Calibrated Mechanism Design

## D. 2 Optimal Calibrated Auction

Proof of Proposition 5. The pointwise solution to the Myersonian problem allocates the good to agents in $N^{*}(\theta, \omega)=\arg \max _{i \in[N] \cup\{0\}}\left[w_{i}(\theta, \omega)+J_{i}\left(\theta_{i}\right) \omega_{i}+\omega_{0 i}\right]$ where $i=0$ corresponds to an outside option with $w_{0} \equiv J_{0} \equiv \omega_{00} \equiv 0$. The conditions of the proposition ensure that for all $i \in N$ and $j \in[N] \cup\{0\}$,

$$
\frac{d}{d \theta_{i}}\left(w_{i}(\theta, \omega)+J_{i}\left(\theta_{i}, F_{i}\right) \omega_{i}+\omega_{0 i}\right) \geq \frac{d}{d \theta_{i}}\left(w_{j}(\theta, \omega)+J_{j}\left(\theta_{j}, F_{j}\right) \omega_{j}+\omega_{0 j}\right) .
$$

Thus, an optimal selection $q^{*}(\theta, \omega)$ exists such that for each $i, \theta_{-i}$, and $\omega, q^{*}\left(\theta_{i}, \theta_{-i}, \omega\right)$ is nondecreasing in $\theta_{i}$ (e.g., one that uniformly randomizes over $N^{*}(\theta, \omega)$ ).

Denote by $Q_{\text {full }}$ the set of allocation rules implementable under full state disclosure. These are the rules such that for all $i$ and $\omega, \mathbb{E}_{F_{-i}}\left[q_{i}\left(\theta_{i}, \theta_{-i}, \omega\right)\right]$ is non-decreasing in $\theta_{i}$. It follows that $q^{*} \in Q_{\mathrm{full}}$, and hence $q^{*} \in Q_{\mathrm{My}}$. Thus, $q^{*}$ solves the Myersonian problem and can also be implemented by fully disclosing the state to the agents and conducting an optimal mechanism state-by-state. By revenue

[^35]equivalence, the expected revenue of such implementation is the same as under no disclosure, and thus the designer obtains payoff $W_{\text {My }}$. $\square$


Calibrated Mechanism Design

Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.

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

## D. 3 Technical results from Appendix C

Proof of Lemma C.1. The set of continuous bounded functions on $\tilde{\Omega}$ is separable and hence it has a countable dense subset $\left\{g_{k}\right\}_{k \in \mathbb{N}} \subset C_{b}(\tilde{\Omega})$. It is immediate to see that $\mu_{n} \xrightarrow{w^{*}} \mu$ if and only if for all $k \in \mathbb{N}$ $\int g_{k} d \mu_{n} \rightarrow \int g_{k} d \mu$.

For each $k \in \mathbb{N}$ define a real-valued, bounded, martingale on ( $\mathcal{H}^{\infty}, \mathcal{B}_{\mathcal{H}^{\infty}}, \mathbb{P}_{\sigma}$ ) as follows:

$$
M_{t}^{k}\left(\tilde{\omega}, h^{\infty}\right)=\int_{\tilde{\Omega}} g_{k}\left(\omega^{\prime}, \varepsilon\right) d \mu_{t}\left(\tilde{\omega}, h^{\infty}\right)\left(\omega^{\prime}, \varepsilon\right)
$$

Doob's martingale convergence theorem implies that $M_{t}^{k}\left(\tilde{\omega}, h^{\infty}\right)=\mathbb{E}\left[g_{k} \mid h^{t}\right] \rightarrow M_{\infty}^{k}\left(\tilde{\omega}, h^{\infty}\right)=\mathbb{E}\left[g_{k} \mid h^{\infty}\right]$ $\mathbb{P}_{\sigma}$-a.s. Let $E_{k}$ denote the subset of $\mathcal{H}^{\infty}$ where convergence happens, and note that $\mathbb{P}_{\sigma}\left(E_{k}\right)=1$.

Let $E=\cap_{k} E_{k}$ and note that $\mathbb{P}_{\sigma}(E)=1$. Then, on $E$, we have that for all $k \in \mathbb{N}$,

$$
\int_{\tilde{\Omega}} g_{k}\left(\omega^{\prime}, \varepsilon\right) d \mu_{t}\left(\tilde{\omega}, h^{\infty}\right)\left(\omega^{\prime}, \varepsilon\right) \rightarrow M_{\infty}^{k}\left(\tilde{\omega}, h^{\infty}\right)
$$

Fix now a terminal history ( $\tilde{\omega}, h^{\infty}$ ). Because $\Delta(\tilde{\Omega})$ is compact (Aliprantis and Border, 2006, Theorem 15.11), the sequence $\left(\mu_{t}\left(\tilde{\omega}, h^{\infty}\right)\right)_{t \in \mathbb{N}}$ has a convergent subsequence $\mu_{t_{j}}\left(\tilde{\omega}, h^{\infty}\right) \xrightarrow{w^{*}} \tilde{\mu}$. Passing the limit along $t_{j}$ in Equation D. 1 we have that for all $k \in \mathbb{N}$

$$
\int_{\tilde{\Omega}} g_{k}\left(\omega^{\prime}, \varepsilon\right) d \mu_{t}\left(\tilde{\omega}, h^{\infty}\right)\left(\omega^{\prime}, \varepsilon\right) \rightarrow \int_{\tilde{\Omega}} g_{k}\left(\omega^{\prime}, \varepsilon\right) d \tilde{\mu}
$$

Because the set $\left\{g_{k}: k \in \mathbb{N}\right\}$ determines the convergent subsequences, any subsequential limits must be equal. Hence $\mu_{t}(\cdot)$ converges on $E$ and call this limit $\tilde{\mu}_{\infty}$. Hence on $E$ we have that $\mu_{t}\left(h^{\infty}\right) \xrightarrow{w^{*}} \tilde{\mu}_{\infty}\left(h^{\infty}\right)$.

Now, for each $k, \int g_{k} d \mu_{\infty}=\mathbb{E}\left[g_{k} \mid h^{\infty}\right]$ almost surely. Hence, $\tilde{\mu}_{\infty}$ is a version of the law of $\tilde{\Omega}$ conditional on $h^{\infty}$. This is $\mathbb{P}_{\sigma}\left(\cdot \mid h^{\infty}\right)$, completing the proof. $\square$

Proof of Lemma C.2. Fix a continuous and bounded function $g \in C_{b}(A \times \Theta \times \Omega \times \Delta(\Omega))$ and define for each terminal history $\left(\omega, h^{\infty}\right)$

$$
\Delta_{t}\left(\omega, h^{\infty}\right)=g\left(a_{t}\left(h^{\infty}\right), \theta_{t}\left(h^{\infty}\right), \omega, \mu_{t}\left(h^{\infty}\right)\right)-g\left(a_{t}\left(h^{\infty}\right), \theta_{t}\left(h^{\infty}\right), \omega, \mu_{t+1}\left(h^{\infty}\right)\right) .
$$

As in Lemma C.1, let $E$ denote the probability-1 subset of $\mathcal{H}^{\infty}$ on which $\mu_{t} \xrightarrow{w^{*}} \mu_{\infty} .{ }^{54}$ Then, on $E$, $\Delta_{t}\left(\omega, h^{\infty}\right) \rightarrow 0$, and hence,

$$
\mathbb{E}_{\sigma}\left[\Delta_{t}\left(\omega, h^{\infty}\right)\right] \rightarrow 0,
$$

[^36]as $t \rightarrow \infty$. Now, for every $T$,
$$
D_{T}(g) \equiv \mathbb{E}_{\bar{v}_{\sigma}^{T, 1}}[g]-\mathbb{E}_{\bar{v}_{\sigma}^{T, 2}}[g]=\frac{1}{T} \sum_{t=1}^{T} \mathbb{E}_{\sigma}\left[\Delta_{t}\right]
$$
and hence the left-hand side goes to 0 as $T \rightarrow \infty$.
Now, let $T_{n}$ be such that $\bar{v}_{\sigma}^{T_{n}, 2} \xrightarrow{w^{*}} \bar{v}$ for some $\bar{v} \in \Delta(A \times \Theta \times \Omega \times \Delta(\Omega))$. Note that
$$
\mathbb{E}_{\bar{v}_{\sigma}^{T n, 1}}[g]=\mathbb{E}_{\bar{v}_{\sigma}^{T n, 2}}[g]+D_{T_{n}}[g] \rightarrow \mathbb{E}_{\bar{v}}[g]+0,
$$
so a subsequential limit of $\bar{v}_{\sigma}^{T_{n}, 2}$ is a subsequential limit of $\bar{v}_{\sigma}^{T_{n}, 1}$. Switching the role of 1 and 2, we obtain the opposite set inclusion and the result follows. $\square$

Proof of Lemma C.3. Fix a continuous function $g$ on $\Delta(\tilde{\Omega})$. Then, we want to show that

$$
\mathbb{E}_{\tau_{T}}[g] \rightarrow \mathbb{E}_{\mathbb{P}_{\sigma} \circ \mu_{\infty}^{-1}}[g] .
$$

By Lemma C.1, $\mu_{t} \xrightarrow{w^{*}} \mu_{\infty} \mathbb{P}_{\sigma}$-almost surely and $g$ is continuous, we have that $\mathbb{E}_{\sigma}\left[g\left(\mu_{t}\right)\right] \rightarrow \mathbb{E}_{\sigma}\left[g\left(\mu_{\infty}\right)\right]$ by dominated convergence theorem. ${ }^{55}$ Because eventually constant sequences have Cesàro limits, we have that

$$
\frac{1}{T} \sum_{t=1}^{T} \mathbb{E}_{\sigma}\left[g\left(\mu_{t}\right)\right] \rightarrow \mathbb{E}_{\sigma}\left[g\left(\mu_{\infty}\right)\right] .
$$

And now we are basically done, because

$$
\mathbb{E}_{\tau_{T}}[g]=\int_{H^{\infty}} \frac{1}{T} \sum_{t=1}^{T} g\left(\mu_{t}\left(h^{\infty}\right)\right) \mathbb{P}_{\sigma}\left(d h^{\infty}\right)=\frac{1}{T} \sum_{t=1}^{T} \mathbb{E}_{\sigma}\left[g\left(\mu_{t}\right)\right] \rightarrow \mathbb{E}_{\sigma}\left[g\left(\mu_{\infty}\right)\right]=\int g d\left(\mathbb{P}_{\sigma} \circ \mu_{\infty}^{-1}\right) .
$$

In other words, the occupation measure on beliefs induced by the strategy (and the prior, the type distribution, and the mechanism) is the push-forward measure $\left(\mathbb{P}_{\sigma} \circ \mu_{\infty}^{-1}\right)$. In particular, that $\mathbb{P}_{\sigma}$ is a measure implies that $\left(\mathbb{P}_{\sigma} \circ \mu_{\infty}^{-1}\right)$ is a measure itself (Bogachev, 2007, Chapter 3.6). $\square$


Calibrated Mechanism Design

Proof of Lemma C.4. We now show the agent can ensure the payoff $U\left(\tau_{\phi}\right)$ in Equation C.21, which corresponds to the agent's maximum payoff under the calibrated information structure $\pi_{\phi}$. Recall that this information structure is the one that corresponds to the partition of $\tilde{\Omega}, \mathscr{P}$, defined as follows: $\tilde{\omega}, \tilde{\omega}^{\prime}$ in the same cell $P$ of $\mathscr{P}$ if for all $(a, m), \phi(a \mid m, \tilde{\omega})=\phi\left(a \mid m, \tilde{\omega}^{\prime}\right)$. Conditional on cell $P$, the associated posterior is $\mu(\mid P) \in \Delta(\tilde{\Omega})$. Let $\tau_{\phi} \in \Delta(\Delta(\tilde{\Omega}))$ denote the induced belief distribution (with mean $\mu_{0} \otimes \eta$ ).

To prove the result, we consider the strategy $\sigma_{N}^{\prime}$ parameterized by a number $N$ and defined as follows. In the exploration phase of $\sigma_{N}^{\prime}$, the agent plays each message $m$ for $N$ rounds. Let $\mu_{N|M|}\left(h^{N|M|}\right)$ denote the agent's beliefs as a function of the realized sequence of allocations implied by $h^{N|M|}$. For each

[^37]history that succeeds $h^{N|M|}$, the agent of type $\theta$ plays the message $m$ that solves
$$
\max _{m \in M} \sum_{\tilde{\omega}} \mu_{N|M|}\left(h^{N|M|}\right)(\tilde{\omega}) \sum_{a \in A} \phi(a \mid m, \tilde{\omega}) u(a, \theta, \omega) .
$$
It is immediate to verify that the agent's (limit) average payoff under $\sigma_{N}^{\prime}$ is given by:
$$
\begin{aligned}
U\left(\sigma_{N}^{\prime}\right) & =\mathbb{E}_{\sigma_{N}^{\prime}}\left[\sum_{\theta \in \Theta} f(\theta) \max _{m \in M} \sum_{\tilde{\omega}} \mu_{N|M|}\left(h^{N|M|}\right)(\tilde{\omega}) \sum_{a \in A} \phi(a \mid m, \tilde{\omega}) u(a, \theta, \omega)\right]= \\
& =\sum_{h^{N|M|} \in H^{N|M|}} \mathbb{P}_{\sigma_{N}^{\prime}}^{N|M|}\left(h^{N|M|}\right) u^{*}\left(\mu_{N|M|}\left(h^{N|M|}\right)\right),
\end{aligned}
$$
where $u^{*}$ is as in Equation C.20.
Below, we show that the distribution of beliefs under $\sigma_{N}^{\prime}, \mathbb{P}_{\sigma_{N}^{\prime}} \circ \mu_{N|M|}^{-1} \xrightarrow{w^{*}} \tau_{\phi}$. Consequently, as $u^{*}$ is continuous and bounded on $\Delta(\tilde{\Omega})$, for any $\delta>0$, we can choose $N_{\delta}$ so that for all $N \geq N_{\delta}$,
$$
\left|U\left(\sigma_{N}^{\prime}\right)-U\left(\tau_{\phi}\right)\right| \leq \delta .
$$
Consequently, the agent's payoff under $\sigma$ must be $U\left(\tau_{\phi}\right)$ because by definition for all $\delta>0^{56}$
$$
\liminf _{T \rightarrow \infty} \mathbb{E}_{\sigma}\left[U_{T}\right] \geq \limsup _{T \rightarrow \infty} \mathbb{E}_{\sigma_{N_{\delta}}^{\prime}}\left[U_{T}\right]=U\left(\sigma_{N_{\delta}}^{\prime}\right) \geq U\left(\tau_{\phi}\right)-\delta,
$$
and hence,
$$
\liminf _{T \rightarrow \infty} \mathbb{E}_{\sigma}\left[U_{T}\right] \geq \lim _{\delta \rightarrow 0} U\left(\tau_{\phi}\right)-\delta=U\left(\tau_{\phi}\right) .
$$
which completes the proof.
We now complete the missing step:

The law of $\mu_{N|M|}$ converges to $\tau_{\phi}$ It is useful to write the bottom line of Equation D. 2 as follows:

$$
\sum_{\tilde{\omega} \in \tilde{\Omega}}\left(\mu_{0} \otimes \eta\right)(\tilde{\omega}) \mathbb{E}_{\mathbb{P}_{\sigma_{N}^{\prime}}^{N|M|}(\cdot \mid \tilde{\omega})}\left[u^{*}\left(\mu_{N|M|}\right)\right] .
$$

We show that the conditional law of $\mu_{N|M|}, \mathbb{P}_{\sigma_{N}^{\prime}}(\cdot \mid \tilde{\omega})$, converges to the Dirac measure on $\mu(\cdot \mid P(\tilde{\omega}))$. Noting that $\tau_{\phi}=\sum_{\tilde{\omega} \in \tilde{\Omega}}\left(\mu_{0} \otimes \eta\right)(\tilde{\omega}) \delta_{\mu(\cdot \mid P(\tilde{\omega}))}$ completes the proof.

For the exploration block, define for each $m \in M$, the empirical frequency $\hat{\phi}_{N, m}:(\Theta \times M \times A)^{N|M|} \rightarrow$ $\Delta(A)$, as follows

$$
\hat{\phi}_{N, m}\left(h^{N|M|}\right)(a)=\frac{1}{N} \sum_{n=1}^{N} \mathbb{1}\left[A_{m, n}=a\right],
$$

where $A_{m, n}$ is the $n^{\text {th }}$ draw from $A$ when the message is $m$. Let $\hat{\phi}_{N}: H^{N|M|} \rightarrow \Delta(A)^{M}$ denote the vector

[^38]of empirical frequencies. The agent's belief at history $h^{N|M|}$ is given by:
$$
\mu_{N|M|}\left(h^{N|M|}\right)(\tilde{\omega})=\frac{\left(\mu_{0} \otimes \eta\right)(\tilde{\omega}) \prod_{m \in M} \prod_{a \in A} \phi(a \mid m, \tilde{\omega})^{N \hat{\phi}_{N, m}\left(h^{N|M|}\right)(a)}}{\sum_{\tilde{\omega}^{\prime}}\left(\mu_{0} \otimes \eta\right)\left(\tilde{\omega}^{\prime}\right) \prod_{m \in M} \prod_{a \in A} \phi\left(a \mid m, \tilde{\omega}^{\prime}\right)^{N \hat{\phi}_{N, m}\left(h^{N|M|}\right)(a)}} .
$$
Denote by $\tau_{N|M|, \tilde{\omega}} \in \Delta(\Delta(\tilde{\Omega}))$ the law of $\mu_{N|M|}$ conditional on $\tilde{\omega}$, i.e., $\tau_{N|M|, \tilde{\omega}}=\mathbb{P}_{\sigma_{N}^{\prime}}(\cdot \mid \tilde{\omega}) \circ \mu_{N|M|}^{-1}$. Below, we show that $\tau_{N|M|, \tilde{\omega}}$ converges weakly to $\delta_{\mu(\cdot \mid P(\tilde{\omega}))}$.

Suppose the true state is $\tilde{\omega}^{\star}$. Then, $\left(A_{m, 1}, \ldots, A_{m, N}\right)$ are drawn i.i.d. from distribution $\phi\left(\cdot \mid m, \tilde{\omega}^{\star}\right)$. Fix a continuous and bounded function $g$ on $A$. Then, almost surely, ${ }^{57}$

$$
\int_{A} g d \hat{\phi}_{N}(\cdot \mid m)=\frac{1}{N} \sum_{n=1}^{N} g\left(A_{m, n}\right) \rightarrow \mathbb{E}_{\phi}\left[g\left(A_{m, 1}\right)\right]=\sum_{a \in A} g(a) \phi\left(a \mid m, \tilde{\omega}^{\star}\right),
$$

by the strong law of large numbers applied to the i.i.d random variables $\left(g\left(A_{m, 1}\right), \ldots, g\left(A_{m, N}\right)\right)$. Because this holds for all $g$, then $\hat{\phi}_{N}(\cdot \mid m) \xrightarrow{w^{*}} \phi\left(\cdot \mid m, \tilde{\omega}^{\star}\right)$ almost surely when the true state is $\tilde{\omega}^{\star}$.

Fix an arbitrary state $\tilde{\omega}$ and consider the ratio of the right-hand side of Equation D. 3 at $\tilde{\omega}$ and $\tilde{\omega}^{\star}$ :


Calibrated Mechanism Design

$$
\frac{\mu_{N|M|}\left(h^{N|M|}\right)(\tilde{\omega})}{\mu_{N|M|}\left(h^{N|M|}\right)\left(\tilde{\omega}^{\star}\right)}=\frac{\left(\mu_{0} \otimes \eta\right)(\tilde{\omega})}{\left(\mu_{0} \otimes \eta\right)\left(\tilde{\omega}^{\star}\right)} \prod_{a \in A} \prod_{m \in M}\left(\frac{\phi(a \mid m, \tilde{\omega})}{\phi\left(a \mid m, \tilde{\omega}^{\star}\right)}\right)^{N \hat{\phi}_{N, m}\left(h^{N|M|}\right)(a)}
$$

Suppose $\tilde{\omega} \notin P\left(\tilde{\omega}^{\star}\right)$. By definition of the partition $\mathscr{P}$, a message $m \in M$ and allocation $a \in A$ exist such that $\phi(a \mid m, \tilde{\omega}) \neq \phi\left(a \mid m, \tilde{\omega}^{\star}\right)$. Taking logarithm on both sides of Equation D. 4 and dividing by $N$,

$$
\frac{1}{N} \log \left(\frac{\mu_{N|M|}\left(h^{N|M|}\right)(\tilde{\omega})}{\mu_{N|M|}\left(h^{N|M|}\right)\left(\tilde{\omega}^{\star}\right)}\right)=\frac{1}{N} \log \left(\frac{\left(\mu_{0} \otimes \eta\right)(\tilde{\omega})}{\left(\mu_{0} \otimes \eta\right)\left(\tilde{\omega}^{\star}\right)}\right)+\sum_{a^{\prime} \in A} \sum_{m^{\prime} \in M} \hat{\phi}_{N, m^{\prime}}\left(h^{N|M|}\right)\left(a^{\prime}\right) \log \left(\frac{\phi\left(a^{\prime} \mid m^{\prime}, \tilde{\omega}\right)}{\phi\left(a^{\prime} \mid m^{\prime}, \tilde{\omega}^{\star}\right)}\right) .
$$

Because $\hat{\phi}_{N}(\cdot \mid m) \xrightarrow{w^{*}} \phi\left(\cdot \mid m, \tilde{\omega}^{\star}\right)$ almost surely when the true state is $\tilde{\omega}^{\star}$,

$$
\lim _{N \rightarrow \infty} \frac{1}{N} \log \left(\frac{\mu_{N|M|}\left(h^{N|M|}\right)(\tilde{\omega})}{\mu_{N|M|}\left(h^{N|M|}\right)\left(\tilde{\omega}^{\star}\right)}\right)=-\sum_{m \in M} \mathrm{D}_{\mathrm{KL}}\left(\phi\left(\cdot \mid m, \tilde{\omega}^{\star}\right) \mid \phi(\cdot \mid m, \tilde{\omega})\right),
$$

where $\mathrm{D}_{\mathrm{KL}}$ is the Kullback-Leibler divergence. Note that at least one of the terms in the KL-divergence is positive as $\phi(\cdot \mid m, \tilde{\omega}) \neq \phi\left(\cdot \mid m, \tilde{\omega}^{\star}\right)$. Hence,

$$
\lim _{N \rightarrow \infty} \log \left(\frac{\mu_{N|M|}\left(h^{N|M|}\right)(\tilde{\omega})}{\mu_{N|M|}\left(h^{N|M|}\right)\left(\tilde{\omega}^{\star}\right)}\right)=-\infty,
$$

meaning that $\mu_{N|M|}\left(h^{N|M|}\right)(\tilde{\omega}) / \mu_{N|M|}\left(h^{N|M|}\right)\left(\tilde{\omega}^{\star}\right) \rightarrow 0$.
Suppose now that $\tilde{\omega} \in P\left(\tilde{\omega}^{\star}\right)$. Then, Equation D. 4 reduces to

$$
\frac{\mu_{N|M|}\left(h^{N|M|}\right)(\tilde{\omega})}{\mu_{N|M|}\left(h^{N|M|}\right)\left(\tilde{\omega}^{\star}\right)}=\frac{\left(\mu_{0} \otimes \eta\right)(\tilde{\omega})}{\left(\mu_{0} \otimes \eta\right)\left(\tilde{\omega}^{\star}\right)},
$$

for all $N$.

[^39]Collecting both cases, we conclude that conditional on the true state being $\tilde{\omega}^{\star}$,

$$
\sum_{\tilde{\omega} \in P\left(\tilde{\omega}^{\star}\right)} \mu_{N|M|}(\tilde{\omega}) \rightarrow_{N \rightarrow \infty} 1,
$$

and moreover, within the cell, the fixed-ratio property implies the law $\tau_{N|M|, \tilde{\omega}^{\star}} \xrightarrow{w^{*}} \delta_{\mu\left(\cdot \mid P\left(\tilde{\omega}^{\star}\right)\right)}$. We conclude that the unconditional belief distribution, $\sum_{\tilde{\omega} \in \tilde{\Omega}}\left(\mu_{0} \otimes \eta\right)(\tilde{\omega}) \tau_{N|M|, \tilde{\omega}} \xrightarrow{w^{*}} \sum_{\tilde{\omega} \in \tilde{\Omega}}\left(\mu_{0} \otimes \eta\right)(\tilde{\omega}) \delta_{\mu(P(\tilde{\omega}))}=$ $\tau_{\phi}$. In particular,

$$
U\left(\sigma_{N}^{\prime}\right)=\mathbb{E}_{\mathbb{P}_{\sigma_{N}^{\prime}} \circ \mu_{N|M|}^{-1}}\left[u^{*}(\mu)\right] \rightarrow \mathbb{E}_{\tau_{\phi}}\left[u^{*}(\mu)\right]
$$

completing the proof. $\square$

Lemma D.1. Suppose $u: \Delta(\Omega) \rightarrow \mathbb{R}$ satisfies that

$$
\int_{\Delta(\Omega)} u(\mu) \tau(d \mu)<\int_{\Delta(\Omega)} \max \{u(\mu), 0\} \tau(d \mu)
$$

then a set $B \subseteq \Delta(\Omega)$ open relative to $\Delta(\Omega)$ exists such that

$$
\int_{B} u(\mu) \tau(d \mu)<0
$$

Proof. Define the positive and negative parts of $u$ :

$$
u_{+}(\mu):=\max \{u(\mu), 0\}, \quad u_{-}(\mu):=\max \{-u(\mu), 0\} .
$$

Then $u=u_{+}-u_{-}$pointwise. Integrating and using the assumed strict inequality,

$$
\int u d \tau=\int u_{+} d \tau-\int u_{-} d \tau<\int u_{+} d \tau \Rightarrow \int u_{-} d \tau>0 .
$$

Hence the set $N:=\{\mu \in \Delta(\Omega): u(\mu)<0\}$ has strictly positive mass under $\tau$.
Embed $\Delta(\Omega) \subset \mathbb{R}^{|\Omega|}$. Extend $\tau$ to a finite Borel measure $\tilde{\tau}$ on $\mathbb{R}^{d}$ by

$$
\tilde{\tau}(A):=\tau(A \cap \Delta(\Omega)) \quad\left(A \subseteq \mathbb{R}^{d} \text { Borel }\right),
$$

and extend $u$ to $\tilde{u}: \mathbb{R}^{d} \rightarrow[-1,1]$ by $\tilde{u}=u$ on $\Delta(\Omega)$ and $\tilde{u}=0$ on $\mathbb{R}^{d} \backslash \Delta(\Omega)$. Then $\tilde{u} \in L^{1}(\tilde{\tau})$ and $\tilde{\tau}(N)=\tau(N)>0$.

Let $B_{\mathbb{R}|\Omega|}(x, r)$ denote the ball in $\mathbb{R}^{|\Omega|}$ with center $x$ and radius $r$. By the Lebesgue differentiation theorem for finite Borel measures on $\mathbb{R}^{d}$, there is a $\tilde{\tau}$-full-measure set $D \subseteq \mathbb{R}^{d}$ such that for every $x \in D$,

$$
\lim _{r \downarrow 0} \frac{1}{\tilde{\tau}\left(B_{\mathbb{R}^{d}}(x, r)\right)} \int_{B_{\mathbb{R}^{d}}(x, r)} \tilde{u} d \tilde{\tau}=\tilde{u}(x),
$$

whenever $\tilde{\tau}\left(B_{\mathbb{R}^{d}}(x, r)\right)>0$ (and this positivity holds for all sufficiently small $r$ for $\tilde{\tau}$-a.e. $x$ ). Since $\tilde{\tau}(N \cap D)>0$, choose $\mu_{0} \in N \cap D$. Then $\tilde{u}\left(\mu_{0}\right)=u\left(\mu_{0}\right)<0$. Therefore the above limit is strictly negative,
so there exists $r_{0}>0$ such that for all $0<r<r_{0}$,

$$
\int_{B_{\mathbb{R}^{d}}\left(\mu_{0}, r\right)} \tilde{u} d \tilde{\tau}<0
$$

For such an $r$, let $B:=B_{\Delta}\left(\mu_{0}, r\right)=\Delta(\Omega) \cap B_{\mathbb{R}^{d}}\left(\mu_{0}, r\right)$, which is an open ball in $\Delta(\Omega)$. Using the definitions of $\tilde{\tau}$ and $\tilde{u}$,

$$
\int_{B} u d \tau=\int_{B_{\mathbb{R}^{d}}\left(\mu_{0}, r\right)} \tilde{u} d \tilde{\tau}<0
$$

This proves the claim. $\square$


Calibrated Mechanism Design

## D.3.1 Revelation principle for limit of means preferences

We show in this section that when the designer uses dynamic mechanisms, it is without loss of generality for the designer to employ direct dynamic mechanisms that (i) implement the outside option at all histories after the agent first exercises her option not to participate in the mechanism, and (ii) for which the agent's best response is to always participate and truthfully report her type. This justifies the class of mechanisms we employ in the analysis of Section 5.2.

Histories, mechanisms, and strategies As in the main text, to simplify notation, we do not include the agent's decision to participate in the mechanism in the histories of the game. Instead, we follow the convention that if the agent does not participate, it is as if she reported $\varnothing$ and the allocation is $a_{\varnothing}$. Formally, let $M A_{\varnothing}=(M \times A) \cup\left\{\left(\varnothing, a_{\varnothing}\right)\right\}$. With this notation, a history through period $t$ is an element of $\hat{H}_{M}^{t} \equiv\left(M A_{\varnothing}\right)^{t-1}$ and let $\hat{\mathcal{H}}_{M}^{t}=\Omega \times \hat{H}_{M}^{t} .{ }^{58}$

A mechanism is a collection $\varphi \equiv\left(\varphi_{t}\right)_{t=1}^{\infty}$ such that the mechanism in period $t$ is a mapping $\varphi_{t}$ : $\hat{\mathcal{H}}_{M}^{t} \times M \rightarrow \Delta(A)$.

Let $H_{M}^{t}=\left(\Theta \times M A_{\varnothing}\right)^{t-1}=\Theta^{t-1} \times \hat{H}_{M}^{t}$. The agent's strategy, ( $p, \sigma$ ), is given by her participation strategy $p_{t}: H_{M}^{t} \times \Theta \rightarrow[0,1]$, and conditional on participating, her reporting strategy $\sigma_{t}: H_{M}^{t} \times \Theta \rightarrow \Delta(M)$.

The distribution over terminal histories To obtain the complete description of the paths on the tree we need to append $\Omega$ to $H_{M}^{t}$; hence the paths through period $t-1$ are $\Omega \times H_{M}^{t} \equiv \mathcal{H}_{M}^{t}$. The distributions over states and agent's types, the agent's strategy, and the mechanism induce a distribution over the terminal histories $\mathcal{H}_{M}^{\infty} \equiv \Omega \times H_{M}^{\infty}$, which we denote by $\mathbb{P}_{\varphi,(p, \sigma)} \in \Delta\left(\Omega \times H_{M}^{\infty}\right)$, as it is now useful to keep track of the mechanism. We denote by $\mathbb{E}_{(p, \sigma)}$ the expectation under this measure. The distribution $\mathbb{P}_{\varphi,(p, \sigma)} \in \Delta\left(\Omega \times H_{M}^{\infty}\right)$ is the unique distribution that satisfies that for all $t \in \mathbb{N}, \tilde{\mathcal{H}}_{M}^{t} \subset \Omega \times H_{M}^{t}$,

$$
\mathbb{P}_{\varphi,(p, \sigma)}\left(\tilde{\mathcal{H}}_{M}^{t} \times \prod_{s=t+1}^{\infty}\left(\Theta \times M A_{\varnothing}\right)\right)=\mathbb{P}_{\varphi,(p, \sigma)}^{t}\left(\tilde{\mathcal{H}}_{M}^{t}\right),
$$

[^40]where the distributions $\left(\mathbb{P}_{\varphi,(p, \sigma)}^{t}\right)_{t \in \mathbb{N}}$ satisfy
$$
\begin{aligned}
\mathbb{P}_{\varphi,(p, \sigma)}^{t+1}\left(\omega, h_{M}^{t}, \theta, m, a\right) & =\mathbb{P}_{\varphi,(p, \sigma)}^{t}\left(\omega, h_{M}^{t}\right) f(\theta) p_{t}\left(h_{M}^{t}, \theta\right) \sigma_{t}\left(h_{M}^{t}, \theta\right)(m) \varphi_{t}\left(\omega, \hat{h}_{M}^{t}, m\right)(a), \\
\mathbb{P}_{\varphi,(p, \sigma)}^{t+1}\left(\omega, h_{M}^{t}, \theta, \varnothing, a\right) & =\mathbb{P}_{\varphi,(p, \sigma)}^{t}\left(\omega, h_{M}^{t}\right) f(\theta)\left(1-p_{t}\left(h_{M}^{t}, \theta\right)\right) \mathbb{1}\left[a=a_{\varnothing}\right] .
\end{aligned}
$$

Outcome distribution Our interest is in the distribution over payoff-relevant outcomes, $\Omega \times(\Theta \times A)^{\infty}$, and hence on the marginal of $\mathbb{P}_{\varphi,(p, \sigma)}$ on $\Omega \times(\Theta \times A)^{\infty}$, which we denote by $\overline{\mathbb{P}}_{\varphi,(p, \sigma)}$.

Best response We say that strategy $(p, \sigma)$ is a best response for the agent if for all alternative strategies $\left(p^{\prime}, \sigma^{\prime}\right)$, we have that

$$
\liminf _{T \rightarrow \infty} \mathbb{E}_{(p, \sigma)}\left[U_{T}\right] \geq \limsup _{T \rightarrow \infty} \mathbb{E}_{\left(p^{\prime}, \sigma^{\prime}\right)}\left[U_{T}\right],
$$

where recall $U_{T}$ is the agent's average payoff until period $T$.

Direct and full participation mechanisms A special case of the above game is that in which $M=\Theta$, and whenever the agent does not participate, the mechanism chooses $a_{\varnothing}$ with probability 1 for any message in all continuation histories. We call these mechanisms direct and full participation mechanisms. Below, when $M=\Theta$, we drop the dependence of the set histories on $M$.

Formally, let $\hat{\mathcal{H}}_{\varnothing}^{t}$ denote the subset of $\hat{\mathcal{H}}^{t}$ such that at some point the sequence ( $\varnothing, a_{\varnothing}$ ) appears. We define mechanisms

$$
\tilde{\varphi}_{t}: \hat{\mathcal{H}}^{t} \times \Theta \rightarrow \Delta(A),
$$

such that $\tilde{\varphi}_{t}\left(\omega, \hat{h}^{t}, \cdot\right)=\mathbb{1}\left[a=a_{\varnothing}\right]$ whenever $\left(\omega, \hat{h}^{t}\right) \in \hat{\mathcal{H}}_{\varnothing}^{t}$.
Theorem D.1. Suppose that $(p, \sigma)$ is a best response to mechanism $\varphi$. Then, a direct and full participation mechanism $\tilde{\varphi}$ exists such that

1. Participation with probability 1 and truthtelling are a best response for the agent,
2. The distribution over $\Omega \times(\Theta \times A)^{\infty}$ induced by $(\varphi,(p, \sigma))$ is the same as that induced by $\tilde{\varphi}$ under participation and truthtelling.

Proof. Fix a mechanism $\varphi=\left(\varphi^{t}\right)_{t \geq 1}$ and a best response ( $p, \sigma$ ) for the agent in the sense of Equation 13. Let $\mathbb{P}_{\varphi,(p, \sigma)}$ denote the induced distribution over $\mathcal{H}^{\infty}$. We write $\left(\omega,\left(\theta_{t}, m_{t}, a_{t}\right)_{t \geq 1}\right)$ for a generic realization, where $m_{t}=\varnothing \Rightarrow a_{t}=a_{\varnothing}$.


Calibrated Mechanism Design

The proof proceeds in three steps. In the first step, we construct a direct (but not full participation) mechanism $\tilde{\varphi}$, which under participation and truthtelling after every history on path implements the same outcome distribution as $(\varphi,(p, \sigma))$. In the second step, we verify that participation and truthtelling after every history on path is a best response to $\tilde{\varphi}$. In the third step, we construct a direct and full participation mechanism from $\tilde{\varphi}$. That the agent can always quit the mechanism at each step and obtain $a_{\varnothing}$ and Step 3 implies that participation and truthtelling after every history is also a best response to the full participation mechanism obtained from $\tilde{\varphi}$.

Step 1: We first construct the direct mechanism $\tilde{\varphi}_{t}: \hat{\mathcal{H}}^{t} \times \Theta \rightarrow \Delta(A)$. Define a collection of transition probabilities $\kappa_{t}: \mathcal{H}_{M}^{t} \times \Theta \rightarrow \Delta(M \cup\{\varnothing\})$ as follows:

$$
\kappa_{t}\left(m_{t} \mid h_{M}^{t}, \theta_{t}\right)=\left(1-p_{t}\left(h_{M}^{t}, \theta_{t}\right)\right) \mathbb{1}\left[m_{t}=\varnothing\right]+p_{t}\left(h_{M}^{t}, \theta_{t}\right) \sigma_{t}\left(h_{M}^{t}, \theta_{t}\right)\left(m_{t}\right) .
$$

We construct $\tilde{\varphi}$ recursively. In period 1, if the state is $\omega$ and the report is $\theta_{1}$, the designer draws fictitious $m_{1}$ from $\kappa_{1}\left(\cdot \mid \theta_{1}\right)$, and implements $a_{1}=a_{\varnothing}$ if $m_{1}=\varnothing$, and otherwise draws $a_{1} \sim \varphi_{1}\left(\omega, m_{1}\right)$.

Recursively, for $t \geq 2$, if the sequence of reports, fictitious messages, and allocations is $\left(\theta^{\prime t-1}, m^{t-1}, a^{t-1}\right)=$ $\left(\theta_{s}^{\prime}, m_{s}, a_{s}\right)_{s=1}^{t-1}$ and the agent reports $\theta_{t}$, the designer draws $m_{t}$ from $\kappa_{t}\left(\cdot \mid \theta^{\prime t-1}, m^{t-1}, a^{t-1}, \theta_{t}\right)$ and implements $a_{t}=a_{\varnothing}$ if $m_{t}=\varnothing$, and otherwise draws $a_{t} \sim \varphi_{t}\left(\omega, m^{t-1}, a^{t-1}, m_{t}\right) .{ }^{59}$

It is immediate that under truthtelling and participation the mechanism $\tilde{\varphi}$ implements the same distribution over $\Omega \times(A \times \Theta)^{\infty} .{ }^{60}$

Step 2: Let $\left(p^{*}, \sigma^{*}\right)$ denote the agent's strategy that participates and truthfully reports after every history. We now show that $\left(p^{*}, \sigma^{*}\right)$ is a best response to $\tilde{\varphi}$ in the sense of Equation D.8.

To do so, we show that for any strategy ( $\tilde{p}, \tilde{\sigma}$ ) in the game induced by the direct mechanism $\tilde{\varphi}$, a strategy $\left(p^{\prime}, \sigma^{\prime}\right)$ exists such that

$$
\mathbb{E}_{\tilde{\varphi},(\tilde{p}, \tilde{\sigma})}\left[U_{T}\right]=\mathbb{E}_{\varphi,\left(p^{\prime}, \sigma^{\prime}\right)}\left[U_{T}\right] \quad \text { for all } T,
$$

where $U_{T}$ is the average payoff through period $T$. Given Equation D. 9 and the best-response property of $(p, \sigma)$ to $\varphi$,

$$
\liminf _{T \rightarrow \infty} \mathbb{E}_{\varphi,(p, \sigma)}\left[U_{T}\right] \geq \limsup _{T \rightarrow \infty} \mathbb{E}_{\varphi,\left(p^{\prime}, \sigma^{\prime}\right)}\left[U_{T}\right] \quad \text { for all }\left(p^{\prime}, \sigma^{\prime}\right) .
$$

Using Step 1, we have $\mathbb{E}_{\varphi,(p, \sigma)}\left[U_{T}\right]=\mathbb{E}_{\tilde{\varphi},\left(p^{*}, \sigma^{*}\right)}\left[U_{T}\right]$ for all $T$, and by Equation D. 9 we have $\mathbb{E}_{\varphi,\left(p^{\prime}, \sigma^{\prime}\right)}\left[U_{T}\right]=$ $\mathbb{E}_{\tilde{\varphi},(\tilde{p}, \tilde{\sigma})}\left[U_{T}\right]$ for all $T$. Hence

$$
\liminf _{T \rightarrow \infty} \mathbb{E}_{\tilde{\varphi},\left(p^{*}, \sigma^{*}\right)}\left[U_{T}\right] \geq \limsup _{T \rightarrow \infty} \mathbb{E}_{\tilde{\varphi},(\tilde{p}, \tilde{\sigma})}\left[U_{T}\right] \quad \text { for all }(\tilde{p}, \tilde{\sigma}),
$$

which is exactly the definition of $\left(p^{*}, \sigma^{*}\right)$ being a best response to $\tilde{\varphi}$. It remains to construct $\left(p^{\prime}, \sigma^{\prime}\right)$ and verify (D.9).

We show how the agent can emulate the strategy ( $\tilde{p}, \tilde{\sigma}$ ) in the game induced by the indirect mechanism via strategy $\left(p^{\prime}, \sigma^{\prime}\right)$. Define $\tilde{\kappa}_{t}: H^{t} \times \Theta \rightarrow \triangle(\Theta \cup\{\varnothing\})$ as follows

$$
\tilde{\kappa}_{t}\left(\theta^{\prime} \mid h^{t}, \theta_{t}\right)=\left(1-\tilde{p}\left(h^{t}, \theta_{t}\right)\right) \mathbb{1}\left[\theta^{\prime}=\varnothing\right]+\tilde{p}\left(h^{t}, \theta_{t}\right) \tilde{\sigma}\left(h^{t}, \theta_{t}\right)\left(\theta^{\prime}\right) .
$$

The strategy ( $p^{\prime}, \sigma^{\prime}$ ) privately simulates the report process induced by ( $\tilde{p}, \tilde{\sigma}$ ) in the direct mechanism, and conditional on the fictitious reports, generates the actual message in the indirect mechanism $\varphi$ using the kernel $\kappa_{t}$ in Step 1, evaluated at the fictitious type history.

[^41]Formally, for $t \geq 1$ given the history of types, messages, and allocations through period $t,\left(\theta^{t-1}, m^{t-1}, a^{t-1}\right)$, and the privately tracked fictitious type reports $\theta^{\prime-1}$, ${ }^{61}$ the agent of type $\theta_{t}$ draws a fictitious report $\theta_{t}^{\prime} \sim \tilde{\kappa}_{t}\left(\cdot \mid \theta^{t-1}, \theta^{\prime \text { t }-1}, a^{t-1}, \theta_{t}\right)$. If $\theta_{t}^{\prime}=\varnothing$, then $m_{t}=\varnothing$ (the agent does not participate in period $t$ ). Otherwise, $\theta_{t}^{\prime} \in \Theta$ and $m_{t}$ is drawn from $\kappa_{t}\left(\cdot \mid \theta^{\prime-1}, m^{t-1}, a^{t-1}, \theta_{t}^{\prime}\right)$. The allocation is $a_{\varnothing}$ upon rejection, and $a_{t} \sim \varphi_{t}\left(\omega, m^{t-1}, a^{t-1}, m_{t}\right)$, otherwise.


Calibrated Mechanism Design

It is immediate that Equation D. 9 holds and hence $\left(p^{*}, \sigma^{*}\right)$ is a best response to $\tilde{\varphi}:{ }^{62}$ In the extensive form game induced by $\tilde{\varphi}$, the designer simulates the agent's participation and reporting strategies using $(p, \sigma)$ based on the agent's type reports and determines allocations in the mechanism. When the agent's strategy is given by $(\tilde{p}, \tilde{\sigma})$, the process described in the above paragraph correspond to the designer's simulated participation and reporting strategies, and allocations continued to be determined by $\varphi$. Hence, in the extensive form game induced by $\tilde{\varphi}$, when the agent plays ( $\tilde{p}, \tilde{\sigma}$ ), it is as if she faces mechanism $\varphi$ and plays strategy ( $p^{\prime}, \sigma^{\prime}$ ). The best response property of ( $p, \sigma$ ) implies that playing $\left(p^{\prime}, \sigma^{\prime}\right)$ yields a weakly worse payoff, and hence $(\tilde{p}, \tilde{\sigma})$ is not a profitable deviation from $\left(p^{*}, \sigma^{*}\right)$ in the direct mechanism $\tilde{\varphi}$.

Step 3: Modify $\tilde{\varphi}$ at all histories that include at least one non-participation decision, so that the mechanism implements the outside option $a_{\varnothing}$. With this modification, the mechanism satisfies the full participation property. It is immediate that participation and truthtelling after every history remains a best response. $\square$

[^42]
[^0]:    *The latest version of the paper can be found here. We thank Dirk Bergemann, James Best, Elliot Lipnowski, Emir Kamenica, Navin Kartik, Stephen Morris , Phil Reny, Marzena Rostek, and Refine.ink for valuable comments and suggestions, as well as seminar audiences at EARIE 2024, WUSTL Economic Theory Conference 2024, MIT IDSS Distinguished Speaker Series, the Workshop in Market Design, Bonn, Berlin HU, CERGE-EI, Gerzensee, UCL, Naples, Stanford University, MIT Sloan, and the CEPR Workshop on Contracts, Incentives and Information at Collegio Carlo Alberto. Laura Doval gratefully acknowledges financial support from the Sloan Foundation. Alex Smolin gratefully acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (grant ANR-17-EURE-0010) and the AI Interdisciplinary Institute ANITI (grant ANR-23-IACL-0002). This paper was partly written while the first author was visiting Stanford University and the second author was visiting Northwestern University and Columbia Business School; we are grateful for their hospitality.
    ${ }^{\dagger}$ Columbia Business School and CEPR. E-mail: laura.doval@columbia.edu.
    ${ }^{\ddagger}$ Toulouse School of Economics and CEPR. E-mail: alexey.v.smolin@gmail.com.

[^1]:    ${ }^{1}$ In (generalized) two-stage mechanisms, the designer communicates with the agents before the agents communicate with the mechanism. Whereas this communication is a restriction on the set of implementable outcomes relative to the single-designer Myersonian benchmark, Attar et al. (2025) show that allowing competing principals to first communicate with agents expands the set of implementable outcomes.

[^2]:    ${ }^{2}$ We assume the agent has limit-of-means preferences, so we can pass to the $\delta \rightarrow 1$ limit without approximation.
    ${ }^{3}$ Theorem 3 holds when the agent's type is drawn once at the beginning without further assumptions on the distribution. As we explain in Section 5, we choose the i.i.d. specification for the evolution of the agent's private information to put repeated and dynamic mechanisms on a more equal footing.

[^3]:    ${ }^{4}$ There is also a literature that studies a designer's disclosure of information that must be elicited from the agents (Eső and Szentes, 2007; Bergemann and Pesendorfer, 2007; Li and Shi, 2017; Krähmer, 2020; Bergemann et al., 2022a,b; Smolin, 2023). By contrast, the designer knows the realization of the state and also what information the two-stage mechanism discloses to the agents, so he need not elicit this information.

[^4]:    ${ }^{5}$ Assuming the allocation space is a product space is without loss of generality. Any restriction on the allocations, such as all agents must receive the same allocation, can be incorporated as restrictions on the support of the mechanism.
    ${ }^{6}$ Because we allow for lotteries over allocations, that the allocation space is finite does not preclude the case of transferable utility. Indeed, we could let each $A_{i}=\tilde{A_{i}} \times\{-K, K\}$ for some large enough $K>0$.

[^5]:    ${ }^{7}$ When the agent is infinitely patient as in Section 5, we can exhibit a sequence of strategies under which the agent (approximately) learns this mapping. See the proof of Lemma C. 4 in Appendix D.
    ${ }^{8}$ Gentzkow and Kamenica (2017) highlight that the language in Green and Stokey (2022) allows one to describe the correlation across signal structures. This is exactly what we need to allow the designer to obfuscate the agents' ability to learn. It is again instructive to consider the single-agent case. For each type report $\theta \in \Theta$, the mechanism can be seen as an information structure $\phi(\theta, \cdot): \Omega \times[0,1] \rightarrow \Delta(A)$. Thus, the randomization device allows the designer to control the correlation across these different signal structures, which in turn disciplines what the agent stands to learn when experimenting with different reports.
    ${ }^{9}$ The terminology is by analogy to reduced form auctions where the map from own types to own probabilities of being allocated the good are referred to as the interim allocation.

[^6]:    ${ }^{10}$ When we consider mechanisms with arbitrary message spaces in Appendix D.1, the requirement of calibration is relative to both the mechanism and agents' equilibrium participation and reporting strategies.
    ${ }^{11}$ Thus, we are assuming that $a_{\varnothing}=\left(a_{i \varnothing}\right)_{i \in N}$ is an element of $A$. In Appendix D.1, we consider more general participation decisions, allowing the mechanism to condition on the set of participating agents, but even with this extra generality, it is still without loss to restrict attention to mechanisms that induce full participation.


Calibrated Mechanism Design

[^7]:    ${ }^{12}$ It is common for online platforms to inform agents of the consequences of their choices: marketplaces inform sellers of their probability of sale at different posted prices, and transportation providers inform riders of their probability of receiving a seat upgrade at different bid levels. Even insurance companies provide consumers with projected expenditures under different plan choices.

[^8]:    ${ }^{13}$ Indeed, in the analysis of the dynamic interaction in Section 5, we assume the agent only observes her type and her allocation, but not her payoffs.
    ${ }^{14}$ This assumption is routinely made in dynamic settings. See Pavan et al. (2014) and Cesa-Bianchi et al. (2024) for two examples in the context of agents' behavior within mechanisms.
    ${ }^{15}$ In Section 3, we provide an example in which when agents have access to the calibrated information structure the optimal Myersonian mechanism fails to be incentive compatible.

[^9]:    ${ }^{16}$ A tempting comparison is Maskin and Tirole (1990, Prop. 11): with private values and quasilinear utilities, the informed principal's unique equilibrium payoff coincides with the state-by-state optimum. The authors show this conclusion depends on quasilinearity: absent this assumption, an informed principal can benefit from concealing his information in the case of private values. Instead, Theorem 1 relies neither on quasilinearity nor on the designer's lack of commitment.

[^10]:    ${ }^{17}$ Formally,

    $$
    \mu_{0}(\omega \mid \theta)=\frac{\mu_{0}(\omega) f(\theta \mid \omega)}{\sum_{\omega^{\prime} \in \Omega} \mu_{0}\left(\omega^{\prime}\right) f\left(\theta \mid \omega^{\prime}\right)}
    $$

    ${ }^{18}$ We refer the reader to the appendix for our mathematical conventions, in particular, the definition of the corresponding $\sigma$-algebras.

[^11]:    ${ }^{19}$ Formally, define the belief distribution induced by $\beta, \tau_{\beta}=\mu_{0} \otimes \beta$. The claim is that $\mathbb{E}_{\tau_{\beta}}[\mu]=\mu_{0}$.
    ${ }^{20}$ This is a consequence of Bayes rule: beliefs are a sufficient statistic for $\omega$. Hence, conditional on $(\theta, \mu)$, the allocation rule carries no more information about the state.

[^12]:    ${ }^{21}$ In contrast to the model of Section 2, we are assuming the set of types and allocations to be intervals in the real line. The results in the previous sections go through with richer type and allocation spaces, at the cost of more notation.

[^13]:    ${ }^{22}$ This assumption ensures the Lipschitz continuity of the agent's indirect utility function when the allocation space is infinite. Because $\Omega$ is finite, requiring the condition to hold state-by-state suffices.

[^14]:    ${ }^{23}$ The calibrated mechanism which state-by-state implements the optimal direct mechanism under common knowledge of the state may reveal less than full information about the state, e.g., because at $\omega$ and $\omega^{\prime}$ the same direct mechanism is optimal. The point is that pooling those states does not weaken the incentive constraints of the agent, so it is as if the designer were forced to reveal the state.

[^15]:    ${ }^{24}$ Restricting attention to deterministic mechanisms, Ottaviani and Prat (2001) obtain the optimality of full disclosure without such a linearity assumption. Their model, however, is different from ours and that of the aforementioned papers: the agent's type is not payoff relevant and the agent's type and the state are affiliated. Thus, while related in spirit, Proposition 2 is distinct from their result.
    ${ }^{25}$ See Szabadi (2018) for a similar observation.

[^16]:    ${ }^{26}$ Because payoffs are quasilinear, considering mechanisms that do not randomize on transfers is without loss of generality.
    ${ }^{27}$ The equi-Lipschitz condition on $u$ ensures we can take the derivative inside the integral.

[^17]:    ${ }^{28}$ The designer can be viewed as an online advertising platform and the agent as an advertiser. State $\omega$ represents the click-through rate of an ad slot, and higher $\theta$ corresponds to a larger advertiser willing to pay more for exposure. The designer's payoff captures both the value created by advertising and the disutility from showing ads of large advertisers, e.g., due to user brand fatigue.

[^18]:    ${ }^{29}$ In the statement, $w_{k \theta_{i}}$ denotes the derivative of $w_{k}$ with respect to $\theta_{i}$, for $k \in\{1, \ldots, N\}$.
    ${ }^{30}$ To be sure, Theorem 3 extends to the case in which the agent's type is fully persistent and correlated with the state.

[^19]:    ${ }^{31}$ The Ionescu-Tulcea extension theorem implies this measure is always well-defined for any mechanism and any agent's

[^20]:    strategy. See Appendix C. 1 for details.
    ${ }^{32}$ Throughout this section, limits of measures should always be understood in the weak* sense.
    ${ }^{33}$ Note that in mechanism design one always focuses on mechanisms that have well-defined best responses in single-agent settings, and equilibria in multi-agent ones.

[^21]:    ${ }^{34}$ By assumption, the marginal of $\bar{v}_{\sigma}^{T}$ on $A \times \Theta \times M \times \tilde{\Omega}$ converges to $v_{\sigma}$. Moreover, we show that the marginal of $\bar{v}_{\sigma}^{T}$ on $\Delta(\tilde{\Omega})$ also converges (Lemma C.3). However, this is not enough to ensure the convergence of $\bar{v}_{\sigma}^{T}$.
    ${ }^{35}$ Whereas the above two-stage mechanism is described in terms of beliefs over $\Omega \times \mathcal{E}$, we show in the appendix how to derive from it a two-stage mechanism in terms of beliefs over $\Omega$.

[^22]:    ${ }^{36}$ This notation allows us to keep the definitions of the histories when the agent participates and does not participate symmetric, and saves us on including the agent's participation strategy in the histories.
    ${ }^{37}$ That is, starting from a dynamic mechanism $\varphi$ and a best response strategy ( $p, \sigma$ ), one can construct an alternative mechanism $\varphi^{\prime}$ such that participation and truthtelling are a best response for the agent and preserves the distribution over $(\Omega \times \Theta \times A)^{\infty}$ induced by $(p, \sigma)$ and $\varphi$.


Calibrated Mechanism Design

[^23]:    ${ }^{38}$ This is easily seen in the example after Definition 7. Note the mechanism that allocates the good to the agent if and only if her type is $\theta_{2}$ satisfies monotonicity. Hence, a transfer scheme exists that implements this allocation rule with transfers.
    ${ }^{39}$ In a repeated principal-agent game with communication, Meng (2021) shows that the principal can guarantee in the patient limit his complete information payoff subject to the constraint that her actions satisfy the cyclic monotonicity condition in Rochet (1987). We view the results as complementary: We focus on implementable outcome distributions, instead of payoffs, when the agent has limit of the means preferences, which makes our notion of implementation exact.

[^24]:    ${ }^{40}$ A standard argument implies that if $\vartheta$ satisfies Equation 16, then a finite support $\beta^{\prime}$ exists such that $\vartheta$ satisfies Equation 16 with $\beta^{\prime}$ in place of $\beta$.

[^25]:    ${ }^{41}$ Let $\mu_{0}\left(\cdot \mid \theta_{i}\right) \in \Delta(\Omega)$ denote the prior of the agent with type $\theta_{i}$ and $\mu\left(\cdot \mid s_{i}\right)$ denote the updated belief of an agent with prior

[^26]:    $\mu_{0}$ upon observing signal $s_{i}^{*}$. When the signal is $s_{i}^{*}$, the agent with type $\theta_{i}$ updates her beliefs to:

    $$
    \mu_{i}\left(\cdot \mid \theta_{i}, s_{i}^{*}\right)=\frac{\mu_{0}\left(\cdot \mid \theta_{i}\right) \cdot \frac{\mu\left(\cdot \mid s_{i}^{*}\right)}{\mu_{0}(\cdot)}}{\left\|\mu_{0}\left(\cdot \mid \theta_{i}\right) \cdot \frac{\mu\left(\cdot \mid s_{i}^{*}\right)}{\mu_{0}(\cdot)}\right\|},
    $$

    where the ⋅ and / operations are meant componentwise, and $\|\cdot\|$ is the $l^{1}$-norm.
    ${ }^{42}$ Namely, Equation B. 5 implies that for all $(\theta, \omega), \vartheta(\cdot \mid \theta, \omega) \in \operatorname{clco}\{\alpha(\cdot \mid \theta, \mu): \mu \in \Delta(\Omega)\}$. Rubin and Wesler (1958) implies that our under assumptions $\operatorname{clco}\{\alpha(\cdot \mid \theta, \mu): \mu \in \Delta(\Omega)\}=\operatorname{co}\{\alpha(\cdot \mid \theta, \mu): \mu \in \Delta(\Omega)\}$, and the rest of the claim follows from Carathéodory's theorem.
    ${ }^{43}$ To extend the result to the case in which the agent's type is correlated with the state, note the following. Knowing $\mu_{0}$

[^27]:    updates to $\mu_{k}$ conditional on $s$ is enough to pin down the agent's belief $\mu(\cdot \mid \theta, s)$, with respect to which the agent's incentive compatibility and individual rationality constraints are defined (see footnote 41).

[^28]:    ${ }^{44}$ We could expand the mechanism by allowing the agent to have a message which triggers the outside option, but this is not necessary as $\alpha$ is individually rational.

[^29]:    ${ }^{45}$ Lemma C. 2 implies that $\bar{v}_{\sigma^{\prime}}^{T, 1}$ and $\bar{v}_{\sigma^{\prime}}^{T, 2}$ have the same set of subsequential limits. Indeed, let $g$

[^30]:    denote any continuous bounded function on $\Omega \times \Delta(\Omega) \times \Theta \times \Theta \times A$. Let $D_{T}(g)=\mathbb{E}_{\bar{v}_{\sigma^{\prime}}^{T, 2}}[g]-\mathbb{E}_{\bar{v}_{\sigma^{\prime}}^{T, 1}}[g]=$ $\frac{1}{T} \sum_{t=1}^{T} \mathbb{E}_{\sigma^{\prime}}\left[g\left(a_{t}, \theta_{t}, \theta_{t}^{\prime}, \omega, \mu_{t+1}\right)-g\left(a_{t}, \theta_{t}, \theta_{t}^{\prime}, \omega, \mu_{t}\right)\right]$. The argument in Lemma C. 2 implies that $D_{T}(g) \rightarrow 0$ as $T \rightarrow \infty$ (this does not rely on the existence of a limit, just the convergence of beliefs and the continuity of $g$ ). Now, let $T_{n}$ be such that $\bar{v}_{\sigma^{\prime}}^{T_{n}, 1} \xrightarrow{w^{*}} \tilde{v}$. Note that

    $$
    \int g d \bar{v}_{\sigma^{\prime}}^{T_{n}, 2}=\int g d \bar{v}_{\sigma^{\prime}}^{T_{n}, 1}+D_{T_{n}}(g) \rightarrow \int g d \tilde{v}+0,
    $$

    so a subsequential limit of $\bar{v}_{\sigma^{\prime}}^{T, 1}$ is a subsequential limit of $\bar{v}_{\sigma^{\prime}}^{T, 2}$. Switching the role of 1 and 2, we obtain the opposite set inclusion.

[^31]:    ${ }^{46}$ A set $X \subseteq \Delta(\Omega)$ is open relative to $\Delta(\Omega)$ if an open set $Y \subseteq \mathbb{R}^{|\Omega|}$ exists such that $X=Y \cap \Delta(\Omega)$. The boundary relative to $\Delta(\Omega)$ is analogously defined via open sets relative to $\Delta(\Omega)$.
    ${ }^{47}$ Lemma D. 1 provides an interval of radii $r \in\left(0, r_{0}\right)$ such that $\int_{B(\hat{\mu}, r)} U_{\text {net }}(\mu) \tau_{(p, \sigma)}(d \mu)<0$. Note that only countable many such $r$ can have $\tau_{(p, \sigma)}(\partial B(\hat{\mu}, r))>0$ (the boundaries for different radii are disjoint), so we can always pick $r$ such that $\tau_{(p, \sigma)}(\partial B(\hat{\mu}, r))=0$ and preserve the negative sign.

[^32]:    ${ }^{48}$ The property that $\tau_{(p, \sigma)}(\partial B)=0$ ensures that $\mathbb{1}\left[\mu_{t}\left(h^{\infty}\right) \in B\right]$ is eventually constant almost surely. Let $E=\left\{h^{\infty}: \mu_{\infty}\left(h^{\infty}\right) \notin\right.$ $\partial B\}$. On $E$, either $\mu_{\infty}$ is in the interior of $B$ (relative to $\Delta(\Omega)$ ) or in the interior of $B^{\complement}$ (relative to $\Delta(\Omega)$ ), that is, an $\epsilon>0$ exists such that $\left(B\left(\mu_{\infty}, \epsilon\right) \cap \Delta(\Omega)\right) \subset B$ or $B^{\complement}$. In either case, for each $h^{\infty}$, there exists $N\left(h^{\infty}\right)$ such that for all $t \geq N\left(h^{\infty}\right)$, $\mu_{t}\left(h^{\infty}\right) \in\left(B\left(\mu_{\infty}, \epsilon\right) \cap \Delta(\Omega)\right)$ and hence $\mathbb{1}\left[\mu_{t}\left(h^{\infty}\right) \in B\right]$ is eventually constant. When $\tau_{(p, \sigma)}(\partial B)=0$, we have that $E$ has probability 1 under $\mathbb{P}_{(p, \sigma)}$.
    ${ }^{49}$ Indeed, $\mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=1}^{T}\left(u\left(a_{t}, \theta_{t}, \omega\right)-u\left(a_{\varnothing}, \theta_{t}, \omega\right)\right) \mathbb{1}\left[\mu_{t} \in B\right]\right]=\mathbb{E}_{\bar{v}_{(p, \sigma)}^{T, 1}}\left[\left(u(a, \theta, \omega)-u\left(a_{\varnothing}, \theta, \omega\right)\right) \mathbb{1}[\mu \in B]\right]$ and our previous analysis implies it converges to $\int_{B} U_{\text {net }}(\mu) \tau_{(p, \sigma)}(d \mu)$.
    ${ }^{50}$ Choose $\bar{T}$ so that for all $T \geq \bar{T}$ :


Calibrated Mechanism Design

$$
    \begin{aligned}
    & \left|\mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=1}^{T}\left(u\left(a_{t}, \theta_{t}, \omega\right)-u\left(a_{\varnothing}, \theta_{t}, \omega\right)\right) \mathbb{1}\left[\mu_{t} \in B\right]\right]-\int_{B} U_{\text {net }}(\mu) \tau_{(p, \sigma)}(d \mu)\right| \\
    & +\left|\mathbb{E}\left[-\frac{1}{T} \sum_{t=1}^{L-1}\left(u\left(a_{t}, \theta_{t}, \omega\right)-u\left(a_{\varnothing}, \theta_{t}, \omega\right)\right) \mathbb{1}\left[\mu_{t} \in B\right]\right]\right| \leq \delta / 2 .
    \end{aligned}
    $$

[^33]:    ${ }^{51}$ Anticipating our construction in item 3, for each belief $\mu$ in the support of $\tau$, the allocation rule $\alpha^{\prime}(\cdot \mid \cdot, \mu)$ has no profitable undetectable deviations relative to payoff function $u^{\prime}(a, \theta)=\sum_{\omega \in \Omega} \mu(\omega) u(a, \theta, \omega)$.

[^34]:    ${ }^{52}$ Any $\|.\|_{p}$ would work, but larger $p$ results in weakly shorter adjustment phases.

[^35]:    ${ }^{53}$ In the expressions that follow, recall the full participation mechanism $\phi_{N}^{*}$ already averages over the agents' own randomization devices.

[^36]:    ${ }^{54}$ To be sure, the proof of Lemma C. 1 is written in the context of repeated mechanisms but it extends verbatim to dynamic mechanisms with simple notational adjustments.

[^37]:    ${ }^{55}$ To be sure,

    $$
    \mathbb{E}_{\sigma}\left[g\left(\mu_{t}\right)\right]=\int_{H^{\infty}} g\left(\mu_{t}\left(h^{\infty}\right)\right) \mathbb{P}_{\sigma}\left(d h^{\infty}\right)
    $$

[^38]:    ${ }^{56}$ The argument shows that the agent's equilibrium payoff is at least $U\left(\tau_{\phi}\right)$. However, it is immediate that $U\left(\tau_{\phi}\right)$ is the most the agent can make in the game as $\tau_{\phi}$ extracts all information from the mechanism.

[^39]:    ${ }^{57}$ This almost surely is under the law of $A$ under $\phi\left(\cdot \mid m, \tilde{\omega}^{\star}\right)$.

[^40]:    ${ }^{58}$ We index histories by the messages to distinguish these histories from those when the designer uses direct mechanisms.

[^41]:    ${ }^{59}$ Recall the fictitious reports encode the agent's participation decisions in the original mechanism.
    ${ }^{60}$ In fact, if we kept track of the designer's draws of fictitious messages, the new mechanism implements the same distribution over terminal histories $\mathcal{H}_{M}^{\infty}$, and a fortiori, its marginal over $\Omega \times(A \times \Theta)^{\infty}$ is the same.

[^42]:    ${ }^{61}$ Recall that to minimize notation and make history lengths symmetric across participation and nonparticipation, we record the agent's rejection of the mechanism as the empty message $\varnothing$.
    ${ }^{62}$ In fact, the construction ensures the stronger property that $\mathbb{P}_{\tilde{\varphi},(\tilde{p}, \tilde{\sigma})}^{T}=\mathbb{P}_{\varphi,\left(p^{\prime}, \sigma^{\prime}\right)}^{T}$, when in a slight abuse of notation we keep track of the fictitious messages in $\mathbb{P}_{\tilde{\varphi},(\tilde{p}, \tilde{\sigma})}^{T}$.


Calibrated Mechanism Design

## Citation and provenance

**Authors:** Laura Doval; Alex Smolin

**Canonical citation:** Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.

**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/calibrated-mechanism-design.md

**Source record:** https://alexsmolin.com/#research

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

Provenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt


> Machine-readable author manuscript.
> Authors: Shota Ichihashi; Alex Smolin.
> Canonical citation: Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.
> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt


# Buyer-Optimal Algorithmic Recommendations

**Authors:** Shota Ichihashi; Alex Smolin

**Manuscript date:** 2025-02-15

#### Abstract

In markets where algorithmic data processing is increasingly prevalent, recommendation algorithms can substantially affect trade and welfare. We consider a setting in which an algorithm recommends a product based on its value to the buyer and its price. We characterize an algorithm that maximizes the buyer's expected payoff and show that it strategically biases recommendations to induce lower prices. Revealing the buyer's value to the seller leaves overall payoffs unchanged while leading to more dispersed prices and a more equitable distribution of surplus across buyer types. These results extend to all Pareto-optimal algorithms and to multiseller markets, with implications for AI assistants and e-commerce ranking systems.

[^0]


## 1 Introduction

In today's digital economy, algorithmic decision-making is transforming how consumers navigate markets-from price trackers that seek and pinpoint lower-priced products, to robo-advisors that propose financial securities, to e-commerce platforms where ranking algorithms shape which products capture buyers' attention, and to AI-powered personal assistants that advise individuals on their purchasing decisions. ${ }^{1}$ Algorithms are not merely processing information but are actively influencing consumer and seller behavior.

In this paper, we formalize this dynamic by developing a model of algorithmic recommendations and examining how they can reshape market outcomes and redistribute welfare. Our model emphasizes three key features of recommendation algorithms. First, they operate according to preprogrammed rules. Second, they are capable of locating and processing critical information about a product's existence, inherent value, and price. Third, by influencing when and how recommendations are made, these algorithms can affect the purchasing decisions by the buyers and pricing decisions by the sellers.

We begin by developing a baseline model of bilateral trade. In this setting, a single product is traded between a buyer and a seller in the presence of uncertainty about both trade costs and values. The seller privately observes her production cost-her type- while the buyer is initially uninformed about both the product's existence and its value. An algorithm, however, can discover the product's value or, equivalently, produce its best estimate and issue a recommendation based on this information combined with the seller's posted price. Upon receiving a recommendation, the buyer updates her belief and decides whether to purchase at the offered price; if no recommendation is received, no trade occurs- that is, the algorithm can serve as a gatekeeper. The seller, fully aware of the algorithm's design, sets her price strategically to maximize profit.

The algorithm shapes equilibrium trade and welfare. We abstract away from the source of the algorithm and instead analyze all algorithms with desirable welfare properties. Motivated by the fact that currently most revenue for AI assistants comes from

[^1]their users, we present most of the analysis with the objective being the buyer's expected surplus, thus characterizing a buyer-optimal algorithm. However, our characterization and key results extend, with minimal changes, to any Pareto-optimal algorithm in the space of buyer and seller surpluses. Thus, our findings speak to a variety of market structures that include, for example, two-sided platforms aiming to accommodate both of their sides.

A buyer-optimal algorithm must navigate a trade-off: It needs to reward sellers for offering lower prices by increasing the frequency of recommendations, while ensuring that beneficial trades are not foregone. We show that this balance is optimally achieved by an algorithm that uses a pseudo value threshold-recommending the product when this adjusted metric exceeds the price, even if the true value might not. The equilibrium we characterize (Proposition 1) reveals three key insights. First, although the algorithm's design and the equilibrium prices depend on both the cost and value distributions, the final product allocation is driven solely by the cost distribution. Second, by deliberately deviating from a simple ex post optimal rule-eschewing recommendations at high prices even when the true value is favorable and endorsing products at low prices even when the value falls short-the algorithm increases the buyer's price sensitivity, thereby pressuring the seller to offer lower prices. ${ }^{2}$ Third, compared to standard monopoly pricing with ex post optimal trade, buyer-optimal algorithmic recommendations substitute high-value, high-cost trades with low-value, low-cost trades (Proposition 2).

The complete characterization of an optimal algorithm and the equilibrium strategies allows us to uncover major changes in the welfare implications of third-degree price discrimination compared to standard monopoly pricing. We show that if the seller can distinguish and price discriminate among different buyer segments, the algorithm adapts accordingly and fully mitigates the average effects of market segmentation. As a result, price discrimination does not affect the average price, buyer's total surplus, seller profits, or product allocation (Proposition 3). At the same time, finer segmentation enables the seller to correlate prices with values, and within a class of monotone segmentations, it

[^2]leads to more dispersed equilibrium prices and a more concentrated-and thus more equitable-distribution of consumer surplus (Proposition 4). These results hold broadly and demonstrate that algorithmic recommendations can serve as a powerful consumer protection tool, complementing existing regulatory methods (e.g., Scott Morton et al. (2019)).

Our analysis extends in several dimensions. First, our characterization and market segmentation insights extend to any Pareto-optimal algorithm by simply incorporating Pareto weights into the definition of a pseudo-value (Proposition 5). Second, most of our key results and characterizations generalize to settings with multiple competing sellers (Proposition 6). Notably, despite strategic competition among sellers, the specifics of market segmentation do not affect the average price, total buyer surplus, or seller profits (Proposition 7), and finer segmentation leads to more dispersed prices (Proposition 8). Third, we expand our analysis by allowing the buyer to be informed in advance about either the product's existence or its value, showing how this information can potentially harm the buyer and discussing how algorithm design can mitigate this harm (Proposition 9 and Proposition 10).


Related literature.- Our paper is closest to the recent strand of economic literature that examines methods of empowering buyers in monopolistic settings via information control. Roesler and Szentes (2017) analyze buyer-optimal learning in a bilateral trade setting. Like us, they show that the buyer benefits from ex post imperfect decisions to influence the seller's pricing; that is, full learning about the value is not optimal. Unlike us, they require learning to occur before the price is set, which limits its impact (see further Section 3.1 and Section 5.3). Deb and Roesler (forth.) extend this analysis to the case of a multiproduct monopoly and Bergemann et al. (2023) to auctions; Condorelli and Szentes (2020) analyze the buyer-optimal distribution of values within a given interval. We contribute to this literature by allowing the buyer's information to depend on the price and by allowing the seller to have private information. ${ }^{3}$

[^3]In our study of algorithm design under sellers' private information, we integrate Bayesian persuasion (e.g., Kamenica and Gentzkow (2011)) with mechanism design (Baron and Myerson (1982)). Several works have combined these frameworks in trade settings. Among these, the most closely related studies are Yang (2022), Bergemann and Bonatti (2024), Xu and Yang (2024), and Bergemann et al. (forth.). ${ }^{4}$ We share the perspective that new information technologies enable powerful third parties to process data, extract valuable insights, and shape market outcomes by serving as informational gatekeepers. However, our focus differs in that those studies allow the intermediary to charge transfers and study revenue maximization, whereas we focus on achieving Pareto efficiency. We further differ from Bergemann and Bonatti (2024) and Bergemann et al. (forth.) in the nature of the friction faced by the designer: Whereas they contend with buyers' off-site purchase opportunities, we focus on the private information held by sellers. In this respect, we are closer to Yang (2022) and Xu and Yang (2024), sharing some features of the solution with them (Remark 2).

Our analysis in Section 4 offers a novel perspective on the classic question of the impact of price discrimination based on consumer information, as studied in the market segmentation literature (e.g., Bergemann et al. (2015) and Haghpanah and Siegel (2023)). We show that consumer use of algorithms may introduce a new welfare implication whereby price discrimination results in a more equal distribution of consumer surplus without affecting average welfare outcomes. This finding contributes to the recent literature that explores ways to promote equality and fairness through mechanism design (Kleinberg et al. (2018), Dworczak et al. (2021), Akbarpour et al. (2024)) or information design (Doval and Smolin (2024)).

Finally, our work contributes broadly to recent research strands in the economics of algorithmic decision-making. A significant body of work has examined algorithmic pricing, largely focusing on how such tools bolster seller power. Empirical studies-such as those by Calvano et al. (2020), Asker et al. (2022), and Assad et al. (2024)-document how algorithms facilitate collusion and dynamic pricing, while theoretical work (e.g., Sal-

[^4]cedo (2015), Lamba and Zhuk (2023)) and combined empirical-theoretical approaches (e.g., Brown and MacKay (2023), Johnson et al. (2023)) investigate their strategic impact. In contrast, we explore how algorithmic recommendations can empower buyers by mitigating information asymmetries and counteracting traditional price discrimination. This buyer-centric perspective resonates with early insights from Galbraith (1952), offering an alternative approach to balancing market power. ${ }^{5,6}$


## 2 Baseline Model

There is a buyer and a seller. The buyer may purchase a product but initially knows neither the existence nor the value of the product: The value is denoted by $v$ and is distributed according to distribution $G$ with positive density $g$ over its support [0, 1]. A recommendation algorithm or simply algorithm provides the buyer with information about the product by means of recommendations. ${ }^{7}$ The algorithm is characterized by a function $r:[0,1] \times \mathbb{R}_{+} \rightarrow[0,1]$ such that for any pair $(v, p)$ of a realized value $v \in[0,1]$ and a product price $p \in \mathbb{R}_{+}$, the algorithm recommends that the buyer purchase the product with probability $r(v, p) .{ }^{8}$

The seller can produce one unit of a product at cost $c$, which is her private type. The type distribution $F$ has support $[0,1]$ with positive density $f$.

[^5]Timing For any given algorithm, which is commonly known to both players, the timing of the game is as follows. First, nature draws the seller's type $c$ and the buyer's value $v$. Second, the seller privately observes her type $c$ but not value $v$, and posts a price, $p$. With probability $1-r(v, p)$, the algorithm does not recommend the product, in which case trade does not occur and the game ends. With probability $r(v, p)$, the algorithm recommends the product to the buyer, in which case the buyer observes the recommendation and the price, and then decides whether to buy the product. If trade occurs, the buyer and seller obtain ex post payoffs $v-p$ and $p-c$, respectively. Otherwise, both players obtain zero payoffs.

Given the algorithm, the solution concept is a perfect Bayesian equilibrium. If the product is recommended, the buyer updates the expected value of the product to

$$
\mathbb{E}[v \mid \text { recommended, } p]=\frac{\int_{0}^{1} x r(x, p) g(x) \mathrm{d} x}{\int_{0}^{1} r(x, p) g(x) \mathrm{d} x},
$$

and then purchases the product whenever this value weakly exceeds the price. A pair of an algorithm and a buyer's strategy induces a demand curve, which maps each price to a probability of trade. In equilibrium, each seller type takes this demand curve as given and chooses a price that maximizes her expected profit.

Objective We call the buyer's ex ante expected payoff buyer surplus and the ex ante seller's expected payoff seller profit. An algorithm attains a given buyer surplus if this buyer surplus arises in an equilibrium under this algorithm. For most of the analysis, we focus on the recommendation algorithms that maximize buyer surplus. However, in Section 5.1, we show that the same analysis readily applies to the whole class of Pareto-optimal algorithms.

Definition 1 (Buyer-Optimal Algorithm). A recommendation algorithm is buyeroptimal if it attains a greater buyer surplus than any other recommendation algorithm.

In what follows, it will be useful to distinguish between seller types who trade and those who do not under a given algorithm and their posted prices. Given an algorithm
and an equilibrium, we say that a price is active if it results in a strictly positive trade probability and is inactive otherwise. Similarly, we say that a type is active if she posts an active price with a strictly positive probability and is inactive otherwise.


## 3 Buyer-Optimal Algorithm

In this section, we characterize the buyer-optimal algorithm. We say that an algorithm $r$ is a threshold algorithm if there exists a threshold function $\hat{v}: \mathbb{R}_{+} \rightarrow[0,1]$ such that $r(v, p)=\mathbb{1}(v \geq \hat{v}(p))$, i.e., the algorithm recommends the product with probability 1 if the value exceeds a price-dependent threshold and with probability 0 otherwise.

Lemma 1 (Threshold Algorithms). For any algorithm $r$, there exists a threshold algorithm under which the buyer follows the recommendations and that yields a weakly greater buyer surplus than $r$ and the same seller profit as $r$.

The proofs of this and all other results are in the Appendix. Lemma 1 shows that threshold algorithms span a Pareto frontier in the space of buyer surplus and seller profit. Intuitively, the buyer can be set to follow the recommendations because the algorithm can anticipate and mimic the buyer's response. In turn, the Pareto efficiency of threshold algorithms follows from the observation that each seller type is concerned solely with trade volume whereas the buyer surplus is maximized when the higher values are prioritized. Consequently, if a buyer-optimal algorithm exists, then it can be found in the class of threshold algorithms, and in what follows, we focus on threshold algorithms.

The optimal choice of a threshold function must balance the trade-off between maximizing the trade surplus and incentivizing the seller to lower the price. One natural option is to set $\hat{v}(p)=p$ so that the product is recommended if and only if the value exceeds the price. This ex post optimal algorithm maximizes the buyer's payoff given fixed prices. However, the algorithm fails to maximize buyer surplus because it underuses the opportunity to dampen equilibrium prices.


### 3.1 Known Seller Cost

The simplest case to illustrate the power of algorithmic recommendations is to consider a limit case of our model in which the seller cost is commonly known.

Claim 1 (Known Cost). Suppose the seller cost is commonly known to be $c \in[0,1]$. Then an algorithm that employs a threshold function $\hat{v}$ defined by $\hat{v}(c)=c$ and $\hat{v}(p)>1$ for $p \neq c$ is optimal and extracts the full surplus from the seller.

The condition $\hat{v}(p)>1$ for $p \neq c$ effectively forces the seller to set the price equal to the marginal cost, thereby eliminating any seller rents, while $\hat{v}(c)=c$ ensures that trade is efficient and generates maximal surplus. In equilibrium, the entire surplus is generated and appropriated by the buyer, which confirms the algorithm's optimality.

Claim 1 highlights a key distinction between our setting and Roesler and Szentes (2017). In their model, the cost is commonly known, but the algorithm cannot condition information on price nor exclude the seller from trade. Consequently, even though equilibrium trade is efficient, the seller earns positive rents. In contrast, in our model, the algorithm can condition information on price and exclude the seller from trade by not informing the buyer of the existence of the product. The combination of these properties enables the buyer to extract full surplus when the seller's cost is known.

In Section 5.3.1, we elaborate on the roles of price dependence and gatekeeping: When cost uncertainty is low, gatekeeping is particularly valuable, while price dependence is not, because the algorithm effectively provides information at only a single price. Conversely, when cost uncertainty is high-and we provide precise distributional conditions under which this holds-price dependence is as effective whether or not the buyer could purchase the product without the recommendation.

While the optimal algorithm in Claim 1 is simple, it relies on precise knowledge of seller cost and is fragile to small perturbations: Even an arbitrarily small increase in cost would lead to a complete collapse of trade. In practice, the algorithm may lack the information necessary to identify seller cost. Moreover, this asymmetry can be exacerbated by idiosyncratic cost shocks arising from supply disruptions or alternative
sale opportunities. These observations motivate our subsequent analysis of private seller information.


### 3.2 General Case

To find an optimal algorithm in the general case of an unknown seller cost, we build on Lemma 1 and frame the designer's problem as a nonlinear screening problem in which the recommendation threshold responds to the price. The choice of a threshold at any given price simultaneously determines the expected trade surplus, which is valued by the buyer, and the expected trade volume, which is valued by the seller. We recover the optimal threshold function by adapting the seminal analysis of Baron and Myerson (1982) and confirm that, with this algorithm, the buyer indeed finds it optimal to follow the recommendations.

The optimal algorithm and equilibrium pricing are easier to describe not in terms of price-dependent thresholds for values but in terms of value-dependent thresholds for prices. Specifically, denote the virtual cost function by $\gamma(c) \triangleq c+F(c) / f(c)$, and assume that it is continuous and strictly increasing on $[0,1] .{ }^{9}$ For each $v \in[0,1]$, define the buyer's pseudo value as

$$
y(v) \triangleq \mathbb{E}_{\tilde{v} \sim G}\left[\gamma^{-1}(\tilde{v}) \mid \tilde{v} \geq v\right] .
$$

The pseudo value $y(v)$ is an increasing function of $v$. When the true value is sufficiently low, close to 0, the pseudo value is higher than the true value, $y(v)>v$, because $y(0)=\mathbb{E}_{\tilde{v} \sim G}\left[\gamma^{-1}(\tilde{v})\right]>0$. When the true value is sufficiently high, close to 1 , the pseudo value is below the true value, $y(v)<v$, because $y(1)=\gamma^{-1}(1)<1$. Define $\bar{c} \triangleq \gamma^{-1}(1) .^{10}$

[^6]Proposition 1 (Buyer-Optimal Algorithm). A buyer-optimal algorithm recommends the product if and only if $y(v) \geq p$. Under this algorithm, the seller of type $c \leq \bar{c}$ posts price $p^{*}(c)=y(\gamma(c))$, and the seller of type $c>\bar{c}$ is inactive. Under this algorithm and pricing, the trade occurs if and only if $v \geq \gamma(c)$.

Proposition 1 reveals two notable features. First, the impact of the value and cost distributions can be decoupled: Even though the optimal algorithm and the equilibrium prices depend both on the cost and value distributions, the optimal product allocation depends only on the cost distribution because the product is traded if and only if the value exceeds the virtual cost. This feature is crucial for the market segmentation results in Section 4.

Second, the buyer-optimal algorithm makes two types of ex post errors: When the true value is sufficiently high, the pseudo value is below the true value, and the optimal algorithm never recommends the product when the value is below the price. At the same time, for seller types that satisfy $y(v)<y(\gamma(c))<v$, the algorithm does not recommend the product even though the value exceeds the price. In contrast, when the true value is sufficiently low, the pseudo value is higher than the true value and the algorithm always recommends the product when the value exceeds the price. However, for seller types that satisfy $v<y(\gamma(c))<y(v)$, the algorithm recommends the product even though the value is below the price. As a result, compared with the ex post optimal algorithm, the buyer-optimal algorithm overrecommends at low prices and underrecommends at high prices. These distortions benefit the buyer because they incentivize the seller to set lower prices.

Example 1 (Uniform). We illustrate the buyer-optimal algorithm when $c$ and $v$ are uniformly distributed on [0, 1]. In this case, the virtual cost is $\gamma(c)=2 c$; the pseudo value is $y(v)=\mathbb{E}_{\tilde{v} \sim U[0,1]}\left[\left.\frac{\tilde{v}}{2} \right\rvert\, \tilde{v} \geq v\right]=(1+v) / 4$; and the associated threshold function $\hat{v}(p)$ is equal to 0 for $p<1 / 4$, to $4 p-1$ for $p \in[1 / 4,1 / 2]$ and to 1 for $p>1 / 2$. The equilibrium price posted by active type $c$ is $p^{*}(c)=y(\gamma(c))=(1+2 c) / 4$ for $c \in[0,1 / 2]$. Types $c>1 / 2$ are inactive and post, for example, $p^{*}(c)=1 / 2$. A buyer who receives a recommendation to purchase at price $p \in[1 / 4,1 / 2]$ infers that the product's expected

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 1: Optimal recommendation algorithm (left) and the resulting equilibrium pricing strategy and trade region (right). $v \sim U[0,1], c \sim U[0,1]$.

value is $(4 p-1+1) / 2=2 p>p$ and is thus strictly willing to purchase it.
The left side of Figure 1 depicts the optimal recommendation threshold (solid line) along with the ex post optimal recommendation threshold (dashed line). As we discussed above, the ex ante optimal algorithm is suboptimal ex post in two ways: If the product price is low, i.e., $p<1 / 3$, it recommends the product even when the value is below the price; if the product price is high, i.e., $p>1 / 3$, the algorithm does not recommend the product even when the value is above the price.

The right side of Figure 1 depicts the resulting equilibrium pricing and trade: The price $p^{*}(c)$ posted by the seller of type $c$, the region of values and types in which the trade occurs (filled area), and the efficient trade region (area encircled by dashed lines). In accordance with Proposition 1, under an optimal algorithm, trade occurs whenever the buyer value is greater than the seller's virtual cost. Type $c=0$ always trades; all higher types post progressively higher prices and serve progressively fewer buyers. Types $c>1 / 2$ never trade. Equilibrium active prices span the interval [1/4, 1/2].

We compare the equilibrium outcomes under the buyer-optimal algorithm and the ex post optimal algorithm. Under the ex post optimal algorithm, the seller's problem is a standard monopoly problem with the optimal price $p^{m}(c)=\frac{1+c}{2}$, and trade occurs if

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 2: Comparison of the equilibrium outcomes under the ex ante optimal algorithm and the ex post optimal algorithm in terms of the seller's pricing strategies (left) and trade regions (right).


and only if $v \geq p^{m}(c)$. As indicated by the left panel of Figure 2, as we move from the ex post optimal algorithm to the buyer-optimal algorithm, each seller type $c \leq 0.5$-who transacts with a positive probability under both algorithms-decreases its price by 0.25; thus, the buyer-optimal algorithm induces a uniform downward shift of the equilibrium prices across all seller types. (In this example, the price change is associated with the uniform decrease of the seller profits: The interim profit for seller type $c$ under the ex post optimal algorithm is $\frac{(1-c)^{2}}{4}$, whereas that under the buyer-optimal algorithm is $\frac{(1-2 c)^{2}}{4}$ if $c \leq \frac{1}{2}$ and 0 otherwise. However, in general, some seller types may prefer the buyer-optimal algorithm due to the overrecommendations.)

Complementary, the right panel of Figure 2 depicts the sets of pairs of value $v$ and cost $c$ under which transaction occurs. Under the ex post optimal algorithm, transaction occurs when $v \geq p^{m}(c)$. The figure indicates that compared to the ex post optimal algorithm, the buyer-optimal algorithm reduces transactions between high-value buyers and high-cost sellers (the red region) and increases transactions between low-value buyers and low-cost sellers (the blue region). $\square$

The last observation of the previous example turns out to be more general. To describe the result, let $\Delta^{*}$ denote the set of $(c, v)$ pairs such that trade occurs under the buyer-optimal algorithm, and let $\Delta^{m}$ denote the set of $(c, v)$ pairs such that trade occurs under the ex post optimal algorithm.

Proposition 2 (Allocation Substitution). If $F$ has a strictly decreasing reversed hazard rate $\frac{f}{F}$ and $G$ has a strictly increasing hazard rate $\frac{g}{1-G}$, then for any $(c, v) \in$ $\Delta^{*} \backslash \Delta^{m}$ and $\left(c^{\prime}, v^{\prime}\right) \in \Delta^{m} \backslash \Delta^{*}$ such that $(c, v) \neq\left(c^{\prime}, v^{\prime}\right)$, we have $c<c^{\prime}$ and $v<v^{\prime}$.

Proposition 2 shows that in regular environments, switching from monopoly pricing to algorithmic recommendations leads to a systematic shift in equilibrium trade: Highvalue, high-cost transactions are replaced by low-value, low-cost ones. Intuitively, algorithmic recommendations push prices down by under-recommending high-value products and over-recommending low-value ones. This, in turn, reduces trade with high-cost sellers while increasing trade with low-cost sellers.

Remark 1 (Comparison to Full Commitment Benchmark). As our proof reveals, the optimal algorithm in Proposition 1 attains the same outcome as when the buyer is aware of the product's existence, can observe the product's value, and has full commitment power over purchasing decisions and monetary transfers at different product values. As a result, even though the algorithm serves only information, it effectively transfers market power from the seller to the buyer. This observation has two consequences. First, the same algorithm remains optimal in the case of fully automated trade, i.e., if it could execute transactions without having the buyer in a loop, or if the algorithm could charge monetary transfers to the seller, e.g., referral or commission fees. Second, the same outcome would be optimal even if the seller could employ more general trade protocols than a posted price. In that case, a buyer-optimal algorithm would recommend products sold via posted prices according to the characterization in Proposition 1 and would never recommend products sold via alternative protocols.

Remark 2 (Optimal Product Allocation). The findings in Proposition 1 are related to those of Yang (2022) and Xu and Yang (2024). They study revenue-maximizing
intermediation by a platform that can assess consumer value and can charge payments to both sides but is uncertain about seller costs. Our paper and their papers differ in the designer's objective, the proof techniques, and the optimal mechanisms. At the same time, the equilibrium product allocations coincide-the good is traded if and only if the value exceeds virtual costs. This outcome is not obvious a priori and suggests that a revenue-maximizing platform would be more aligned, in terms of the resulting product allocation, with the side about which it has more information.


## 4 Algorithm Design and Market Segmentation

Algorithmic recommendations have major consequences for third-degree price discrimination. To demonstrate this, we allow the seller to observe and base prices on signal $\mathcal{I}=(S, \pi)$ informative about the buyer's value. The signal consists of a set $S$ of signal realizations $s$ and a family of probability distributions $\{\pi(\cdot \mid v)\}_{v \in[0,1]}$ over $S$. We write $\hat{\pi}(s)$ for the marginal probability of signal realization $s \in S$ and $G_{s}$ for the posterior value distribution conditional on $s$. Each signal can be viewed as a market segmentation, with $\hat{\pi}$ capturing the relative frequency of buyer segments and $G_{s}$ capturing the distribution of buyer values within each segment (cf. Bergemann et al. (2015)). The signal is exogenous, and the signal realization is independent of the seller's type.

We assume that the algorithm can perfectly distinguish different market segments so that the cost information remains the only private information of the seller. ${ }^{11}$ As the seller can set different prices in different segments, the optimal algorithm's recommendations should depend on the segment as well. In fact, the buyer-optimal segmentdependent algorithm must be buyer-optimal in each segment and is thus characterized in each segment $s$ by Proposition 1, with the value distribution being $G_{s}:{ }^{12}$ For each $s \in S$, the optimal algorithm recommends the product if and only if the corresponding

[^7]pseudo value exceeds the price, i.e.,
$$
y_{s}(v) \triangleq \mathbb{E}_{\tilde{v} \sim G_{s}}\left[\gamma^{-1}(\tilde{v}) \mid \tilde{v} \geq v\right] \geq p .
$$
In equilibrium, the seller with type $c$ posts a price of
$$
p_{s}^{*}(c) \triangleq y_{s}(\gamma(c))=\mathbb{E}_{\tilde{v} \sim G_{s}}\left[\gamma^{-1}(\tilde{v}) \mid \tilde{v} \geq \gamma(c)\right]
$$
and transacts whenever $v \geq \gamma(c)$.
Importantly, in contrast to the recommendation function or the seller's pricing, the product allocation, when viewed as a function of value and cost, is the same across all segments. This enables us to derive sharp implications of finer market segmentation. For any active type $c$, we define the distribution of prices of type $c$ as the distribution of $p_{s}^{*}(c)$ when we fix $c$ and draw $s$ from distribution $\hat{\pi}$. The corresponding profit of type $c$ is
$$
\pi(c) \triangleq \mathbb{E}_{s, v}\left[\left(p_{s}^{*}(c)-c\right) \mathbb{1}\left(y_{s}(v) \geq p_{s}^{*}(c)\right) \mid c\right],
$$
where the expectation is taken with respect to $v \sim G$ and $s \sim \pi(\cdot \mid v)$.
Proposition 3 (Segmentation Neutrality). For any signal $\mathcal{I}$ available to the seller, the buyer-optimal algorithm induces the same ex post product allocation, the same expected price and profit of each seller type, and the same ex ante buyer surplus.

Proof Outline. Consider the buyer-optimal algorithms with and without the seller's signal $\mathcal{I}$. Under either algorithm, in equilibrium, the trade occurs if and only if $v \geq \gamma(c)$, and the highest type, $c=1$, earns zero profit as she never trades. We can view the two algorithms as indirect mechanisms that induce the same allocation rule and yield the same profit for type $c=1$ when the seller prices in an incentive-compatible way. A version of the revenue equivalence theorem (Lemma 2 in the appendix) implies that the individual profit of each seller type with and without the seller's signal $\mathcal{I}$ must coincide. Consequently, the buyer surplus, which is the total surplus minus the seller's ex ante
profit, is also identical in the two settings. $\square$

Proposition 3 establishes in a stark manner that no seller types benefit from having more information about the buyer value, and the buyer neither benefits from nor is harmed by the release of such information on average, as long as this release is accounted for in the algorithm design.

Despite the neutral aspects highlighted by Proposition 3, a change in market segmentation does affect the optimal algorithm, the equilibrium pricing, and the distribution of payoffs across buyers with different valuations. To analyze the redistribution effect, we define the buyer surplus at value $v$ and type $c, w(v, c)$, as the equilibrium expected payoff of the buyer conditional on his value being $v$ and the seller's type being $c$ :

$$
\left.w(v, c) \triangleq \mathbb{E}_{s}\left[\left(v-p_{s}^{*}(c)\right) \mathbb{1}\left(y_{s}(v)\right) \geq p_{s}^{*}(c)\right) \mid v\right],
$$

where the expectation is taken with respect to signal realization $s \sim \pi(\cdot \mid v)$ conditional on value $v$. Similarly, we define the distribution of buyer surplus at type $c$ as the distribution of $w(v, c)$ with $v \sim G$. Furthermore, we will obtain a cleaner characterization and stronger results for the natural class of monotone partitional signals.

Definition 2 (Monotone Partitional Signal). A signal $\mathcal{I}$ is monotone partitional if there exists a finite partition of $[0,1]$ into intervals $\left\{I_{1}, \ldots, I_{n}\right\}$ such that for each interval $I_{k}$ and each $v \in I_{k}, \pi(\cdot \mid v)$ assigns probability 1 to either (i) $s=v$ or (ii) $s=k$.

Under a monotone partitional signal, each market segment is either a singleton or an interval. Furthermore, different buyer values belong to different segments; thus, the segment dependency in the description of an algorithm, which already conditions on the value, is redundant. The buyer-optimal algorithm and equilibrium pricing can be described segment by segment. Consider a segment $[\underline{v}, \bar{v}]$, which either equals $I_{k}$ for some $k$ or satisfies $\underline{v}=\bar{v}$. By Proposition 1, for all $v \in[\underline{v}, \bar{v}]$, the buyer-optimal algorithm recommends the product if and only if the pseudo value $y(v)=\mathbb{E}_{\tilde{v} \sim G}\left[\gamma^{-1}(\tilde{v}) \mid \bar{v} \geq \tilde{v} \geq v\right]$ is above the price. Given this algorithm, in the segment $[\underline{v}, \bar{v}]$, the seller of type $c$ posts
a price


$$
p_{[\underline{v}, \bar{v}]}^{*}(c)= \begin{cases}\mathbb{E}_{\tilde{v} \sim G}\left[\gamma^{-1}(\tilde{v}) \mid \bar{v} \geq \tilde{v} \geq \gamma(c)\right] & \text { if } \quad \underline{v} \leq \gamma(c) \leq \bar{v}, \\ \mathbb{E}_{\tilde{v} \sim G}\left[\gamma^{-1}(\tilde{v}) \mid \bar{v} \geq \tilde{v} \geq \underline{v}\right] & \text { if } \quad \gamma(c)<\underline{v} .\end{cases}
$$

In particular, if the buyer value is revealed to be $v$ (i.e., $\underline{v}=\bar{v}=v$ ), the algorithm recommends the product if and only if the price is below $\gamma^{-1}(v)$, and any seller with a cost below this value posts price $\gamma^{-1}(v)$.

Proposition 4 (Segmentation Redistribution). If signal $\mathcal{I}_{H}$ is Blackwell more informative than $\mathcal{I}_{L}$, then the distribution of prices set by each seller type under $\mathcal{I}_{H}$ is a mean-preserving spread of that under $\mathcal{I}_{L}$. Furthermore, if signals $\mathcal{I}_{H}$ and $\mathcal{I}_{L}$ are monotone partitional, then the distribution of individual buyer surplus at any seller type under $\mathcal{I}_{H}$ is a mean-preserving contraction of that under $\mathcal{I}_{L}$.

When the seller faces a more informative signal, the posterior beliefs on values conditional on signal realizations and the event $v \geq \gamma(c)$ become a mean-preserving spread of the posterior beliefs when the seller faces a less informative signal. By Equation 2, the equilibrium prices are linear in these beliefs; thus, the prices undergo a mean-preserving spread as the signal becomes more informative.

To gain intuition about the buyer surplus at different values, compare the seller who has no information and the seller who perfectly observes the value. When the seller has no information, the seller of type $c$ posts a price of $\mathbb{E}_{\tilde{v} \sim G}\left[\gamma^{-1}(\tilde{v}) \mid \tilde{v} \geq \gamma(c)\right]$ regardless of the value, and any buyer with value $v \geq \gamma(c)$ trades at that price. When the seller has full information, the price depends on the segment, and for the buyer with value $v$, the seller of type $c$ posts a price of $\gamma^{-1}(v)$. The buyer trades at this price, which is increasing in value. As a result, the seller's information increases the prices set for buyers with higher values, whereas the average remains the same by the first part of the argument. Consequently, the seller's information decreases the individual surplus of buyers with high values and increases the individual surplus of buyers with low values, leading to a more equalized surplus distribution.

The intuition behind the general monotone partitional signals is similar. In fact, the proof of Proposition 4 establishes an additional result: For any monotone partitional signal $\mathcal{I}$ and type $c$, a cutoff $v(c, \mathcal{I})$ exists such that the buyer surplus at value $v$ and type $c$ is greater with signal $\mathcal{I}$ than under no information if and only if $v \leq v(c, \mathcal{I}) .{ }^{13}$

Example 1 (Continued). Let $v$ and $c$ be uniformly distributed on [ 0,1 ]. Suppose that the seller has access to a monotone partitional signal with a uniform grid, i.e., each $I_{k}$ is an interval between $\frac{k-1}{n}$ and $\frac{k}{n}$. All values in interval $I_{k}$ are pooled into signal $k$. Let $I(v)$ denote the interval to which value $v$ belongs.

We fix any type $c<\frac{1}{2}$ and examine how the equilibrium price and buyer surplus at $v$ and $c$ vary across $v \geq 2 c$. Let $k=1, \ldots, n$ satisfy $2 c \in I_{k}$. If $v \in I_{m}$, the seller will observe signal realization $m$ and infer that the value is uniformly distributed between $\frac{m-1}{n}$ and $\frac{m}{n}$. As a result, the price posted by type $c$ against value $v$ is

$$
p(c, v)=\mathbb{E}_{\tilde{v} \sim G}\left[\left.\frac{\tilde{v}}{2} \right\rvert\, \tilde{v} \geq 2 c, \tilde{v} \in I(v)\right]= \begin{cases}\frac{2 c n+k}{4 n}, & \text { if } v \in I_{k}, \\ \frac{2 m-1}{4 n}, & \text { if } v \in I_{m}, m>k .\end{cases}
$$

The buyer's surplus at $v \geq 2 c$ is $w(v, c)=v-p(c, v)$.
Figure 3 depicts the optimal algorithm thresholds and the buyer surplus $\mathbb{E}_{c \sim F}[w(\cdot, c)]$ at different values for the uninformative signal, the signal represented by binary partition $\{[0,0.5],(0.5,1]\}$ and the fully informative signal. As the market segmentation becomes finer, the equilibrium price responds more to the buyer value, which redistributes the surplus from higher to lower values. As a result, the finer market segmentation, or more buyer information provided to the seller, makes the distribution of buyer surplus more equalized across different values.

Proposition 4 generalizes this observation for Blackwell-comparable market segmen-

[^8]

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 3: Buyer-optimal algorithm thresholds (left) and buyer surplus at different values (right). Computed at no segmentation (solid), binary partition (dashed), and full segmentation (dotted). $v \sim U[0,1], c \sim U[0,1]$.

tations, which, in the case of uniform monotone partitions, correspond to partitions with $n_{H}$ and $n_{L}$ elements such that $n_{H} / n_{L} \in \mathbb{N}$. $\square$

An important takeaway from our analysis in this section is that consumers can win the technological race against sellers. Advances in information technology not only drive the proliferation of algorithmic recommendations but also enable sellers to engage in third-degree price discrimination. Our neutrality results show that the benefits of buyer-optimal algorithmic consumption can fully offset the harm caused by price discrimination. However, the choice of the algorithm is crucial for this, e.g., an ex-post optimal algorithm could erode all buyer welfare under perfect price discrimination.


## 5 Extensions