nThe following distinction proves helpful. The interim beliefs of the two types are said to be congruent if both types would choose the same action without additional information. If we adopt the convention that the high type chooses action $a_{1}$ under his prior information (i.e., $\\theta^{H}>\\theta^{*}$ ), then beliefs (and corresponding types) are congruent if $\\theta^{*}<\\theta^{H}<\\theta^{L}$ and noncongruent if $\\theta^{L}<\\theta^{*}<\\theta^{H}$.\n\nWe first establish two familiar properties: \"no distortion at the top,\" i.e., for the high type and \"no rent at the bottom,\" i.e., for the low type.","text_sha256":"13fdeab9e246dc59564edef636a08dc13f0a25a2a3212dcf0bc6c3681dc246dc"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0018","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proposition 3 (Binding Constraints)","text":"## Proposition 3 (Binding Constraints)\n\nIn an optimal menu:\n\n1. type $\\theta^{H}$ purchases the fully informative experiment $\\bar{E}$;\n2. the participation constraint of type $\\theta^{L}$ binds;\n3. the incentive-compatibility constraint of type $\\theta^{H}$ binds.\n\nWe note that these properties of the two-type environment hold for any number of states and actions under arbitrary payoffs, and we shall revisit them in Section 6. The description of the optimal menu is then completed by characterizing the experiment $E\\left(\\theta^{L}\\right)$ purchased by the low type. Here, it is productive to distinguish between congruent and noncongruent beliefs.\n\nCongruent Beliefs. In the case of congruent beliefs $\\left(\\theta^{*}<\\theta^{H}<\\theta^{L}\\right)$, the argument is related to the classic monopoly pricing problem. By Proposition 2, we know that the optimal experiment $E\\left(\\theta^{L}\\right)$ is concentrated. With congruent beliefs, both types would choose action $a_{1}$ absent any additional information. The data seller does not want to reduce the information relative to the outside option and, thus, sets $\\pi_{1}\\left(\\theta^{L}\\right)=1$. The issue is then how much information to provide about state $\\omega_{2}$, i.e.,\n\n| $E\\left(\\theta^{L}\\right)$ | $s_{1}$ | $s_{2}$ |\n| :--- | :--- | :--- |\n| $\\omega_{1}$ | 1 | 0 |\n| $\\omega_{2}$ | $1-\\pi_{2}\\left(\\theta^{L}\\right)$ | $\\pi_{2}\\left(\\theta^{L}\\right)$ |\n\nNow, any partially informative experiment with $\\pi_{2}\\left(\\theta^{L}\\right) \\in(0,1)$ geared towards the low type is also valuable to the high type. Using the expression for the value of information given by (15), we can write the incentive constraint for the high type as follows:\n\n$$\n\\theta^{H} u_{1}+\\left(1-\\theta^{H}\\right) u_{2}-t\\left(\\theta^{H}\\right) \\geq \\theta^{H} u_{1}+\\left(1-\\theta^{H}\\right) \\pi_{2}\\left(\\theta^{L}\\right) u_{2}-t\\left(\\theta^{L}\\right),\n$$\n\nsince $\\pi_{1}\\left(\\theta^{H}\\right)=\\pi_{2}\\left(\\theta^{H}\\right)=1$ and $\\pi_{1}\\left(\\theta^{L}\\right)=1$. The incentive constraint for the high type thus reduces to:\n\n$$\n\\underbrace{\\left(1-\\theta^{H}\\right)}_{\\operatorname{Pr}\\left(\\omega_{2}\\right)} \\cdot \\underbrace{\\left(1-\\pi_{2}\\left(\\theta^{L}\\right)\\right)}_{\\text {additional precision }} \\cdot u_{2} \\geq t\\left(\\theta^{H}\\right)-t\\left(\\theta^{L}\\right) .\n$$\n\nHence, we observe that both the objective and the constraints in the seller's problem are linear in the choice variable $\\pi_{2}$. We can therefore appeal to the no-haggling result of Riley and Zeckhauser (1983) that establishes the optimality of an extremal policy. Such a policy consists of either allocating the object (here, the information) with probability one or not allocating it at all, hence, $\\pi_{2} \\in\\{0,1\\}$. As in the single-good monopolist's problem, the optimal policy depends on the distribution of buyer types. In particular, the low type receives the fully informative experiment if and only if the probability $\\gamma$ of the high type is sufficiently small, or\n\n$$\n\\left(1-\\theta^{L}\\right) u_{2} \\geq \\gamma\\left(1-\\theta^{H}\\right) u_{2} \\Longleftrightarrow \\gamma \\leq \\frac{1-\\theta^{L}}{1-\\theta^{H}} .\n$$\n\nNoncongruent Beliefs. In the case of noncongruent beliefs $\\left(\\theta^{L}<\\theta^{*}<\\theta^{H}\\right)$, both the argument and the result are distinct from those of the classic monopoly problem. In the absence of additional information, the two types choose different actions. The seller can then provide information in a format that has positive value to one type but zero value to the other type. For example, suppose that $\\pi_{2}\\left(\\theta^{L}\\right)=1$ and $\\pi_{1}\\left(\\theta^{L}\\right)$ is chosen such that after receiving signal $s_{2}$, the high type $\\theta^{H}$ is indifferent between actions $a_{1}$ and $a_{2}$ (i.e., his posterior belief is $\\theta^{*}$ ):\n\n$$\n\\theta^{H}\\left(1-\\pi_{1}^{\\prime}\\right) u_{1}=\\left(1-\\theta^{H}\\right) u_{2} \\Longleftrightarrow \\pi_{1}^{\\prime}=\\frac{u_{1} \\theta^{H}-u_{2}\\left(1-\\theta^{H}\\right)}{u_{1} \\theta^{H}} \\in(0,1) .\n$$\n\nBecause type $\\theta^{H}$ chooses action $a_{1}$ without additional information, the experiment\n\n| $E^{\\prime}$ | $s_{1}$ | $s_{2}$ |\n| :--- | :--- | :--- |\n| $\\omega_{1}$ | $\\pi_{1}^{\\prime}$ | $1-\\pi_{1}^{\\prime}$ |\n| $\\omega_{2}$ | 0 | 1 |\n\ndoes not lead to a strict improvement in the decision making (or utility) of the high type $\\theta^{H}$. By contrast, the low type $\\theta^{L}$ assigns a positive value to experiment $E^{\\prime}$. After all, signal $s_{1}$ would lead him to match his action to state $\\omega_{1}$, which he would never achieve without additional information. Thus, the seller can offer partial information to the low type without incurring any implicit cost in terms of surplus extraction vis-à-vis the high type. ${ }^{7}$\n\nThe argument in support of a partially informative experiment is illustrated for the case $u_{1}=u_{2}=1$ in Figure 3, which depicts the value of two experiments net of the price as a function of the buyer's type $\\theta \\in[0,1]$. In this example, we set $\\theta^{L}=1 / 5<1 / 2<7 / 10=\\theta^{H}$.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 3: Suboptimal Menu: $\\left(\\pi_{1}, \\pi_{2}\\right) \\in\\{(1,1),(4 / 7,1)\\}$\n\nThe net value of the fully informative experiment $\\bar{E}$ is depicted by the solid line at a price $t\\left(\\theta^{H}\\right)=1-\\theta^{H}$ that leaves the type $\\theta^{H}$ indifferent between buying and not buying the fully informative experiment. The dashed line depicts the partial information experiment given by $\\pi_{1}^{\\prime}$, as described by (18). The associated partially informative experiment $E^{\\prime}$ leaves the low type $\\theta^{L}$ indifferent between buying and not buying. As for the high type $\\theta^{H}$, this experiment offers zero value at a positive price, leaving his incentive constraint slack. Correspondingly, for the high type, the net value of the partially informative experiment $E^{\\prime}$ is strictly below the net value of the fully informative experiment.","text_sha256":"6684c84326a7f3e3c35293973b43cc407a5f8f9bb698231986f85b6b90ae6edf"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0019","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proposition 3 (Binding Constraints)","text":"As the incentive-compatibility constraints are slack given this specific menu of experiments for both types, the seller can offer a more informative experiment $E^{\\prime \\prime}$ to the low type $\\theta^{L}$ while still satisfying the incentive constraint for the high type $\\theta^{H}$. As the partially informative experiment $E^{\\prime \\prime}$ has a positive price tailored to type $\\theta^{L}$, i.e., $t\\left(\\theta^{L}\\right)=\\theta^{L} \\pi_{1}^{\\prime \\prime} u_{1}$, the high type can be made indifferent between experiments $\\bar{E}$ and $E^{\\prime \\prime}$ by making sure that incentive\n\n[^6]constraint (17) is binding:\n$$\n\\underbrace{\\theta^{H}}_{\\operatorname{Pr}\\left(\\omega_{1}\\right)} \\cdot \\underbrace{\\left(1-\\pi_{1}^{\\prime \\prime}\\right)}_{\\text {additional precision }} \\cdot u_{1}=\\underbrace{\\left(1-\\theta^{H}\\right) u_{2}}_{t\\left(\\theta^{H}\\right)}-\\underbrace{\\theta^{L} \\pi_{1}^{\\prime \\prime} u_{1}}_{t\\left(\\theta^{L}\\right)} \\Longleftrightarrow \\pi_{1}^{\\prime \\prime}=\\frac{u_{1} \\theta^{H}-u_{2}\\left(1-\\theta^{H}\\right)}{u_{1}\\left(\\theta^{H}-\\theta^{L}\\right)} .\n$$\nThe experiment $E^{\\prime \\prime}$ characterized in (20) is as informative as possible while satisfying the high type's incentive compatibility constraint and both participation constraints with equality. Experiments $\\bar{E}$ and $E^{\\prime \\prime}$ are illustrated in Figure 4, again for the case $u_{1}=u_{2}=1$.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 4: Optimal Menu: $\\left(\\pi_{1}, \\pi_{2}\\right) \\in\\{(1,1),(4 / 5,1)\\}$\n\nThis example highlights the horizontal aspect of selling information that increases the scope of screening: the high type $\\theta^{H}$ buys the perfectly informative experiment; the low type $\\theta^{L}$ buys a partially informative experiment; and the seller extracts the entire surplus (which, however, falls short of the socially efficient surplus). The relative frequency of each buyer type determines the shape of the optimal menu. In particular, the partially informative experiment $E^{\\prime \\prime}$ is replaced by the fully informative experiment when there is a high proportion of low types, i.e., when $\\gamma<\\theta^{L} / \\theta^{H}$ for all payoffs $\\left(u_{1}, u_{2}\\right)$.\n\nWe have thus shown the following results for the fully binary model.","text_sha256":"e5a617cf0f422d75309b6ab5c05575a289d1c9afb6e1e360fb34334c8dd6740b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0020","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proposition 4 (Partial Information)","text":"## Proposition 4 (Partial Information)\n\n1. With congruent priors, the low type receives either zero or complete information.\n2. With noncongruent priors, the low type receives either partial or complete information.\n\nWith congruent priors, both types receive complete information if they are sufficiently similar or if the low type is relatively frequent. With noncongruent priors, the optimal menu offers the fully informative experiment to both types if they are sufficiently similar in their prior information, or if the low type is sufficiently frequent. Offering two experiments\nbecomes optimal if the two types are sufficiently different in their level of informativeness. Importantly, the low type always receives some information in that case and is not excluded. ${ }^{8}$ Furthermore, the high type receives positive rents only if he is pooled with the low type. Otherwise, the seller extracts the entire surplus that is generated.\n\nWe now examine the comparative statics of the optimal menu. Because the high type buys the fully informative experiment $\\bar{E}$ and the low type's experiment is concentrated with $\\pi_{2}=1$, the optimal experiment $E\\left(\\theta^{L}\\right)$ can be described by its first diagonal entry $\\pi_{1}$.","text_sha256":"3ca9d1367fa3e0331a66a36b8f9fd567a8d784b88628661104f56804e31c14c8"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0021","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proposition 5 (Comparative Statics)","text":"## Proposition 5 (Comparative Statics)\n\nThe informativeness $\\pi_{1}$ of the optimal experiment $E\\left(\\theta^{L}\\right)$ is\n\n1. decreasing in the probability $\\gamma$ of the high type;\n2. decreasing in the precision of the low type's prior belief $\\left|\\theta^{L}-\\theta^{*}\\right|$;\n3. increasing in the precision of the high type's prior belief $\\left|\\theta^{H}-\\theta^{*}\\right|$ when priors are congruent or the menu is discriminatory.\n\nThus, even though the shape of the optimal menu depends on whether types have congruent or noncongruent priors, the comparative statics of the optimal experiment are robust across the different scenarios. The rent extraction vs. efficiency trade-off is resolved at the expense of the low type as (i) the fraction of high types increases, or (ii) the low type's willingness to pay for the complete experiment decreases.\n\nFinally, as the high type's willingness to pay for information decreases (his prior becomes more precise), the optimal menu may switch from offering full to partial information to the low type. This occurs because separating the two types becomes profitable whenever the gap in their prior beliefs is sufficiently wide. Once partial information is offered, however, the optimal distortions decrease with the precision of the high type. This accounts for the qualifying statement in Proposition 5.3.","text_sha256":"4092766b4ff9dbde9c78433623ecb4767e7cdc21533cf02fc39f69bf9523396f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0022","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.3 Continuum of Types","text":"### 4.3 Continuum of Types\n\nWe now complete the analysis of the binary-action environment. We denote the interim belief of the data buyer by $\\theta \\triangleq \\operatorname{Pr}\\left(\\omega=\\omega_{1}\\right)$ and consider a continuum of types $\\theta \\in[0,1]$ on the unit interval, with a distribution $F(\\theta)$ and associated density $f(\\theta)$. We shall show that\n\n[^7]many qualitative properties of the two-type case-including the cardinality of the optimal menu-hold in this setting.\n\nRecall the value of information was described in (15):\n\n$$\nV(E, \\theta)=\\max \\left\\{\\theta \\pi_{1} u_{1}+(1-\\theta) \\pi_{2} u_{2}-\\max \\left\\{\\theta u_{1},(1-\\theta) u_{2}\\right\\}, 0\\right\\} .\n$$\n\nWe can rewrite the value of information as\n\n$$\nV(E, \\theta)=\\max \\left\\{\\theta\\left(\\pi_{1} u_{1}-\\pi_{2} u_{2}\\right)+\\pi_{2} u_{2}-\\max \\left\\{\\theta u_{1},(1-\\theta) u_{2}\\right\\}, 0\\right\\},\n$$\n\nand we capture the value of experiment $E$ for type $\\theta$ via a one-dimensional variable\n\n$$\nq(\\theta) \\triangleq \\pi_{1}(\\theta) u_{1}-\\pi_{2}(\\theta) u_{2} \\in\\left[-u_{2}, u_{1}\\right],\n$$\n\nwhich describes the differential informativeness of the experiment. The endpoints of the interval $\\left[-u_{2}, u_{1}\\right]$ identify two extreme experiments $q \\in\\left\\{-u_{2}, u_{1}\\right\\}$ that are attained when either one of the two signals occurs with probability one in both states. In either case, the resulting experiment reveals no information to the data buyer. Conversely, the fully informative experiment is given by $q=u_{1}-u_{2}$. Because Proposition 2 establishes that either $\\pi_{1}(\\theta)=1$ or $\\pi_{2}(\\theta)=1$ (or both) for each type $\\theta$, we know that $q(\\theta)>u_{1}-u_{2}$ implies $\\pi_{1}(\\theta)=1$, and that $q(\\theta)<u_{1}-u_{2}$ implies $\\pi_{2}(\\theta)=1$. Collecting terms, we can rewrite the value of an experiment in terms of the one-dimensional variable $q$ as follows:\n\n$$\nV(q, \\theta)=\\max \\left\\{\\theta q+u_{2}+\\min \\left\\{u_{1}-u_{2}-q, 0\\right\\}-\\max \\left\\{\\theta u_{1},(1-\\theta) u_{2}\\right\\}, 0\\right\\} .\n$$\n\nThe value of information in (21) illustrates the main properties of our screening problem: (i) the buyer has a type-dependent participation constraint; (ii) the ex ante indifferent type $\\theta^{*}$ has the highest willingness to pay for any experiment $q$; (iii) the experiment $q=u_{1}-u_{2}$ is the most valuable for all types $\\theta$; (iv) different types $\\theta$ rank partially informative experiments differently; and $(v)$ the utility function $V(q, \\theta)$ has the single-crossing property in $(\\theta, q)$.\n\nThe single-crossing property indicates that types with a higher $\\theta$, those who believe that state $\\omega_{1}$ is more likely, assign a higher value to experiments with a higher $q$. In turn, these experiments contain a signal that delivers stronger evidence regarding state $\\omega_{2}$, which they deem less likely ex ante. As in the binary-type case, the vertical dimension (quality of the information) and the horizontal dimension (position of the information) cannot be chosen separately by the seller. In particular, it is not possible to change the differential informativeness of the experiment (i.e., to choose a very high or a very low $q$ ) without\nreducing its overall informativeness.\nWe know from Proposition 1 that we can focus on responsive menus. In the case of binary actions, an experiment is responsive if and only if the value of following both signals' recommendations is non-negative. Therefore, equation (21) implies that experiment $q(\\theta)$ offered to type $\\theta$ is responsive if and only if\n\n$$\nq(\\theta) \\in Q(\\theta) \\triangleq\\left\\{\\begin{array}{cl}\n{\\left[-u_{2}, \\frac{u_{1}}{1-\\theta}-u_{2}\\right]} & \\text { for } \\quad \\theta \\leq \\theta^{*}, \\\\\n{\\left[u_{1}-\\frac{u_{2}}{\\theta}, u_{1}\\right]} & \\text { for } \\quad \\theta \\geq \\theta^{*} .\n\\end{array}\\right.\n$$\n\nIn other words, the restriction $q(\\theta) \\in Q(\\theta)$ allows us to eliminate the first max operator from (21) when computing the value $V(q(\\theta), \\theta)$. We thus obtain a characterization of implementable and responsive menus of experiments.","text_sha256":"2a0b7cc430a711b8c3bc12ea35ac888947d63edb4faf242a288d49f7a06d3a47"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0023","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Lemma 1 (Implementable and Responsive Menus)","text":"## Lemma 1 (Implementable and Responsive Menus)\n\nA menu $\\{q(\\theta)\\}_{\\theta \\in \\Theta}$ is implementable and responsive if and only if\n\n$$\nq(\\theta) \\in\\left[-u_{2}, u_{1}\\right] \\text { is non-decreasing }\n$$\n\nand\n\n$$\n\\int_{0}^{1} q(\\theta) d \\theta=u_{1}-u_{2}\n$$\n\nTo obtain some intuition for constraint (24), observe that the interior type $\\theta^{*}$ assigns the highest value to any experiment $q$, and that the utility function $V(q, \\theta)$ has a downward kink at $\\theta=\\theta^{*}$. In order to compute the buyer's rent function $V$, we apply the envelope theorem to the two intervals on $\\left[0, \\theta^{*}\\right]$ and $\\left[\\theta^{*}, 1\\right]$ separately. Because information has zero value for types $\\theta \\in\\{0,1\\}$, we know that $V(0)=V(1)=0$. We thus obtain two (potentially different) expressions for $V\\left(\\theta^{*}\\right)$. Finally, the buyer's gross value of an experiment (21) is jointly continuous in $(q, \\theta)$, and hence, the rent function $V$ is continuous by the Maximum Theorem. ${ }^{9}$ Incentive compatibility then imposes the following restriction on responsive allocations:\n\n$$\nV\\left(\\theta^{*}\\right)=\\int_{0}^{\\theta^{*}} V_{\\theta}(q, \\theta) \\mathrm{d} \\theta=-\\int_{\\theta^{*}}^{1} V_{\\theta}(q, \\theta) \\mathrm{d} \\theta\n$$\n\nComputing the buyer's marginal rent from (21) and using the definition of $\\theta^{*}$ to simplify (??) yields the integral condition in the Lemma. ${ }^{10}$ Notably, condition (22) does not appear\n\n[^8]in the statement of Lemma 1 because it is implied by the monotonicity condition (23) and by the integral constraint (24).\n\nWith this result in place, the transfer $t(\\theta)$ associated with every experiment $q(\\theta)$ can be computed from the envelope formula on the $\\left[0, \\theta^{*}\\right]$ and $\\left[\\theta^{*}, 1\\right]$ separately. Using integral constraint (24) to simplify further, the seller's problem can be written as\n\n$$\n\\max _{q(\\cdot)} \\int_{0}^{1}\\left[(\\theta f(\\theta)+F(\\theta)) q(\\theta)+\\min \\left\\{\\left(u_{1}-u_{2}-q(\\theta)\\right) f(\\theta), 0\\right\\}\\right] \\mathrm{d} \\theta,\n$$\n\nsubject to constraints (23) and (24).\nObserve that the seller's objective (25) is piecewise linear (and concave) in the experiment $q(\\theta)$. Thus, absent the integral constraint, the optimal experiments take values at the kinks, i.e., $q^{*}(\\theta) \\in\\left\\{-u_{2}, u_{1}-u_{2}, u_{1}\\right\\}$ for all $\\theta$. This corresponds to a menu containing the fully informative experiment only. While such a menu is, in fact, optimal under some conditions, the seller can sometimes do better by offering (at most) one additional experiment.","text_sha256":"3e88d9c3ed4347418d0a9b121ce2e3da5ec78bb897ffa8ca5a5e2d1923346e9c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0024","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proposition 6 (Cardinality of the Optimal Menu)","text":"## Proposition 6 (Cardinality of the Optimal Menu)\n\nAn optimal menu consists of at most two experiments.\nTo establish this property of optimal menus, we reduce the seller's problem (25) to a linear program with equality and non-negativity constraints. An application of the Fundamental Theorem of Linear Programming (e.g., Theorem 8.4 in Chvatal (1983)) implies that the solution is an increasing step function with at most three jumps. ${ }^{11}$ Because it is optimal to set $q(0)=-u_{2}$ and $q(1)=u_{1}$, this means that the optimal menu contains at most two informative experiments $q \\in\\left(-u_{2}, u_{1}\\right)$. When offered, the second experiment contains a signal that perfectly reveals one state. However, the linearity of the environment prevents the seller from offering more than one distorted experiment, i.e., no further versioning is optimal. We formalize this intuition in the following subsection.","text_sha256":"8da9dc27d38a3f08031d81b90a284db5c6f91bae2fef9c70121de9f299de0ae2"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0025","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.4 Single-Item vs. Discriminatory Pricing","text":"### 4.4 Single-Item vs. Discriminatory Pricing\n\nTo obtain further insights into the properties of the optimal menu, we refine our approach to the seller's problem. We combine Lagrange methods, as in the type-dependent participation\n\n[^9]constraints model of Jullien (2000), with the ironing procedure developed by Toikka (2011) extending that in Myerson (1981). This approach allows us to overcome two difficulties posed by our problem: (i) integral constraint (24) and objective (25) have generically different weights, $\\mathrm{d} \\theta$ and $\\mathrm{d} F(\\theta)$; hence, (ii) the problem is non-separable in the type $\\theta$ and the experiment $q(\\theta)$, which interact in two different terms. In particular, the \"virtual values\" $\\phi(\\theta, q)$-defined as the partial derivative of the integrand in (25) with respect to $q$-are a non-constant function of the experiment.\n\nWe derive the solution to the seller's problem by maximizing the virtual values. Because the integrand in objective (25) is piecewise linear in $q$, its partial derivative $\\phi(\\theta, q)$ takes on only two values. We then define the following two functions\n\n$$\n\\begin{aligned}\n\\phi^{-}(\\theta) & \\triangleq \\theta f(\\theta)+F(\\theta) \\\\\n\\phi^{+}(\\theta) & \\triangleq(\\theta-1) f(\\theta)+F(\\theta)\n\\end{aligned}\n$$\n\nthat describe the virtual value $\\phi(\\theta, q)$ for $q<u_{1}-u_{2}$ and $q \\geq u_{1}-u_{2}$, respectively. Heuristically, the two virtual values represent the marginal benefit to the seller of increasing each type's experiment $q$ from $-u_{2}$ to $u_{1}-u_{2}$ and from $u_{1}-u_{2}$ to $u_{1}$. If ironing à la Myerson is required, we denote the ironed virtual values as $\\bar{\\phi}^{-}(\\theta)$ and $\\bar{\\phi}^{+}(\\theta)$.\n\nFinally, we say that a menu satisfies the pooling property if it is constant on any interval where the relevant (ironed) virtual value is constant.","text_sha256":"2b80cd3be552ca6a921332744f99a50629c4d7a3a3b9d66c584ef0b0ffb17152"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0026","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proposition 7 (Optimal Menu)","text":"## Proposition 7 (Optimal Menu)\n\nThe menu $\\left\\{q^{*}(\\theta)\\right\\}_{\\theta \\in \\Theta}$ is optimal if and only if the following conditions hold:\n\n1. there exists $\\lambda^{*}>0$ such that, for all $\\theta$,\n$$\nq^{*}(\\theta)=\\arg \\max _{q \\in\\left[-u_{2}, u_{1}\\right]}\\left[\\bar{\\phi}^{-}(\\theta) \\min \\left\\{q, u_{1}-u_{2}\\right\\}+\\bar{\\phi}^{+}(\\theta) \\max \\left\\{0, q-\\left(u_{1}-u_{2}\\right)\\right\\}-\\lambda^{*} q\\right]\n$$\n2. $\\left\\{q^{*}(\\theta)\\right\\}_{\\theta \\in \\Theta}$ has the pooling property and satisfies integral constraint (24).\n\nWe now illustrate the optimal menu under flat and discriminatory pricing separately.","text_sha256":"93b542aaa21e2f0c32c2eeb91045a24f8a69d60f9b608b52c3d2a9ce6bd9a72b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0027","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Corollary 1 (Single-Item Menu)","text":"## Corollary 1 (Single-Item Menu)\n\nThe optimal menu contains a single item whenever any of the following hold:\n\n1. almost all types have congruent priors, i.e., $F\\left(\\theta^{*}\\right) \\in\\{0,1\\}$;\n2. the monopoly price for experiment $\\bar{E}$ is equal on $\\left[0, \\theta^{*}\\right]$ and $\\left[\\theta^{*}, 1\\right]$;\n3. both virtual values $\\phi^{-}(\\theta)$ and $\\phi^{+}(\\theta)$ are strictly increasing.\n\nTo grasp the intuition behind these results, consider a relaxed problem where the seller contracts separately with two groups of buyers, $\\theta<\\theta^{*}$ and $\\theta \\geq \\theta^{*}$. Part (1) essentially addresses this case. Because of the linearity of the problem, the optimal mechanism is a cutoff mechanism: the seller offers only the fully informative experiment to each group, at generically different prices. Part (2) states that, if the cutoff types in the two subproblems have the same willingness to pay, then those cutoffs also solve the unrestricted problem. For instance, when $u_{1}=u_{2}$, any distribution that is symmetric around $\\theta=1 / 2$ satisfies this condition. Part (3) identifies regularity conditions on the distribution of types that rule out, for example, most bimodal distributions. Under these conditions, the seller prefers to offer the fully informative experiment to all buyers at an intermediate price.\n\nWe note that only condition (3) can be stated independently of the matching values $\\left(u_{1}, u_{2}\\right)$. For example, an optimal menu contains a single item whenever the values are uniformly distributed, irrespective of the payoffs. Figure 5 describes the optimal menu for the case of constant match values, $u_{1}=u_{2}=1$, and uniformly distributed types. Because both virtual values are strictly increasing, they cross the threshold level $\\lambda^{*}$ only once. Thus, the optimal menu is a step function $q(\\theta) \\in\\{-1,0,1\\}$ assigning the fully informative experiment $q=0$ to types $\\theta \\in[1 / 4,3 / 4]$ at a price $t=1 / 4$, and no information to all other types.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 5: Optimal Allocation with Uniform Distribution\n\nIf the conditions of Corollary 1 fail, however, the seller may choose to offer a second (distorted) experiment to one group in order to maintain a high price for the fully informative experiment. Furthermore, Corollary 1.3 implies that the seller offers a second experiment only if the distribution of types requires ironing of the virtual values. When types correspond to interim beliefs, it is natural to consider bimodal densities that fail the strong regularity conditions and, therefore, introduce the need for ironing. This is the case, for example, if most buyers are well informed ex ante (with most types close to 0 or 1 ). In other words, ironing is not a technical curiosity in our case but rather a technique that becomes unavoidable because of the properties of the information environment. Figure 6 (left) illustrates the ironed virtual\nvalues for a bimodal probability density function (drawn on a different scale). ${ }^{12}$ Figure 6 (right) illustrates the resulting optimal menu, again for the case $u_{1}=u_{2}$.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 6: Probability Density Function and Optimal Allocation\n\nAs in the optimal menu in the binary-type setting, the partially informative experiment $q \\approx-0.21$ contains one signal that perfectly reveals state $\\omega_{1}$. This experiment is relatively unattractive for higher types, and it allows the monopolist to increase the price for the large mass of types located around $\\theta \\approx 0.7$. Note that the ex ante least informed type $\\theta^{*}$ need not purchase the fully informative experiment, despite having the highest value of information and obtaining the highest rent $V\\left(\\theta^{*}\\right)$. Indeed, the type that is indifferent between the two experiments in Figure 6 is $\\theta \\approx 0.55$. From the seller's perspective, incentivizing type $\\theta=1 / 2$ to purchase the fully informative experiment would require further distortions (and, hence, a lower price) for the second experiment. This leads to a loss of revenue from the types around $\\theta \\approx 0.2$. Because such types are quite frequent, this loss more than offsets the gain in revenue from the types around $\\theta=1 / 2$.\n\nFinally, a sufficient condition for the optimality of two-item menus can be obtained by continuity with the binary-type case. In that setting, when the two types are equally likely, symmetric about 1/2 and sufficiently well informed (i.e., $\\theta^{H}>2 / 3$ ), the optimal menu is discriminatory for all $u_{1} \\neq u_{2}$.","text_sha256":"1a721063f54f966ec46444964a9cfd96a085540ee9a8793de9035558090a9de5"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0028","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Implications for Data Pricing","text":"## 5 Implications for Data Pricing\n\nWe discuss how to bring our model's results to bear on the design of real-world information products. We focus on offline and online brokers of big data-firms such as Acxiom, Nielsen, and Oracle-that sell information about individual consumers to business customers.\n\n[^10]Data obtained from brokers is typically used to facilitate marketing efforts and to mitigate risks. ${ }^{13}$ Information used for marketing purposes is typically sold through data appends and marketing lists. Data appends reveal supplemental information about a firm's existing or potential customers, allowing the firm to place them into more precise market segments. For instance, all major data brokers offer data management platforms (DMPs)-customized software that enables websites to track their users and integrate their own data with 3rdparty data. Most risk-mitigation data products offered by credit rating agencies are also of this kind. Conversely, marketing lists facilitate targeted advertising to new consumers. Advertisers can choose whether to acquire standard lists of potential consumers with prespecified sets of characteristics, or to customize their list of desired consumer attributes.\n\nOur model of selling supplemental information is best suited to analyzing data appends. ${ }^{14}$ We therefore describe these information products in greater detail, and we comment on marketing lists and other data services at the end of this section. For concreteness, consider the following examples of data appends.\n\n- Oracle ID-Graph tracks firms' customers across several devices, augmenting the data collected on the firms' websites with behavioral observations from different sources. ${ }^{15}$\n- Email Intelligence by TowerData attaches demographic, income, intent, and purchase information to a merchant's own list of email addresses. ${ }^{16}$\n- The credit reporting agency Equifax offers its business customers (e.g., banks and credit card companies) a risk-mitigation product called Undisclosed Debt Monitoring. This product tracks an individual borrower to identify new negative information that arrives between the original loan approval and the closing date. ${ }^{17}$\n\nEach of these products is available in several versions, which differ mainly in terms of the number of informative variables the seller discloses to the buyer. For instance, ID-Graph and Email Intelligence allow buyers to customize their queries to the database (e.g., a consumer's age group, income level, interests, and intent). Similarly, the versions of Undisclosed Debt\n\n[^11]Monitoring differ in terms of the number of potentially negative events (e.g., late payments, credit inquiries, bankruptcy filings) that are monitored and disclosed.\n\nThe most common buyers of data products are marketing and advertising firms, lenders and financial services firms, and retail companies (Federal Trade Commission, 2014). The data buyers differ along several dimensions, including their ability to process data and the richness of their action space. ${ }^{18}$ Any of these dimensions of heterogeneity can lead to interesting sorting patterns of buyers into product versions. Here we focus on the differences in the strength of the buyers' priors, i.e., in the availability of initial information.\n\nOur main structural results (Propositions 1 and 2) derive the properties of the distribution of states and signals that are associated with monopolistic screening. In particular, our results impose restrictions on the support of the conditional distribution of signals in a given state. In order to leverage these insights to evaluate and inform the design of data appends, it is useful to rephrase the design of a statistical experiment in terms of hypothesis testing. Our structural results identify the types of statistical errors incurred by data buyers as a consequence of market power in the sale of information. ${ }^{19}$\n\nFor the present purpose, consider a data buyer, such as an advertiser or a lender. The data buyer wishes to test the null hypothesis \"target with an ad\" or \"grant a loan\" for a specific consumer, i.e., to distinguish a null hypothesis $H_{0}$ from an alternative hypothesis $H_{1}$. The data buyer is a Bayesian decision maker with a prior distribution over the hypotheses. He can take one of two actions, each of which is optimal under the respective hypothesis.\n\nThe data seller has a test statistic whose distribution, conditional on the true state of the world, is given by $H_{0}$ or $H_{1}$ as in Figure 7. We assume that $H_{0}$ and $H_{1}$ satisfy the monotone likelihood ratio property. While the data seller could potentially disclose arbitrary information about the distribution of the test statistic, suppose she chooses to inform the data buyer if the statistic is above or below a certain threshold. The data buyer then chooses the corresponding action. We can represent this experiment as a statistical test of the null hypothesis $H_{0}=\\left\\{\\omega_{2}\\right\\}$,\n\n| $E$ | $s_{1}$ | $s_{2}$ |\n| :--- | :--- | :--- |\n| $\\omega_{1}$ | $1-\\beta$ | $\\beta$ |\n| $\\omega_{2}$ | $\\alpha$ | $1-\\alpha$ |\n\n[^12]\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 7: Conditional Distributions of the Test Statistic\n\nwhere $\\alpha$ and $\\beta$ denote the probability of a type I and type II statistical error, respectively. ${ }^{20}$\nGiven the information contained in the database of the seller in Figure 7, the set of feasible statistical tests is described by the area between the blue curve and the red line in the left panel of Figure 8. The blue curve identifies the loci of the least type I and type II statistical errors given the data available. As the data seller can always introduce noise into the test statistic, the set of feasible statistical tests is given by the entire area. In our model, we assumed that the data seller has complete information, and thus, the boundary is given by the blues lines in the right panel of Figure 8, which coincide with the $\\alpha$ and $\\beta$ axes. ${ }^{21}$\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 8: Feasible Information Structures","text_sha256":"0e3ce72eee486e616443d44ae2ccc20077acbb254cce47ab72e68f562b055612"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0029","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Implications for Data Pricing","text":"The central issue for the data seller is that she does not know the data buyer's prior beliefs and, hence, the buyer's willingness to pay for this information. She must therefore employ a richer mechanism to screen heterogeneous buyers. In particular, the seller offers the buyer a menu of binary (\"pass/fail\") tests. Each test reports the outcome \"pass\" when\n\n[^13]the test statistic is below a particular threshold. Each test yields a different combination of type I and type II errors $(\\alpha, \\beta)$.\n\nOur main structural result (Proposition 2) identifies systematic patterns in the optimal design of partially informative statistical tests. With binary states and actions, and no constraints on $(\\alpha, \\beta)$, the seller induces some buyers to make either type I or type II errors, but not both. The logic underlying this result extends to the case of a partially informed seller: an identical argument establishes the stronger result that all optimal tests lie on the lower boundary of the feasible set in Figure 8 (left). Separation in the optimal menu is then supported by the differences in the error structure of each test and by the buyers' heterogeneous preferences over statistical errors.\n\nA concrete implication of the optimality of non-dispersed experiments is that no product in an optimal menu should add unbiased noise to the seller's information. Adding idiosyncratic noise can be useful when multiple buyers compete in a downstream market, as in Admati and Pfleiderer (1986), or when it is important to preserve the anonymity of the data, as in the differential privacy literature (Dwork, 2008). In the absence of these concerns, our results suggest that this practice reduces the seller's revenues. Instead, optimal experiments should minimize the type-II error for any level of type-I error (as in Figure 8). For example, if the data buyer faces a binary advertising decision, the optimal menu should lead to excessively broad or to excessively narrow campaigns, i.e., advertising to a subset of high-value consumers but not to low-value consumers, or to all high-value and to some low-value consumers.\n\nInterestingly, in practice, none of the data products described earlier appear to introduce noise into the data. For example, credit rating agencies do not offer both precise and noisy versions of the same information (e.g., computing a consumer's credit score on the basis of more or less detailed data). Instead, information is degraded by revealing only a portion of the available data to the buyer. We now discuss how omitting explanatory variables can be seen as implementing our optimal mechanism under specific conditions on the buyer's decision problem and on the seller's data.\n\nConsider the following stylized description of the above-mentioned Equifax product Undisclosed Debt Monitoring. A credit rating agency collects $K$ binary characteristics $x_{k} \\in\\{0,1\\}$ for each potential borrower. Each realization $x_{k}=1$ corresponds to a \"red flag\" on the borrower's record. Suppose, for simplicity, that the data buyer's (the lender's) payoff-relevant state is $\\omega=\\mathbf{1}\\left[\\Sigma_{k=1}^{K} x_{k}=0\\right]$, i.e., it is optimal to grant a loan if and only if there are no red flags. The data seller knows the borrower's vector of characteristics $\\left(x_{1}, \\ldots, x_{K}\\right)$, while the data buyer privately knows the (identical) probability distribution of each binary characteristic $x_{k}$. In this case, hiding the realized value of any characteristic $x_{k}$ induces the data\nbuyer to grant a loan to an unqualified borrower with positive probability, but no qualified borrower is ever turned down.\n\nIn this example, the data buyer incurs only one type of statistical error. Thus, omitting variables can be part of an optimal design if subsets of the seller's data contain conclusive evidence against at least one state. In other settings, withholding data may introduce fullsupport noise into the experiment. This is the case any time a hidden variable could have overturned the signals based on the revealed data and changed the buyer's action in any direction. For example, it is not difficult to construct examples in which a broker (e.g., Oracle or Nielsen) provides demographic, income and interest information about consumers, but using demographic and income data as proxies for interest leads the data buyer to incur both types of statistical errors.\n\nData brokers offer several other information products in addition to data appends. For instance, marketing lists are queries to a broker's database of individual consumer records. Such queries allow advertisers to identify potential customers with a set of desired characteristics. ${ }^{22}$ Thus, a marketing list can be viewed as a statistical experiment where each signal indicates (possibly with error) a pre-specified consumer segment. From a modeling standpoint, however, selling a list is very different from appending data to the buyer's existing information. In particular, when selling a list, the seller is able to charge a price contingent on the signal realization. ${ }^{23}$ Marketing lists can also be sold contextually to advertising space. This occurs when a data provider (e.g., Bluekai) partners with a publisher of space (Doubleclick) and charges the advertiser a price per impression in addition to the cost of the advertising space. Thus the price of the data augments the cost of the ads. In other words, the buyer's advertising decision is contractible. ${ }^{24}$","text_sha256":"4e0d0ae493031dad3e1eb58f6a5fa0ec19c41953f6302b445069b32b0ff8e0fe"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0030","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Optimal Menu with Many Actions","text":"## 6 Optimal Menu with Many Actions\n\nWe now extend the analysis of the optimal menu to environments with many actions and many states. In order to make progress in this richer environment, we restrict our attention to the case of binary types and matching payoffs defined in (2). We then provide a suitable generalization of the optimal menu derived in Section 4. We continue to define the low\n\n[^14]and the high type $\\theta \\in\\left\\{\\theta^{L}, \\theta^{H}\\right\\}$ such that the high type values the completely informative experiment more than the low type, as in (16).\n\nThe construction of the optimal menu now proceeds differently than in Section 4. We first solve a relaxed problem wherein we require the high type $\\theta^{H}$ to take action $a_{i}$ upon observing signal $s_{i}$ even when buying the experiment destined for the low type $\\theta^{L}$. We then show that the solution to the relaxed problem-which disregards the obedience constraints-indeed satisfies the original constraints; hence, it also solves the full problem. In other words, we first guess and then verify that the optimal mechanism is obedient on and off the equilibrium path. In Example 3, we show that this relaxed approach fails to deliver a valid solution with more than two types. Thus, a different approach is required if we want to consider problems with arbitrary cardinality in both the action/state space and the type space.\n\nIn the relaxed problem, we replace the high type's incentive-compatibility constraint\n\n$$\nV\\left(\\theta^{H}\\right) \\geq V\\left(E\\left(\\theta^{L}\\right), \\theta^{H}\\right)-t\\left(\\theta^{L}\\right)=\\sum_{j=1}^{I} \\max _{i}\\left\\{\\theta_{i}^{H} u_{i} \\pi_{i j}\\left(\\theta^{L}\\right)\\right\\}-\\max _{i}\\left\\{\\theta_{i}^{H} u_{i}\\right\\}-t\\left(\\theta^{L}\\right),\n$$\n\nwith the weaker constraint\n\n$$\nV\\left(\\theta^{H}\\right) \\geq \\sum_{i=1}^{I} \\theta_{i}^{H} u_{i} \\pi_{i i}\\left(\\theta^{L}\\right)-\\max _{i}\\left\\{\\theta_{i}^{H} u_{i}\\right\\}-t\\left(\\theta^{L}\\right) .\n$$\n\nThe relaxed version of the constraint drops the max operator and simply asks type $\\theta^{H}$ to accept the recommendation $a_{i}$ implicit in signal $s_{i}$.\n\nProposition 3 established that both the low type's participation constraint and the high type's incentive constraint bind. This reduces the seller's problem to choosing the diagonal entries of the low-value type's experiment $\\pi_{i i}\\left(\\theta^{L}\\right) \\in[0,1]$. Substituting $t\\left(\\theta^{L}\\right)=V\\left(E\\left(\\theta^{L}\\right), \\theta^{L}\\right)$ in the right-hand side of (26) above, the seller maximizes\n\n$$\n(1-\\gamma) \\underbrace{\\left(\\sum_{i=1}^{I} \\theta_{i}^{L} u_{i} \\pi_{i i}\\left(\\theta^{L}\\right)-\\max _{i} \\theta_{i}^{L} u_{i}\\right)}_{t\\left(\\theta^{L}\\right)=V\\left(E\\left(\\theta^{L}\\right), \\theta^{L}\\right)}+\\gamma \\underbrace{\\left(\\sum_{i=1}^{I} \\theta_{i}^{H} u_{i}-\\max _{i} \\theta_{i}^{H} u_{i}-V\\left(\\theta^{H}\\right)\\right)}_{t\\left(\\theta^{H}\\right)=V\\left(\\bar{E}, \\theta^{H}\\right)-V\\left(\\theta^{H}\\right)}\n$$\n\nsubject to the high type's participation constraint\n\n$$\nV\\left(\\theta^{H}\\right)=\\sum_{i=1}^{I}\\left(\\theta_{i}^{H}-\\theta_{i}^{L}\\right) u_{i} \\pi_{i i}\\left(\\theta^{L}\\right)-\\max _{i}\\left\\{\\theta_{i}^{H} u_{i}\\right\\}+\\max _{i}\\left\\{\\theta_{i}^{L} u_{i}\\right\\} \\geq 0 .\n$$\n\nThe relaxation of the incentive constraints-fixing a mapping from signals to actions for both types and experiments-turns the seller's problem into a linear program.\n\nIntuitively, the seller's choice of a partially informative experiment $E\\left(\\theta^{L}\\right)$ is guided by the disagreement in the beliefs over states of the two types. The seller is most willing to\nintroduce noise into signals about states that the low type considers relatively less likely than the high type. The resulting distortions in the decisions facilitate screening without sacrificing too much of the surplus. To formalize this intuition, we re-order the states $\\omega_{j}$ by the likelihood ratios of the two types' beliefs. In particular, let\n\n$$\n\\frac{\\theta_{1}^{L}}{\\theta_{1}^{H}} \\leq \\cdots \\leq \\frac{\\theta_{i}^{L}}{\\theta_{i}^{H}} \\leq \\cdots \\leq \\frac{\\theta_{I}^{L}}{\\theta_{I}^{H}} .\n$$\n\nA basic application of the Lagrange multiplier method yields the following characterization result, which generalizes the optimal experiment in Section 4.2.","text_sha256":"4184619f1d7aba40d03250ea61baae726cebb516cf24df6332c35bd6f359822e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0031","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proposition 8 (Optimal Menu with Two Types)","text":"## Proposition 8 (Optimal Menu with Two Types)\n\nThere exists a critical state $i^{*}<I$ such that an optimal experiment $E\\left(\\theta^{L}\\right)$ has $\\pi_{i i}=0$ for all $i<i^{*}$ and $\\pi_{i i}=1$ for all $i>i^{*}$.\n\nBecause, without loss of generality, the low type chooses action $a_{i}$ when observing signal $s_{i}$, a nil diagonal entry $\\pi_{i i}=0$ means signal $s_{i}$ is never sent by experiment $E\\left(\\theta^{L}\\right)$. Unlike in the binary action case, the partially informative experiment $E\\left(\\theta^{L}\\right)$ may thus contain fewer signals than the available actions, as the seller drops some signals to reduce the information rent of the high type. Furthermore, Proposition 8 implies that the optimal experiment $E\\left(\\theta^{L}\\right)$ has a lower triangular shape, with at most one strictly interior diagonal entry $\\pi_{i^{*} i^{*}} \\in(0,1)$ :\n\n| $E\\left(\\theta^{L}\\right)$ | $s_{1}$ | ⋯ |  | $s_{i^{*}}$ | ⋯ |  | $s_{I}$ |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | 0 | ⋯ | 0 | $\\pi_{1 i^{*}}$ | ⋯ |  | $\\pi_{1 I}$ |\n| ⋮ |  |  |  | ⋮ |  |  | ⋮ |\n| $\\omega_{i^{*}}$ | ⋮ |  | ⋮ | $\\pi_{i^{*} i^{*}}$ | ... |  | $\\pi_{i^{*} I}$ |\n|  |  |  |  | 0 | 1 |  | 0 |\n| ⋮ |  |  |  | ⋮ |  | ⋱ |  |\n| $\\omega_{I}$ | 0 | ... | 0 | 0 | 0 |  | 1 |\n\nIn other words, the seller chooses a subset of \"targeted\" states $i \\in\\left\\{i^{*}, \\ldots, I\\right\\}$ in which the buyer takes the correct action with positive probability. Conversely, when a \"residual\" state $i \\in\\left\\{1, \\ldots, i^{*}-1\\right\\}$ is realized, the buyer never takes the correct action. Residual states are used to degrade the information revealed about the targeted states. The seller chooses how to partition states based on two factors: the two types' relative beliefs; and the relative frequency of each type.\n\nIn order to construct the optimal experiment $E\\left(\\theta^{L}\\right)$, we substitute the expression for the information rent $V\\left(\\theta^{H}\\right)$ from (28) into the seller's objective (27). Up to an additive constant,\nthe seller's profits are given by\n\n$$\n\\sum_{i=1}^{I}\\left(\\theta_{i}^{L}-\\gamma \\theta_{i}^{H}\\right) u_{i} \\pi_{i i}\\left(\\theta^{L}\\right) .\n$$\n\nThe shape of the optimal experiment then depends on whether the participation constraint of the high type binds. In particular, if (28) is slack, the seller's profits (31) are maximized by assigning only extreme values to the diagonal, $\\pi_{i i} \\in\\{0,1\\}$. Because the likelihood ratios $\\theta_{i}^{L} / \\theta_{i}^{H}$ are increasing, we have $\\pi_{i i}=1$ for all $i \\geq i_{s}$, where\n\n$$\ni_{s} \\triangleq \\min \\left\\{i: \\gamma \\leq \\theta_{i}^{L} / \\theta_{i}^{H}\\right\\} .\n$$\n\nTo determine whether the participation constraint of the high type is satisfied by this solution, define the following function\n\n$$\nR(j) \\triangleq \\sum_{i=j}^{I}\\left(\\theta_{i}^{H}-\\theta_{i}^{L}\\right) u_{i}-\\max _{i}\\left\\{\\theta_{i}^{H} u_{i}\\right\\}+\\max _{i}\\left\\{\\theta_{i}^{L} u_{i}\\right\\} .\n$$\n\nThis expression corresponds to the rent of the high type if the seller drops all signals $s_{i}$ (and, hence, recommended actions $a_{i}$ ) with $i<j$ and sets $\\pi_{i i}=1$ for all $i \\geq j$. In particular, $R(1)$ is the difference in the value of full information between the high and the low type. By construction, it is positive. We further note that $R(j)$ is strictly decreasing in $j$ if $\\theta_{j}^{H}>\\theta_{j}^{L}$ and increasing otherwise. Furthermore, because of the likelihood ratio order on states $i$, it attains its minimum at $i_{c} \\triangleq \\min \\left\\{i: 1 \\leq \\theta_{i}^{L} / \\theta_{i}^{H}\\right\\}$, where $i_{c} \\geq i_{s}$. In the proof of Proposition 9, we show that $R\\left(i_{c}\\right)<0$. Therefore, if $R\\left(i_{s}\\right) \\geq 0$, the constraint (28) is slack at the optimum.\n\nConversely, if $R\\left(i_{s}\\right)<0$, then the solution to unconstrained problem (31) violates the high type's participation constraint. Therefore, constraint (28) must bind at the optimum. In particular, the seller chooses a state $i_{b}$ and a diagonal entry $\\pi_{i_{b} i_{b}} \\in[0,1]$ to satisfy (28) with equality when, in addition, $\\pi_{i i}=1$ for all states $i>i_{b}$ and $\\pi_{i i}=0$ for all $i<i_{b}$. By the properties of $R(j)$, this critical state $i_{b}$ is given by the unique solution to\n\n$$\nR\\left(i_{b}\\right)>0>R\\left(i_{b}+1\\right) .\n$$\n\nIntuitively, state $i_{b}$ corresponds to the minimum number of signals (and corresponding action recommendations) that must be eliminated in order to satisfy the high type's incentive constraint while extracting all the rent. Proposition 9 establishes that the participation constraint of the high type binds if and only if $i_{b}<i_{s}$.","text_sha256":"5b9260a393c8ab67ec6293995b79b3bfae1fd4d13eaf75142858fa213193b32d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0032","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proposition 9 (Information Rents)","text":"## Proposition 9 (Information Rents)\n\n1. The critical state $\\omega_{i^{*}}$ in experiment (30) is given by $i^{*}=\\min \\left\\{i_{s}, i_{b}\\right\\}$.\n2. If $i_{b} \\geq i_{s}$, then $E\\left(\\theta^{L}\\right)$ has $\\pi_{i^{*} i^{*}}=1$ and $V\\left(\\theta^{H}\\right)>0$.\n3. If $i_{b}<i_{s}$, then $E\\left(\\theta^{L}\\right)$ has $\\pi_{i^{*} i^{*}}=\\pi_{i^{*}}^{B}$ and $V\\left(\\theta^{H}\\right)=0$.\n\nTo grasp the intuition, note that the definition of state $i_{b}$ does not depend on the distribution of types, while state $i_{s}$ is increasing in the fraction of high types $\\gamma$. The fraction of high types represents the shadow cost of providing information to the low types. When this opportunity cost is low, the monopolist prefers to limit distortions and concede rents to the high type. As $\\gamma$ increases, the informativeness of the low type's experiment decreases. For example, if $\\gamma<\\theta_{1}^{L} / \\theta_{1}^{H}$ (i.e., $i_{s}=1$ ), Proposition 9 implies that both types purchase the fully informative experiment $\\bar{E}$, and that the high type obtains positive rents. Conversely, if $\\gamma>\\theta_{i}^{L} / \\theta_{i}^{H}$ for all $i$ such that $\\theta_{i}^{L} / \\theta_{i}^{H}<1$, then $i_{s}>i_{b}$ and the high type obtains no rent.\n\nTo complete the description of the optimal experiment $E\\left(\\theta^{L}\\right)$, we need to specify the distribution of signals $\\pi_{i j}$ for $i \\leq i^{*}$. We construct an off-diagonal assignment procedure that induces both types to follow the recommendation of every signal. In particular, for every state $i \\leq i^{*}$, we begin with the last signal $s_{I}$ and assign the off-diagonal probabilities $\\pi_{i I}$ such that the high type is indifferent between actions $a_{i}$ and $a_{I}$. We then proceed backward to signal $s_{I-1}$ preserving indifference and placing the residual probability, if any, on $\\pi_{i i^{*}}$. We show that the high type prefers action $a_{i^{*}}$ to any action $a_{i}$ with $i<i^{*}$. Therefore, the solution to the relaxed problem satisfies the original constraints. We illustrate the construction of the optimal experiment and the implications for information rents in Example 1 in the Supplemental Appendix.\n\nEarlier, we defined any two types $\\theta^{L}$ and $\\theta^{H}$ as congruent if they shared the same optimal action $a_{i}$ for some $i$ given their interim beliefs:\n\n$$\n\\underset{a_{i} \\in A}{\\arg \\max }\\left\\{\\sum_{j=1}^{J} \\theta_{j}^{L} u_{i j}\\right\\}=\\underset{a_{i} \\in A}{\\arg \\max }\\left\\{\\sum_{j=1}^{J} \\theta_{j}^{H} u_{i j}\\right\\}=a_{i} .\n$$\n\nWe now define two types to be strongly congruent if the posterior belief of $\\theta^{L}$ can also be represented as a convex combination of $\\theta^{H}$ and the vertex of the probability simplex identifying state $\\omega_{i}$, i.e., if (33) holds and there exists $\\lambda \\in[0,1]$ such that\n\n$$\n\\theta^{L}=\\lambda \\theta^{H}+(1-\\lambda)(0, \\ldots, 0,1,0, \\ldots, 0) .\n$$\n\nIn other words, if $\\theta^{L}$ and $\\theta^{H}$ are strongly congruent, then $\\theta^{L}$ lies on a ray that goes through the vertex $(0, \\ldots, 0,1,0, \\ldots, 0)$ and $\\theta^{H}$. An implication of this geometric condition is that the types $\\theta^{L}$ and $\\theta^{H}$ have a constant likelihood ratio $\\theta_{j}^{L} / \\theta_{j}^{H}=\\lambda$ across all states $\\omega_{j} \\neq \\omega_{i}$.\n\nWith more than two states, discriminatory pricing can now be profitable even if the two types are congruent, as long as they are not strongly congruent.","text_sha256":"c940398825dcae0725a41d1568674050a452974ebbff378cdcd4a5bbc8dbaa0c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0033","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Proposition 10 (Partial Information)","text":"## Proposition 10 (Partial Information)\n\n1. With strongly congruent priors, type $\\theta^{L}$ receives zero or full information.\n2. Without strongly congruent priors, type $\\theta^{L}$ receives zero, partial or full information.\n\nProposition 10 thus strengthens and generalizes Proposition 4. With more than two states, the seller can exploit disagreement along any dimension and extract all the surplus through discriminatory pricing. Example 2 in the Supplemental Appendix illustrates how congruent, but not strongly congruent beliefs, allow for surplus extraction.\n\nWith more than two types (and more than two actions and states), our relaxed approach is not always valid, i.e., the optimal menu leads different types to choose different actions in response to the same signal realizations. In particular, type $\\theta$ need not follow the recommendation of every signal in all experiments $E\\left(\\theta^{\\prime}\\right)$, for $\\theta \\neq \\theta^{\\prime}$. We provide an instance of this additional issue in Example 3 of the Supplemental Appendix.","text_sha256":"e038d9d500b0add92b25d11d93a9343b1bbe5b1d3b8544583f14605838896a3c"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0034","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"7 Conclusion","text":"## 7 Conclusion\n\nWe have studied a monopolist who sells supplemental information to privately informed buyers. The resulting screening problem reflects several key properties of information goods that set it apart from traditional models of price discrimination.\n\nFirst, the Bayesian nature of the buyers' decision making is fundamental to the seller's problem. Differences in the buyers' private beliefs introduce a novel aspect of horizontal differentiation that widens the seller's scope for price discrimination.\n\nSecond, information is inherently rich and can be modified in many ways. Even onedimensional data, such as a consumer's credit score, can be turned into a rich set of information products, as some data buyers may want to identify consumers with excellent scores, while other data buyers may be concerned with avoiding consumers with very low scores.\n\nThird, ultimately, instrumental information is useful as long as it guides the decision of the data buyer. Information therefore enters as an input into the buyer's decision problem. Thus, buyers with different private beliefs may act differently upon receiving the same supplemental information. We have shown that the buyer's ability to adjust his behavior in response to new information complicates the seller's screening problem by introducing the possibility of double deviations. In this respect, selling information is akin to selling multiple inputs\nto heterogeneous buyers who can combine them in different ways, according to a privately known production technology.\n\nIn this paper, we have deliberately relied solely on belief heterogeneity to motivate sales of supplemental information. In practice, however, buyers of information may differ along several alternative or additional dimensions, including their cost of choosing specific actions, their ability to process data, or their preferences for timely information. ${ }^{25}$ Each of these extensions can be implemented within the framework we have outlined. Combining different sources of heterogeneity appears more challenging but promises to yield additional insights. Finally, we have focused on the packaging or versioning problem of a seller who is free to acquire and degrade information. Thus, our results represent only a first pass at understanding the trade-offs involved in selling information products. A richer model would distinguish the fixed cost of acquiring the information (e.g., building a database) from the variable cost of duplicating, distributing, and potentially degrading the available information.\n\n[^15]","text_sha256":"58861b2f0373325fb13f6091d5ea60716b22d56da7700415e8a86401be5cb544"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0035","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Appendix","text":"## A Appendix\n\nProof of Proposition 1. Consider any type $\\theta$ and experiment $E=(S, \\pi)$. Without loss of generality, let the type choose a single action after each signal. ${ }^{26}$ Let $S^{a}$ denote the set of signals in experiment $E$ that induces type $\\theta$ to choose action $a$. Thus, $\\cup_{a \\in A} S^{a}=S$. Construct experiment $E^{\\prime}=\\left(S^{\\prime}, \\pi^{\\prime}\\right)$ as a recommendation for type $\\theta$ based on experiment $E$, with signal space $S^{\\prime}=\\left\\{s_{a}\\right\\}_{a \\in A}$ and\n\n$$\n\\pi^{\\prime}\\left(s_{a} \\mid \\omega\\right)=\\int_{S^{a}} \\pi(s \\mid \\omega) \\mathrm{d} s \\quad \\omega \\in \\Omega, a \\in A .\n$$\n\nBy construction, $E^{\\prime}$ and $E$ induce the same outcome distribution for type $\\theta$; hence, $V\\left(E^{\\prime}, \\theta\\right)=$ $V(E, \\theta)$. Moreover, $E^{\\prime}$ is a garbling of $E$. By Blackwell's theorem, we have $V\\left(E^{\\prime}, \\theta^{\\prime}\\right) \\leq$ $V\\left(E, \\theta^{\\prime}\\right)$ for all $\\theta^{\\prime}$. Therefore, for any incentive-compatible and individually rational direct mechanism $\\{E(\\theta), t(\\theta)\\}$, we can construct another direct mechanism $\\left\\{E^{\\prime}(\\theta), t(\\theta)\\right\\}$ whose experiments lead type $\\theta$ to take action $a$ after observing signal $s_{a} \\in S^{\\prime}(\\theta)$ that is also incentive compatible and individually rational, thus yielding weakly larger profits. ■\n\nProof of Proposition 2. (1.) The argument is given in the text.\n(2.) Let $\\mathcal{M}=\\{E(\\theta), t(\\theta)\\}$ be an individually rational and incentive-compatible direct mechanism. Fix an experiment $E \\in \\mathcal{M}$, let $\\pi_{i j}$ denote the conditional probability of signal $s_{j}$ in state $\\omega_{i}$, and suppose $\\pi_{i j}>0$ for all $i$ and $j$. We argue the seller can improve her profits by replacing $E$ with a non-dispersed experiment. By Proposition 1, we can restrict attention to responsive experiments with $J$ signals. Hence, the value of experiment $E$ for any obedient type $\\theta$ is given by\n\n$$\nV_{\\mathrm{ob}}(E, \\theta)=\\sum_{i=1}^{I} \\theta_{i} \\sum_{j=1}^{J} \\pi_{i j} u_{i j}-u(\\theta) .\n$$\n\nFor each state $\\omega_{i}$, define the worst action $a_{j(i)}$ and the corresponding signal $s_{j(i)}$, where $j(i) \\in \\arg \\min _{j} u_{i j}$. Now let\n\n$$\n\\varepsilon_{i} \\triangleq \\frac{\\eta}{u_{i i}-u_{i j(i)}} \\text { with } \\eta \\triangleq \\min _{i}\\left\\{\\left(u_{i i}-u_{i j(i)}\\right) \\pi_{i j(i)}\\right\\},\n$$\n\nand construct a new experiment $E^{\\prime}$ where $\\pi_{i i}^{\\prime}=\\pi_{i i}+\\varepsilon_{i}$ and $\\pi_{i j(i)}^{\\prime}=\\pi_{i j(i)}-\\varepsilon_{i}$ for all $i$.\n\n[^16]Experiment $E^{\\prime}$ is non-dispersed by construction, i.e., $\\pi_{i^{*} j\\left(i^{*}\\right)}=0$, where\n\n$$\ni^{*} \\in \\arg \\min _{i}\\left\\{\\left(u_{i i}-u_{i j(i)}\\right) \\pi_{i j(i)}\\right\\} .\n$$\n\nWe now argue that the seller can improve his profits and relax the incentive constraints. Using (34), we write the incremental value of experiment $E^{\\prime}$ for an obedient type $\\theta$ as follows:\n\n$$\nV_{\\mathrm{ob}}\\left(E^{\\prime}, \\theta\\right)-V_{\\mathrm{ob}}(E, \\theta)=\\sum_{i=1}^{I} \\theta_{i}\\left(u_{i i}-u_{i j(i)}\\right) \\varepsilon_{i}=\\sum_{i=1}^{I} \\theta_{i} \\eta=\\eta .\n$$\n\nThe seller can therefore increase the price of experiment $E$ by exactly $\\eta$ and leave the net utility of any truth-telling type $\\theta$ unchanged.\n\nNow consider any other type $\\theta^{\\prime}$ who chooses a different action $a_{j} \\neq a_{j(i)}$ after signal $s_{j(i)}$ from experiment $E$. The marginal benefit to type $\\theta^{\\prime}$ in state $\\omega_{i}$ from the shift in probability from $\\pi_{i j(i)}$ to $\\pi_{i i}$ is given by\n\n$$\nu_{i i}-u_{i j} \\leq u_{i i}-u_{i j(i)}\n$$\n\nby definition of action $j(i)$. Furthermore, if the discrete shift in probabilities causes type $\\theta^{\\prime}$ to change his action in response to a given signal, the average benefit of such a shift will be a convex combination of $u_{i i}-u_{i j}$ and $u_{i i}-u_{i j(i)}$. Therefore, the value of experiment $E^{\\prime}$ for type $\\theta^{\\prime}$ exceeds the value of $E$ by at most by $\\eta$.\n\nFinally, suppose the original experiment was intended for type $\\theta$, i.e., $E=E(\\theta)$. The direct mechanism $\\mathcal{M}^{\\prime}$, where $E^{\\prime}(\\theta)$ replaces $E(\\theta)$ and $t(\\theta)+\\eta$ replaces $t(\\theta)$, is individually rational and incentive compatible. Moreover, experiment $E^{\\prime}(\\theta)$ is non-dispersed by construction, and all transfers are weakly greater than in the original mechanism $\\mathcal{M}$.\n(3.) For some utility functions $u_{i j}$, multiple actions can be critical for any given state. The uniform-improvement procedure can then be applied to an experiment as long as it assigns positive probabilities to critical actions in every state. As a result, every optimal experiment will contain one row with a number of zero entries greater than or equal to $\\min _{i}\\left|\\arg \\min _{j} u_{i j}\\right|$. In the case of matching utility, $\\left|\\arg \\min _{j} u_{i j}\\right|=J-1$ for all $i$; hence, every optimal experiment is concentrated. ■","text_sha256":"a24c9bcf3827dd3c562681fa374dd86a8781f26f1125893d76ba07466f91fe9d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0036","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Appendix","text":"Proof of Proposition 3. (1.) We know from Proposition 2 that at least one type must buy the fully informative experiment $\\bar{E}$. Suppose only type $\\theta^{L}$ buys $\\bar{E}$ as part of the optimal menu. Then the price of $\\bar{E}$ is at most $V\\left(\\bar{E}, \\theta^{L}\\right)$. By incentive compatibility, if the high type $\\theta^{H}$ purchases $E \\neq \\bar{E}$, it must be that $t\\left(\\theta^{H}\\right)<V\\left(\\bar{E}, \\theta^{L}\\right)$. Therefore, eliminating experiment $E\\left(\\theta^{H}\\right)$ from the menu strictly improves the seller's profits, yielding a contradiction.\n(2.) The participation constraint of $\\theta^{L}$ must bind. Indeed, some participation constraint must bind, otherwise the seller could increase both prices. Suppose the constraint of $\\theta^{H}$ is\nbinding and that of $\\theta^{L}$ is not. Since $\\theta^{H}$ is served by $\\bar{E}$, then $t\\left(\\theta^{H}\\right)=V\\left(\\bar{E}, \\theta^{H}\\right) \\geq V\\left(\\bar{E}, \\theta^{L}\\right)$. Hence, we can increase the price $t\\left(\\theta^{L}\\right)$ without violating incentive compatibility.\n(3.) The incentive constraint of type $\\theta^{H}$ must bind. Suppose not and then consider two cases: if the participation constraint of type $\\theta^{H}$ does not bind, we can increase $t\\left(\\theta^{H}\\right)$; if the participation constraint of type $\\theta^{H}$ does bind, then it must be that $E\\left(\\theta^{L}\\right)$ is not equal to $\\bar{E}$. Since payoffs are continuous in $\\pi_{i j}$ we can increase both the informativeness of $E\\left(\\theta^{L}\\right)$ and the price $t\\left(\\theta^{L}\\right)$. ■\n\nProof of Proposition 4. (1.) Recall the definition $\\theta^{*}=u_{2} /\\left(u_{1}+u_{2}\\right)$, and consider the case of congruent priors $\\theta^{L}>\\theta^{H}>\\theta^{*}$. It follows from Proposition 9 that an optimal menu contains only the fully informative experiment $\\bar{E}$. The value of experiment $\\bar{E}$ is given by $\\left(1-\\theta^{H}\\right) u_{2}$ and $\\left(1-\\theta^{L}\\right) u_{2}$ for the high type and the low type, respectively. The profits from selling to one or both types are given by $\\gamma\\left(1-\\theta^{H}\\right) u_{2}$ and $\\left(1-\\theta^{L}\\right) u_{2}$, respectively. Thus, it is optimal to serve both types if and only if $\\gamma \\leq\\left(1-\\theta^{L}\\right) /\\left(1-\\theta^{H}\\right)$.\n(2.) Consider the case of noncongruent priors, $\\theta^{L}<\\theta^{*}<\\theta^{H}$. Let $q \\triangleq \\pi_{11} u_{1}-\\pi_{22} u_{2}$ and denote $q^{L}=q\\left(\\theta^{L}\\right), q^{H}=q\\left(\\theta^{H}\\right)$. It follows from Proposition 9 that in an optimal menu, we have $q^{L} \\leq q^{H}=u_{1}-u_{2}$. If $\\gamma<\\theta^{L} / \\theta^{H}$, we wish to show that flat pricing is optimal, i.e., $q^{L}=q^{H}=u_{1}-u_{2}$ and $t_{1}=t_{2}=\\left(1-\\theta^{L}\\right) u_{1}$. Fix an incentive-compatible menu $\\left(q^{H}, q^{L}, t^{H}, t^{L}\\right)$ with $q^{L}<u_{1}-u_{2}$ and define the following modification:\n\n$$\n\\left(q^{H \\prime}, t^{H \\prime}, q^{L \\prime}, t^{L \\prime}\\right)=\\left(q^{H}, t^{H}-\\varepsilon\\left(\\theta^{H}-\\theta^{L}\\right), q^{L}+\\varepsilon, t^{L}+\\varepsilon \\theta^{L}\\right) .\n$$\n\nIf $q^{L}<u_{1}-u_{2}$, modification (35) with $\\varepsilon \\in\\left(0, u_{1}-u_{2}-q_{L}\\right)$ preserves incentive compatibility. Furthermore, because $\\gamma<\\theta^{L} / \\theta^{H}$, this improves profits by at least\n\n$$\n\\varepsilon \\theta^{L}(1-\\gamma)-\\varepsilon\\left(\\theta^{H}-\\theta^{L}\\right) \\gamma>0,\n$$\n\nwhich yields a contradiction. Finally, we argue that if $\\gamma>\\theta^{L} / \\theta^{H}$, then discriminatory pricing is optimal. First, notice that the individual rationality of the high type must bind, i.e., $q^{H}=u_{1}-u_{2}$ and $t^{H}=\\left(1-\\theta^{H}\\right) u_{2}$; otherwise, modification (35) would be profitable for some $\\varepsilon<0$. Second, $q^{L}$ and $t^{L}$ maximize the payment of the low type, subject to his individual-rationality constraint and to the high type's incentive-compatibility constraint. At the optimum, both constraints bind, and the solution is given by\n\n$$\nq^{L}=\\frac{\\theta^{H} u_{1}-\\left(1-\\theta^{L}\\right) u_{2}}{\\theta^{H}-\\theta^{L}}, t^{L}=\\frac{\\left(\\theta^{H} u_{1}-\\left(1-\\theta^{H}\\right) u_{2}\\right) \\theta^{L}}{\\theta^{H}-\\theta^{L}} .\n$$\n\nSubstituting the definition of $q$ yields expression (20) in the text. ■\n\nProof of Proposition 5. If priors are noncongruent and the optimal menu is discriminatory, experiment $E\\left(\\theta^{L}\\right)$ has\n\n$$\n\\hat{\\pi}_{1}=\\frac{u_{1} \\theta^{H}-u_{2}\\left(1-\\theta^{H}\\right)}{u_{1}\\left(\\theta^{H}-\\theta^{L}\\right)} \\leq 1 .\n$$\n\nThe informativeness $\\pi_{1}$ is increasing in types: $\\partial \\hat{\\pi}_{1} / \\partial \\theta^{H} \\geq 0$ and $\\partial \\hat{\\pi}_{1} / \\partial \\theta^{L} \\geq 0$. The proposition then follows directly from the optimal menu characterization presented in the text.\n(1.) If priors are congruent then $\\pi_{1}=1$ for $\\gamma \\leq\\left(1-\\theta^{L}\\right) /\\left(1-\\theta^{H}\\right)$ and $\\pi_{1}=0$ otherwise. If priors are noncongruent then $\\pi_{1}=1$ for $\\gamma \\leq \\theta^{L} / \\theta^{H}$ and $\\pi_{1}=\\hat{\\pi}_{1}$ otherwise.\n(2.) If priors are congruent then $\\left|\\theta^{L}-\\theta^{*}\\right|=\\theta^{L}-\\theta^{*}$. The optimal menu has $\\pi_{1}=1$ if $\\theta^{L} \\leq 1-\\gamma\\left(1-\\theta^{H}\\right)$ and $\\pi_{1}=0$ otherwise. If priors are noncongruent then $\\left|\\theta^{L}-\\theta^{*}\\right|=\\theta^{*}-\\theta^{L}$. The optimal menu has $\\pi_{1}=0$ if $\\theta^{L}<\\gamma \\theta^{H}$ and $\\pi_{1}=\\hat{\\pi}_{1}$, which is increasing in $\\theta^{L}$, otherwise. (3.) By the normalization of a high type $\\left|\\theta^{H}-\\theta^{*}\\right|=\\theta^{H}-\\theta^{*}$. If priors are congruent then the optimal menu is $\\pi_{1}=0$ for $\\theta^{H}<1-\\left(1-\\theta^{L}\\right) / \\gamma$ and $\\pi_{1}=1$ otherwise. If priors are noncongruent and the menu is discriminatory, then $\\pi_{1}=\\hat{\\pi}_{1}$, which is increasing in $\\theta^{H}$. ■\n\nProof of Lemma 1. We begin with necessity. Consider a responsive menu $\\{q(\\theta)\\}_{\\theta \\in \\Theta}$ and any two types $\\theta_{2}>\\theta_{1}$ who follow the recommendations of signals in both experiments $q_{1}$ and $q_{2}$. We can then rewrite the net utility of experiment $q$ in (21) as\n\n$$\nV(q, \\theta)=\\theta q+u_{2}+\\min \\left\\{u_{1}-u_{2}-q, 0\\right\\}-\\max \\left\\{\\theta u_{1},(1-\\theta) u_{2}\\right\\} .\n$$\n\nBecause the menu is implementable, the incentive-compatibility constraints imply","text_sha256":"a9c55fd17f3354dfc7f335d8bd45cbab06e1c13a4515d3da68ac78bc396400aa"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0037","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Appendix","text":"$$\n\\begin{aligned}\nV\\left(q_{1}, \\theta_{1}\\right)-t_{1} & \\geq V\\left(q_{2}, \\theta_{1}\\right)-t_{2}, \\\\\nV\\left(q_{2}, \\theta_{2}\\right)-t_{2} & \\geq V\\left(q_{1}, \\theta_{2}\\right)-t_{1}, \\\\\nV\\left(q_{2}, \\theta_{2}\\right)-V\\left(q_{1}, \\theta_{2}\\right) & \\geq t_{2}-t_{1} \\geq V\\left(q_{2}, \\theta_{1}\\right)-V\\left(q_{1}, \\theta_{1}\\right) .\n\\end{aligned}\n$$\n\nThe strict single-crossing property of $V(q, \\theta)$ in (36) implies $q_{2} \\geq q_{1}$; hence, $q(\\theta)$ is increasing. Because the buyer's rent is differentiable with respect to $\\theta$ on $\\left[0, \\theta^{*}\\right]$ and $\\left[\\theta^{*}, 1\\right]$ respectively, we can compute the function $V(\\theta)$ on these two intervals separately. By inspection of (36), the rent is increasing on the former interval and decreasing on the latter. Furthermore, because the utility function $V(q, \\theta)$ is continuous in $\\theta$, the rent function $V$ is continuous by the Maximum Theorem. We thus obtain the expression in the text:\n\n$$\nV\\left(\\theta^{*}\\right)=V(0)+\\int_{0}^{\\theta^{*}} V_{\\theta}(q, \\theta) \\mathrm{d} \\theta=V(1)-\\int_{\\theta^{*}}^{1} V_{\\theta}(q, \\theta) \\mathrm{d} \\theta .\n$$\n\nBy the envelope theorem, $V_{\\theta}(q, \\theta)=q+u_{2}$ for $\\theta<\\theta^{*}$, and $V_{\\theta}(q, \\theta)=q-u_{1}$ for $\\theta>\\theta^{*}$. Finally, because $V(0)=V(1)$, we can simplify the equation above and obtain\n\n$$\n\\int_{0}^{1} q(\\theta) \\mathrm{d} \\theta=u_{1}-u_{2}\n$$\n\nWe now turn to sufficiency. Suppose the menu $\\{q(\\theta)\\}_{\\theta \\in \\Theta}$ is increasing and satisfies the integral constraint (24). Then, construct the following transfers:\n\n$$\nt(\\theta)= \\begin{cases}\\theta q(\\theta)+u_{2}+\\min \\left\\{u_{1}-u_{2}-q(\\theta), 0\\right\\}-\\int_{0}^{\\theta}\\left(q(x)+u_{2}\\right) \\mathrm{d} x-(1-\\theta) u_{2} & \\text { if } \\quad \\theta<\\theta^{*}, \\\\ \\theta q(\\theta)+u_{2}+\\min \\left\\{u_{1}-u_{2}-q(\\theta), 0\\right\\}+\\int_{\\theta}^{1}\\left(q(x)-u_{1}\\right) \\mathrm{d} x-\\theta u_{1} & \\text { if } \\quad \\theta \\geq \\theta^{*} .\\end{cases}\n$$\n\nBecause the menu satisfies the integral constraint, we have\n\n$$\n\\int_{0}^{\\theta} q(x) \\mathrm{d} x=u_{1}-u_{2}-\\int_{\\theta}^{1} q(x) \\mathrm{d} x,\n$$\n\nand we can express all transfers $t(\\theta)$ in (37) as\n\n$$\nt(\\theta)=\\theta q(\\theta)+\\min \\left\\{u_{1}-u_{2}-q(\\theta), 0\\right\\}-\\int_{0}^{\\theta} q(x) \\mathrm{d} x .\n$$\n\nUnder these transfers, the expected utility of type $\\theta$ from reporting type $\\theta^{\\prime}$ is given by\n\n$$\nV\\left(q\\left(\\theta^{\\prime}\\right), \\theta\\right)-t\\left(\\theta^{\\prime}\\right)=\\left(\\theta-\\theta^{\\prime}\\right) q\\left(\\theta^{\\prime}\\right)+\\int_{0}^{\\theta^{\\prime}} q(x) \\mathrm{d} x+u_{2}-\\max \\left\\{\\theta u_{1},(1-\\theta) u_{2}\\right\\}\n$$\n\nBecause $q$ is monotone, the expression on the right-hand side is maximized at $\\theta^{\\prime}=\\theta$, and hence, the incentive constraints are satisfied. Because the rent $V(\\theta) \\triangleq V(q(\\theta), \\theta)-t(\\theta)$ is non-negative for all $\\theta \\in[0,1]$, the participation constraints are also satisfied.\n\nFinally, note that the integral constraint (24) and the monotonicity condition imply the responsiveness condition (22). The set $Q(\\theta)$ of responsive experiments is described in Figure 9 below. Suppose to the contrary, that $q(\\theta) \\notin Q(\\theta)$ and, in particular, that $q(\\theta)=\\hat{q}<u_{1}-u_{2} / \\theta$ for some $\\theta>\\theta^{*}$. Then, by monotonicity, we would have\n\n$$\n\\int_{0}^{1} q(x) \\mathrm{d} x \\leq \\hat{q} \\theta+u_{1}(1-\\theta)<u_{1}-u_{2}\n$$\n\nwhich yields a contradiction. ■\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 9: Bounds on Responsive Experiments $Q(\\theta)$\n\nProof of Proposition 6. The Fundamental Theorem of Linear Programming applies to settings with $b \\in \\mathbb{R}^{m}$ and $c \\in \\mathbb{R}^{n}$, with $m<n$. Let $A$ be an $m \\times n$ matrix with the $m$ rows being linearly independent. If the linear problem\n\n$$\n\\begin{aligned}\n& \\max _{x \\in \\mathbb{R}^{n}} c \\cdot x \\\\\n& \\text { s.t. } A x=b \\text { and } x \\geq 0\n\\end{aligned}\n$$\n\nhas a solution, then it has a solution with all but $m$ entries being zero.\nIn this form, the Fundamental Theorem cannot be directly applied to the monopolist's problem (25): in this problem, (i) the number of choice variables is a continuum; (ii) the objective function is nonlinear; and (iii) monotonicity constraints are absent in the canonical representation of the theorem. To address (i), we discretize the state space into a fine grid with a radius $\\varepsilon,[0, \\varepsilon, 2 \\varepsilon, \\ldots, 1]$. To address (ii), we recall from Proposition 2 that an optimal menu contains the fully informative experiment. Hence, for any optimal menu $\\{q(\\theta)\\}_{\\theta \\in \\Theta}$, there is some type $\\theta^{I}$ such that $q\\left(\\theta^{I}\\right)=u_{1}-u_{2}$. Consequently, any optimal $q$ must solve the problem with the additional constraint $q\\left(\\theta^{I}\\right)=u_{1}-u_{2}$ for an appropriately chosen $\\theta^{I}$. Because the menu is monotone, the objective can be written linearly with $\\min \\left\\{u_{1}-u_{2}-q(\\theta), 0\\right\\}=0$ for all $\\theta<\\theta^{I}$ and $\\min \\left\\{u_{1}-u_{2}-q(\\theta), 0\\right\\}=u_{1}-u_{2}-q(\\theta)$ for all $\\theta>\\theta^{I}$. Finally, to address (iii), we change variables from $q(\\theta)$ to its increment $\\hat{q}(\\theta) \\triangleq q(\\theta)-q(\\theta-\\varepsilon)$, where $\\hat{q}(0) \\triangleq 0$. Finally, we substitute $q(\\theta)=-u_{2}+\\sum_{x=0}^{\\theta} \\hat{q}(x)$ and rewrite the monotonicity constraint as $\\hat{q}(\\theta) \\geq 0$.\n\nThe monopolist's problem (25) can be restated as\n\n$$\n\\begin{aligned}\n& \\max _{\\hat{q}(\\theta)} \\sum_{\\theta=0}^{1} \\hat{q}(\\theta)\\left(\\sum_{x=\\theta}^{1}(x f(x)+F(x))-\\sum_{x=\\theta^{I}}^{1} f(x)\\right) \\\\\n& \\text { s.t. } \\sum_{\\theta=0}^{\\theta^{I}} \\hat{q}(\\theta)=u_{1}, \\\\\n& \\quad \\sum_{\\theta=\\theta^{I}}^{1} \\hat{q}(\\theta)=u_{2}, \\\\\n& \\quad \\sum_{\\theta=0}^{1}\\left(-u_{2}+\\sum_{x=0}^{\\theta} \\hat{q}(x)\\right)=u_{1}-u_{2}, \\\\\n& \\quad \\hat{q}(\\theta) \\geq 0 .\n\\end{aligned}\n$$","text_sha256":"75d7d6f6d735b05fa920b8d89aaf022308b559dcdd3862fd2714f6e07c18b2e8"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0038","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Appendix","text":"This is a canonical linear programming problem with three linearly independent equality constraints. The first two correspond to the total change from the experiment $q(0)=-u_{2}$ to the fully informative experiment $q\\left(\\theta^{I}\\right)=u_{1}-u_{2}$ to $q(1)=u_{1}$. The last equality is integral constraint (24). Hence, by the Fundamental Theorem, for any $\\theta^{I}$ there is an optimal menu of experiments with at most three positive entries. Since $q(1)=u_{1}$ corresponds to an uninformative experiment, this implies there are at most two informative experiments in any optimal menu.\n\nFinally, consider an arbitrary discrete distribution that converges in distribution to a continuous distribution. The profits converge for any fixed menu, and hence, optimal profits converge as well. Furthermore, the set of profits that can be achieved by two-item menus is compact. Therefore, there exists an optimal menu with continuous types that has at most two items. ■\n\nProof of Proposition 7. We first derive the seller's objective in the usual way. Using (38) to write the expected transfers and integrating by parts, we obtain\n\n$$\n\\int_{0}^{1} t(\\theta) \\mathrm{d} F(\\theta)=\\int_{0}^{1}\\left(\\theta q(\\theta)+\\min \\left\\{u_{1}-u_{2}-q(\\theta), 0\\right\\}-\\frac{1-F(\\theta)}{f(\\theta)} q(\\theta)\\right) \\mathrm{d} F(\\theta) .\n$$\n\nUsing the integral constraint, we obtain (up to an additive constant)\n\n$$\n\\int_{0}^{1} t(\\theta) \\mathrm{d} F(\\theta)=\\int_{0}^{1}\\left[(\\theta f(\\theta)+F(\\theta)) q(\\theta)+\\min \\left\\{u_{1}-u_{2}-q(\\theta), 0\\right\\} f(\\theta)\\right] \\mathrm{d} \\theta .\n$$\n\nWe now establish that the solution to the seller's problem (25) can be characterized through Lagrangian methods. For necessity, note that the objective is concave in the experiment; the set of non-decreasing functions is convex, and the integral constraint can be\nweakened to the real-valued inequality constraint\n\n$$\n\\int_{0}^{1} q(\\theta) \\mathrm{d} \\theta \\leq u_{1}-u_{2} .\n$$\n\nNecessity of the Lagrangian then follows from Theorem 8.3.1 in Luenberger (1969). Sufficiency follows from Theorem 8.4.1 in Luenberger (1969). In particular, any solution maximizer of the Lagrangian $q(\\theta)$ with\n\n$$\n\\int_{0}^{1} q(\\theta) \\mathrm{d} \\theta=\\bar{q}\n$$\n\nmaximizes the original objective subject to the inequality constraint\n\n$$\n\\int_{0}^{1} q(\\theta) \\mathrm{d} \\theta \\leq \\bar{q} .\n$$\n\nThus, any solution to the Lagrangian that satisfies the constraint solves the original problem.\nBecause the Lagrangian approach is valid, we apply the results of Toikka (2011) to solve the seller's problem for a given value of the multiplier $\\lambda$ on the integral constraint. Write the Lagrangian as\n\n$$\n\\int_{0}^{1}\\left[(\\theta f(\\theta)+F(\\theta)) q(\\theta)+\\min \\left\\{u_{1}-u_{2}-q(\\theta), 0\\right\\} f(\\theta)+\\lambda\\left(u_{1}-u_{2}-q(\\theta)\\right)\\right] \\mathrm{d} \\theta .\n$$\n\nIn order to maximize the Lagrangian subject to the monotonicity constraint, consider the generalized virtual surplus\n\n$$\n\\Phi(\\theta, q):=\\int_{-u_{2}}^{q}\\left(\\bar{\\phi}(\\theta, x)-\\lambda^{*}\\right) \\mathrm{d} x\n$$\n\nwhere $\\bar{\\phi}(\\theta, x)$ denotes the ironed virtual value for experiment $x$. (Up to a constant, $\\Phi(\\theta, q)$ is a general formulation of the virtual value in the statement of the proposition.) Note that the virtual surplus $\\Phi(\\theta, q)$ is weakly concave in $q$. Because the multiplier $\\lambda$ shifts all virtual values by a constant, the result in Proposition 7 follows from Theorem 4.4 in Toikka (2011). Finally, note that the optimal value of $\\lambda^{*}$ is strictly positive. If not, the pointwise maximizer of $\\bar{\\phi}(\\theta, q)$ would be weakly above $u_{1}-u_{2}$ and strictly so for some $\\theta$, leading to a violation of the integral constraint. Therefore, the inequality constraint (39) must bind. ■\n\nProof of Proposition 9. We know from Proposition 3 that the high type $\\theta^{H}$ purchases the fully informative experiment. We now derive the optimal experiment $E\\left(\\theta^{L}\\right)$. Suppose (as we later verify) that both types $\\theta^{H}$ and $\\theta^{L}$ choose action $a_{i}$ after observing signal $s_{i}$ from\nexperiment $E\\left(\\theta^{L}\\right)$. The seller's relaxed problem can be written as\n\n$$\n\\begin{aligned}\n\\max _{0 \\leq \\pi_{i i} \\leq 1, V\\left(\\theta^{H}\\right)} & (1-\\gamma) \\sum_{i=1}^{I} \\theta_{i}^{L} u_{i} \\pi_{i i}-\\gamma V\\left(\\theta^{H}\\right), \\\\\n& \\text { s.t. } V\\left(\\theta^{H}\\right) \\geq \\sum_{i=1}^{I} \\pi_{i i} u_{i}\\left(\\theta_{i}^{H}-\\theta_{i}^{L}\\right)-\\max _{i} \\theta_{i}^{H} u_{i}+\\max _{i} \\theta_{i}^{L} u_{i} \\geq 0 .\n\\end{aligned}\n$$\n\nwhere the latter inequality ensures that the high type $\\theta^{H}$ achieves a non-negative payoff when misreporting his type and following every signal's recommendation.\n\nWe now simplify the problem as follows. By Proposition 3, the incentive-compatibility constraint of type $\\theta^{H}$ binds and the relaxed problem can be rewritten as\n\n$$\n\\begin{aligned}\n& \\max _{0 \\leq \\pi_{i i} \\leq 1} \\sum_{i=1}^{I} \\pi_{i i} u_{i}\\left(\\theta_{i}^{L}-\\gamma \\theta_{i}^{H}\\right) \\\\\n& \\quad \\text { s.t. } \\sum_{i=1}^{I} \\pi_{i i} u_{i}\\left(\\theta_{i}^{H}-\\theta_{i}^{L}\\right)-\\max _{i}\\left\\{\\theta_{i}^{H} u_{i}\\right\\}+\\max _{i}\\left\\{\\theta_{i}^{L} u_{i}\\right\\} \\geq 0\n\\end{aligned}\n$$\n\nNow, arrange the states such that\n\n$$\n\\frac{\\theta_{1}^{L}}{\\theta_{1}^{H}} \\leq \\cdots \\leq \\frac{\\theta_{I}^{L}}{\\theta_{I}^{H}},\n$$\n\nand consider the function $R(i)$ defined in the text\n\n$$\nR(j) \\triangleq \\sum_{i=j}^{I} u_{i}\\left(\\theta_{i}^{H}-\\theta_{i}^{L}\\right)-\\max _{i}\\left\\{\\theta_{i}^{H} u_{i}\\right\\}+\\max _{i}\\left\\{\\theta_{i}^{L} u_{i}\\right\\} .\n$$\n\nThe function $R(j)$ is decreasing in $j$ for $j<i_{c}$ and increasing in $j$ for $j>i_{c}$, where\n\n$$\ni_{c} \\triangleq \\min \\left\\{i: 1 \\leq \\theta_{i}^{L} / \\theta_{i}^{H}\\right\\} .\n$$\n\nWe now show that $\\min _{j} R(j)=R\\left(i_{c}\\right)<0$. This is immediate if $\\max _{i}\\left\\{\\theta_{i}^{H} u_{i}\\right\\} \\geq \\max _{i}\\left\\{\\theta_{i}^{L} u_{i}\\right\\}$, as we have $\\sum_{i=i_{c}}^{I} u_{i}\\left(\\theta_{i}^{H}-\\theta_{i}^{L}\\right)<0$ by construction. Conversely, if $\\max _{i}\\left\\{\\theta_{i}^{H} u_{i}\\right\\}<\\max _{i}\\left\\{\\theta_{i}^{L} u_{i}\\right\\}$, then we let $i_{L} \\triangleq \\arg \\max _{i}\\left\\{\\theta_{i}^{L} u_{i}\\right\\}$, and we derive the following bound:","text_sha256":"352880d67b7b47e7e1489d3ae501dfdffece602306941864fc5707e25b149ee1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0039","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"A Appendix","text":"$$\nR\\left(i_{c}\\right) \\leq R\\left(i_{L}\\right) \\leq \\sum_{i=i_{L}}^{I} u_{i}\\left(\\theta_{i}^{H}-\\theta_{i}^{L}\\right)-\\theta_{i_{L}}^{H} u_{i_{L}}+\\theta_{i_{L}}^{L} u_{i_{L}}=\\sum_{i=i_{L}+1}^{I} u_{i}\\left(\\theta_{i}^{H}-\\theta_{i}^{L}\\right)<0\n$$\n\nwhere the latter inequality follows from the fact that $\\max _{i}\\left\\{\\theta_{i}^{H} u_{i}\\right\\}<\\max _{i}\\left\\{\\theta_{i}^{L} u_{i}\\right\\}$ implies $\\theta_{i_{L}}^{L}>\\theta_{i_{L}}^{H}$, which means $i_{L} \\geq i_{c}$.\n\nWe now turn to the solution to the seller's problem. Notice that (40)-(41) is a linear problem that attains a monotone solution. In particular, any solution must be a monotone sequence: $\\pi_{i i}=0$ for $i<\\hat{\\imath}$ and $\\pi_{i i}=1$ for $i>\\hat{\\imath}$, for some $\\hat{\\imath} \\in\\{1, \\ldots, I\\}$. (This structure can\nbe seen immediately from the Lagrange multiplier method, as the solution must maximize some linear combination of the objective function and the constraint.)\n\nSuppose that the constraint does not bind at the optimum. Then, the solution assigns $\\pi_{i i}=1$ if and only if $i \\geq i_{s} \\triangleq \\min \\left\\{i: \\gamma \\leq \\theta_{i}^{L} / \\theta_{i}^{H}\\right\\}$. Note that $i_{s} \\leq i_{c}$, which was defined in (42). Therefore, if $R\\left(i_{s}\\right) \\geq 0$ the constraint is satisfied at the unconstrained optimum. If, instead, we have $R\\left(i_{s}\\right)<0$, then by the definition of the critical state $i_{b}$,\n\n$$\nR\\left(i_{b}\\right)>0>R\\left(i_{b}+1\\right),\n$$\n\nwe have $i_{b}<i_{s}$. In this case, the solution is given by $\\pi_{i i}=0$ for $i<i_{b}, \\pi_{i i}=1$ for $i>i_{b}$ and by setting $\\pi_{i_{b} i_{b}}$ to satisfy (41) with equality, i.e.,\n\n$$\n\\pi_{i_{b} i_{b}}=\\frac{\\sum_{j=i_{b}+1}^{I}\\left(\\theta_{j}^{L}-\\theta_{j}^{H}\\right) u_{j}-\\max _{j} \\theta_{j}^{L} u_{j}+\\max _{j} \\theta_{j}^{H} u_{j}}{\\left(\\theta_{i_{b}}^{H}-\\theta_{i_{b}}^{L}\\right) u_{i_{b}}} .\n$$\n\nCombining these two arguments, we conclude that the optimal cutoff state $\\hat{\\imath}$ is given by $i^{*}=\\min \\left\\{i_{b}, i_{s}\\right\\}$ and that $\\pi_{i^{*} i^{*}} \\in\\left\\{\\pi_{i_{b} i_{b}}, 1\\right\\}$ depending on whether $i_{b} \\lessgtr i_{s}$.\n\nTo complete the menu, we now need to specify the off-diagonal entries $\\pi_{i j}$ to ensure both types $\\theta^{H}$ and $\\theta^{L}$ choose action $a_{i}$ when observing signal $s_{i}$. This requires\n\n$$\n\\pi_{i i} \\theta_{i} u_{i} \\geq \\pi_{j i} \\theta_{j} u_{j}\n$$\n\nfor both types and for all $j<i$, because the signal matrix can be taken to be lower triangular. In particular, we need to ensure that, for all $j<i^{*}$,\n\n$$\n\\begin{aligned}\n& \\pi_{i^{*} i^{*}} u_{i^{*}} \\theta i_{i^{*}}^{H} \\geq \\pi_{j i^{*}} u_{j} \\theta_{j}^{H} \\\\\n& \\pi_{i^{*} i^{*}} u_{i^{*}} \\theta_{i^{*}}^{L} \\geq \\pi_{j i^{*}} u_{j} \\theta_{j}^{L}\n\\end{aligned}\n$$\n\nBecause $\\theta_{j}^{L} / \\theta_{j}^{H} \\leq \\theta_{i}^{L} / \\theta_{i}^{H}$ for $j<i$, it suffices to satisfy the constraint of type $\\theta^{H}$.\nWe proceed as follows. Fix an alternative action $a_{j}$ with $j<i^{*}$. For any $i>i^{*}$, we make type $\\theta^{H}$ indifferent between following the recommendation of signal $i$ and choosing action $a_{j}$; we do so beginning with $\\pi_{j I}$ and proceeding backward as long as required. If this procedure assigns positive weight to $\\pi_{j i^{*}}$, then it must be that\n\n$$\n\\pi_{j i^{*}}=1-\\sum_{i=i^{*}+1}^{I} \\frac{\\theta_{i}^{H} u_{i}}{\\theta_{j}^{H} u_{j}} .\n$$\n\nWe argue that type $\\theta^{H}$ has strict incentives to follow the recommendation of signal $i^{*}$ when\n$\\pi_{i^{*} i^{*}}=\\pi_{i^{*}}^{B}$. (A fortiori, type $\\theta^{H}$ has strict incentives to choose action $a_{i}$ following any signal $s_{i}$ with $i \\geq i^{*}$ if $\\pi_{i i}=1$.) Recall the definition\n\n$$\n\\pi_{i^{*} i^{*}} \\theta_{i^{*}}^{H} u_{i^{*}}=\\frac{\\sum_{i=i^{*}+1}^{I} u_{i}\\left(\\theta_{i}^{L}-\\theta_{i}^{H}\\right)-\\max _{i} \\theta_{i}^{L} u_{i}+\\max _{i} \\theta_{i}^{H} u_{i}}{\\theta_{i^{*}}^{H}-\\theta_{i^{*}}^{L}} \\theta_{i^{*}}^{H} .\n$$\n\nLet $i_{L}=\\arg \\max _{i} \\theta_{i}^{L} u_{i}$ and $i_{H}=\\arg \\max _{i} \\theta_{i}^{H} u_{i}$, and consider the following two cases.\n(1.) If $i_{L}>i^{*}$, then we know that\n\n$$\n\\sum_{i=i^{*}+1}^{I} \\theta_{i}^{L} u_{i}>\\max _{i} \\theta_{i}^{L} u_{i},\n$$\n\nand we bound $\\pi_{i^{*} i^{*}} \\theta_{i^{*}}^{H} u_{i^{*}}$ by\n\n$$\n\\pi_{i^{*} i^{*}} \\theta_{i^{*}}^{H} u_{i^{*}}>\\frac{\\theta_{i^{*}}^{H}}{\\theta_{i^{*}}^{H}-\\theta_{i^{*}}^{L}}\\left(\\theta_{j}^{H} u_{j}-\\Sigma_{i=i^{*}+1}^{I} \\theta_{i}^{H} u_{i}\\right)>\\theta_{j}^{H} u_{j}-\\Sigma_{i=i^{*}+1}^{I} \\theta_{i}^{H} u_{i}=\\pi_{j i^{*}} \\theta_{j}^{H} u_{j} .\n$$\n\n(2.) If $i_{L} \\leq i^{*}$, then from (43) and (44), we know that the difference $\\pi_{i^{*} i^{*}} \\theta_{i^{*}}^{H} u_{i^{*}}-\\pi_{j i^{*}} \\theta_{j}^{H} u_{j}$ is proportional to\n\n$$\n\\begin{aligned}\n& \\sum_{i=i^{*}+1}^{I} u_{i}\\left(\\theta_{i}^{L}-\\theta_{i}^{H}\\right)-\\max _{i} \\theta_{i}^{L} u_{i}+\\max _{i} \\theta_{i}^{H} u_{i}-\\left(1-\\frac{\\theta_{i^{*}}^{L}}{\\theta_{i^{*}}^{H}}\\right)\\left(\\theta_{j}^{H} u_{j}-\\Sigma_{i=i^{*}+1}^{I} \\theta_{i}^{H} u_{i}\\right) \\\\\n= & \\sum_{i=i^{*}+1}^{I} u_{i}\\left(\\theta_{i}^{L}-\\frac{\\theta_{i^{*}}^{L}}{\\theta_{i^{*}}^{H}} \\theta_{i}^{H}\\right)-\\max _{i} \\theta_{i}^{L} u_{i}+\\max _{i} \\theta_{i}^{H} u_{i}-\\left(1-\\frac{\\theta_{i^{*}}^{L}}{\\theta_{i^{*}}^{H}}\\right) \\theta_{j}^{H} u_{j} .\n\\end{aligned}\n$$\n\nNotice that every term in the sum is positive because the likelihood ratio is increasing in $i$. The remaining terms can be written as\n\n$$\n\\begin{aligned}\n-\\max _{i} \\theta_{i}^{L} u_{i}+\\max _{i} \\theta_{i}^{H} u_{i}-\\left(1-\\frac{\\theta_{i^{*}}^{L}}{\\theta_{i^{*}}^{H}}\\right) \\theta_{j}^{H} u_{j} & =-\\frac{\\theta_{i_{L}}^{L}}{\\theta_{i_{L}}^{H}} \\theta_{i_{L}}^{H} u_{i_{L}}+\\theta_{i_{H}}^{H} u_{i_{H}}-\\left(1-\\frac{\\theta_{i^{*}}^{L}}{\\theta_{i^{*}}^{H}}\\right) \\theta_{j}^{H} u_{j} \\\\\n& \\geq \\theta_{i_{H}}^{H} u_{i_{H}}\\left(1-\\frac{\\theta_{i_{L}}^{L}}{\\theta_{i_{L}}^{H}}\\right)-\\left(1-\\frac{\\theta_{i^{*}}^{L}}{\\theta_{i^{*}}^{H}}\\right) \\theta_{j}^{H} u_{j} .\n\\end{aligned}\n$$\n\nHere, the likelihood ratios are ranked (because $i_{L} \\leq i^{*}$ in this case), and $\\theta_{i_{H}}^{H} u_{i_{H}} \\geq \\theta_{j}^{H} u_{j}$ by the definition of $i_{H}$. ■","text_sha256":"b6f85da2a25992a397ea76e7f9c990390023f40ed43d601fe8c8c58a844902d0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0040","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B Supplemental (Online) Appendix","text":"## B Supplemental (Online) Appendix","text_sha256":"7fb4ccebf1617c035a8dbd2dd9651eb4a6c50d16144ed9810c471f0fb09a2130"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0041","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B. 1 Many Actions and States","text":"## B. 1 Many Actions and States\n\nWe illustrate the construction of the optimal experiment as determined by Proposition 8 and the implications for information rents in the following example.\n\nExample 1 (Noncongruent Types) Consider uniform match values ( $u_{i}=1$ for all $i=$ 1,2,3) and two types, $\\theta^{L}=(1 / 10,1 / 10,8 / 10)$ and $\\theta^{H}=(4 / 10,3 / 10,3 / 10)$. These types are noncongruent: without additional information, $\\theta^{H}$ would choose action $a_{1}$ and $\\theta^{L}$ would choose $a_{3}$. The likelihood ratios $\\theta_{i}^{L} / \\theta_{i}^{H}$ are (1/4, 1/3, 8/3). This implies $i_{b}=2$, whereas $i_{s} \\in\\{1,2\\}$ depending on the prior probability $\\gamma$ of the high type. For $\\gamma \\in[0,1 / 4]$ and $\\gamma \\in[1 / 4,1 / 3]$, Proposition 9,2 applies, and the high type obtains positive rents. Furthermore, for $\\gamma \\geq 1 / 4$, the partially informative experiment $E\\left(\\theta^{L}\\right)$ involves dropping signal $s_{1}$. The optimal experiment $E\\left(\\theta^{L}\\right)$ as a function of $\\gamma$ is given by\n\n| $E\\left(\\theta^{L}\\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | 1 | 0 | 0 |\n| $\\omega_{2}$ | 0 | 1 | 0 |\n| $\\omega_{3}$ | 0 | 0 | 1 |\n|  | if | < | 1/4, |\n\n| $E\\left(\\theta^{L}\\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | 0 | 1/4 | 3/4 |\n| $\\omega_{2}$ | 0 | 1 | 0 |\n| $\\omega_{3}$ | 0 | 0 | 1 |\n|  | if | $\\in[1$ | 1/4, 1/3], |\n\n| $E\\left(\\theta^{L}\\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | 0 | 1/4 | 3/4 |\n| $\\omega_{2}$ | 0 | 1/2 | 1/2 |\n| $\\omega_{3}$ | 0 | 0 | 1 |\n|  | if | > 1 | 1/3. |\n\nExample 2 illustrates how congruent, but not strongly congruent beliefs, allow for surplus extraction. In the example, the two types deem state $\\omega_{2}$ the most likely. Thus the types are congruent but not strongly congruent, as they disagree on the relative likelihood of states $\\omega_{1}$ and $\\omega_{3}$.\n\nExample 2 (Congruent Priors) Consider uniform match values ( $u_{i}=1$ for all $i=$ $1,2,3)$ and two types, $\\theta^{L}=(5 / 10,1 / 10,4 / 10)$ and $\\theta^{H}=(4 / 10,3 / 10,3 / 10)$. Because $i_{b}=1$, the optimal experiment $E\\left(\\theta^{L}\\right)$ as a function of $\\gamma$ is given by\n\n| $E\\left(\\theta^{L}\\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | 1 | 0 | 0 |\n| $\\omega_{2}$ | 0 | 1 | 0 |\n| $\\omega_{3}$ | 0 | 0 | 1 |\n|  | if | $\\leq$ | 1/3, |\n\n| $E\\left(\\theta^{L}\\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | 1 | 0 | 0 |\n| $\\omega_{2}$ | 0 | 1/2 | 1/2 |\n| $\\omega_{3}$ | 0 | 0 | 1 |\n|  | if | > 1 | 1/3, |\n\nand the high type obtains positive rents only if $\\gamma<1 / 3$.\n\nIn Example 3 we illustrate that the relaxed approach is not valid with many types. In the example below with three types, no experiment $E\\left(\\theta^{1}\\right)$ can lead both types $\\theta^{2}$ and $\\theta^{3}$ to follow the action recommended by every signal. Thus, the profits in the relaxed problem are strictly greater than those in original problem.\n\nExample 3 (Many Types and Actions) Consider uniform match values ( $u_{i}=1$ for all $i=1,2,3)$ and three types, $\\theta^{1}=(1 / 6,1 / 6,4 / 6), \\theta^{2}=(1 / 2,1 / 2,0)$, and $\\theta^{3}=(1 / 2,0,1 / 2)$, which are all equally likely. In the relaxed problem, the monopolist sells the fully informative experiment to types $\\theta^{2}$ and $\\theta^{3}$. Type $\\theta^{1}$ is offered the partially informative experiment\n\n| $E\\left(\\theta^{1}\\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | 1/2 | 0 | 1/2 |\n| $\\omega_{2}$ | 0 | 1 | 0 |\n| $\\omega_{3}$ | 0 | 0 | 1 |\n\nand the seller's revenues are equal to 5/12. However, if type $\\theta^{2}$ purchased experiment $E\\left(\\theta^{1}\\right)$, he would choose action $a_{1}$ when observing signal $s_{3}$. In the solution to the full problem, which we can construct by a guess-and-verify approach, the optimal experiment $E\\left(\\theta^{1}\\right)$ consists of\n\n| $E\\left(\\theta^{1}\\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | 1/2 | 0 | 1/2 |\n| $\\omega_{2}$ | 1/2 | 1/2 | 0 |\n| $\\omega_{3}$ | 0 | 0 | 1 |\n\nwhich yields revenues of 1/3, i.e., revenues are strictly lower in the relaxed program.","text_sha256":"f317831a65a052000538fc5b08b91e67c233088776f0a4b135ce07820051f135"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0042","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B. 2 More Actions than States","text":"## B. 2 More Actions than States\n\nWe consider a setting with two types, and we relax the assumption of matching state-action payoffs. In particular, we consider the following example with two types, two states, and three actions. The data buyer's payoff is given by\n\n| $u(\\omega, a)$ | $a_{1}$ | $a_{2}$ | $a_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | 1 | 0 | 4/5 |\n| $\\omega_{2}$ | 0 | 1 | 4/5 |\n\nThus, action $a_{i}$ is the optimal action in state $\\omega_{i}$, but action $a_{3}$ provides a lower bound on the payoffs that is uniform across states-an insurance action. Let the two types be given by $\\theta^{L}=(1 / 10,9 / 10)$ and $\\theta^{H}=(6 / 10,4 / 10)$. As the results of Proposition 3 do not rely\non matching payoffs, we know type $\\theta^{H}$ receives full information, $E\\left(\\theta^{H}\\right)=\\bar{E}$, his incentive constraint binds, and the participation constraint of type $\\theta^{L}$ binds.\n\nFor the case $\\gamma \\triangleq \\operatorname{Pr}\\left(\\theta=\\theta^{H}\\right)=3 / 4$, an optimal menu contains the experiment\n\n| $E\\left(\\theta^{L}\\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |\n| :--- | :--- | :--- | :--- |\n| $\\omega_{1}$ | 1/3 | 2/15 | 8/15 |\n| $\\omega_{2}$ | 0 | 4/5 | 1/5 |\n\nat a price $t\\left(\\theta^{L}\\right)=1 / 25$ and the experiment $\\bar{E}$ at a price $t\\left(\\theta^{H}\\right)=1 / 5$. The optimal menu is illustrated in Figure 10.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFigure 10: Optimal Menu\n\nThis menu has the following notable properties: (i) the seller extracts all the surplus from both types; (ii) type $\\theta^{L}$ follows the recommendation of every signal in $E\\left(\\theta^{L}\\right)$; (iii) type $\\theta^{H}$, if purchasing experiment $E\\left(\\theta^{L}\\right)$, is indifferent between action $a_{2}$ and action $a_{3}$ when observing signal $s_{2}$ as well as between $a_{1}$ and $a_{3}$ when observing signal $s_{3}$; and (iv) the optimal profits are strictly lower than those in the relaxed problem.\n\nIndeed, ignoring the off-path obedience constraints, the optimal menu is discriminatory, and the seller extracts all the rents by offering the experiment\n\n| $E\\left(\\theta^{L}\\right)$ | $s_{1}$ | $s_{2}$ |\n| :--- | :--- | :--- |\n| $\\omega_{1}$ | 2/5 | 3/5 |\n| $\\omega_{2}$ | 0 | 1 |\n\nat a price $t\\left(\\theta^{L}\\right)=1 / 25$ and the experiment $\\bar{E}$ at a price $t\\left(\\theta^{H}\\right)=2 / 5$.\nIn the full problem, the seller cannot turn experiment $E\\left(\\theta^{L}\\right)$ in (46) into the more informative one in (47). If she did, buyer type $\\theta^{H}$ would choose action $a_{3}$ after deviating\nand observing signal $s_{2}$. ${ }^{27}$ In other words, the seller extracts the surplus, but at the cost of additional distortion-notably, there is no \"1\" entry in (46).\n\nInterestingly, action $a_{3}$ may not be induced in an optimal menu yet still restrict the seller. Indeed, if we modify the above example by setting $\\gamma=2 / 3$, an optimal menu can be calculated to contain the experiment\n\n| $E\\left(\\theta^{L}\\right)$ | $s_{1}$ | $s_{2}$ |\n| :--- | :--- | :--- |\n| $\\omega_{1}$ | 5/6 | 1/6 |\n| $\\omega_{2}$ | 0 | 1 |\n\nat a price $t\\left(\\theta^{L}\\right)=1 / 12$ and the experiment $\\bar{E}$ at a price of $t\\left(\\theta^{H}\\right)=11 / 60<1 / 5=V\\left(\\theta^{H}, \\bar{E}\\right)$. As before, type $\\theta^{H}$ is indifferent between $a_{2}$ and $a_{3}$ after deviating to $E\\left(\\theta^{L}\\right)$ and observing $s_{2}$. Contrary to our earlier examples, the high type makes positive rents despite the seller's discriminatory menu offering.\n\nTo emphasize, action $a_{3}$ is not chosen in an optimal menu by either type, yet it looms large and prevents the seller from extracting the full surplus. In particular, the seller would like to reduce $\\pi_{11}$ in order to relax the high type's incentive constraint and increase $t^{H}$. However, by doing so she would induce type $\\theta^{H}$ to choose action $a_{3}$ after $s_{2}$. This means that the high type's marginal benefit from a probability shift from $\\pi_{11}$ to $\\pi_{12}$ is $\\theta_{1}^{H}(-1+4 / 5)=-3 / 25$, while the corresponding marginal change in price $t^{L}$ is $-\\theta_{1}^{L}=-1 / 10$. Therefore, $t^{H}$ can only increase at rate 1/50, which is not profitable for the seller when the fraction of high types is $\\gamma=2 / 3$. If both types were instead required to follow the signals' recommendations, the high type's misreporting value would change at rate $-\\theta_{1}^{H}=-3 / 5$, allowing the seller to increase $t^{H}$ at the profitable rate of 1/2. The kink in the \"exchange rate\" when the high type is indifferent among several actions prevents the seller from making the modification.\n\nTo reinforce the point, maintain the assumption $\\gamma=2 / 3$ but exclude action $a_{3}$ from the set of available actions. The optimal menu is again given by the experiment in (47), which is now less informative than (48), but allows the seller to extract all the rents.","text_sha256":"c544ed4ffbfa527f956c601864002c270e6360ba352a61f0eedad7a1a62b511f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0043","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"B. 3 Sequential Design","text":"## B. 3 Sequential Design\n\nWe show that sequential design of experiments can increase the seller's revenues in our leading binary-type example. We focus on the simplest instance of a dynamic protocol, whereby the seller first releases a free informative experiment to the buyer and then, without observing\n\n[^17]the realized signal, offers a menu of (experiment, price) pairs from which to choose.\nLet $\\Omega=\\left\\{\\omega_{1}, \\omega_{2}\\right\\}, A=\\left\\{a_{1}, a_{2}\\right\\}$, and assume uniform match values, i.e.,\n$$\nu\\left(\\omega_{i}, a_{j}\\right)=\\mathbb{I}_{[i=j]} .\n$$\nConsider two equally likely types with interim beliefs $\\theta^{L}=1 / 8$ and $\\theta^{H}=1 / 4$,respectively, where $\\theta \\triangleq \\operatorname{Pr}\\left[\\omega_{1}\\right]$.\n\nBecause the two types are congruent, an optimal static mechanism (Proposition 4) contains only the fully informative experiment. In the current example, the seller is indifferent between charging prices $t=1 / 8$ and $t=1 / 4$. In either case, the monopoly profits are\n\n$$\n\\pi_{\\text {static }}^{*}=1 / 8 .\n$$\n\nConsider the following sequential scheme. First, the seller reveals an outcome of the following experiment $E_{0}$ at no cost to the buyer\n\n| $E_{0}$ | $s_{1}$ | $s_{2}$ |\n| :--- | :--- | :--- |\n| $\\omega_{1}$ | 1 | 0 |\n| $\\omega_{2}$ | 1/3 | 2/3 |\n\nAfter observing signal $s_{2}$, the buyer is convinced that the state is $\\omega=\\omega_{2}$, which confirms his prior, and does not buy further information. After realization $s_{1}$, however, the buyer's beliefs are updated to\n\n$$\n\\theta^{L}\\left(s_{1}\\right)=3 / 10, \\quad \\theta^{H}\\left(s_{1}\\right)=1 / 2 .\n$$\n\nAt this point, the seller offers the fully informative experiment at a price $\\bar{t}=3 / 10$.\nThe key observation is that signal $s_{1}$ under experiment $E_{0}$ is more likely be realized for the high type $\\theta^{H}=1 / 4$ than for the low type $\\theta^{L}=1 / 8$. In particular, the signal distribution is given by\n\n$$\n\\operatorname{Pr}\\left[s_{1} \\mid \\theta^{L}\\right]=\\frac{5}{12}, \\operatorname{Pr}\\left[s_{1} \\mid \\theta^{H}\\right]=\\frac{1}{2} .\n$$\n\nAs a consequence, the monopolist's profit is given by\n\n$$\n\\pi_{\\mathrm{dyn}}^{*} \\triangleq \\bar{t}\\left(\\gamma \\operatorname{Pr}\\left[s_{1} \\mid \\theta^{H}\\right]+(1-\\gamma) \\operatorname{Pr}\\left[s_{1} \\mid \\theta^{L}\\right]\\right)=11 / 80 .\n$$\n\nThus, the sequential sale outperforms the static sale in this example, i.e.,\n\n$$\n\\pi_{\\mathrm{dyn}}^{*}=11 / 80>1 / 8=\\pi_{\\mathrm{static}}^{*} .\n$$\n\nTaking a step back, it is clear that the seller would ideally like to condition payments on the realized states. In this case, she could charge a payment of 1 upon realization of state $\\omega_{1}$, which is the state less likely for either type. Both types would accept such a contract, and the seller achieves the first-best profits. As we do not allow for the payments to be made contingent on the realization of the state, a sequential mechanism essential represents a costly instrument to (partially) circumvent this restriction.\n\nIn essence, the proposed sequential scheme charges a constant price $\\bar{t}=3 / 10$ upon realization of signal $s_{1}$. Because the signal is correlated with the state under experiment $E_{0}$, it occurs more frequently for the higher type, allowing the seller to effectively price discriminate without ever giving the buyer a choice of experiment.\n\nFinally, note that the seller could do better within the simple class of mechanisms that initially release a free experiment, followed by a menu.\n\nIntuitively, as the correlation between state and signal $s_{1}$ becomes more precise (i.e., as $s_{1}$ becomes more informative), the seller's ability to condition payments on states improves. Ultimately, however, the seller must balance the ability to correlate payments with the willingness to pay for supplemental information after observing signal $s_{1}$ (e.g., the signal cannot be arbitrarily precise).\n\nTo formalize the intuition, consider offering free experiments of the following form\n\n| $E(x)$ | $s_{1}$ | $s_{2}$ |\n| :--- | :--- | :--- |\n| $\\omega_{1}$ | 1 | 0 |\n| $\\omega_{2}$ | $1-x$ | $x$ |\n\nThese experiments lead to posterior beliefs\n\n$$\n\\begin{aligned}\n\\theta^{L}(x) & \\triangleq \\operatorname{Pr}\\left[s_{1} \\mid \\theta^{L}\\right]=\\frac{1}{8-7 x}, \\\\\n\\theta^{H}(x) & \\triangleq \\operatorname{Pr}\\left[s_{1} \\mid \\theta^{H}\\right]=\\frac{1}{4-3 x} .\n\\end{aligned}\n$$\n\nThese beliefs satisfy the condition $1 / 2=\\gamma \\geq \\theta^{L}(x) / \\theta^{H}(x)$ for all $x$. Therefore, after releasing experiment $E(x)$ the seller optimally offers the fully informative experiment $\\bar{E}$ at a price\n\n$$\n\\bar{t}(x)=\\min \\left\\{\\theta^{L}(x), 1-\\theta^{H}(x)\\right\\} .\n$$\n\nFinally, a straightforward calculation reveals that the seller's profits are maximized by choosing $x$ such that $\\theta^{L}(x)<1 / 2<\\theta^{H}(x)$. In particular, it is optimal for the seller to induce\nthe two types to have identical willingness to pay for the full information, i.e.,\n\n$$\n\\theta^{L}\\left(x^{*}\\right)=1-\\theta^{H}\\left(x^{*}\\right) .\n$$\n\nThe optimal experiment has\n\n$$\nx^{*}=1-1 / \\sqrt{21} \\approx 0.781,\n$$\n\nwhich is larger than $x=2 / 3$, as used in the initial example, and yields profits $\\pi^{*}=(7+$ $2 \\sqrt{21}) / 112 \\approx 0.144$ that exceed 11/80, as computed above.","text_sha256":"ab60af7154665f11eed460530a10df10fddc0fcfacf0fce8d39a663b358fbca0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0044","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"## References\n\nAdmati, A. R., and P. Pfleiderer (1986): \"A monopolistic market for information,\" Journal of Economic Theory, 39(2), 400-438.\n\n- (1990): \"Direct and indirect sale of information,\" Econometrica, 58(4), 901-928.\n\nAnderson, E. T., and D. Simester (2013): \"Advertising in a Competitive Market: The Role of Product Standards, Customer Learning and Switching Costs,\" Journal of Marketing Research, 50(4), 489-504.\n\nBabaioff, M., R. Kleinberg, and R. Paes Leme (2012): \"Optimal Mechanisms for Selling Information,\" in Proceedings of the 13th ACM Conference on Electronic Commerce, EC '12, pp. 92-109.\n\nBalestrieri, F., and S. Izmalkov (2014): \"Informed seller in a Hotelling market,\" Discussion paper, HP Labs and New Economic School.\n\nBergemann, D., and A. Bonatti (2015): \"Selling Cookies,\" American Economic Journal: Microeconomics, 7(3), 259-294.\n\nBergemann, D., and M. Pesendorfer (2007): \"Information Structures in Optimal Auctions,\" Journal of Economic Theory, 137(1), 580-609.\n\nBrusco, S., and H. Hopenhayn (2007): \"Deregulation with consensus,\" Economic Theory, 32(1), 223-250.\n\nCelik, L. (2014): \"Information unraveling revisited: disclosure of horizontal attributes,\" Journal of Industrial Economics, 62(1), 113-136.\n\nChvatal, V. (1983): Linear Programming. Freeman and Co.\nCremer, J., and R. McLean (1988): \"Full Extraction of the Surplus in Bayesian and Dominant Strategy Auctions,\" Econometrica, 56, 1247-1258.\n\nDaskalakis, C., A. Deckelbaum, and C. Tzamos (2017): \"Strong Duality for a Multiple-Good Monopolist,\" Econometrica, 85(3), 735-767.\n\nDwork, C. (2008): \"Differential Privacy: A Survey of Results,\" in International Conference on Theory and Applications of Models of Computation, pp. 1-19. Springer.\n\nEső, P., and B. Szentes (2007a): \"Optimal information disclosure in auctions and the handicap auction,\" Review of Economic Studies, 74(3), 705-731.\n(2007b): \"The price of advice,\" Rand Journal of Economics, 38(4), 863-880.\nFederal Trade Commission (2014): Data Brokers: a Call for Transparency and Accountability.\n\nFuchs, W., and A. Skrzypacz (2015): \"Government interventions in a dynamic market with adverse selection,\" Journal of Economic Theory, 158, 371-406.\n\nHörner, J., and A. Skrzypacz (2016): \"Selling information,\" Journal of Political Economy, 124(6), 1515-1562.\n\nJohnson, J. P., and D. P. Myatt (2006): \"On the Simple Economics of Advertising, Marketing, and Product Design,\" American Economic Review, 96(3), 756-784.\n\nJullien, B. (2000): \"Participation Constraints in Adverse Selection Models,\" Journal of Economic Theory, 93(1), 1-47.\n\nKamenica, E., and M. Gentzkow (2011): \"Bayesian Persuasion,\" American Economic Review, 101(6), 2590-2615.\n\nKoessler, F., and V. Skreta (2016): \"Informed seller with taste heterogeneity,\" Journal of Economic Theory, 165, 456-471.\n\nKolotilin, A., M. Li, T. Mylovanov, and A. Zapechelnyuk (2015): \"Persuasion of a Privately Informed Receiver,\" Discussion paper, University of New South Wales.\n\nKrähmer, D., and R. Strausz (2015): \"Ex post information rents in sequential screening,\" Games and Economic Behavior, 90, 257-273.\n\nLi, H., and X. Shi (2015): \"Discriminatory Information Disclosure,\" Discussion paper, University of British Columbia and University of Toronto.\n\nLizzeri, A. (1999): \"Information revelation and certification intermediaries,\" Rand Journal of Economics, 30(2), 214-231.\n\nLuenberger, D. G. (1969): Optimization by Vector Space Methods. John Wiley \\& Sons.\nManelli, A. M., and D. R. Vincent (2006): \"Bundling as an optimal selling mechanism for a multiple-good monopolist,\" Journal of Economic Theory, 127(1), 1-35.\n\nMyerson, R. (1981): \"Optimal Auction Design,\" Mathematics of Operations Research, 6, 58-73.\n\n(1982) : \"Optimal Coordination Mechanism in Generalized Principal-Agent Problems,\" Journal of Mathematical Economics, 10, 67-81.\n\nMylovanov, T., and T. Tröger (2014): \"Mechanism Design by an Informed Principal: Private Values with Transferable Utility,\" Review of Economic Studies, 81(4), 1668-1707.\n\nOttaviani, M., and A. Prat (2001): \"The value of public information in monopoly,\" Econometrica, 69(6), 1673-1683.\n\nPavlov, G. (2011a): \"Optimal Mechanism for Selling Two Goods,\" The BE Journal of Theoretical Economics, 11(1), 1-33.\n\n- (2011b): \"A Property of Solutions to Linear Monopoly Problems,\" The B.E. Journal of Theoretical Economics, 11(1), 1-16.\n\nPycia, M. (2006): \"Stochastic vs Deterministic Mechanisms in Multidimensional Screening,\" Discussion paper, UCLA.\n\nRayo, L., and I. Segal (2010): \"Optimal Information Disclosure,\" Journal of Political Economy, 118(5), 949-987.\n\nRiley, J., and R. Zeckhauser (1983): \"Optimal Selling Strategies: When to Haggle, When to Hold Firm,\" Quarterly Journal of Economics, 98, 267-290.\n\nRochet, J., and J. Thanassoulis (2015): \"Stochastic Bundling,\" Discussion paper, University of Zürich.\n\nSamuelson, W. (1984): \"Bargaining under Asymmetric Information,\" Econometrica, 54(4), 995-1005.\n\nShapiro, C., and H. R. Varian (1999): Information Rules: A Strategic Guide to the Network Economy. Harvard Business Press.\n\nStiglitz, J. E. (1977): \"Monopoly, Non-Linear Pricing and Imperfect Information: The Insurance Market,\" Review of Economic Studies, 44(3), 407-430.\n\nToikka, J. (2011): \"Ironing without Control,\" Journal of Economic Theory, 146(6), 2510-2526.\n\n[^0]:    *We thank the co-editor, Jeff Ely, and three anonymous referees for their productive suggestions. We are grateful for conversations with Ben Brooks, Giacomo Calzolari, Gabriel Carroll, Gonzalo Cisternas, Jacques Crémer, Andrei Hagiu, Teck Ho, Bruno Jullien, Emir Kamenica, Alessandro Lizzeri, Alessandro Pavan, Mike Powell, Phil Reny, Mike Riordan, Maher Said, Jean Tirole, Juuso Toikka, Alex Wolitzky, and to seminar participants at Berkeley, Bocconi, Bologna, Carnegie Mellon, Chicago, Eief, Harvard, Mannheim, Northwestern, NYU, Oxford, QMUL, Toulouse, UCL, Vienna, Warwick, Yale, and the 2015 World Congress of the Econometric Society. We also thank Ian Ball for excellent research assistance.\n    ${ }^{\\dagger}$ Yale University, 30 Hillhouse Ave., New Haven, CT 06520, USA, dirk.bergemann@yale.edu.\n    ${ }^{\\ddagger}$ MIT Sloan School of Management, 100 Main Street, Cambridge, MA 02142, USA bonatti@mit.edu.\n    ${ }^{§}$ University of Bonn, Lennéstraße 37, 53113, Bonn, Germany alexey.v.smolin@gmail.com.","text_sha256":"8f90e8cca1c1a3602415ed5cbca699cb1b11c15e3f99f688d376593634afed2e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0045","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"[^1]:    ${ }^{1}$ In addition, a number of more recent papers, including Balestrieri and Izmalkov (2014), Celik (2014), Koessler and Skreta (2016), and Mylovanov and Tröger (2014), analyze this question from an informed principal perspective.\n\n[^2]:    ${ }^{2}$ For example, see Manelli and Vincent (2006), Pycia (2006), Pavlov (2011a), and Rochet and Thanassoulis (2015). In particular, Daskalakis, Deckelbaum, and Tzamos (2017) construct an example wherein the types follow a Beta distribution, and the optimal menu contains a continuum of stochastic allocations.\n\n[^3]:    ${ }^{3}$ As usual, the model allows for the alternative interpretations of a single buyer and a continuum of buyers.\n\n[^4]:    ${ }^{4}$ The value of information for the data buyer differs from a consumer's value for multiple goods or bundles of characteristics. In particular, the first max operator in (6) and (7) corresponds to the optimality condition for the buyer's action given the available information. The second max operator reflects the type-dependent nature of participation constraints. Both elements are missing from the multiproduct monopolist's problem (Pavlov, 2011a b; Daskalakis, Deckelbaum, and Tzamos, 2017).\n    ${ }^{5}$ The information structure in the right panel is given by the following stochastic matrix:\n\n[^5]:    ${ }^{6}$ The fully informative experiment is part of every optimal menu if all types assing positive probability to all states. It need not be part of every optimal menu if some types assign zero probability to some states. In that case, an optimal menu may contain a partially informative experiment that, combined with some type's prior, fully reveals all states that have positive probability, and so remains fully informative in this weaker sense.\n\n[^6]:    ${ }^{7}$ Discriminatory menus that do not shut down the low-value type are optimal in nonlinear screening models, such as the monopolistic insurance market studied in Stiglitz (1977). The novel element of selling information to buyers with noncongruent priors is that the seller can more easily extract (possibly all) the information rents.\n\n[^7]:    ${ }^{8}$ In Section 6, we establish a stronger result: with many actions and states, there always exists a distribution $\\gamma$ such that partial information is provided to the low type unless the two types agree on both the most likely state and the relative likelihood of any other two states. In the binary-state model, the latter condition is vacuously satisfied. However, with more than two states, the latter condition fails generically.\n\n[^8]:    ${ }^{9}$ Intuitively, if the rent function had a downward jump at $\\theta^{*}$, some nearby type $\\theta^{*}+\\varepsilon$ could purchase the experiment intended for type $\\theta^{*}-\\varepsilon$. This yields a payoff that is arbitrarily close to $V\\left(\\theta^{*}-\\varepsilon\\right)$, and hence, gives $\\theta^{*}+\\varepsilon$ an incentive to misreport his type.\n    ${ }^{10}$ Our environment with type-dependent participation constraints is an instance of the \"high convexity\"\n\n[^9]:    case in Jullien (2000). As such, the integral condition for implementability (24) differs from budget, capacity, or enforceability constraints because the distribution $F(\\theta)$ does not appear in the integrand. A similar condition, for different reasons and with different implications, appears as a persuasion budget constraint in Kolotilin, Li, Mylovanov, and Zapechelnyuk (2015).\n    ${ }^{11}$ This result is related to linear mechanism-design problems with budget constraints, such as Samuelson (1984), Brusco and Hopenhayn (2007), and Fuchs and Skrzypacz (2015). In our setting, the reduction to a linear program is obtained through a change of variable from $q(\\theta)$ to its increments, and by imposing the additional constraint that some type $\\theta^{*} \\in(0,1)$ purchases the fully informative experiment.\n\n[^10]:    ${ }^{12}$ The distribution in Figure 6 (left) represents a \"perturbation\" of the two-type example in Section 4.2 It is a mixture (with equal weights) of two Beta distributions with parameters $(8,30)$ and (60, 30).\n\n[^11]:    ${ }^{13}$ Marketing and risk-mitigation products generate 46\\% and 41\\% of data brokers' revenues, respectively (Federal Trade Commission, 2014).\n    ${ }^{14}$ Many brokers offer both kinds of products. For example, Nielsen Total Audience identifies two ways of packaging its data: a DMP to \"expand, optimize, segment and activate your customer data across all marketing channels and platforms\" and Data as a Service (DaaS) to \"find your target audience segment, or customize your own based on the characteristics that are most important to you.\" See http://www.nielsen.com/us/en/solutions/capabilities/total-audience.html for more details.\n    ${ }^{15}$ https://www.oracle.com/marketingcloud/products/data-management-platform/id-graph.html\n    ${ }^{16}$ http://www.towerdata.com/email-intelligence/pricing\n    ${ }^{17}$ https://www.equifax.com/business/undisclosed-debt-monitoring\n\n[^12]:    ${ }^{18}$ For instance, a local bank deciding whether to give a mortgage at the prevailing rate has a coarser action space than a major credit card company deciding on a new account's credit limit and interest rate.\n    ${ }^{19}$ Any data broker enjoys some degree of market power to the extent that its data sources are not perfectly correlated with those of other brokers or it has a superior ability to process information. Furthermore, it can be shown that the structural properties of optimal menus (Proposition 2) hold even in an imperfectly competitive setting with differentiated products (e.g., competition in nonlinear prices among sellers with partially correlated databases). In that case, non-dispersed and concentrated experiments are the most profitable instruments through which to provide any given utility level to the data buyer.\n\n[^13]:    ${ }^{20}$ A type I error leads the decision maker to reject the null hypothesis even though it is true. By contrast, a type II error leads the decision maker to accept the null hypothesis even though it is false.\n    ${ }^{21}$ Because distributions $H_{0}$ and $H_{1}$ satisfy the monotone likelihood ratio property, the feasibility frontier is spanned by threshold tests. As the informativeness of the statistic increases, the feasible set expands and approaches our unrestricted setting.","text_sha256":"d2b167dc4aa5209e5ec86944375e262b19ad036c9162e3d046c7893cd41722f4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0046","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"References","text":"[^14]:    ${ }^{22}$ For example, the online invitation website Evite sells lists of attendees of events at specific locations; the share-buttons provider AddThis sells lists of internet users' tastes for news; and Mailways sells lists of physical mailing addresses at the mail carrier route level (Anderson and Simester, 2013).\n    ${ }^{23}$ Babaioff, Kleinberg, and Paes Leme (2012) provide further insight into the role of realization-contingent pricing. In our model, contracting on signal realization leads to the first-best profits for the seller.\n    ${ }^{24}$ In our setting, contracting on the buyer's action leads to the first-best profits when the payoff of matching state and action is constant, though the same is not true in more general environments. Data providerpublisher partnerships can be analyzed more comprehensively in the setting of Eső and Szentes (2007b).\n\n[^15]:    ${ }^{25}$ For example, the Consumer Sentiment Index (released by Thomson-Reuters and the University of Michigan) screens buyers of time-sensitive data by offering information products that differ only in the timing of their availability.\n\n[^16]:    ${ }^{26}$ Any signal inducing a mixed action could first be split (independently of $\\omega$ ) into subsignals that each induce one of the pure actions in the support the mixed action. The resulting experiment still satisfies obedience and truth-telling and induces the same outcome.\n\n[^17]:    ${ }^{27}$ Similarly, one can show that type $\\theta^{H}$ must be indifferent after signal $s_{3}$ in experiment (46). The argument here is by contradiction: if he strictly preferred action $a_{3}$, the seller could make the experiment more valuable for $\\theta^{L}$ without changing its value for $\\theta^{H}$; and if he strictly preferred $a_{1}$, the seller could rearrange all signals and relax the incentive-compatibility constraint.","text_sha256":"024ba84c364c716eb1c5bf5880880e48146830d75ab8f5b05fa793853124b857"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-design-and-price-of-information:2017-06-03:0047","work_id":"alex-smolin:the-design-and-price-of-information","paper_id":"alex-smolin:the-design-and-price-of-information:2017-06-03","title":"The Design and Price of Information","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2017-06-03","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md","source_record":"https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf","doi":"https://doi.org/10.1257/aer.20161079","citation":"Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Dirk Bergemann; Alessandro Bonatti; Alex Smolin\n\n**Canonical citation:** Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md\n\n**Source record:** https://alexsmolin.com/files/the-design-and-price-of-information-working-paper.pdf\n\n**Published record:** https://doi.org/10.1257/aer.20161079\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"84e70910599457f042ece02513cc40ded865af05701abb77c1562aaf15126876"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0001","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Dirk Bergemann; Alessandro Bonatti; Andreas Haupt; Alex Smolin.\n> Canonical citation: Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"65aeb5affd8070951958a9ebecdf2d5d1ee2b587d8d8c10dd8f9b84541152349"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0002","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"The Optimality of Upgrade Pricing","text":"# The Optimality of Upgrade Pricing\n\n**Authors:** Dirk Bergemann; Alessandro Bonatti; Andreas Haupt; Alex Smolin\n\n**Manuscript date:** 2021-12-02\n\n#### Abstract\n\nWe consider a multiproduct monopoly pricing model. We provide sufficient conditions under which the optimal mechanism can be implemented via upgrade pricing-a menu of product bundles that are nested in the strong set order. Our approach exploits duality methods to identify conditions on the distribution of consumer types under which (a) each product is purchased by the same set of buyers as under separate monopoly pricing (though the transfers can be different), and (b) these sets are nested. We exhibit two distinct sets of sufficient conditions. The first set of conditions weakens the monotonicity requirement of types and virtual values but maintains a regularity assumption, i.e., that the product-by-product revenue curves are single-peaked. The second set of conditions establishes the optimality of upgrade pricing for type spaces with monotone marginal rates of substitution (MRS)-the relative preference ratios for any two products are monotone across types. The monotone MRS condition allows us to relax the earlier regularity assumption. Under both sets of conditions, we fully characterize the product bundles and prices that form the optimal upgrade pricing menu. Finally, we show that, if the consumer's types are monotone, the seller can equivalently post a vector of single-item prices: upgrade pricing and separate pricing are equivalent.\n\nKeywords: Revenue maximization; mechanism design; strong duality; upgrade pricing.","text_sha256":"01dc0094989bfb07c274c3465516917cddaa122587a5c7b79d79d8a4597f9162"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0003","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1 Introduction","text":"## 1 Introduction","text_sha256":"18f8b993fceb1a7535f9d0a0f4b58ea721117844a80f48a2844eba5cc8c9a8ef"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0004","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1.1 Motivation and Results","text":"### 1.1 Motivation and Results\n\nPricing multiple goods with market power is a canonical problem in the theory of mechanism design. It is also a challenge of growing importance and complexity for online retailers and service providers, such as Amazon and Netflix. Both in theory and in practice, designing the optimal mixed bundling mechanism, (i.e., pricing every subset of products) becomes exceedingly complex in the presence of a large number of goods.\n\nA natural question is then whether simpler pricing schemes are optimal under suitable demand conditions. A simple, commonly used mechanism consists of upgrade pricing, whereby the available options are ranked by set inclusion, i.e., some goods are only available as add-ons, Ellison (2005). For example, many online streaming services use a tiered subscription model, whereby users can pay to upgrade to a \"premium package\"-a subscription with a larger selection of the provider's content relative to the \"basic package\", Philips (2017).\n\nIn this paper, we obtain sufficient conditions under which upgrade pricing maximizes the seller's revenue. Our approach consists of first identifying conditions under which the consumer's types can be ordered in terms of their absolute or relative willingness to pay for the seller's goods, and then ranking the goods themselves by the profitability of selling them to larger sets of consumer types. Our sufficient conditions not only establish the optimality of some upgrade pricing menu: they also show that the optimal bundles are deterministic, and they reveal the order in which they are ranked in the menu. That is, we identify all the nested bundles that appear in the seller's menu, and the profit-maximizing price for each one.\n\nOur results consist of two distinct sets of conditions. The first set of conditions (Theorem 1) illustrates the essence upgrade pricing optimality in what we label as \"regular\" settings. While these conditions are reminiscent of regularity in one dimension, they are in fact weaker than the monotonicity of the buyer's multidimensional types and of the (item by item) Myersonian virtual values. What we require is for the consumer's types to be ranked in such a way that the virtual values for each item are negative over an initial and positive over a final segment. Furthermore, we require any consumer with a positive virtual value for an item to also have a larger value for that item, relative to any type with a negative virtual value. At the optimal prices, the lowest type buying each good is indifferent between buying it and not buying it. Finally, the sets of types buying each item are nested under the weak monotonicity property, which implies the optimal allocation can be implemented via upgrade pricing.\n\nThe second set of conditions (Theorem 2) describes our best attempt at extending our approach to non-regular distribution of types. In order to further weaken the regularity requirement, we restrict attention to type spaces for which the relative preference ratios for any two goods are monotone across types. An example of ordered relative preferences is if higher types have a stronger prefer-\nence for good 2 over good 1 . We refer to such a condition as \"monotone marginal rates of substitution\" (monotone MRS).\n\nThe intuition for our two results can be grasped by considering the demand functions for each good separately. Under monotonicity and monotone MRS, the optimal monopoly prices for each of the goods are ranked. In the special case where the Myersonian virtual values for our ordered types\n\n$$\n\\phi_{i}^{k}=\\theta_{i}^{k}-\\frac{1-F_{i}}{f_{i}}\\left(\\theta_{i+1}^{k}-\\theta_{i}^{k}\\right)\n$$\n\nare also monotone for each item $k$, the first set of conditions applies.\nWhen virtual values are not monotone, however, they can cross zero more than once. In that case, the result still holds, but the proof requires the right ironing procedure. Our ironing procedure relaxes the standard approach of ? and the literature up to Haghpanah and Hartline (2020). Specifically, we do not iron with the goal of monotone virtual values, which corresponds to a concave revenue curve. Rather we iron towards single-crossing virtual values which leads to a quasiconcave revenue curve. We then use the structure implied by monotone MRS to derive a dual certificate of optimality.\n\nUnder either set of conditions, each good is purchased by the same set of buyers that would buy it if that were the seller's only product. We further show (Theorem 3) that, if the consumer's types are (not weakly) monotone, the seller can equivalently post the vector of single-item monopoly prices-i.e., bundling is redundant. For example, in the case of two goods sold separately, monotone type spaces mean that no consumer type will buy good 2 without also buying good 1. More generally, the seller benefits from restricting the set of bundles the consumer can purchase through a proper menu of options with the upgrade property. However, examples also show that implementability through separate pricing is neither necessary nor sufficient for the optimality of upgrade pricing.","text_sha256":"8207f49a95c539b4f9401965e46ef472c213a397b6c8a586e8eee6b88d6e653f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0005","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1.2 Related Literature","text":"### 1.2 Related Literature\n\nFirst and foremost, our paper contributes to the economics literature on product bundling. The profitability of mixed bundling relative to separate pricing was first examined by Adams and Yellen (1976), and further generalized by McAfee et al. (1989). More recently, a number of contributions have studied the optimal selling mechanisms in the case of two or three goods, and derived conditions for the optimality of pure bundling (see, for example, Manelli and Vincent (2006) and Pavlov (2011)). Daskalakis et al. (2017) use duality methods to characterize the solution of the multiproduct monopolist's problem, and show how the optimal mechanism may involve a continuum of lotteries over items. Bikhchandani and Mishra (2020) derive conditions under which the optimal mechanism is deterministic when the buyer's utility is not necessarily additive. Finally, Ghili (2021) establishes conditions for the optimality of pure bundling when buyers' values are interdependent. Relative to all these papers, we focus on a specific class of simple mechanisms, which includes pure bundling as a special case.\n\nHart and Nisan (2017) and Babaioff et al. (2014) also study the properties of simpler schemes. The former derives a lower bound on the revenue obtained from separate item pricing. The latter obtains an upper bound on the revenue of the optimal mechanism, relative to the better of pure bundling and separate pricing.\n\nIn the context of nonlinear pricing, ? suggested a \"demand profile\" approach that determines the price of each incremental unit by treating it as separate market. This approach is particularly attractive in settings where there is a natural ordering over the items. This in particular is the case when there is a homogeneous good that is offered in various quantities, such as in energy markets for electricity or water. This approach naturally generates a sequence of upgrade prices. The demand profile approach, and in particular the incremental pricing rule implied by it, does not always yield an optimal mechanism as consumers may wish to obtain earlier units in order to obtain the later units. Thus, a contribution of the current paper is to determine when upgrade pricing is exactly optimal and then to find the upgrade prices as solutions to the global revenue maximization problem rather than the incremental item problem. Other papers make assumptions that make sure that a demand profile-type approach yields an optimal mechanism. In Johnson and Myatt (2003), buyers have unit demand and sellers offer different varieties of a single good. The approach in their paper is to assume a quality ranking on the varieties and to solve for the upgrade prices-the additional payments required to buy a better variety. The survey of the nonlinear pricing literature by Armstrong (2016) covers related approaches that optimize upgrades separately.\n\nOur formulation of the dual problem follows Cai et al. (2016), who present a general duality approach to Bayesian mechanism design. Cai et al. (2016) formulate virtual valuations in terms of dual variables, state the weak and the strong duality results, and use them to establish lower bounds for relative performance of simple mechanisms. An important contribution by Haghpanah and Hartline (2020) exploits the duality machinery to provide sufficient conditions for the exact optimality of a specific, simple mechanism-pure bundling-consisting of offering a maximal bundle at a posted price. Under their sufficient conditions, the dual variables can be recovered from a single-dimensional problem in which the seller is restricted to bundle all items together.\n\nWe follow the approach of Haghpanah and Hartline (2020) by leveraging the duality approach to provide sufficient conditions for the optimality of a particular class of mechanisms. Haghpanah and Hartline (2020) gave a characterization of the optimality of the grand bundle, we provide a characterization for upgrade pricing. As upgrade pricing allows multiple bundles to be present in the menu, we cannot assign the dual variables by solving a one-dimensional problem. Instead, we develop a novel ironing algorithm that generates these variables by ironing different item's revenue curves for different types. Under our sufficient conditions, the so-constructed virtual surplus is maximized by an element-wise monotone allocation that can be implemented by upgrade pricing; by complementary slackness, this certifies the optimality of upgrade pricing. Because pure\nbundling is one instance of upgrade pricing, our conditions differ from those of Haghpanah and Hartline (2020).\n\nOur ironing differs from existing ironing approaches using duality and tackles a more general problem. In comparison to Haghpanah and Hartline (2020), we prove optimality for mechanisms with menu size surpassing two. Fiat et al. (2016) studies a two-parameter model, and uses an ironing approach that leads from the revenue curves to their concave closure. Devanur et al. (2020) generalizes Fiat et al. (2016) to more general orders on the second parameter. Our approach tackles optimality for an arbitrary finite number of items and varies the ironing procedure. On a technical level, our ironing procedure yields quasi-concave ironed revenue curves, whereas the ironed revenue curves in Haghpanah and Hartline (2020); Fiat et al. (2016); Devanur et al. (2020) are concave.\n\nOur results also feed into a literature specifying optimal finite mechanisms for multi-dimensional types. (Daskalakis et al., 2017, section 7) for example characterizes the optimal mechanisms for the two-good monopolist problem if the optimal mechanism has a particular structure. While Daskalakis et al. (2017) requires that the region of the type space that is not allocated any item is not adjacent to all regions getting specific constant allocations, upgrade pricing mechanisms consistently break this requirement.","text_sha256":"2ed228f7a244023b6dcf1b0713a14cd3511fde9d8e2870c9cc828af6f7315811"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0006","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1.3 Structure of the Paper","text":"### 1.3 Structure of the Paper\n\nThe model is introduced in section 2. The first set of sufficient condition is presented in section 3. In section 4, we present our results for monotone MRS type spaces. In section 5, we discuss the relationship between separate pricing and upgrade pricing. We conclude in section 6.","text_sha256":"5eef7d8de55a2ce05d87b6c92109a7e9026895f85b813a2b18964be2109f4905"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0007","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2 Model","text":"## 2 Model\n\nWe consider a standard multiple-good monopoly setting. There is a single seller of $d \\geq 1$ goods and a single buyer. The seller's marginal costs of production are normalized to zero. The buyer's utility function is additive across goods. We refer to the vector of marginal utilities $\\theta_{i} \\in \\mathbb{R}^{d}$ as the buyer's type. Therefore, the utility of buyer type $\\theta_{i}$ from the consumption vector $q \\in[0,1]^{d}$ is given by\n\n$$\nU\\left(\\theta_{i}, q\\right)=\\sum_{k=1}^{d} \\theta_{i}^{k} q^{k}\n$$\n\nWe also adopt the shorthand notation $\\left\\langle\\theta_{i}, q\\right\\rangle:=\\sum_{k=1}^{d} \\theta_{i}^{k} q^{k}$. As a convention, we denote types by subscripts and items by superscripts. The buyer's utility is quasi-linear in transfers and her outside option is also normalized to zero.\n\nThe buyer knows her type. From the seller's perspective, the buyer's type is distributed over a finite set $\\Theta \\subseteq \\mathbb{R}_{+}^{d}$, with $|\\Theta|=n$, according to the distribution $f \\in \\Delta(\\Theta)$. For any positive integer $n$, we adopt the convention that $[n]:=$\n$\\{1,2, \\ldots, n\\}$, and we index types by $i \\in[n]$. We let $f_{i}:=f\\left(\\theta_{i}\\right)$ and denote the cumulative distribution sequence by $F_{i}=\\sum_{j=1}^{i} f_{j}, i \\in[n]$.\n\nThe seller aims to maximize revenue. By the revelation principle, we can focus on direct mechanisms $(q, t)=\\left(q_{i}, t_{i}\\right)_{i \\in\\{0\\} \\cup[n]}$. These mechanisms can be interpreted as menus with $n+1$ items so that item $i$ delivers consumption vector $q_{i}$ at price $t_{i}$ and item $\\left(q_{0}, t_{0}\\right):=(0,0)$ captures the buyer's outside option.\n\nWe call a menu upgrade pricing if $\\left\\{q_{0}, q_{1}, \\ldots, q_{n}\\right\\}$ can be ordered in the component-wise partial order on $\\mathbb{R}^{d}$ given by $q \\leq q^{\\prime} \\Leftrightarrow \\forall k \\in[d]: q^{k} \\leq\\left(q^{\\prime}\\right)^{k}$. Our main goal is to provide conditions under which upgrade pricing maximizes the seller's revenue among all direct mechanisms.","text_sha256":"ed4cf35f2af49c9fc8956c2b40b28d49f7b336dd033921aa0c1077a484fa41f4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0008","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3 Optimal Mechanisms for Regular Distributions","text":"## 3 Optimal Mechanisms for Regular Distributions\n\nWe will make prominent use of the (partial) Lagrangian duality-based certificate of optimality used by Cai et al. (2016). We state the underlying duality result to fix notation.","text_sha256":"e66ce71c578ad2697e49100c092170b089b037fafe95d4ee76913bf33aa4b4a4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0009","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.1 Duality","text":"### 3.1 Duality\n\nIn what follows, we will associate with $\\lambda_{j i}$ the Lagrange multiplier of the incentive compatibility constraint of type $\\theta_{j}$ deviating to type $\\theta_{i}, j \\in[n], i \\in\\{0\\} \\cup[n]$ :\n\n$$\n\\left\\langle q_{j}, \\theta_{j}\\right\\rangle-t_{j} \\geq\\left\\langle q_{i}, \\theta_{j}\\right\\rangle-t_{i} .\n$$\n\nWe note that the incentive constraints corresponding to $\\lambda_{j 0}, j \\in[n]$ are type $j$ 's individual rationality constraints. As a main tool in our analysis, we define the multi-dimensional virtual values associated with Lagrange multipliers $\\lambda \\in$ $\\mathbb{R}^{n} \\times \\mathbb{R}^{n+1}$ as\n\n$$\n\\phi_{i}^{\\lambda}:=\\theta_{i}-\\frac{1}{f_{i}} \\sum_{j=1}^{n} \\lambda_{j i}\\left(\\theta_{j}-\\theta_{i}\\right) .\n$$\n\nLemma 1. A mechanism $\\left(q_{i}, t_{i}\\right)_{i \\in\\{0\\} \\cup[n]}$ maximizes revenue if and only if there exist multipliers $\\lambda_{j i}, j \\in[n], i \\in\\{0\\} \\cup[n]$ such that\n\n1. $\\lambda_{j i} \\geq 0$ (Non-Negativity)\n2. $\\left(q_{i}\\right)_{i \\in[n]}$ optimizes $\\max _{\\left(q_{i}\\right)_{i \\in[n]} \\in[0,1]^{n}} \\sum_{i=1}^{n} f_{i}\\left\\langle q_{i} \\cdot \\phi_{i}^{\\lambda}\\right\\rangle$ (Virtual Welfare Maximization)\n3. $f_{i}=\\sum_{j=0}^{n} \\lambda_{i j}-\\sum_{j=1}^{n} \\lambda_{j i}$ for all $i \\in[n]$ (Feasibility of Flow)\n4. $\\lambda_{j i}\\left(\\left\\langle q_{j}, \\theta_{j}\\right\\rangle-t_{j}-\\left\\langle q_{i}, \\theta_{j}\\right\\rangle-t_{i}\\right)=0$ for all $j \\in[n], i \\in\\{0\\} \\cup[n]$ (Complementary Slackness)\n5. There are transfers $t$ such that $(q, t)$ is incentive compatible and individually rational (Implementability)\n\nWe call the dual variables $\\lambda_{j i}, j \\in[n], i \\in[n] \\cup\\{0\\}$ flows from type $j$ to type $i$ whenever they are non-negative and satisfy Lemma 1item 3. This name is inspired by flow conservation constraints from the maximum flow and minimum cost flow problem in discrete mathematics (Korte and Vygen, 2011).\n\nProof of Lemma 1. Slater's condition for affine inequality constraints (Boyd and Vandenberghe, 2004, p. 227) allows us to write revenue maximization subject to the incentive compatibility and individual rationality constraints as an unconstrained optimization problem for $\\left(q_{i}, t_{i}\\right)_{i \\in[n]}$ subject to complementary slackness and non-negativity of dual variables. The Lagrangian reads:\n\n$$\n\\begin{aligned}\n\\mathcal{L} & =\\sum_{i=1}^{n} f_{i} t_{i}+\\sum_{j=1}^{n} \\sum_{i=0}^{n} \\lambda_{j i}\\left(\\left\\langle q_{j}, \\theta_{j}\\right\\rangle-t_{j}-\\left\\langle q_{i}, \\theta_{j}\\right\\rangle-t_{i}\\right) \\\\\n& =\\sum_{i=1}^{n} t_{i}\\left(f_{i}-\\sum_{j=0}^{n} \\lambda_{i j}+\\sum_{j=1}^{n} \\lambda_{j i}\\right)+\\sum_{j=1}^{n} \\sum_{i=0}^{n} \\lambda_{j i}\\left\\langle q_{j}, \\theta_{j}\\right\\rangle-\\sum_{j=1}^{n} \\sum_{i=0}^{n} \\lambda_{j i}\\left\\langle q_{i}, \\theta_{j}\\right\\rangle \\\\\n& =\\sum_{j=1}^{n} \\sum_{i=0}^{n} \\lambda_{j i}\\left\\langle q_{j}, \\theta_{j}\\right\\rangle-\\sum_{j=1}^{n} \\sum_{i=0}^{n} \\lambda_{j i}\\left\\langle q_{i}, \\theta_{j}\\right\\rangle \\\\\n& =\\sum_{j=1}^{n}\\left(\\left(\\sum_{i=1}^{n} \\lambda_{i j}-\\sum_{i=0}^{n} \\lambda_{j i}\\right)\\left\\langle q_{j}, \\theta_{j}\\right\\rangle-\\sum_{i=0}^{n} \\lambda_{j i}\\left(\\left\\langle q_{i}, \\theta_{j}\\right\\rangle-\\left\\langle q_{j}, \\theta_{j}\\right\\rangle\\right)\\right) \\\\\n& =\\sum_{j=1}^{n}\\left(f_{j}\\left\\langle q_{j}, \\theta_{j}\\right\\rangle-\\sum_{i=0}^{n} \\lambda_{j i}\\left(\\left\\langle q_{i}, \\theta_{j}\\right\\rangle-\\left\\langle q_{j}, \\theta_{j}\\right\\rangle\\right)\\right)=\\sum_{j=1}^{n} f_{j}\\left\\langle q_{j}, \\phi_{j}\\right\\rangle .\n\\end{aligned}\n$$\n\nClearly, it is necessary for an optimal mechanism to be implementable. To conclude the proof, we need to show that virtual welfare maximization and feasibility of flow are equivalent to maximizing the Lagrangian. Assume virtual welfare maximization and feasibility of flow. Then the above equalities show that the Lagrangian is maximized and certify the optimality of the mechanism $\\left(q_{i}, t_{i}\\right)_{i \\in[n]}$. Conversely, assume that the Lagrangian is maximized by $\\left(q_{i}, t_{i}\\right)_{i \\in[n]}$. If the flow $\\lambda_{j i}$ were not feasible, then choosing $t_{i}$ arbitrarily large or small would lead to a higher value for the Lagrangian, which yields a contradiction. Given that this is zero, the Lagrangian equals virtual welfare, and virtual welfare maximization follows from optimality. $\\square$","text_sha256":"ab6ceb1b48109b71fc07557cf20d94133f93d41b3f9768ceae3229932c982db6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0010","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3.2 A Sufficient Condition for Regular Distributions","text":"### 3.2 A Sufficient Condition for Regular Distributions\n\nOur first set of sufficient conditions for upgrade pricing optimality consists of a weak monotonicity condition and a regularity condition.\n\nWe call a type distribution $F$ weakly monotone with cutoffs $i^{1}, i^{2}, \\ldots, i^{d} \\in[n]$ if for any $i, j \\in[n]$ and $k \\in\\{1,2, \\ldots, d\\}$,\n\n$$\ni \\leq i^{k} \\leq j \\Longrightarrow \\theta_{i}^{k} \\leq \\theta_{j}^{k} .\n$$\n\nNote that weak monotonicity is strictly weaker than monotonicity: for each item, only order comparisons with respect to a cutoff type need to hold, whereas types above or below the cutoff can be arbitrarily ordered.\n\nSimilarly, a type distribution $F$ is regular with respect to cutoffs $i^{1}, i^{2}, \\ldots, i^{d} \\in$ $[n]$ if for any $i, j \\in[n]$ and $k \\in\\{1,2, \\ldots, d\\}$,\n\n$$\ni \\leq i^{k} \\leq j \\Longrightarrow \\phi_{i}^{k} \\leq 0 \\leq \\phi_{j}^{k},\n$$\n\nwhere $\\phi_{i}$ denotes the initial $d$-dimensional virtual values\n\n$$\n\\phi_{i}:=\\theta_{i}-\\frac{1-F_{i}}{f_{i}}\\left(\\theta_{i+1}-\\theta_{i}\\right) .\n$$\n\nThe initial $d$-dimensional virtual values can be seen as multi-dimensional versions of the virtual values in ?.\n\nWe say that a type distribution $F$ is compatibly weakly monotone and regular if it is both weakly monotone and regular with respect to the same set of cutoffs. When such cutoffs $i^{k}$ exist, they are essentially unique except between contiguous types of vanishing virtual value $\\phi_{i}^{k}$ and monotone types $\\theta_{i}^{k}, i \\in[n], k \\in[d]$. Subfigure 1 a illustrates a type distribution with this property.\n\nOur regularity condition can be equivalently stated in terms of the pseudorevenues\n\n$$\nR_{i}^{k}:=\\left(1-F_{i-1}\\right) \\theta_{i}^{k} .\n$$\n\nSubfigure 1 clepicts pseudo-revenues. We call (4) pseudo-revenue because, without an assumption that the values are monotone with respect to the componentwise partial order, the pseudo-revenue does not correspond to the revenue from sales of item $k$ at a posted price of $\\theta_{i}^{k}$. In particular, because we have\n\n$$\n\\begin{aligned}\n\\frac{R_{i}^{k}-R_{i+1}^{k}}{f_{i}} & =\\frac{\\left(1-F_{i-1}\\right) \\theta_{i}^{k}-\\left(1-F_{i}\\right) \\theta_{i+1}^{k}}{f_{i}} \\\\\n& =\\frac{f_{i} \\theta_{i}^{k}-\\left(1-F_{i}\\right)\\left(\\theta_{i+1}^{k}-\\theta_{i}^{k}\\right)}{f_{i}}=\\theta_{i}^{k}-\\frac{1-F_{i}}{f_{i}}\\left(\\theta_{i+1}^{k}-\\theta_{i}^{k}\\right)=\\phi_{i}^{k}\n\\end{aligned}\n$$\n\nimposing regularity with respect to the cutoffs $i^{k}$ is equivalent to requiring that $R_{i}^{k}$ is single-peaked with peak $i^{k}$. While pseudo-revenues do not have immediate economic meaning, they are an important technical tool, in particular for our analysis of non-regular distributions in section 4.\n\nTheorem 1. If the type distribution $F$ is compatibly weakly monotone and regular with respect to cutoffs $\\left(i^{k}\\right)_{k \\in[d]}$, then upgrade pricing is optimal. In particular, the following mechanism is optimal:\n\n$$\nq_{i}^{k}:=\\left\\{\\begin{array}{cc}\n1 & i \\geq i^{k} \\\\\n0 & \\text { else. }\n\\end{array}, \\quad i \\in[n], k \\in[d] .\\right.\n$$\n\nProof of Theorem 1. Define the dual variables\n\n$$\n\\hat{\\lambda}_{j i}= \\begin{cases}1-F_{i} & \\text { if } j=i+1 \\\\ 0 & \\text { else. }\\end{cases}\n$$\n\nObserve that, by definition, $\\hat{\\lambda}$ induces the initial virtual values, $\\phi_{i}=\\phi_{i}^{\\hat{\\lambda}}$.\nWe check the properties of Lemma 1. Virtual welfare maximization, condition 2. follows from\n\n$$\nq_{i}^{k}=1 \\stackrel{(6)}{\\Longleftrightarrow} R_{i}^{k} \\geq R_{i+1}^{k} \\stackrel{(5)}{\\Longleftrightarrow} \\phi_{i}^{k} \\geq 0 .\n$$\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFig. 1: Types, virtual values and pseudo-revenues for type space $\\Theta=\\left\\{\\left(\\frac{9}{128}, \\frac{27}{64}\\right),\\left(\\frac{1}{4}, \\frac{3}{2}\\right),\\left(\\frac{1}{2}, 2\\right),(1,1)\\right\\}$ and type distribution $f=\\left(\\frac{7}{16}, \\frac{3}{16}, \\frac{1}{8}, \\frac{1}{4}\\right)$. The optimal mechanism sells good 2 at a price of 1 and good 1 as an upgrade, also at a price of 1 . All types except $\\theta_{1}$ buy good 2, and only type $\\theta_{4}$ buys good 1.\n\nFor flow preservation, condition 3, observe that\n\n$$\n\\sum_{j=1}^{n} \\hat{\\lambda}_{i j}-\\sum_{j=0}^{n} \\hat{\\lambda}_{j i}=1-F_{i-1}-\\left(1-F_{i}\\right)=f_{i} .\n$$\n\nThe mechanism is implementable, condition 5. by assumption of compatible weak monotonicity and regularity.\n\nFinally, we need to check that complementary slackness (condition 4) holds. Observe that $\\hat{\\lambda}_{i j}>0$ implies $j=i-1$. Hence, all types must be indifferent between their allocation and payment and the allocation and payment of the next lower type. If the next lower type has the same allocation and payment, this is clearly satisfied. Otherwise, this is the first type buying an upgrade. If this type were not indifferent between buying it and not buying it, the price of the upgrade could be raised, and the revenue increased, without affecting other types' incentives. Thus, this type must be indifferent between their allocation (and payment) and the next lower type's allocation. $\\square$\n\nOur assumptions of regularity and weak monotonicity relax the monotonicity of types and Myersonian virtual values by allowing for permutations above and below the monopoly price. These assumptions nonetheless require that the set of types that buy each object remains an upper selection, and conversely the set of types that do not buy remains a lower selection. The intuition for why this works is similar to the idea that the monopoly price does not depend on the valuations of types that are not marginally buying, just as long as they do not become marginal buyers.\n\nThese assumptions depend on the fixed order of types we have introduced in the model. Thus, if there exists an order that satisfies these assumptions, upgrade pricing is optimal. Furthermore, multiple orders of types might satisfy the theorem's conditions for a given type distribution $F$. In this case, the theorem can be used to certify optimality of mechanism (6), based on the different orders.\n\nAs optimality of a mechanism for a distribution $F$ does not depend on the order on types, the revenue of (6) must be the same for all orders with which the conditions of Theorem 1 are satisfied.\n\nOur next set of conditions imposes similar requirements, strengthened appropriately to allow for non-regular type distributions, which require ironing.","text_sha256":"c0ee26eb723f33440534b7e7625258a4deaf03f5aa649a7cc8d17ae14c4762c0"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0011","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Optimal Mechanisms for Non-Regular Distributions","text":"## 4 Optimal Mechanisms for Non-Regular Distributions\n\nWe now establish the optimality of an upgrade pricing mechanism in settings without regularity. The weaker sufficient conditions will replace the regularity condition and will allow for ironing to be part of the optimal mechanism. The new sufficient conditions will serve to allow us to perform the ironing procedure item-by-item, and limit the interaction of constraints across items. We say that a type space $\\Theta$ has monotone marginal rates of substitution if\n\n$$\n1 \\leq i \\leq j \\leq n \\text { and } 1 \\leq k \\leq l \\leq d \\Longrightarrow \\frac{\\theta_{i}^{l}}{\\theta_{i}^{k}} \\leq \\frac{\\theta_{j}^{l}}{\\theta_{j}^{k}} .\n$$\n\nfor any $i, j \\in[n], l, k \\in[d]$.\nRecall that pseudo-revenue is given by $R_{i}^{k}=\\left(1-F_{i}\\right) \\theta_{i}^{k}$.\nWe call a scalar sequence $\\left(R_{i}\\right)_{i \\in[n]}$, quasi-concave if there is a cutoff $i^{\\prime} \\in[n]$ such that $i^{\\prime} \\leq i \\leq j$ or $j \\leq i \\leq i^{\\prime}$ implies $R_{i} \\geq R_{j}$. We call the point-wise smallest quasi-concave sequence that point-wise dominates $\\left(R_{i}\\right)_{i \\in[n]}$ its quasiconcave closure and denote it by $\\left(\\bar{R}_{i}\\right)_{i \\in[n]}$.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFig. 2: Type space and pseudo-revenues for type space $\\Theta=$ $\\{(57 / 64,1),(1,5 / 4),(2,3),(9 / 4,5)\\}$ and type distribution $f=\\left(3 / 8, \\frac{1}{4}, \\frac{1}{8}, \\frac{1}{4}\\right)$. The optimal mechanism sells good 1 at a price of $\\frac{57}{64}$, and good 2 as an upgrade at a price of 5. All types buy good 1, and only type $\\theta_{4}$ buys good 2.\n\nWe will make regular use of the sequence $\\left(\\bar{R}_{i}^{k}\\right)_{i \\in[n]}$, the quasi-concave closure of the pseudo-revenue for item $k$.\n\nTo allow for our construction of a dual certificate of optimality, we need additional assumptions. These will be formulated in terms of candidate ironing\nintervals. For a pseudo-revenue $R$, we call a set of contiguous types $I \\subseteq[n]$ with\n\n$$\n\\bar{R}_{i}^{k} \\neq R_{i}^{k}\n$$\n\nfor all $i \\in I$ such that there is no superset of contiguous types $I^{\\prime} \\supseteq I$ such that (8) holds for all $i \\in I^{\\prime}$, a candidate ironing interval for item $k$. (With slight abuse of language, we refer to discrete sets of contiguous types as intervals.) Every item $k$ may have several candidate ironing intervals, and every type can be contained in a candidate ironing interval for different items.\n\nWe relax the regularity assumption on pseudo-revenues $R_{i}^{k}$. Instead of assuming regularity, i.e. $R_{i}^{k}$ to be single-peaked with peak $i^{k}$, we assume two properties that are in combination weaker than regularity. We call a type distribution $F$ mostly regular if for some cutoffs $i^{k} \\in \\arg \\max _{i \\in[n]} R_{i}^{k}$ and any $i$ such that $i^{k}<i \\leq i^{k+1}$, the following hold:\n\n1. (No partial overlap) If $I$ is a candidate ironing interval of item $k$ and $J$ is a candidate ironing interval of item $k+1$, then either $I \\cap J=\\emptyset$ and there is $i \\in[n]$ such that $I<i<J$ or $J<i<I$, or one of $I, J$ is a subset of the other excluding its endpoints.\n2. (No ironing on neighboring maxima) For any ironing candidate interval $I$ of item $k, i^{k}, i^{k+1} \\notin I$.\n3. (Not too shuffled) For any candidate ironing interval $I \\subseteq\\left\\{i^{k}+1, i^{k}+\\right.$ $\\left.2, \\ldots, i^{k+1}-1\\right\\}$ and $i \\in I$,\n$$\n\\theta_{\\min I}^{k+1} \\leq \\theta_{i}^{k+1} \\quad \\theta_{\\max I}^{k} \\leq \\theta_{\\max I+1}^{k}\n$$\nFinally, we call a distribution compatibly weakly monotone and mostly regular if it is weakly monotone and mostly regular with respect to the same cutoffs $i^{k}$, $k \\in[d]$.\n\nNote that monotone MRS by itself is not a restrictive assumption. For example, in two dimensions, every type set can be ordered in order of monotone MRS. In combination with compatible weak monotonicity and mostly regularity, this assumption becomes stronger.\n\nSubfigure 2a shows the type space of a compatibly weakly monotone and mostly regular type distribution, and subfigure 2b its pseudo-revenues.\n\nConversely, Figure 3 shows an instance of a distribution over a monotone MRS type space that is not mostly regular. In particular, this example fails the first condition, because it involves overlapping candidate ironing intervals. We will use our assumptions to construct dual variables $\\left(\\lambda_{i j}\\right)_{i, j \\in[n]}$ by ironing pseudo-revenues for each item. In our proof that there is an optimal mechanism with an upgrade pricing allocation, we will use monotone MRS to show that for each type, ironing is only needed for two items, the lowest item in the MRS order that the type bought, and the highest item in the MRS order that she didn't buy. We will use the first two conditions of mostly regularity to show that from these two items, we can select a single item to iron at a time, while not changing the other item's virtual values in a way that will break virtual welfare maximization of the allocation. As in Theorem 1 , weak monotonicity ensures implementability\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFig. 3: Failure of no overlap: $\\{2,3\\}$ is a candidate ironing interval for item 2, \\{3, 4\\} is a candidate ironing interval for item 1.\n\nof an upgrade pricing allocation, i.e., the existence of a price vector $\\left(t_{i}\\right)_{i \\in[n]}$ such that the mechanism $(q, t)$ is incentive compatible and individually rational. To allow for our ironing procedure to work, we also need a mild requirement on the monotonicity of types beyond weak monotonicity. While weak monotonicity was a requirement that could be formulated item-by-item, this requirement links the type order of neighboring items.\n\nTheorem 2. Let $\\Theta$ have monotone marginal rates of substitution. If the type distribution $F$ is compatibly weakly monotone and mostly regular with respect to cutoffs $\\left(i^{k}\\right)_{k \\in[d]}$, then upgrade pricing is optimal. In particular, the following mechanism is optimal:\n\n$$\nq_{i}^{k}:=\\left\\{\\begin{array}{ll}\n1 & i \\geq i^{k} \\\\\n0 & \\text { else },\n\\end{array} \\quad i \\in[n], k \\in[d] .\\right.\n$$\n\nNote that the allocation (9) is the allocation that arises from separate monopoly pricing. ${ }^{5}$","text_sha256":"fe08354b89a68e0545403ec825080224d6d6c51a8d446dcf6175b6622f437e7e"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0012","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Optimal Mechanisms for Non-Regular Distributions","text":"To prove Theorem 2, we will construct a sequence of flows $\\lambda_{i}$ from $i=n$ down to $i=1$, starting with $\\hat{\\lambda}$, the initial flow that induces Myersonian multidimensional virtual values. Given a definition of pseudo-revenue implied by a flow, our Ironing Algorithm will, for each type $i$ and for at least one item $k$, iron to match induced pseudo-revenue with the quasi-concave closure of multidimensional Myersonian pseudo-revenue, $\\bar{R}_{i}^{k}$. This is illustrated in Figure 4\n\nThe main steps in this proof are to show that the ironing is well-defined in that such implied pseudo-revenue is attainable with a non-negative and feasible flow (Lemma 6 and Lemma 4, respectively). The most technical part of the proof consists of showing that the Ironing Algorithm produces dual variables that maximize virtual welfare (Lemma 3 and Lemma 7 (a)), and satisfy complementary slackness (Lemma 7 (b)).\n\n[^0]Our first lemma is a main structural tool to link different items' virtual values and is tightly connected to monotone MRS. For $k \\in[d], i \\in[n]$, and flow $\\lambda$, denote the normalized virtual value by\n\n$$\n\\nu_{i}^{\\lambda, k}:=\\frac{\\phi_{i}^{k, \\lambda}}{\\theta_{i}^{k}}\n$$\n\nThe property that we will use repeatedly is that $\\nu_{i}^{\\lambda, k}$ has the same sign as $\\phi_{i}^{\\lambda, k}$. We call a flow downward if $\\lambda_{j i}>0$ for $i, j \\in[n]$ implies that $j>i$.\n\nLemma 2. Let $\\Theta$ have monotone MRS. For any non-negative downward flow $\\lambda, \\nu_{i}^{\\lambda, k} \\geq \\nu_{i}^{\\lambda, l}$ for any $1 \\leq k \\leq l \\leq d$ and $i \\in[n]$.\n\nProof. It follows from definitions and monotone marginal rates of substitution that\n\n$$\n\\begin{aligned}\n\\frac{\\phi_{i}^{\\lambda, k}}{\\theta_{i}^{k}} & =\\frac{\\theta_{i}^{k}-\\frac{1}{f_{i}} \\sum_{j=1}^{n} \\lambda_{j i}\\left(\\theta_{j}^{k}-\\theta_{i}^{k}\\right)}{\\theta_{i}^{k}}=1+\\frac{1}{f_{i}} \\sum_{j=i}^{n} \\lambda_{j i}-\\frac{1}{f_{i}} \\sum_{j=i}^{n} \\lambda_{j i} \\frac{\\theta_{j}^{k}}{\\theta_{i}^{k}} \\\\\n& \\geq 1+\\frac{1}{f_{i}} \\sum_{j=i}^{n} \\lambda_{j i}-\\frac{1}{f_{i}} \\sum_{j=i}^{n} \\lambda_{j i} \\frac{\\theta_{j}^{l}}{\\theta_{i}^{l}}=\\frac{\\theta_{i}^{l}-\\frac{1}{f_{i}} \\sum_{j=1}^{n} \\lambda_{j i}\\left(\\theta_{j}^{l}-\\theta_{i}^{l}\\right)}{\\theta_{i}^{l}}=\\frac{\\phi_{i}^{\\lambda, l}}{\\theta_{i}^{l}} .\n\\end{aligned}\n$$ $\\square$\n\nThe next Lemma shows that virtual welfare maximization reduces to virtual welfare maximization for the neighboring items, i.e., the last item that a type buys and the first item that a type does not buy-with respect to the MRS order.\n\nLemma 3. Assume $\\Theta$ has monotone MRS and mostly regular and that there exists a non-negative downward flow $\\lambda$ such that for any $i \\in[n]$ such that $i^{k} \\leq$ $i \\leq i^{k+1}$, we have $\\phi_{i}^{\\lambda, k} \\geq 0$ and $\\phi_{i}^{\\lambda, k+1} \\leq 0$. Then, the allocation in (9) maximizes virtual welfare.\n\nProof. Fix $i \\in[n]$ such that $i^{k} \\leq i \\leq i^{k+1}$. Note that as $\\phi_{i}^{\\lambda, k}$ and $\\nu_{i}^{\\lambda, k}$ are positive multiples of each other, Lemma 2 implies the implications\n\n$$\n\\begin{aligned}\n\\phi_{i}^{\\lambda, k+1} \\leq 0 & \\Longrightarrow \\phi_{i}^{\\lambda, l} \\leq 0, \\quad l \\geq k+1 \\\\\n\\phi_{i}^{\\lambda, k} \\geq 0 & \\Longrightarrow \\phi_{i}^{\\lambda, l} \\geq 0, \\quad l<k .\n\\end{aligned}\n$$\n\nTherefore, the assumption implies that $\\phi_{i}^{\\lambda, l} \\leq 0$ for any $l>k$ and $\\phi_{i}^{\\lambda, l} \\geq 0$ for any $l \\leq k$, which ensures virtual welfare maximization of (9). $\\square$\n\nFor $k=0$ and $k=d$ this Lemma reduces virtual welfare maximization for all items, and ironing for all items, to virtual welfare maximization for the first resp. last item. Finding a flow that maximizes virtual welfare reduces to ironing the (one-dimensional) virtual values $\\phi_{i}^{1}$ and $\\phi_{i}^{d}$. For types $i \\leq i^{1}$ and $i \\geq i^{d}$,\nwe can hence use techniques from one-dimensional ironing and iron the pseudorevenue to its concave closure in a discrete variant of ?'s procedure. From now, our discussion therefore focuses on $k \\in[d-1]$ and $i \\in\\left[i^{k}+1, i^{k+1}\\right]$, i.e. types where an ironing that ensures virtual welfare maximization for both item $k$ and item $k+1$ is needed.\n\nThe following algorithm will make use of $\\hat{\\lambda}$ as defined in (7), the initial flow and of a generalization of the pseudo-revenue. The pseudo-revenue associated to a flow $\\lambda, R_{i}^{\\lambda, k}$ is\n\n$$\nR_{i}^{\\lambda, k}=\\sum_{j=i}^{n} f_{j} \\phi_{j}^{\\lambda, k}\n$$\n\nThis generalization is intuitive, as virtual values are, as in (5), slopes of pseudorevenues\n\n$$\n\\frac{R_{i}^{\\lambda, k}-R_{i+1}^{\\lambda, k}}{f_{i}}=\\frac{\\sum_{j=i}^{n} f_{j} \\phi_{j}^{\\lambda, k}-\\sum_{j=i+1}^{n} f_{j} \\phi_{j}^{\\lambda, k}}{f_{i}}=\\phi_{i}^{\\lambda, k} .\n$$\n\nOur algorithm will adjust a flow by raising one point in a revenue sequence at a time, from right to left. We will prove that this will yield slopes of revenue sequences-i.e. virtual values-which have the correct sign for virtual welfare maximization of (9). This is non-trivial, as pseudo-revenues for different items might not move in the same direction when dual variables are changed.\n\n$$\n\\begin{aligned}\n& \\lambda \\leftarrow \\hat{\\lambda} ; \\\\\n& \\text { for } i=n \\text { to } 1 \\text { do } \\\\\n& \\qquad \\begin{array}{l}\n\\text { Let } \\gamma_{i} \\in[0,1] \\text { be maximal such that for } \\\\\n\\lambda_{j i}^{\\prime} \\leftarrow \\gamma_{i} \\lambda_{j i}, \\quad \\forall j: n>j>i \\\\\n\\lambda_{j(i-1)}^{\\prime} \\leftarrow \\lambda_{j(i-1)}+\\left(1-\\gamma_{i}\\right) \\lambda_{j i}, \\quad \\forall j: n>j>i \\\\\n\\lambda_{i(i-1)}^{\\prime} \\leftarrow \\lambda_{i(i-1)}-\\left(1-\\gamma_{i}\\right) \\sum_{i^{\\prime}=i}^{n} \\lambda_{i^{\\prime} i} \\\\\nR_{i}^{\\lambda^{\\prime}, \\kappa(i)}=\\bar{R}_{i}^{\\kappa(i)} \\text { holds; } \\\\\n\\lambda \\leftarrow \\lambda^{\\prime} ;\n\\end{array}\n\\end{aligned}\n$$\n\nReturn $\\lambda^{\\prime}$;\nAlgorithm: Ironing, parameterized by an ironing mapping $\\kappa:[n] \\rightarrow[d]$\nThe flow (11) was used earlier in Haghpanah and Hartline (2020). An important difference is that Haghpanah and Hartline (2020) choose $\\gamma_{i}$ to iron the revenue sequence of the grand bundle to the concave closure of pseudo-revenue. Instead, we iron to the quasi-concave closure of (their equivalent of) pseudorevenue of an item $\\kappa(i)$. The parameter $\\gamma_{i}$ can be found as solution to a system of linear equations. We show that a solution $\\gamma_{i} \\in[0,1]$ exists in Lemma 6.","text_sha256":"d517f256b23f3ba1b1b4bb5e5e3f21b12dfae8018acbce6396e86aadcad65655"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0013","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Optimal Mechanisms for Non-Regular Distributions","text":"We first observe that the Ironing Algorithm outputs a flow which is nonnegative and feasible.\n\nLemma 4. The output of the Ironing Algorithm is a flow, i.e. non-negative and satisfies flow preservation, Lemma 1 item 3.\n\nProof. We prove the claim by induction from $i=n$ to $i=1$. Note that $\\hat{\\lambda}$ is feasible as argued in the proof of Theorem 1, which starts the induction. Fix an arbitrary iteration $i \\in[n]$, and assume that $\\lambda$ is feasible. We check that the difference in excess flow, i.e. incoming and outgoing flow, cancel out for $j>i$. For $i$ and $i+1$ similar calculations yield the result. We omit these. For any $j>i$,\n\n$$\n\\lambda_{j i}^{\\prime}-\\lambda_{j i}+\\lambda_{j(i-1)}^{\\prime}-\\lambda_{j(i-1)}=\\gamma \\lambda_{j i}-\\lambda_{j i}+\\lambda_{j(i-1)}+\\left(1-\\gamma_{i}\\right) \\lambda_{j i}-\\lambda_{j(i-1)}=0 .\n$$\n\nNow consider non-negativity. Each $\\lambda_{i j}$ reduces at most once during the course of the Ironing Algorithm. More specifically, only if $j=i-1$ and during iteration $i$. In this iteration,\n\n$$\n\\lambda_{i(i-1)}^{\\prime}=\\lambda_{i(i-1)}-\\left(1-\\gamma_{i}\\right) \\sum_{i^{\\prime}=i}^{n} \\lambda_{i^{\\prime} i} \\geq \\lambda_{i(i-1)}-\\sum_{i^{\\prime}=i}^{n} \\lambda_{i^{\\prime} i}=f_{i} \\geq 0,\n$$\n\nwhere we used that $\\gamma_{i} \\leq 1, \\lambda_{i r}=\\hat{\\lambda}_{i r}, r<i-1$, which in particular implies that $\\lambda_{i r}=0$, and feasibility of the flow. $\\square$\n\nNext observe that in the Ironing Algorithm, iteration $i$ changes the revenue (for any item $k$ ) only for type $i$. Hence, our ironing algorithm raises pseudorevenue for one type at a time.\n\nLemma 5. For any iteration $i, R_{j}^{\\lambda^{\\prime}, k}=R_{j}^{\\lambda, k}$ for any $j \\neq i$. In particular, $\\phi_{j}^{\\lambda^{\\prime}, k}=\\phi_{j}^{\\lambda, k}$ for $j \\notin\\{i-1, i\\}$.\n\nProof. First note that as the in-flow for higher types remains unchanged in iteration $i, \\lambda_{r j}^{\\prime}=\\lambda_{r j}, r \\in[n], j>i$, the revenue does not change, $R_{j}^{\\lambda^{\\prime}, k}=R_{j}^{\\lambda, k}$. For types $j<i$, we check that the changes to virtual welfare on the types whose inflows do change, $i$ and $i-1$, cancel out. By definition of virtual values,\n\n$$\n\\begin{aligned}\n\\phi_{i}^{\\lambda^{\\prime}, k} & =\\phi_{i}^{\\lambda, k}+\\frac{1-\\gamma_{i}}{f_{i}} \\sum_{j=i}^{n} \\lambda_{j i}\\left(\\theta_{j}^{k}-\\theta_{i-1}^{k}\\right) \\\\\n\\phi_{i-1}^{\\lambda^{\\prime}, k} & =\\phi_{i-1}^{\\lambda, k}+\\frac{1-\\gamma_{i}}{f_{i-1}} \\sum_{j=i}^{n} \\lambda_{j i}\\left(\\theta_{j}^{k}-\\theta_{i-1}^{k}\\right)-\\frac{1-\\gamma_{i}}{f_{i-1}} \\sum_{j=i}^{n} \\lambda_{j i}\\left(\\theta_{j}^{k}-\\theta_{i-1}^{k}\\right) \\\\\n& =\\phi_{i}^{\\lambda, k}-\\frac{1-\\gamma_{i}}{f_{i}} \\sum_{j=i}^{n} \\lambda_{j i}\\left(\\theta_{j}^{k}-\\theta_{i-1}^{k}\\right) .\n\\end{aligned}\n$$\n\nHence,\n\n$$\nf_{i-1} \\phi_{i-1}^{\\lambda^{\\prime}, k}+f_{i} \\phi_{i}^{\\lambda^{\\prime}, k}=f_{i-1} \\phi_{i-1}^{\\lambda, k}+f_{i} \\phi_{i}^{\\lambda, k}\n$$\n\nThe statement on the virtual values follows from (10). $\\square$\n\nBefore showing that $\\gamma_{i}$ in the algorithm always exists, we define the ironing function $\\kappa(i)$.\n\nBy no ironing on neighboring maxima, each candidate ironing interval $I$ must be contained in an interval $\\left\\{i^{k}, i^{k}+1, \\ldots, i^{k+1}\\right\\}$. By this condition, in addition to\nno partial overlap, for each type $i$, there is a unique inclusion maximal candidate among the candidate ironing intervals for items $k$ and $k+1$. We let $\\kappa(i)$ denote the item this interval is a candidate ironing interval for. If $i$ is not part of any ironing interval, we set $\\kappa(i)$ arbitrarily in $\\{k, k+1\\}$. We call $\\kappa(i)$ the ironed item for type $i$ and piece-wise constant intervals of $\\kappa$ ironing intervals.\n\nLemma 6. Assume that $F$ is mostly regular. Then, for each $i \\in[n], \\gamma_{i}$ such that $R_{i}^{\\lambda_{i}\\left(\\gamma_{i}\\right), \\kappa(i)}=\\bar{R}_{i}^{\\kappa(i)}$ exists. In particular, the Ironing Algorithm is well-defined.\n\nProof. We prove this statement by induction from $i=n$ down to 1 . Let $i \\in[n]$ and assume that $R_{i+1}^{\\lambda, \\kappa(i)}=\\bar{R}_{i+1}^{\\lambda, \\kappa(i)}$. If $i$ is not part of an ironing interval, then by definition of ironing intervals and Lemma $5, R_{i}^{\\lambda, k}=\\bar{R}_{i}^{\\lambda, k}$, and the induction step is trivial by choosing $\\gamma_{i}=1$, yielding $R_{i}^{\\lambda^{\\prime}(1), k}=\\bar{R}_{i}^{\\lambda^{\\prime}(1), k}$. Otherwise, $i$ is in an ironing interval. Let $\\kappa(i)=k$. By no partial overlap, if $i+1$ is part of an ironing interval, it must be part of the same ironing interval, in particular must have been ironed for item $k$. Hence, by the induction hypothesis, $R_{i+1}^{\\lambda, k}=\\bar{R}_{i+1}^{\\lambda, k}$.\n\nDenote\n\n$$\n\\bar{\\phi}_{i}^{k}=\\frac{\\bar{R}_{i}^{k}-\\bar{R}_{i+1}^{k}}{f_{i}}\n$$\n\nthe slope of the quasi-concave closure of pseudo-revenue of item $k$ at type $i$. By definition of the quasi-concave closure, the slope of the revenue curve must be non-positive,\n\n$$\n\\bar{\\phi}_{i}^{k} \\leq 0 .\n$$\n\nAs all types are non-negative, we get that\n\n$$\n\\overline{\\phi_{i}^{k}} \\leq 0 \\leq \\theta_{i}^{k}=\\phi_{i}^{\\lambda^{\\prime}(0), k} \\text {. }\n$$\n\nAgain by Lemma 5, $\\bar{R}_{i+1}^{k}=R_{i+1}^{\\lambda^{\\prime}(0), k}$. Therefore\n\n$$\n\\begin{aligned}\n\\bar{R}_{i}^{k} & =f_{i} \\overline{\\phi_{i}^{k}}+\\bar{R}_{i+1}^{k}=f_{i} \\overline{\\phi_{i}^{k}}+R_{i+1}^{\\lambda^{\\prime}(0), k} \\\\\n& \\leq f_{i} \\phi_{i}^{\\lambda^{\\prime}(0), k}+R_{i+1}^{\\lambda^{\\prime}(0), k}=R_{i}^{\\lambda_{i}(0), k}\n\\end{aligned}\n$$\n\nIn particular, $\\bar{R}_{i}^{k} \\leq R_{i}^{\\lambda_{i}(0), k}$.\nAlso, by Lemma 5 and the definition of the quasi-concave closure, $R_{i}^{\\lambda^{\\prime}(1), k}=$ $R_{i}^{\\lambda, k} \\leq \\bar{R}_{i}^{k}$. As $\\gamma \\mapsto R_{i}^{\\lambda^{\\prime}}(\\gamma), k$ is a continuous function, the existence of the desired $\\gamma \\in[0,1]$ follows from the Intermediate Value Theorem. $\\square$\n\nThe last lemma before the proof of Theorem 2 shows that the output of the algorithm satisfies complementary slackness and the condition of Lemma 3, which is sufficient for virtual welfare maximization.\n\nLemma 7. Assume that $\\Theta$ is has monotone MRS, and that $F$ is mostly regular. Then, $q$ maximizes virtual welfare and satisfies the requirements of Lemma 3 with respect to $\\lambda^{\\prime}$, the output of the Ironing Algorithm.","text_sha256":"824de71ff7f2b5790c2fbc967c507f1833f1753e84a912f50b03db1455243f3b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0014","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4 Optimal Mechanisms for Non-Regular Distributions","text":"> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFig. 4: Ironing of virtual values and corresponding pseudo-revenues.\n\nProof of Lemma 7. We first show that $\\lambda^{\\prime}$ satisfies the requirements of Lemma 7, i.e. that for any $i^{k} \\leq i \\leq i^{k+1}$ we have that $\\phi_{i}^{\\lambda, k} \\geq 0$ and $\\phi_{i}^{k+1, \\lambda} \\leq 0$. Let $i \\in[n]$. If $i$ is not in an ironing interval, and there is no ironing interval $I$ such that $i=\\min I-1$, the claim follows from the definition of the quasi-concave closure and our definition of ironing intervals, as well as Lemma 5.\n\nWe consider the remaining cases $i=\\max I, i=\\min I-1$, and $i \\in I \\backslash\\{\\max I\\}$ separately. For the third case, by definition of the ironing and the quasi-concave closure,\n\n$$\n\\phi_{i}^{\\lambda^{\\prime}, \\kappa(i)}=\\nu_{i}^{\\lambda^{\\prime}, \\kappa(i)}=0 .\n$$\n\nFor $\\kappa(i)=k, \\nu_{i}^{k+1, \\lambda} \\leq 0$ by Lemma 2 and hence $\\phi_{i}^{k+1, \\lambda} \\leq 0$ as $\\nu_{i}^{k+1, \\lambda}$ and $\\phi_{i}^{k+1, \\lambda}$ are positive multiples of each other. Similarly, we have for $\\kappa(i)=k+1$ that $\\nu_{i}^{k, \\lambda} \\geq 0$ by Lemma 2 and hence $\\phi_{i}^{k, \\lambda} \\geq 0$. It remains to consider $i=\\min I-1$ and $i=\\max I$. We consider these separately for $\\kappa(i)=k$ and $\\kappa(i)=k+1$, for a total of four cases.\n\nCase 1. $i=\\max I$ and $\\kappa(i)=k$. By definition of the quasi-concave closure, $\\bar{R}_{i}^{k}=R_{i}^{k}$, the algorithm chooses $\\gamma_{i}=1$ and hence $\\phi_{i}^{\\lambda^{\\prime}, k}=\\phi_{i}^{k}=0$ and hence by arguments as above, $\\phi_{i}^{\\lambda^{\\prime}, k+1}=\\phi_{i}^{k+1} \\leq 0$.\n\nCase 2. $i=\\min I-1$ and $\\kappa(i)=k+1$. By definition of the quasi-concave closure, $\\overline{R_{i}^{k+1}}=R_{i}^{k+1}$, the algorithm chooses $\\gamma_{i}=1$ and hence, by an argument similar to case $1, \\phi_{i}^{\\lambda, k}=\\phi_{i}^{k} \\geq 0$.\n\nCase 3. $i=\\max I$ and $\\kappa(i)=k+1$.\nThe derivative of the virtual value of the right end of the ironing interval, type $i$ is given by\n\n$$\n\\frac{\\partial \\phi_{i}^{\\lambda^{\\prime}, k}}{\\partial \\gamma}=-\\frac{1}{f_{i}} \\sum_{j=i+1}^{n} \\lambda_{j i}\\left(\\theta_{j}^{k}-\\theta_{i}^{k}\\right)=-\\frac{1}{f_{i}} \\lambda_{(i+1) i}\\left(\\theta_{i+1}^{k}-\\theta_{i}^{k}\\right) .\n$$\n\nBy no partial overlap, $0 \\leq \\phi_{i}^{k}$. Also, by not too shuffledness, (13) is nonpositive. Hence, $\\phi_{i}^{k}=\\phi_{i}^{\\gamma^{\\prime}(0), k} \\leq \\phi_{i}^{\\lambda^{\\prime}, k}$, and the algorithm chooses $\\gamma_{i}<1$. Combining these observations, we obtain $\\phi_{i}^{k, \\lambda^{\\prime}} \\geq 0$.\n\nBecause the algorithm chooses $\\gamma_{i}<1$, this implies\n\n$$\n\\phi_{i}^{\\lambda^{\\prime}, k} \\geq \\phi_{i}^{k} \\geq 0 .\n$$\n\nCase 4. $i=\\min I-1$ and $\\kappa(i)=k$. The derivative of the virtual value of the next item at the left end of the ironing interval, type $i$ is given by\n\n$$\n\\begin{aligned}\n\\frac{\\partial \\phi_{i}^{\\lambda^{\\prime}, k+1}}{\\partial \\gamma} & =\\frac{1}{f_{i}} \\sum_{j=i+1}^{n} \\lambda_{j i}\\left(\\theta_{j}^{k+1}-\\theta_{i}^{k+1}\\right)-\\frac{1}{f_{i}} \\sum_{j=i+1}^{n} \\lambda_{j i}\\left(\\theta_{i+1}^{k+1}-\\theta_{i}^{k+1}\\right) \\\\\n& =\\frac{1}{f_{i}} \\sum_{j=i+1}^{n} \\lambda_{j i}\\left(\\theta_{j}^{k+1}-\\theta_{i+1}^{k+1}\\right) \\geq 0,\n\\end{aligned}\n$$\n\nwhere the last inequality follows from not-too-shuffledness (note that $i+1=$ $\\min I)$. By no partial overlap, $0 \\geq \\phi_{i}^{k+1}$. Moreover, $\\phi_{i}^{k+1}=\\phi_{i}^{\\gamma^{\\prime}(0), k} \\geq \\phi_{i}^{\\lambda^{\\prime}, k+1}$ because of (14) and because the algorithm chooses $\\gamma_{i}<1$. Combining these observations, we obtain $\\phi_{i}^{k+1, \\lambda} \\leq 0$.\n\nTo show complementary slackness, observe that whenever $i$ is not in an ironing interval, as $\\gamma_{i}$ is chosen maximal such that $\\bar{R}_{i}^{\\kappa(i)}=R_{i}^{\\kappa(i)}$, the Algorithm chooses $\\gamma_{i}=1$, which implies that for $j>i>r$ and for $j>i+1$ and $i=r$, $\\lambda_{j r}^{\\prime}=0$. By no ironing over maxima, this implies that for $j>i^{k}>r, \\lambda_{j r}=0$. Moreover, as an invariant of the algorithm the flow $\\lambda$ is downward. Hence, the only dual variables that are tight are within types that get the same allocation and payment (corresponding to $\\lambda_{i j}$ such that $i^{k} \\leq i, j \\leq i^{k+1}, k \\in[d]$ ) or local downward constraints (corresponding to $\\lambda_{(i+1) i}, i \\in[n-1]$ ). The former incentive constraints clearly bind, the latter bind by weak monotonicity, as for the marginally buying type (which, by compatibility of weak monotonicity with mostly regularity must also be the first type in the MRS order), the price could be raised if she were not indifferent between her allocation and payment and the allocation and payment of the next lower type. $\\square$\n\nHaving this result, we are ready to finish the proof of Theorem 2.\nProof of Theorem 2. Implementability follows from weak monotonicity and the definition of the optimal mechanism, (9). Non-negativity and feasibility of flow are properties of the Ironing Algorithm shown in Lemma 4. Virtual welfare maximization and complementary slackness have been shown in Lemma 7. $\\square$","text_sha256":"39a7af15248e2adc0c78e1dcc19bc3901c74aed952c6099a2ad6a5584cb64265"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0015","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"5 Upgrade Pricing and Separate Pricing","text":"## 5 Upgrade Pricing and Separate Pricing\n\nIn both Theorem 1 and Theorem 2, we established the optimality of an upgrade pricing mechanism that yields the same allocation as separate (item by item) monopoly pricing, though not necessarily the same transfers. We will show in this section that, under monotonicity with respect to the component-wise partial order, separate pricing and upgrades become equivalent-upgrade pricing is redundant.\n\nWe say that the type space $\\Theta$ is monotone if $\\theta_{i}^{k} \\leq \\theta_{j}^{k}$ for any $i<j \\in[n]$ and $k \\in[d]$.\n\nWe call a mechanism separate pricing if a type separately chooses whether to buy each item $k$ at a price $p_{k}$. Formally, a mechanism satisfies separate pricing if it can be written as:\n\n$$\nq_{i}^{k}=\\left\\{\\begin{array}{ll}\n1 & \\theta_{i}^{k} \\geq p_{k} \\\\\n0 & \\text { else, }\n\\end{array} \\quad t_{i}=\\sum_{k=1}^{d} p_{k} \\mathbb{1}_{q_{i}^{k}=1}\\right.\n$$\n\nTheorem 3. If the type space $\\Theta$ is monotone, then the outcome of any upgrade pricing mechanism can be implemented via separate pricing, and conversely. When the type space is not monotone, neither implication needs to hold.\n\nProof. We first assume types are monotone and show that the allocation and revenue of any upgrade pricing mechanism can be obtained through a separate pricing mechanism, and vice versa.\n\nLet $\\theta_{1} \\leq \\cdots \\leq \\theta_{i} \\leq \\cdots \\leq \\theta_{n}$, and fix an upgrade pricing mechanism $\\mathcal{M}$. This mechanism admits an indirect representation as (a) a collection of bundles ranked by set inclusion $\\left\\{b_{k}\\right\\}_{k=0}^{K}$, with $b_{0}=\\varnothing$ and $K \\leq d$, and (b) a vector of prices $t_{k}$ that are increasing in $k$, with $t_{0}=0$. Let $\\underline{\\theta}_{k}$ and $\\bar{\\theta}_{k}$ denote the lowest and highest types who choose bundle $b_{k}$ under mechanism $\\mathcal{M}$. Because types are monotone, buyer self-selection implies $\\underline{\\theta}_{k} \\geq \\bar{\\theta}_{k-1}$.\n\nWe now construct a separate pricing mechanism, i.e., a vector of prices $\\left\\{p_{j}\\right\\}_{j=1}^{d}$ that yields the same allocation and payments as our upgrade pricing mechanism. To do so, define the collection of upgrades $u_{k}:=b_{k} \\backslash b_{k-1}$ and the upgrade prices $\\tau_{k}:=t_{k}-t_{k-1}$. For each upgrade bundle $k$ and every good $j \\in u_{k}$, let the single-item prices $p_{j}$ satisfy\n\n$$\np_{j} \\in\\left[\\bar{\\theta}_{k-1}^{j}, \\underline{\\theta}_{k}^{j}\\right] \\quad \\text { and } \\quad \\sum_{j \\in u_{k}} p_{j}=\\tau_{k} .\n$$\n\nUnder monotonicity, such a vector of prices always exists. By consumer selfselection in the original mechanism $\\mathcal{M}$, we have\n\n$$\n\\begin{aligned}\n\\bar{\\theta}_{k-1} b_{k}-t_{k} & \\leq \\bar{\\theta}_{k-1} b_{k-1}-t_{k-1}, \\\\\n\\underline{\\theta}_{k} b_{k-1}-t_{k-1} & \\leq \\underline{\\theta}_{k} b_{k}-t_{k} .\n\\end{aligned}\n$$\n\nIn turn, this implies\n\n$$\n\\bar{\\theta}_{k-1} u_{k} \\leq \\tau_{k} \\leq \\underline{\\theta}_{k} u_{k} .\n$$\n\nWith the prices so constructed, each type purchases the same goods as under $\\mathcal{M}$ and pays the same total price. Notice first that each type's choice from the original mechanism $\\mathcal{M}$ is still available at the same price, i.e., each bundle $b_{k}$ can still be purchased for a total price $t_{k}$. Moreover, by monotonicity, no type $\\theta$ who buys bundle $b_{k}$ under the upgrade pricing mechanism $\\mathcal{M}$ derives positive net surplus from any object $j \\in u_{k^{\\prime}}$ with $k^{\\prime}>k$ under the separate prices\nconstructed above. And finally, no such type $\\theta$ derives positive net surplus by removing any object $j \\in u_{k^{\\prime}}$ with $k^{\\prime} \\leq k$ from her consumption bundle.\n\nThe other direction of this result is immediate: if types are monotone, the goods purchased by two different types under any separate pricing mechanism are ranked by set inclusion. Thus, replacing the separate pricing mechanism with the resulting upgrade pricing mechanism yields the same outcome.\n\nFinally, we show by means of two counterexamples that, without type monotonicity, separate pricing is not equivalent to upgrade pricing.\n\nIn particular, there exist type spaces and vectors of separate prices that do not induce an upgrade pricing allocation. For example, let\n\n$$\n\\Theta=\\{(1,1),(1,3),(3,3),(4,1)\\}\n$$\n\nand consider the separate prices $p=(2,2)$ : type $\\theta_{2}$ buys good 2 only, type $\\theta_{3}$ buys both goods, and type $\\theta_{4}$ buys good 1 only.\n\nLikewise, for the same type space, consider the upgrade pricing mechanism where $q=(0,1)$ is sold for $t=2$ and $q=(1,1)$ is sold for $t=4$, i.e., good $j=1$ is only sold as an upgrade, for an additional price $\\tau=2$. Under this mechanism, type $\\theta_{2}$ buys good 2 only, while types $\\theta_{3}$ and type $\\theta_{4}$ buy both goods. However, as we saw above, the vector of separate prices $p=(2,2)$ yields a different allocation (and a lower revenue for the seller). $\\square$\n\nWhenever an upgrade pricing mechanism implements the allocation of optimal separate pricing, each marginal type $\\underline{\\theta}_{k}$ is indifferent by construction between the two consecutive bundles $b_{k-1}$ and $b_{k}$. Theorem 3 then implies that the outcome of this mechanism can be implemented by the separate monopoly prices.\n\nCorollary 1. If $\\Theta$ is monotone, $q$ is an allocation of an optimal upgrade pricing mechanism, and $q$ is the allocation of separate monopoly pricing, then separate monopoly pricing is optimal.\n\nAdding a monotonicity condition to both of our main theorems, Theorem 1 and Theorem 2, we hence obtain two sets of sufficient conditions under which separate monopoly pricing is optimal.\n\nCorollary 2. If $\\Theta$ is monotone and $F$ is regular, separate monopoly pricing is optimal.\n\nCorollary 3. If $\\Theta$ is monotone and has a monotone marginal rates of substitution, and $F$ is mostly regular, then separate monopoly pricing is optimal.","text_sha256":"44cd6584b70257f18ab580e317682d8466d62d46906258194ec9708842b5b636"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0016","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"6 Conclusion","text":"## 6 Conclusion\n\nIt is a common practice for a seller to offer bundles of products or services that are ordered in a way that more expensive bundles contain all items from less expensive bundles as well as some extra items. In this paper, we provide\nsufficient conditions under which such \"upgrade pricing\" schemes are exactly optimal for a monopolist seller.\n\nThere are several ways in which the current analysis could be extended. First, our conditions could be relaxed to account for richer type spaces and type distributions, such as a continuum of types in the d-dimensional space. One natural extension can be obtained immediately: assume that a type distribution can be split into several type cohorts, in fact quantized type space, such that each type cohort satisfies the conditions of our theorems. Our results imply that the optimal mechanisms in each respective cohort are upgrade pricing. In this respect, Bergemann et al. (2021) show that in nonlinear pricing problems, the revenue of the continuous type space is generally well approximated by a finite quantized type space.\n\nSecond, our sufficient conditions for the optimality of upgrade pricing may be complemented by necessary conditions. In doing so, one may want to distinguish between conditions on type distributions and type spaces. For example, one may ask which type spaces guarantee that upgrade pricing is optimal irrespective of the type distribution.\n\nFinally, throughout the paper we highlight the interplay between optimality of different pricing schemes: bundling, upgrade pricing, and separate sales. It would be instructive to provide a more complete characterization of the cases in which one of these schemes strictly outperforms another.\n\nAcknowledgements We thank Mark Armstrong and the seminar audience at MIT for helpful comments. Bergemann and Bonatti acknowledge financial support through NSF SES 1948336. Smolin acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d'Avenir) program (grant ANR-17-EURE-0010).","text_sha256":"c01afcce3836bbf540e7dc2796c48639e740a8487388217d4c2fa9fb3b8b3592"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0017","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Bibliography","text":"## Bibliography\n\nW. J. Adams and J. L. Yellen. Commodity Bundling and the Burden of Monopoly. Quarterly Journal of Economics, 90(3):475-498, 1976.\nM. Armstrong. Nonlinear pricing. Annual Review of Economics, 8:583-614, 2016.\nM. Babaioff, N. Immorlica, B. Lucier, and S. M. Weinberg. A Simple and Approximately Optimal Mechanism for an Additive Buyer. 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 21-30, 2014.\nD. Bergemann, E. Yeh, and J. Zhang. Nonlinear pricing with finite information. Games and Economic Behavior, 130:62-84, 2021.\nS. Bikhchandani and D. Mishra. Selling two identical objects. arXiv:2009.11545, 2020.\nS. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004.\nY. Cai, N. R. Devanur, and S. M. Weinberg. A Duality Based Unified Approach to Bayesian Mechanism Design. Proceedings of the Annual ACM Symposium on Theory of Computing, 2016.\nC. Daskalakis, A. Deckelbaum, and C. Tzamos. Strong Duality for a MultipleGood Monopolist. Econometrica, 85(3):735-767, 2017.\nN. R. Devanur, K. Goldner, R. R. Saxena, A. Schvartzman, and S. M. Weinberg. Optimal mechanism design for single-minded agents. In Proceedings of the 21st ACM Conference on Economics and Computation, pages 193-256, 2020.\nG. Ellison. A Model of Add-On Pricing. Quarterly Journal of Economics, 120 (2):585-637, 2005.\nA. Fiat, K. Goldner, A. R. Karlin, and E. Koutsoupias. The fedex problem. In Proceedings of the 2016 ACM Conference on Economics and Computation, pages 21-22, 2016.\nS. Ghili. A Characterization for Optimal Bundling of Products with Interdependent Values. Technical report, Yale University, 2021.\nN. Haghpanah and J. Hartline. When Is Pure Bundling Optimal? The Review of Economic Studies, 88(3):1127-1156, 082020.\nS. Hart and N. Nisan. Approximate revenue maximization with multiple items. Journal of Economic Theory, 172:313-347, 2017.\nJ. P. Johnson and D. P. Myatt. Multiproduct quality competition: Fighting brands and product line pruning. American Economic Review, 93(3):748-774, 2003.\nB. H. Korte and J. Vygen. Combinatorial Optimization. Springer, 2011.\nA. M. Manelli and D. R. Vincent. Bundling as an optimal selling mechanism for a multiple-good monopolist. Journal of Economic Theory, 127(1):1-35, 2006.\nR. P. McAfee, J. McMillan, and M. D. Whinston. Multiproduct Monopoly, Commodity Bundling, and Correlation of Values. Quarterly Journal of Economics, 104(2):371-383, 1989.\nG. Pavlov. Optimal mechanism for selling two goods. The BE Journal of Theoretical Economics, 11(1), 2011.\nJ. Philips. Don't look now, but the great unbundling has spun into reverse. New York Times, 2017.\n\n[^0]:    ${ }^{5}$ In section 5, we further explore the relationship between upgrade pricing and separate pricing, by showing conditions under which the allocation (9) can be implemented by a vector of single-item prices.","text_sha256":"00e625d62c7deb95c2931612112e029161011e595536c316ad687ae92cf0b227"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02:0018","work_id":"alex-smolin:the-optimality-of-upgrade-pricing","paper_id":"alex-smolin:the-optimality-of-upgrade-pricing:2021-12-02","title":"The Optimality of Upgrade Pricing","authors":[{"name":"Dirk Bergemann","url":"https://campuspress.yale.edu/dirkbergemann/"},{"name":"Alessandro Bonatti","url":"https://mitmgmtfaculty.mit.edu/abonatti/"},{"name":"Andreas Haupt","url":"https://www.andyhaupt.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2021-12-02","language":"en","version_type":"author-manuscript","canonical_url":"https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md","source_record":"https://arxiv.org/abs/2107.10323","doi":"https://doi.org/10.1007/978-3-030-94676-0_3","citation":"Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Dirk Bergemann; Alessandro Bonatti; Andreas Haupt; Alex Smolin\n\n**Canonical citation:** Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/the-optimality-of-upgrade-pricing.md\n\n**Source record:** https://arxiv.org/abs/2107.10323\n\n**Published record:** https://doi.org/10.1007/978-3-030-94676-0_3\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"d316f7b4bb49eaad4985fb0ffefa23814e93c0b2cac1c98269a2c26a5902075f"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0001","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Document metadata","text":"> Machine-readable author manuscript.\n> Authors: Laura Doval; Alex Smolin.\n> Canonical citation: Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.\n> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"0068c61787e62aa4e9ab13086c266536863550ca16b0d209957b750402ee87a6"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0002","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"The Welfare Impact of Recommendation Algorithms","text":"# The Welfare Impact of Recommendation Algorithms\n\n**Authors:** Laura Doval; Alex Smolin\n\n**Manuscript date:** 2025\n\n#### Abstract\n\nIn this letter, we summarize our recent work on the welfare impact of recommendation algorithms and propose questions for further study. We model recommendation algorithms as an information structure, which shapes how a third party takes actions that affect the welfare of different individuals in a population. Each recommendation algorithm thus induces a welfare profile, describing the expected payoffs of different individuals when the third party takes actions following the algorithm. Our framework allows us to characterize and compute the set of all such profiles, which we dub the Bayes welfare set. The Bayes welfare set allows us to reduce society's choice of an algorithm to the choice of a Bayes welfare profile. Our framework complements that of the algorithmic fairness literature which remains agnostic about the population's payoffs, focusing instead on statistical properties of algorithms, such as accuracy, parity, or fairness.\n\nCategories and Subject Descriptors: [Social and Behavioral Sciences]: Economics-\nGeneral Terms: Economics, Theory\nAdditional Key Words and Phrases: Recommendation algorithms, fairness, persuasion, information structures","text_sha256":"4e336430cfa35604aed6123bd3b6b8e19b713865c39144877caba343c68fbcca"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0003","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"1. INTRODUCTION","text":"## 1. INTRODUCTION\n\nInformation has increasingly become a tool for shaping society's choices in highstakes domains. While this phenomenon is not exclusive to the \"big data\" economy-consider the role of scores and ratings in school placement, promotions, and credit allocation-the rise of algorithmic recommendations has highlighted society's growing reliance on information in policy-relevant domains. Consider, for instance, the role of algorithms in recommending who obtains bail [Angwin et al. 2016], credit [Jagtiani and Lemieux 2019], who is hired [Raghavan et al. 2020; Li et al. 2020], and which health treatments to prescribe [Obermeyer et al. 2019], and more recently, generative AI models, which users can leverage as virtual consultants [Immorlica et al. 2024].\n\nThe ever-increasing role of recommendation algorithms in high-stakes domains and their obvious welfare impact has caught the attention of the Computer Science and Economics communities. For instance, [Kearns and Roth 2019; 2020] underscore the importance of having portable definitions of privacy or fairness, which can be coupled with the training model's objective, to produce algorithms with desirable outcomes. Whereas the literature on differential privacy and algorithmic fairness is agnostic about how to measure individuals well-being or the objective of the algorithm designer, [Mullainathan 2018; Kleinberg et al. 2018; Rambachan et al. 2020] argue for letting the social planner's objective determine the properties of algorithms.\n\nIn [Doval and Smolin 2021; 2024], we provide a framework to study the welfare impact of recommendation algorithms on a population of heterogeneous individuals. Our framework marries welfare economics and information design. It integrates welfare economics because a primitive of our environment is a measure of individual welfare, which could represent the actual utility function of individuals in the society, or the social planner's perception of this utility. It also draws from information design because recommendation algorithms fundamentally operate as information structures, which provide noisy signals about an underlying state of the world to a decision maker who ultimately takes actions on behalf of the individuals. As such, recommendation algorithms are inherently bounded in their ability to generate welfare, the same way an information designer is bounded in their ability to persuade a receiver to take a given action [Dughmi 2017].\n\nOur primary goal in this note is to introduce the readers to our framework, based on illustrations of the results in [Doval and Smolin 2021] and [Doval and Smolin 2024]. Section 2 introduces the simplest version of our framework to lay down the concepts in the simplest terms. In Section 3, we extend the framework so that it is closest to that in algorithmic fairness. We conclude by pointing out applications of our framework to information design and directions for future research at the intersection of Computer Science and Economics.","text_sha256":"0a4b37dadf42727aff04e48ed5b8f65ef705f157ffa19bed7027027fe8537700"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0004","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2. BASIC FRAMEWORK","text":"## 2. BASIC FRAMEWORK\n\nIn the basic model, a unit mass population of individuals have types in a finite set $\\Theta=\\left\\{\\theta_{1}, \\ldots, \\theta_{N}\\right\\}$, drawn from a full support prior distribution, $\\mu_{0} \\in \\Delta(\\Theta)$. Each individual's welfare depends on her type $\\theta$ and an (unmodeled) outside observer's\nbelief about her type. We represent this by a welfare function $w: \\Delta(\\Theta) \\times \\Theta \\rightarrow \\mathbb{R}$, representing for each belief $\\mu$ and type $\\theta$, the welfare of individuals of type $\\theta$ when the outside observer's belief is $\\mu$. For instance, if individuals' welfare depends on the actions of the outside observer, the welfare function captures in reducedform how the outside observer's action, and hence welfare, changes as the outside observer's beliefs about $\\Theta$ changes. Alternatively, the welfare function may capture that the population's welfare may be driven by image or reputation concerns, like in [Bénabou and Tirole 2006], or psychological motives, as in [Lipnowski and Mathevet 2018].\n\nWe model algorithms as information structures. An information structure $\\Pi=$ $(\\pi, S)$ consists of a countable set of labels $S$, and a mapping $\\pi$, which associates to each type $\\theta$ a distribution over signals $\\pi(\\cdot \\mid \\theta) \\in \\Delta(S)$. Let $\\mu_{s}$ denote the posterior belief given signal $s \\in S$. An information structure induces two kinds of distribution over posterior beliefs $\\left\\{\\mu_{s}: s \\in S\\right\\}$. First, for each $\\theta$, the signal distribution $\\pi(\\cdot \\mid \\theta)$ induces a distribution over posterior beliefs conditional on an individual's type being $\\theta$. Second, the prior $\\mu_{0}$ and the signal distribution induce an unconditional distribution over posterior beliefs. We denote them by $\\langle\\Pi \\mid \\theta\\rangle$ and $\\langle\\Pi\\rangle$, respectively.\n\nThe welfare function $w$ together with an information structure, $\\Pi$, defines a welfare profile, $w_{\\Pi}: \\Theta \\mapsto \\mathbb{R}$, as\n\n$$\nw_{\\Pi}(\\theta)=\\mathbb{E}_{\\langle\\Pi \\mid \\theta\\rangle}[w(\\mu, \\theta)]=\\sum_{s \\in S} \\pi(s \\mid \\theta) w\\left(\\mu_{s}, \\theta\\right) .\n$$\n\nWe denote such a profile, a Bayes welfare profile, and the set of all Bayes welfare profiles, the Bayes welfare set. Formally, the Bayes welfare set is defined as:\n\n$$\n\\mathrm{W} \\equiv\\left\\{\\mathrm{w} \\in \\mathbb{R}^{N}: \\exists \\Pi \\text { s.t. } \\mathrm{w}_{i}=w_{\\Pi}\\left(\\theta_{i}\\right) \\forall i \\in\\{1, \\ldots, N\\}\\right\\} .\n$$\n\nFrom the point of view of welfare economics, the Bayes welfare set admits a classical interpretation: It is the utility possibility set in an economy in which information structures take the role of allocations.\n\nAn apparent difficulty when characterizing the Bayes welfare set is that the Bayes welfare profiles depend on the conditional distributions over posterior beliefs induced by the information structure (cf. Equation (1)). However, we show any Bayes welfare profile satisfies the following:\n\n$$\nw_{\\Pi}(\\theta)=\\mathbb{E}_{\\langle\\Pi \\mid \\theta\\rangle}[w(\\mu, \\theta)]=\\mathbb{E}_{\\langle\\Pi\\rangle}\\left[\\frac{\\mu(\\theta)}{\\mu_{0}(\\theta)} w(\\mu, \\theta)\\right]=\\mathbb{E}_{\\langle\\Pi\\rangle}[\\hat{w}(\\mu, \\theta)] .\n$$\n\nThat is, the expectation of $w$ under $\\Pi$ conditional on $\\theta$ can be expressed as the unconditional expectation of the truth-adjusted welfare function, $\\hat{w}$, under $\\Pi$. The truth-adjusted welfare function, $\\hat{w}$, is the welfare function $w$ adjusted by the truth$d$ rift $\\mu(\\theta) / \\mu_{0}(\\theta)$. For any given posterior belief $\\mu$, the likelihood ratio $\\mu(\\theta) / \\mu_{0}(\\theta)$ measures the representation of type $\\theta$ under $\\mu$ relative to its ex ante representation under $\\mu_{0}$.\n\nIt follows that the Bayes welfare set can be characterized by studying the convex hull of the graph of the vector-valued function, $\\hat{\\mathrm{w}}: \\Delta(\\Theta) \\mapsto \\mathbb{R}^{N}$, where for each $i \\in\\{1, \\ldots, N\\}, \\hat{\\mathrm{w}}_{i}(\\mu) \\equiv \\hat{w}\\left(\\mu, \\theta_{i}\\right)$. Indeed, we have the following:\n\nTheorem 2.1 [Doval and Smolin 2024, Theorem 1]. The Bayes welfare set W satisfies the following:\n\n$$\n\\mathrm{W}=\\left\\{\\mathrm{w} \\in \\mathbb{R}^{N}:\\left(\\mu_{0}, \\mathrm{w}\\right) \\in \\operatorname{co}(\\text { graph } \\hat{\\mathrm{w}})\\right\\},\n$$\n\nwhere co denotes the convex hull operator.\nTheorem 2.1 provides a geometric characterization of the set W : it is the section at the prior of the convex hull of the graph of the truth-adjusted welfare function $\\hat{\\mathrm{w}}$. We illustrate Theorem 2.1 through an example:\n\nExample 2.2 Online marketplace. An online marketplace wants to design a recommendation algorithm, directing consumers to buy from sellers in the platform. For simplicity, assume sellers may be of one of two equally likely types: low quality $\\theta_{1}$, and high quality $\\theta_{2}$. Consumers prefer to buy from high quality sellers. Thus, each seller's profit in the marketplace depends on the likelihood $\\mu$ the consumer attaches to the seller being of high quality. In particular, we assume the sellers' profits as a function of consumers' beliefs are as follows:\n\n$$\nw(\\mu, \\theta)=\\left\\{\\begin{array}{ll}\n0 & \\text { if } \\mu \\in[0,1 / 3) \\\\\n1 / 2 & \\text { if } \\mu \\in[1 / 3,2 / 3) \\\\\n1 & \\text { if } \\mu \\in[2 / 3,1]\n\\end{array} .\\right.\n$$\n\nIn this example, the set W then represents the set of profit profiles sellers with different qualities can attain in the marketplace under some information structure.\n\nFigure 1 illustrates the convex hull of the graph of $\\hat{\\mathrm{w}}$ (Figure 1a) and the Bayes welfare set (Figure 1b) for the online marketplace example. For instance, fully revealing or concealing the sellers' quality is always feasible, so that the full and no-disclosure profiles, $\\mathrm{w}^{F D}$ and $\\mathrm{w}^{N D}$ are feasible. We highlight some properties of the Bayes welfare set:","text_sha256":"e8295eb9a3e5771fe08a847f9e6456893540480f32cfcff6665f5838ba691dc1"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0005","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"2. BASIC FRAMEWORK","text":"- Despite the welfare function being symmetric across seller types, the Bayes welfare set is not symmetric because the adjusted-welfare function is not symmetric. By Bayes rule, when consumers are optimistic about the seller's quality being high, it is more likely they are facing a high rather than a low quality seller.\n- In particular, the Bayes welfare set lies above the 45° line: the only Bayes welfare profile equalizing seller profits is the no disclosure one, but it is not Pareto efficient. In other words, fairness-measured by welfare parity-may be at odds with Pareto efficiency.\n- The Pareto frontier of the Bayes welfare set is given by its north-east boundary. In particular, the flat segment at the top shows the profits of low-quality sellers can be increased without decreasing those of high-quality sellers.\n- The points on the decreasing part of the Pareto frontier can only be generated with at least three signals. By contrast, in standard Bayesian persuasion, two signals are enough in the case of two states. Formally, the analogue of the W in Bayesian persuasion has dimension $N$, whereas the W has dimension $2 N-1$.\n\nBecause the Bayes welfare set is convex, it can be alternatively described by its supporting hyperplanes. [Doval and Smolin 2024, Theorem 2] shows the frontier of\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFig. 1: Constructing the Bayes welfare set in Example 2.2; $\\mathrm{w}^{F D}$ and $\\mathrm{w}^{N D}$ denote the profit profiles under full and no information, respectively.\n\nthe Bayes welfare set can be obtained as the solution to series of Bayesian persuasion problems as in [Kamenica and Gentzkow 2011], in which a utilitarian planner takes the role of the information designer. Concretely, consider the supporting hyperplane of the W in direction $\\lambda \\in \\mathbb{R}^{N} \\backslash\\{0\\}$. Then, the Bayes welfare profiles on the boundary of the W in direction $\\lambda$ can be obtained by solving the Bayesian persuasion problem of a sender with indirect utility\n\n$$\n\\hat{v}_{\\lambda}(\\mu)=\\sum_{\\theta \\in \\Theta} \\mu(\\theta) \\frac{\\lambda(\\theta)}{\\mu_{0}(\\theta)} w(\\mu, \\theta) .\n$$\n\nWe have found this result very useful in computing the Bayes welfare set in applications (see also [Corrao and Dai 2023] for an application to strategic communication).","text_sha256":"9086f183929ce0f1cc6bd36096bd71e53eae044e2e5175196aac4b48019028e7"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0006","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3. BEYOND THE BASIC MODEL: GROUPS AND DATA","text":"## 3. BEYOND THE BASIC MODEL: GROUPS AND DATA\n\nTwo assumptions are implicit in the analysis so far. First, we assume the variable the unmodeled outside observer cares about is the same variable on which we condition the payoffs. Consider, however, an employer making hiring decisions based on a candidate's ability. If candidates belong to different groups, basing hiring recommendations on ability impacts the welfare of candidates across different groups. Second, we assume the information structure can arbitrarily condition on an individual's payoff-relevant type. However, because regulation may prevent the disclosure of protected characteristics, such as gender or race, considering algorithms that respect these restrictions is natural whenever $\\theta$ includes such characteristics.\n\nFormally, we extend the basic model as follows. We now distinguish between three random variables: an individual's group $g \\in G$, the state $\\omega \\in \\Omega$, and data $d \\in D$. The first is the variable we condition payoffs on; the second is the variable of interest to the outside observer; the third allows us to capture limits on the information provided. We let $\\mathbb{P} \\in \\Delta(G \\times \\Omega \\times D)$ denote the joint distribution over group-state-\ndata pairs, and in a slight abuse of notation we denote by $\\mathbb{P}(\\cdot \\mid g)$ and $\\mathbb{P}(\\cdot \\mid d)$ the prior distribution conditional on the individual's group and the data realization, respectively. Below, we denote the marginal of $\\mathbb{P}$ on $D$ by $\\eta_{0} \\in \\Delta(D)$. In a slight abuse of notation, we define the welfare function as $w: \\Delta(\\Omega) \\times \\Omega \\times G \\mapsto \\mathbb{R}$, with its first argument being the (unmodeled) outside observer's belief $\\mu$ about the state $\\omega, \\mu \\in \\Delta(\\Omega)$. The basic model corresponds to the case in which $\\Theta=G=\\Omega=D$ and $\\mathbb{P}(\\omega, d \\mid g)=\\mathbb{1}[g=\\omega=d]$.\n\nTo capture the limits data imposes on information provision, an information structure is now defined as a tuple $(\\pi, S)$, where $\\pi: D \\mapsto \\Delta(S)$. Given the information policy, belief updating about $(g, \\omega, d)$, and hence about $\\omega$, depends only on the updated belief about $d$. Specifically, letting $\\eta_{s}$ denote the updated belief starting from $\\eta_{0}$, after observing signal $s \\in S$, the updated belief on $(g, \\omega, d)$ is given by $\\mathbb{P}(g, \\omega \\mid d) \\eta_{s}(d)$. Without loss of generality, we can write the welfare function as $w_{\\dagger}(\\eta, \\omega, g) \\equiv w(\\mu(\\eta), \\omega, g)$.\n\nGiven an information structure $(\\pi, S)$, the welfare of individuals of group $g$ is:\n\n$$\nw_{\\Pi}(g)=\\sum_{\\eta \\in \\operatorname{supp}(\\Pi)} \\sum_{s \\in S: \\eta_{s}=\\eta} \\sum_{(\\omega, d)} \\mathbb{P}(\\omega, d \\mid g) \\pi(s \\mid d) w_{\\dagger}(\\eta, \\omega, g),\n$$\n\nand the Bayes welfare set continues to be defined as the set of Bayes welfare profiles.\nThe characterization of the Bayes welfare set in the basic model extends verbatim to the more general model, once we observe (the analogue of) the truth-adjusted welfare function now takes the form:\n\n$$\n\\hat{w}_{\\dagger}(\\eta, g)=\\sum_{(\\omega, d)} \\mathbb{P}(\\omega, d \\mid g) \\frac{\\eta(d)}{\\eta_{0}(d)} w_{\\dagger}(\\eta, \\omega, g) .\n$$\n\nEquation (7) allows us to provide further insight into the adjusted welfare function in the basic model. The likelihood correction is now based on the variable $d$, highlighting that it corresponds to the variable on which information is being provided. In addition, the presence of additional uncertainty requires averaging over $\\omega$ and $d$ using weights $\\mathbb{P}(\\omega, d \\mid g)$. Thus we can immediately extend Theorem 2.1 as:\n\nTheorem 3.1 [Doval and Smolin 2021, Theorem 4]. The Bayes welfare set can be calculated as:\n\n$$\n\\mathrm{W}=\\left\\{\\mathrm{w} \\in \\mathbb{R}^{|G|}:\\left(\\eta_{0}, \\mathrm{w}\\right) \\in \\operatorname{co}\\left(\\operatorname{graph} \\hat{\\mathrm{w}}_{\\dagger}\\right)\\right\\} .\n$$\n\nWe note again that it is the prior on data, $\\eta_{0}$, and not on the states which determines the constraint on how much information can be provided about the state of the world, and hence the limits on how much welfare can be generated via information.\n\nExample 3.2 Data Regulation in Hiring. Consider two equally likely groups of workers, labeled $A$ and $B$. Workers can have one of two ability levels $\\Omega=\\{0,1\\}$. In each group, half of the workers are high ability and half are low ability. Suppose these workers face a competitive job market: if the market's perceived likelihood that their ability is 1 equals $\\mu \\equiv \\mu(1)$, they receive wage equal to $\\mu$. Equating workers' welfare to their wages, this means that $w(\\mu, \\omega, g)=\\mathbb{E}_{\\mu}[\\omega]$.\n\n> [Figure omitted from this text-only corpus; refer to the source manuscript.]\nFig. 2: Bayes welfare set under different data regimes in Example 3.2: The red circle corresponds to the Bayes welfare set under data regimes (i)-(iii); the blue line is the Bayes welfare set in regime (iv).\n\nRather than assuming a fixed data structure, we compare the Bayes welfare sets in this setting across two data regimes which can be interpreted as different data regulation policies that limit how much information can be revealed about a worker's ability: data reveals ability, but not group (i.e., $D=\\Omega$ ), and data reveals both group and ability (i.e., $D=G \\times \\Omega$ ). Figure 2 illustrates the Bayes welfare set in each of these regimes.\n\nWhereas in the first regime we can provide meaningful information about ability, that the distribution of ability is independent across groups together with the martingale property of beliefs implies that on average the posterior belief about the ability remains the same as under no information. It follows that in this case the Bayes welfare set consists of the no disclosure profile, $\\mathrm{W}=\\{(1 / 2,1 / 2)\\}$.\n\nConsider now the second regime and an information structure that pools low-ability workers from group A with high-ability workers from group B and fully reveals all other workers. We can represent this as an information structure with signals $\\{B 0\\},\\{A 0, B 1\\}$, and $\\{A 1\\}$, and induced posterior expectations of 0, $\\frac{1}{2}$, and 1, respectively. Because different groups induce these signals with different probabilities, each group's welfare is given by:","text_sha256":"db8917393a69035981bf634afd5db55e4d54a1be3a61c14222aa6b91e0084835"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0007","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"3. BEYOND THE BASIC MODEL: GROUPS AND DATA","text":"$$\n\\begin{aligned}\n& \\mathrm{w}_{A}=\\frac{1}{2} \\mathbb{E}[\\mu \\mid\\{A 0, B 1\\}]+\\frac{1}{2} \\mathbb{E}[\\mu \\mid\\{A 1\\}]=\\frac{1}{2} \\frac{1}{2}+\\frac{1}{2} 1=\\frac{3}{4}, \\\\\n& \\mathrm{w}_{B}=\\frac{1}{2} \\mathbb{E}[\\mu \\mid\\{B 0\\}]+\\frac{1}{2} \\mathbb{E}[\\mu \\mid\\{A 0, B 1\\}]=\\frac{1}{2} 0+\\frac{1}{2} \\frac{1}{2}=\\frac{1}{4} .\n\\end{aligned}\n$$\n\nIn fact, this information structure achieves the maximal possible payoff for group A: It never pools workers from group A with the low ability workers of group B, it never pools the high ability workers from group A with workers from group B, and it pools all high ability workers from group B with the workers from group A. As such, the maximal welfare $\\mathrm{w}_{A}$ is $\\frac{3}{4}$.\n\nWe note, however, that the average payoff across groups is the same across all information structures:\n\n$$\n\\frac{1}{2} \\mathrm{w}_{A}+\\frac{1}{2} \\mathrm{w}_{B}=\\frac{1}{2} \\mathbb{E}[\\mu \\mid g=A]+\\frac{1}{2} \\mathbb{E}[\\mu \\mid g=B]=\\mathbb{E}[\\mu]=\\frac{1}{2} .\n$$\n\nIn other words, information merely redistributes welfare across the groups. Consequently, the information structure that maximizes the welfare of group $A$ minimizes that of group $B$.\n\nThese observations together with the symmetry of the setting imply that in the fourth regime the Bayes welfare set is given by:\n\n$$\n\\mathrm{W}=\\left\\{\\left(\\mathrm{w}_{A}, \\mathrm{w}_{B}\\right) \\in[1 / 4,3 / 4]^{2}: \\mathrm{w}_{A}+\\mathrm{w}_{B}=1\\right\\} .\n$$","text_sha256":"f164d61ec520fff9a019140d9ca87f209679c4aa52dc93cb1b11c09645df80d4"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0008","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4. FINAL REMARKS","text":"## 4. FINAL REMARKS\n\nWe conclude by describing alternative applications of our framework as well as some directions for further research.","text_sha256":"1850d60ee896ddad148cb95f65e0be80b25c6b0fbee692928c4153ec3fb7797d"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0009","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.1 Applications to information design","text":"### 4.1 Applications to information design\n\nBy interpreting our welfare function as an individual's type-dependent payoff function, the Bayes welfare set is also the object of interest in more standard information design applications. For instance, the types may represent the private information of an informed principal who can commit to an information structure only after observing her type, as in [Perez-Richet 2014] and [Koessler and Skreta 2023]. Similar constraints appear in the studies of information design without commitment, as in [Lipnowski and Ravid 2020], [Drakopoulos et al. 2022], and [Corrao and Dai 2023]. Thus, the Bayes welfare set can be viewed as a unifying concept that underlies the incentive constraints the equilibrium information structure must satisfy. As we show in our first working paper version, [Doval and Smolin 2021], our tools also open the door to the study of new problems in this literature such as communication equilibrium payoffs in Bayesian persuasion with transparent motives and Bayesian persuasion with an ambiguity averse sender. ${ }^{1}$","text_sha256":"a3eebb23af2a8c28eea1d3f540245ad8fb54d3f6cb26734fca7cabbffdc37598"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0010","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"4.2 Further research","text":"### 4.2 Further research\n\nWe conclude with three (non-exhaustive) suggestions for future research:\nIt is well-known that various statistical notions of fairness, such as equalized odds and calibration, are incompatible with each other (cf. [Chouldechova 2017; Kleinberg et al. 2016]). Furthermore, this incompatibility remains even when considering relaxations [Pleiss et al. 2017]. Yet, the Bayes welfare set provides another way to visualize the trade-offs among these competing notions. For instance, one could use the Bayes welfare set to understand which group is hurt the most when imposing either calibration or equalized odds. Similarly, one could consider information structures that preserve some form of privacy-e.g., the algorithm recommendations do not reveal information about group membership-and study the Bayes\n\n[^0]welfare profiles consistent with such restrictions (cf. [Gopalan et al. 2021; Strack and Yang 2024]).\n\nSince the seminal work of [Dughmi and Xu 2016], the computer science literature has made incredible progress in algorithmic Bayesian persuasion (see, e.g., [Babichenko and Barman 2016; Arieli and Babichenko 2019; Banerjee et al. 2024]). Most of this work is concerned with the computational aspects of achieving the sender's preferred payoff, whereas our work focuses on the cross-sectional implications of different information structures for which the sender's average payoff may not be a sufficient statistic.\n\nIn many applications, considering constraints on the information structures the planner has access to is natural. The model in Section 3 puts limits on how much information can be provided about the payoff-relevant state. Constraints such as those arising from differential privacy are relevant in many applications and understanding how they shape the choice out of the Bayes welfare set is of interest.","text_sha256":"aeb0b5b062540a1e04e7c2a16a3f8ffa7a2cb33a14310357e94f08f357b6d1af"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0011","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"REFERENCES","text":"## REFERENCES\n\nAngwin, J., Larson, J., Mattu, S., and Kirchner, L. 2016. Machine bias: There's software used across the country to predict future criminals. and it's biased against blacks. ProPublica 23, 77-91.\nArieli, I. and Babichenko, Y. 2019. Private bayesian persuasion. Journal of Economic Theory 182, 185-217.\nBabichenko, Y. and Barman, S. 2016. Computational aspects of private bayesian persuasion. arXiv preprint arXiv:1603.01444.\nBanerjee, S., Munagala, K., Shen, Y., and Wang, K. 2024. Fair price discrimination. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 2679-2703.\nBénabou, R. and Tirole, J. 2006. Incentives and prosocial behavior. American Economic Review 96, 5, 1652-1678.\nChouldechova, A. 2017. Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. Big data 5, 2, 153-163.\nCorrao, R. and Dai, Y. 2023. The bounds of mediated communication. arXiv preprint arXiv:2303.06244.\nDoval, L. and Smolin, A. 2021. Information payoffs: An interim perspective. arXiv preprint arXiv:2109.03061.\nDoval, L. and Smolin, A. 2024. Persuasion and welfare. Journal of Political Economy 132, 7, 2451-2487.\nDrakopoulos, K., Lo, I., and Mulvany, J. 2022. Blockchain mediated persuasion. USC Marshall School of Business Research Paper Sponsored by iORB.\nDughmi, S. 2017. Algorithmic information structure design: a survey. ACM SIGecom Exchanges 15, 2, 2-24.\nDughmi, S. and Xu, H. 2016. Algorithmic bayesian persuasion. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing. 412-425.\nGopalan, P., Kalai, A. T., Reingold, O., Sharan, V., and Wieder, U. 2021. Omnipredictors. arXiv preprint arXiv:2109.05389.\nImmorlica, N., Lucier, B., and Slivkins, A. 2024. Generative ai as economic agents. ACM SIGecom Exchanges 22, 1, 93-109.\nJagtiani, J. and Lemieux, C. 2019. The roles of alternative data and machine learning in fintech lending: Evidence from the lendingclub consumer platform. Financial Management 48, 4, 1009-1029.\n\nKamenica, E. and Gentzkow, M. 2011. Bayesian persuasion. American Economic Review 101, 2590-2615.\nKearns, M. and Roth, A. 2019. The ethical algorithm: The science of socially aware algorithm design. Oxford University Press.\nKearns, M. and Roth, A. 2020. Ethical algorithm design. ACM SIGecom Exchanges 18, 1, 31-36.\nKleinberg, J., Ludwig, J., Mullainathan, S., and Rambachan, A. 2018. Algorithmic fairness. In AEA Papers and Proceedings. Vol. 108. 22-27.\nKleinberg, J., Mullainathan, S., and Raghavan, M. 2016. Inherent trade-offs in the fair determination of risk scores. arXiv preprint arXiv:1609.05807.\nKoessler, F. and Skreta, V. 2023. Informed information design. Journal of Political Economy 131, 11, 3186-3232.\nLi, D., Raymond, L. R., and Bergman, P. 2020. Hiring as exploration. National Bureau of Economic Research.\nLipnowski, E. and Mathevet, L. 2018. Disclosure to a psychological audience. American Economic Journal: Microeconomics 10, 4, 67-93.\nLipnowski, E. and Ravid, D. 2020. Cheap talk with transparent motives. Econometrica 88, 4, 1631-1660.\nMullainathan, S. 2018. Algorithmic fairness and the social welfare function. In Proceedings of the 2018 ACM Conference on Economics and Computation. 1-1.\nObermeyer, Z., Powers, B., Vogeli, C., and Mullainathan, S. 2019. Dissecting racial bias in an algorithm used to manage the health of populations. Science 366, 6464, 447-453.\nPerez-Richet, E. 2014. Interim bayesian persuasion: First steps. American Economic Review 104, 5, 469-74.\nPleiss, G., Raghavan, M., Wu, F., Kleinberg, J., and Weinberger, K. Q. 2017. On fairness and calibration. Advances in neural information processing systems 30.\nRaghavan, M., Barocas, S., Kleinberg, J., and Levy, K. 2020. Mitigating bias in algorithmic hiring: Evaluating claims and practices. In Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency. 469-481.\nRambachan, A., Kleinberg, J., Ludwig, J., and Mullainathan, S. 2020. An economic perspective on algorithmic fairness. In AEA Papers and Proceedings. Vol. 110. 91-95.\nStrack, P. and Yang, K. H. 2024. Privacy preserving signals. Available at SSRN 4467608.\n\n[^0]:    ${ }^{1}$ [Corrao and Dai 2023] fully characterize the set of communication equilibria with transparent motives.","text_sha256":"9ddd456e5f0de34fba956d742e9a6bd085e7f1bfbc5ca0726486692c6802259b"}
{"schema_version":"1.0","chunk_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025:0012","work_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms","paper_id":"alex-smolin:the-welfare-impact-of-recommendation-algorithms:2025","title":"The Welfare Impact of Recommendation Algorithms","authors":[{"name":"Laura Doval","url":"https://www.laura-doval.com/"},{"name":"Alex Smolin","url":"https://alexsmolin.com/","orcid":"https://orcid.org/0000-0003-4740-2376"}],"manuscript_date":"2025","language":"en","version_type":"author-supplied-research-letter","canonical_url":"https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md","source_record":"https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58","citation":"Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.","attribution_guidance":"Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.","provenance_url":"https://alexsmolin.com/corpus/PROVENANCE.txt","section":"Citation and provenance","text":"## Citation and provenance\n\n**Authors:** Laura Doval; Alex Smolin\n\n**Canonical citation:** Doval, Laura, and Alex Smolin. “The Welfare Impact of Recommendation Algorithms.” ACM SIGecom Exchanges 22, no. 2 (2025): 56–65.\n\n**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/the-welfare-impact-of-recommendation-algorithms.md\n\n**Source record:** https://sigecom.org/exchanges/volume_22/2/issue.pdf#page=58\n\n**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.\n\nProvenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt","text_sha256":"d963280267c95facc1e4b5573a2f176b27eb2d0474d641d2a3ae77fbf72a140a"}