Calibrated Mechanism Design

Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.

> Machine-readable author manuscript.
> Authors: Laura Doval; Alex Smolin.
> Canonical citation: Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.
> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt


Calibrated Mechanism Design

# Calibrated Mechanism Design

**Authors:** Laura Doval; Alex Smolin

**Manuscript date:** 2026-02-18

#### Abstract

We 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.

[^0]


Calibrated Mechanism Design

## 1 Introduction

Many 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.

A 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:

|  | $v=1 / 2$ | $v=1$ |
| :--- | :--- | :--- |
| $L$ | 2/3 | 1/3 |
| $H$ | 1/3 | 2/3 |

Table 1: Value distribution conditional on demand state.

Suppose 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.

Suppose first the buyer and the seller interact only once. Table 2 depicts an optimal mechanism for the seller in this case:

|  | $v=1 / 2$ | $v=1$ |
| :--- | :--- | :--- |
| $L$ | $(1,0)$ | $(1,0)$ |
| $H$ | (1,3/2) | (1,3/2) |

Table 2: Trade probabilities and payments as a function of buyer's value and demand state.

In 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,
the 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).

Suppose 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.

This 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.

We 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.


Calibrated Mechanism Design

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
on 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.

In 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.

In 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.

In 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.

In 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.

[^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.


Calibrated Mechanism Design

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.

Related 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.

The 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).

Following 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

[^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). ${ }^{4}$

Motivated 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.

Our 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.

By 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).

Lastly, 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).

[^3]


Calibrated Mechanism Design

## 2 Calibrated Mechanism Design

In 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.

Primitives 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.

Denote 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,

$$
f(\theta \mid \omega)=\prod_{i=1}^{N} f_{i}\left(\theta_{i} \mid \omega\right),
$$

for 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).

Mechanisms We model mechanisms as mappings

$$
\phi: \Theta \times \Omega \times[0,1] \rightarrow \Delta(A),
$$

where $\varepsilon \in[0,1]$ is a uniformly distributed random variable, which we refer to as the randomization device.

Several 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.

[^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.

Lastly, 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.

Calibrated 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 ${ }^{8}$

$$
\pi: \Omega \times[0,1] \rightarrow S_{1}^{*} \times \cdots \times S_{N}^{*}
$$

where $\varepsilon \in[0,1]$ is a uniformly distributed random variable-in fact, it is the same as in the definition of a mechanism-and

$$
S_{i}^{*}=\Delta\left(A_{i}\right)^{\Theta_{i}}
$$

is 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:

Definition 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 $i \in\{1, \ldots, N\}$,


Calibrated Mechanism Design

[^5]all $\theta_{i} \in \Theta_{i}$, and all $a_{i} \in A_{i}$
$$
s_{i}^{*}\left(a_{i} \mid \theta_{i}\right)=\mathbb{E}_{\tilde{\theta}_{-i} \sim f_{-i}(\cdot \mid \omega)}\left[\sum_{a_{-i} \in A_{-i}} \phi\left(\theta_{i}, \tilde{\theta}_{-i}, \omega, \varepsilon\right)\left(a_{i}, a_{-i}\right)\right] .
$$
We denote by $\pi_{\phi}$ the information structure calibrated to mechanism $\phi$.
In 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.

Information 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 $a_{i \varnothing} .{ }^{11}$

Formally, 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:

$$
\theta_{i} \in \arg \max _{\theta_{i}^{\prime} \in \Theta_{i}} \mathbb{E}_{\left(\omega, \varepsilon, \theta_{-i}\right)}\left[u_{i}\left(\phi\left(\theta_{i}^{\prime}, \theta_{-i}, \omega, \varepsilon\right), \theta_{i}, \omega\right) \mid\left(\theta_{i}, s_{i}^{*}\right)\right]
$$

where 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:

$$
\mathbb{E}_{\left(\omega, \varepsilon, \theta_{-i}\right)}\left[u_{i}\left(\phi\left(\theta_{i}, \theta_{-i}, \omega, \varepsilon\right), \theta_{i}, \omega\right)-u_{i}\left(a_{i \phi}, \theta_{i}, \omega\right) \mid\left(\theta_{i}, s_{i}^{*}\right)\right] \geq 0 .
$$

Importantly, 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

[^6]signals, as the mechanism "knows" each agent's signal realization.

Calibrated 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 $\pi_{\phi}$.

Definition 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:

$$
\max _{\phi \in \mathcal{M}_{\mathrm{cal}}} \mathbb{E}_{(\omega, \varepsilon, \theta)}[w(\phi(\theta, \omega, \varepsilon), \theta, \omega)] .
$$

We 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.

Three comments are in order:
First, 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.

Second, 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. ${ }^{12}$


Calibrated Mechanism Design

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.

Lastly, the definition of calibration assumes agents only learn about the state through their allocations

[^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.

We conclude this section by illustrating how our static solution concept captures the dynamics we alluded to in the introductory example:

Example 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.

The 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.

|  | $\nu=1 / 2$ | $v=1$ |  | $(1,0)$ | (1,3/2) |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $\omega=L$ | $(1,0)$ | $(1,0)$ | $\omega=L$ | 1 | 0 |
| $\omega=H$ | (1,3/2) | (1,3/2) | $\omega=H$ | 0 | 1 |

Table 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.

When 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. ${ }^{15}$

[^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 .

|  | $\nu=1 / 2$ | $v=1$ |  | \{(1,1/2)\} | \{(1/2,(0,0)),(1,(1,1))\} |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $\omega=L$ | (1,1/2) | (1,1/2) | $\omega=L$ | 1 | 0 |
| $\omega=H$ | $(0,0)$ | $(1,1)$ | $\omega=H$ | 0 | 1 |

Table 4: Trade probabilities and payments in optimal calibrated mechanism (left); calibrated information structure (right)

Note 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.

Private 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.

Theorem 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 $\omega$.

Theorem 1 (Private values). Under private values, the designer's payoff under the optimal calibrated mechanism is the same payoff he would obtain by choosing $\phi_{\text {full }}$.


Calibrated Mechanism Design

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. ${ }^{16}$

[^9]


Calibrated Mechanism Design

## 3 Two-stage mechanisms

In 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.

Single-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:

$$
\mathbb{E}_{(\omega, \varepsilon)}\left[u\left(\phi\left(\theta^{\prime}, \omega, \varepsilon\right), \theta, \omega\right) \mid\left(\theta, s^{*}\right)\right]=\sum_{a \in A} s^{*}\left(a \mid \theta^{\prime}\right)\left(\sum_{\omega \in \Omega} \mu\left(\omega \mid \theta, s^{*}\right) u(a, \theta, \omega)\right) .
$$

In 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 $\mu\left(\theta, s^{*}\right)$.

The above discussion suggests an alternative representation of a calibrated mechanism, which we dub a two-stage mechanism and define as follows:

Definition 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 $\tilde{\Delta} \subset \Delta(\Omega),{ }^{18}$

$$
\psi(\{a\} \times \tilde{\Delta} \mid \theta, \omega)=\int_{\tilde{\Delta}} \alpha(a \mid \theta, \mu) \beta(d \mu \mid \omega)
$$

We 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 $\mu$.

In 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

[^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.

Lastly, 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.

Theorem 2 shows that (incentive compatible and individually rational) calibrated mechanisms and two-stage mechanisms implement the same distributions over outcomes $\vartheta \in \Delta(A \times \Theta \times \Omega)$ :

Theorem 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

$$
\vartheta(a, \theta, \omega)=\mu_{0}(\omega) f(\theta \mid \omega) \int_{\Delta(\Omega)} \alpha(a \mid \theta, \mu) \beta(d \mu \mid \omega)
$$

for some Bayes plausible $\beta: \Omega \rightarrow \Delta(\Delta(\Omega))$ and incentive compatible and individually rational $\alpha$ : $\Theta \times \Delta(\Omega) \rightarrow \Delta(A)$.


Calibrated Mechanism Design

The proof of this and all results in this section can be found in Appendix B.
In 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.

Two-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

[^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:

Example 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:

$$
\begin{aligned}
W(\mu) \equiv \max _{(q, t): V \rightarrow[0,1] \times \mathbb{R}} \mu\left(\frac{2}{3} t(1)+\frac{1}{3} t(1 / 2)\right)+(1-\mu)\left(\frac{1}{3} t(1)+\frac{2}{3} t(1 / 2)\right) \\
\text { s.t. }\left\{\begin{array}{ll}
(\forall v \in\{1 / 2,1\}) & v q(v)-t(v) \geq 0 \\
\left(\forall v, v^{\prime} \in\{1 / 2,1\}, v \neq v^{\prime}\right) & v q(v)-t(v) \geq v q\left(v^{\prime}\right)-t\left(v^{\prime}\right)
\end{array} .\right.
\end{aligned}
$$

That 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.

The 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

$$
W(\mu)=\max \left\{\frac{1}{2}, \mu \frac{2}{3}+(1-\mu) \frac{1}{3}\right\},
$$

and 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.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 1: Seller's payoff in Example 1.

The 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 $\omega=H$.

Example 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.

Example 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.

|  | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega=L$ | 1 | 2 | 3 |
| $\omega=R$ | 2 | 2 | 1 |

Table 5: Buyer's values.

Suppose 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.

Consider 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 $(\omega=R)$.

|  | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega=L$ | $(0,1)$ | $(1,2)$ | $(1,2)$ |
| $\omega=R$ | $(1,1)$ | $(1,2)$ | $(1,2)$ |


Calibrated Mechanism Design

|  | $\left\{\left(\theta_{1},(0,1)\right),\left(\theta_{2},(1,2)\right),\left(\theta_{3},(1,2)\right)\right\}$ | $\left\{\left(\theta_{1},(1,1)\right),\left(\theta_{2},(1,2)\right),\left(\theta_{3},(1,2)\right)\right\}$ |
| :--- | :--- | :--- |
| $\omega=L$ | 1 | 0 |
| $\omega=R$ | 0 | 1 |

Table 6: Trade probabilities and transfers in the optimal mechanism (top); calibrated information structure (bottom).

The 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.

Under 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.

To 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

$$
\begin{aligned}
W(\mu) & \equiv \max _{(q, t): \Theta \rightarrow[0,1] \times \mathbb{R}} \sum_{\theta \in \Theta} \frac{1}{3} t(\theta) \\
& \text { s.t. }\left\{\begin{array}{ll}
\left(\forall \theta \in\left\{\theta_{1}, \theta_{2}, \theta_{3}\right\}\right) & q(\theta) \mathbb{E}_{\mu} v(\theta, \cdot)-t(\theta) \geq 0 \\
\left(\forall \theta, \theta^{\prime} \in\left\{\theta_{1}, \theta_{2}, \theta_{3}\right\}, \theta^{\prime} \neq \theta\right) & q(\theta) \mathbb{E}_{\mu} v(\theta, \cdot)-t(\theta) \geq q\left(\theta^{\prime}\right) \mathbb{E}_{\mu} v(\theta, \cdot)-t\left(\theta^{\prime}\right)
\end{array} .\right.
\end{aligned}
$$

In 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 $\mu$.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 2: Seller's profit in the two-stage mechanism

The 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.

Another consequence of Theorem 2 is that without loss of generality, we can focus on calibrated mechanisms with finite calibrated information structures:

Corollary 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.

In other words, it is without loss of generality to focus on calibrated mechanisms that induce at most $|\Omega|$ allocation rules.

Multiple 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.

These 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:

Definition 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

$$
\beta: \Omega \rightarrow \Delta\left(\Delta(\Omega)^{N}\right), \quad \alpha_{i}: \Theta_{i} \times \Delta(\Omega) \rightarrow \Delta\left(A_{i}\right), \quad \alpha: \Theta \times \Omega \times \Delta(\Omega)^{N} \rightarrow \Delta(A),
$$

exist such that:


Calibrated Mechanism Design

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
$$
\psi\left(\times_{i=1}^{N} \tilde{\Delta}_{i} \times\{a\} \mid \theta, \omega\right)=\int_{\times_{i=1}^{N} \tilde{\Delta}_{i}} \alpha\left(a \mid \theta, \omega, \mu_{1}, \ldots, \mu_{N}\right) \beta\left(d\left(\mu_{1}, \ldots, \mu_{N}\right) \mid \omega\right)
$$
2. The Blackwell experiment $\beta$ is Bayes plausible,
3. 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 $\left(a_{i}, \theta_{i}, \omega\right) \in A_{i} \times \Theta_{i} \times \Omega$
$$
\int_{\tilde{\Delta} \times \Delta(\Omega)^{N-1}}\left\{\alpha_{i}\left(a_{i} \mid \theta_{i}, \mu_{i}\right)-\mathbb{E}_{f_{-i}(\cdot \mid \omega)}\left[\sum_{a_{-i} \in A_{-i}} \alpha\left(a_{i}, a_{-i} \mid \theta_{i}, \theta_{-i}, \omega, \mu_{i}, \mu_{-i}\right)\right]\right\} \beta\left(d\left(\mu_{i}, \mu_{-i}\right) \mid \omega\right)=0 .
$$
We 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 $\mu_{i}$.

As 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
allocation 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.

As we show in Proposition 1, a calibrated mechanism induces a generalized two-stage mechanism:
Proposition 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.

In 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.


Calibrated Mechanism Design

## 4 Applications

In 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.


Calibrated Mechanism Design

### 4.1 Calibrated Screening

We 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. ${ }^{21}$

[^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.

Our goal is to characterize the designer optimal calibrated mechanism and how its properties depend on how the state affects the order of types.

State-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 $u(q, \theta, \omega)=(\theta+\omega) q$.

By 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 $W$.

Given belief $\mu$, define the agent's and the designer's (expected) payoff at $(q, t, \theta)$ as follows:

$$
u(q, \theta \mid \mu) \equiv \sum_{\omega \in \Omega} \mu(\omega) u(q, \theta, \omega), w(q, \theta \mid \mu) \equiv \sum_{\omega \in \Omega} \mu(\omega) w(q, \theta, \omega)
$$

Thus, conditional on inducing belief $\mu$, the designer's problem can be written as follows:

$$
\begin{aligned}
W(\mu) \equiv \max _{(q, t): \Theta \rightarrow A} \int_{\Theta}[w(q(\theta), \theta, \mu)+t(\theta)] F(d \theta) \\
\text { s.t. }\left\{\begin{array}{ll}
(\forall \theta \in \Theta) & u(q(\theta), \theta \mid \mu)-t(\theta) \geq 0 \\
\left(\forall \theta, \theta^{\prime} \in \Theta\right) & u(q(\theta), \theta \mid \mu)-t(\theta) \geq u\left(q\left(\theta^{\prime}\right), \theta \mid \mu\right)-t\left(\theta^{\prime}\right)
\end{array} .\right.
\end{aligned}
$$

Our 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.

Define the virtual surplus at $(q, \theta, \omega)$ as follows:

$$
J((q, \theta, \omega) ; F)=w(q, \theta, \omega)+u(q, \theta, \omega)-u_{2}(q, \theta, \omega) \frac{1-F(\theta)}{f(\theta)},
$$

where $u_{2}$ 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

[^13]as follows:
$$
W(\mu)=\max _{q \in Q_{\uparrow}} \int_{\Theta} \mathbb{E}_{\mu}[J((q(\theta), \theta, \omega) ; F)] F(d \theta) .
$$
Note 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 $\omega$.

Proposition 2 summarizes the above discussion:
Proposition 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.

By 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.


Calibrated Mechanism Design

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.

First, 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:

$$
Q_{\uparrow, \text { random }}=\left\{\xi: \Theta \rightarrow \Delta([0, \bar{q}]): \mathbb{E}_{\xi(\theta)}[q] \text { is increasing in } \theta\right\} .
$$

In this case, we obtain that the designer cannot do any better than choosing $\phi_{\text {ful }}$, which is the

[^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. ${ }^{24}$

Second, suppose the agent's payoff has the form

$$
u(q, \theta, \omega)=b(\theta) c(\omega) v(q)+k_{1}(q, \omega)+k_{2}(\theta, \omega),
$$

where $b$ 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.

State-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.

Concretely, suppose that $\Omega=\left\{\omega_{1}, \omega_{2}\right\}$. Furthermore, assume

$$
u(q, \theta, \omega)=\left\{\begin{array}{ll}
q \theta & \text { if } \omega=\omega_{2} \\
q(c-b \theta) & \text { otherwise, }
\end{array},\right.
$$

where $c \in \mathbb{R}$, and $b>0$. Identify beliefs with the probability that the state is $\omega_{2}$ and define

$$
\hat{\mu}=\frac{b}{1+b} .
$$

For $\mu<\hat{\mu}$, 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 $\theta$.
Consider 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, $\hat{\mu}, 1\}$.

[^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.

By 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.


Calibrated Mechanism Design

### 4.2 Comparison with Myersonian Mechanism Design

We 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.

Myersonian 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,

$$
\begin{aligned}
& W_{\mathrm{My}} \equiv \max _{(q, t): \Theta \times \Omega \rightarrow[0, \bar{q}] \times \mathbb{R}} \int_{\Theta} \mathbb{E}_{\mu_{0}}[w(\xi(\theta, \omega), \theta, \omega)+t(\theta, \omega)] F(d \theta) \\
& \text { s.t. }\left\{\begin{array}{l}
(\forall \theta \in \Theta) \mathbb{E}_{\mu_{0}}[u(\xi(\theta, \omega), \theta, \omega)-t(\theta, \omega)] \geq 0 \\
\left(\forall \theta, \theta^{\prime} \in \Theta\right) \mathbb{E}_{\mu_{0}}[u(\xi(\theta, \omega), \theta, \omega)-t(\theta, \omega)] \geq \mathbb{E}_{\mu_{0}}\left[u\left(\xi\left(\theta^{\prime}, \omega\right), \theta, \omega\right)-t\left(\theta^{\prime}, \omega\right)\right]
\end{array},\right.
\end{aligned}
$$

where $w(\xi, \theta, \omega)$ and $u(\xi, \theta, \omega)$ are the linear extensions of $w(\cdot, \theta, \omega)$ and $u(\cdot, \theta, \omega)$, respectively.
Program $\mathrm{OPT}_{\mathrm{My}}$ 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): ${ }^{27}$

$$
\left(\forall \theta, \theta^{\prime} \in \Theta\right) \int_{\theta^{\prime}}^{\theta} \int_{\Omega}\left[u_{2}(\xi(s, \omega), s, \omega)-u_{2}\left(\xi\left(\theta^{\prime}, \omega\right), s, \omega\right)\right] d \mu_{0} d s \geq 0
$$

[^16]where recall $u_{2}$ is the derivative of $u$ in its second coordinate.

Comparison 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

$$
Q_{\mathrm{cal}}=\{q: \Theta \times \Omega \rightarrow[0, \bar{q}]:(\forall \omega \in \Omega) q(\cdot, \omega) \text { is increasing }\} .
$$

That 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 $Q_{\mathrm{My}}$.

When $u(q, \theta, \omega)$ 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:

$$
\begin{aligned}
W_{\mathrm{My}}^{D} & =\max _{q \in Q_{\mathrm{My}}} \int_{\Theta} \mathbb{E}_{\mu_{0}}[J(q(\theta, \omega), \theta, \omega ; F)] F(d \theta), \\
W_{\mathrm{cal}}^{D} & =\max _{q \in Q_{\mathrm{cal}}} \int_{\Theta} \mathbb{E}_{\mu_{0}}[J(q(\theta, \omega), \theta, \omega ; F)] F(d \theta),
\end{aligned}
$$

where 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.

By 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:

Observation 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, $W_{\text {cal }}^{D}=W_{M y}^{D}$.

Two 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

$$
W_{\mathrm{rel}}=\max _{q: \Theta \times \Omega \rightarrow[0, \bar{q}]} \int_{\Theta} \mathbb{E}_{\mu_{0}}[J(q(\theta, \omega), \theta, \omega ; F)] F(d \theta),
$$


Calibrated Mechanism Design

satisfies monotonicity state-by-state by Topkis' theorem. Moreover, a stochastic mechanism is
equivalent 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 $q_{\text {rel }}$.

Proposition 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.

By 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:

Example 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:

$$
\begin{aligned}
u(q, \theta, \omega) & =q \theta \omega \\
w(q, \theta, \omega) & =2(1-2 \theta) q .
\end{aligned}
$$

Note 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:

$$
J((q, \theta, \omega) ; F)=\left\{\begin{array}{ll}
(1-2 \theta) q & \text { if } \omega=\omega_{L} \\
(2 \theta-1) q & \text { otherwise }
\end{array} .\right.
$$

In 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:

$$
q_{M y}(\theta, \omega)=\left\{\begin{array}{ll}
1 & \text { if } \omega=\omega_{L} \text { and } \theta<1 / 2 \\
1 & \text { if } \omega=\omega_{H} \text { and } \theta>1 / 2 \\
0 & \text { otherwise }
\end{array} .\right.
$$

The designer's payoff under the Myersonian mechanism is 1/4.
By 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.

Instead, in the optimal calibrated mechanism, the designer sets $q_{\text {cal }}\left(\theta, \omega_{H}\right)=\mathbb{1}[\theta \geq 1 / 2]$ and sets

[^17]$q\left(\theta, \omega_{L}\right)$ to be constant in $\theta$. The designer's payoff under calibration is $W_{\text {cal }}=1 / 8<W_{M y}$.


Calibrated Mechanism Design

### 4.3 Optimal Calibrated Auction

In 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.

Suppose 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 $0 \leq \sum_{i=1}^{N} q_{i} \leq 1$.

We 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.

To 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 $w_{i}$.

As 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
the virtual surplus as

$$
J((q, \theta, \omega) ; F)=\sum_{i=1}^{N} q_{i}(\theta, \omega)\left(w_{i}(\theta, \omega)+\left(\theta_{i}-\frac{1-F_{i}\left(\theta_{i}\right)}{f_{i}\left(\theta_{i}\right)}\right) \omega_{i}+\omega_{0 i}\right) .
$$

Standard arguments imply the individual rationality constraint of $\theta_{i}=0$ binds for all $i$, and an optimal mechanism solves

$$
W_{\mathrm{My}}=\max _{q \in Q_{\mathrm{My}}} \int_{[0,1]^{N}} \mathbb{E}_{\mu_{0}}[J(q(\theta, \omega), \theta, \omega ; F)] f(\theta) d \theta .
$$

Proposition 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, $W_{\text {cal }}=W_{M y}$.

The 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.

Viewed 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.


Calibrated Mechanism Design

## 5 Microfoundation

In 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.

Throughout, 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

[^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.


Calibrated Mechanism Design

### 5.1 Repeated Interactions with a Mechanism

We 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

$$
\phi: M \times \Omega \times \mathcal{E} \rightarrow \Delta(A),
$$

where $M$ 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 $(\omega, \varepsilon) \in \Omega \times \mathcal{E}, \phi\left(m_{\varnothing}, \omega, \varepsilon\right)=\delta_{a_{\varnothing}}$.

Timing 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.

Given 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.

We 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:

$$
U_{T}\left(\left(\theta_{t}, m_{t}, a_{t}\right)_{t=1}^{T}, \omega, \varepsilon\right)=\frac{1}{T} \sum_{t=1}^{T} u\left(a_{t}, \theta_{t}, \omega\right) .
$$

A strategy $\sigma$ is a best response for the agent if for all alternative strategies $\sigma^{\prime}$, we have that

$$
\lim \inf _{T \rightarrow \infty} \mathbb{E}_{\sigma}\left[U_{T}\right] \geq \lim \sup _{T \rightarrow \infty} \mathbb{E}_{\sigma^{\prime}}\left[U_{T}\right],
$$

where $\mathbb{E}_{\sigma}$ 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 $\sigma .{ }^{31}$

[^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 ${ }^{32}$

$$
v_{\sigma}(a, \theta, m, \omega, \varepsilon)=\lim _{T \rightarrow \infty} \frac{1}{T} \mathbb{E}_{\sigma}\left[\sum_{t=1}^{T} \mathbb{1}\left[\left(a_{t}, \theta_{t}, m_{t}, \omega^{\prime}, \varepsilon^{\prime}\right)=(a, \theta, m, \omega, \varepsilon)\right]\right]=\lim _{T \rightarrow \infty} v_{\sigma}^{T}(a, \theta, m, \omega, \varepsilon),
$$

where 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.

Under 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.

Definition 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

$$
\vartheta(a, \theta, \omega)=\sum_{\varepsilon \in \mathcal{E}, m \in M} v_{\sigma}(a, \theta, m, \omega, \varepsilon) .
$$

We 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:

Theorem 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 $(a, \theta, \omega) \in A \times \Theta \times \Omega$,

$$
\vartheta(a, \theta, \omega)=\mu_{0}(\omega) f(\theta) \int_{\Delta(\Omega)} \alpha(a \mid \theta, \mu) \beta(d \mu \mid \omega)
$$

where $\beta: \Omega \rightarrow \Delta(\Delta(\Omega))$ is Bayes plausible and $\alpha(\cdot \mid \cdot, \mu): \Theta \rightarrow \Delta(A)$ is incentive compatible and individually rational on the support of $\mu_{0} \otimes \beta$.

The proof of this and all results in this section can be found in Appendix C.
Theorem 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.

We now provide a proof sketch for Theorem 3, which is also useful to understand the proof of the result