Calibrated Mechanism Design

Note the agent's average payoff through period $T$ under ( $p^{L}, \sigma^{L}$ ) can be written as follows:
$$
\mathbb{E}_{\left(p^{L}, \sigma^{L}\right)}\left[U_{T}\right]=\mathbb{E}_{(p, \sigma)}\left[U_{T}\right]-\mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=1}^{T}\left(u\left(a_{t}, \theta_{t}, \omega\right)-u\left(a_{\varnothing}, \theta_{t}, \omega\right)\right) \mathbb{1}\left[t \geq L \text { and } \mu_{L} \in B\right]\right] .
$$
We show that for sufficiently large $L$, $\left(p^{L}, \sigma^{L}\right)$ is a profitable deviation. For $T \geq L$, write
$$
\begin{aligned}
& \mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=L}^{T}\left(u\left(a_{t}, \theta_{t}, \omega\right)-u\left(a_{\varnothing}, \theta_{t}, \omega\right)\right) \mathbb{1}\left[\mu_{L} \in B\right]\right]-\int_{B} U_{\text {net }}(\mu) \tau_{(p, \sigma)}(d \mu)= \\
& =\mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=1}^{T}\left(u\left(a_{t}, \theta_{t}, \omega\right)-u\left(a_{\varnothing}, \theta_{t}, \omega\right)\right) \mathbb{1}\left[\mu_{t} \in B\right]\right]-\int_{B} U_{\text {net }}(\mu) \tau_{(p, \sigma)}(d \mu) \\
& -\mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=1}^{L-1}\left(u\left(a_{t}, \theta_{t}, \omega\right)-u\left(a_{\varnothing}, \theta_{t}, \omega\right)\right) \mathbb{1}\left[\mu_{t} \in B\right]\right] \\
& +\mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=L}^{T}\left(u\left(a_{t}, \theta_{t}, \omega\right)-u\left(a_{\varnothing}, \theta_{t}, \omega\right)\right)\left(\mathbb{1}\left[\mu_{L} \in B\right]-\mathbb{1}\left[\mu_{t} \in B\right]\right)\right] .
\end{aligned}
$$
Let $K=\max _{\theta, \omega, a}\left|\left(u(a, \theta, \omega)-u\left(a_{\varnothing}, \theta, \omega\right)\right)\right|$, and note that we can bound the term in the last line of Equation C. 35 as follows:
$$
\left|\mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=L}^{T}\left(u\left(a_{t}, \theta_{t}, \omega\right)-u\left(a_{\varnothing}, \theta_{t}, \omega\right)\right)\left(\mathbb{1}\left[\mu_{L} \in B\right]-\mathbb{1}\left[\mu_{t} \in B\right]\right)\right]\right| \leq K \mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=L}^{T}\left|\mathbb{1}\left[\mu_{L} \in B\right]-\mathbb{1}\left[\mu_{t} \in B\right]\right| .\right]
$$

[^31]Because $\mu_{t} \xrightarrow{w^{*}} \mu_{\infty} \mathbb{P}_{(p, \sigma)}$-a.s. (Lemma C.1) and $\tau_{(p, \sigma)}(\partial B)=0$, we conclude: ${ }^{48}$

$$
\lim _{L \rightarrow \infty} \sup _{t \geq L}\left|\mathbb{1}\left[\mu_{L} \in B\right]-\mathbb{1}\left[\mu_{t} \in B\right]\right|=0 \mathbb{P}_{(p, \sigma)} \text {-a.s. }
$$

Then,

$$
\mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=L}^{T}\left|\mathbb{1}\left[\mu_{L} \in B\right]-\mathbb{1}\left[\mu_{t} \in B\right]\right|\right] \leq \mathbb{E}_{(p, \sigma)}\left[\sup _{t \geq L}\left|\mathbb{1}\left[\mu_{L} \in B\right]-\mathbb{1}\left[\mu_{t} \in B\right]\right|\right]
$$

and choose $\bar{L}$ large enough so that for all $L \geq \bar{L}$, we have that:

$$
\mathbb{E}_{(p, \sigma)}\left[\sup _{t \geq L}\left|\mathbb{1}\left[\mu_{L} \in B\right]-\mathbb{1}\left[\mu_{t} \in B\right]\right|\right] \leq \delta / K .
$$

Consider now the term in the third line of Equation C. 35 and note that it is bounded in absolute value by $K(L-1) / T$, which tends to 0 as $T \rightarrow \infty$. Similarly, the term in the second line of Equation C. 35 vanishes as $T \rightarrow \infty$. ${ }^{49}$ Thus, for $L \geq \bar{L}$, we can find $\bar{T}$ such that for all $T \geq \bar{T}^{50}$

$$
\left|\mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=L}^{T}\left(u\left(a_{t}, \theta_{t}, \omega\right)-u\left(a_{\varnothing}, \theta_{t}, \omega\right)\right) \mathbb{1}\left[\mu_{L} \in B\right]\right]-\int_{B} U_{\mathrm{net}}(\mu) \tau_{(p, \sigma)}(d \mu)\right| \leq \frac{3}{2} \delta,
$$

and hence

$$
\mathbb{E}_{(p, \sigma)}\left[\frac{1}{T} \sum_{t=L}^{T}\left(u\left(a_{t}, \theta_{t}, \omega\right)-u\left(a_{\varnothing}, \theta_{t}, \omega\right)\right) \mathbb{1}\left[\mu_{L} \in B\right]\right] \leq-\frac{1}{2} \delta .
$$

We conclude that

$$
\lim \sup _{T \rightarrow \infty} \mathbb{E}_{\left(p^{L}, \sigma^{L}\right)}\left[U_{T}\right] \geq \lim _{T \rightarrow \infty} \mathbb{E}_{(p, \sigma)}\left[U_{T}\right]+\frac{1}{2} \delta,
$$

a contradiction.


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.

## C.2.2 Proof of Theorem 4 (sufficiency)

We now show that all outcome distributions $\vartheta \in \Delta(A \times \Theta \times \Omega)$ that admit the decomposition in Theorem 4 can be implemented via a dynamic mechanism. To this end, let $\tau$ and $\alpha: \Theta \times \Delta(\Omega) \rightarrow \Delta(A)$

[^32]denote the belief distribution and the ex ante individually rational allocation rule without profitable undetectable deviations corresponding to $\vartheta$. That is,
$$
\vartheta(a, \theta, \omega)=\int_{\Delta(\Omega)} \mu(\omega) f(\theta) \alpha(a \mid \theta, \mu) \tau(d \mu)
$$
The proof proceeds as follows:

1. We first consider a fictitious setting in which there is no state uncertainty and we are given an allocation rule $\alpha^{\prime}: \Theta \rightarrow \Delta(A)$ that is ex ante individually rational and lacks profitable undetectable deviations for some utility function $u^{\prime}: A \times \Theta \rightarrow \mathbb{R}$. Proposition C. 1 shows that a dynamic mechanism exists that implements $\alpha^{\prime}$.
2. We then show that if $\vartheta$ satisfies Equation C.39, then a finite support belief distribution $\tau^{\prime}$ exists such that $\vartheta$ and $\alpha$ satisfies Equation C. 39 with $\tau^{\prime}$ instead of $\tau$.
3. Lastly, we use this result to construct a dynamic game that implements $\vartheta$.

Step 1 For this step, we consider a fictitious setting in which there is no state uncertainty and the designer faces a privately informed agent with payoffs $u^{\prime}: A \times \Theta \rightarrow \mathbb{R}$, where $\theta \sim f \in \Delta(\Theta) .{ }^{51}$

Suppose we are given an allocation rule $\alpha^{\prime}: \Theta \rightarrow \Delta(A)$ that admits no profitable undetectable deviations relative to $u^{\prime}$ as in Definition 7 and is individually rational as in Definition 8. We have the following result:

Proposition C.1. Let $\vartheta^{\prime}=f(\theta) \alpha^{\prime}(a \mid \theta) \in \Delta(A \times \Theta)$ such that $\alpha^{\prime}$ lacks profitable undetectable deviations and is ex ante individually rational. Then, a dynamic mechanism exists that implements $\vartheta^{\prime}$.

Proof of Proposition C.1. The proof is constructive. We build on the analysis of Margaria and Smolin (2018) and present a dynamic mechanism that alternates between communication and adjustment phases. In all phases, the mechanism selects allocations using reports $\theta^{\prime}$ according to $\alpha^{\prime}$. In a communication phase, the reports are those sent by the agent. In an adjustment phase, the agent's reports are disregarded; instead, the mechanism simulates reports to guarantee that the occupation measure over reports coincides with $f$ and these simulated reports are used to determine the allocation. The mechanism ensures that under any agent's strategy, the occupation measure over reports and allocations exists and equals $\vartheta^{\prime}$; thus, any strategy corresponds to an undetectable deviation. The length of communication phases grows in time. Thus, under truthtelling the relative length of adjustment phases vanishes in time, and the expected occupation measure over types and allocations exists and equals $\vartheta^{\prime}$. Because $\alpha^{\prime}$ lacks profitable undetectable deviations, it follows that truthtelling is optimal for the agent. Because $\alpha^{\prime}$ is ex ante individually rational, it follows that the participation constraints are satisfied.

Formally, the mechanism consists of sequential blocks, each block starting with a communication phase followed by an adjustment phase. The lengths of communication phases are fixed at $L_{1}, L_{2}, \ldots$

[^33]such that $L_{n} \rightarrow \infty$ and $L_{n} / \sum_{k \leq n} L_{k} \rightarrow 0$, e.g., $L_{n}=n$. The length of adjustment phase $N_{n}$ depends on the agent's reports in the communication phase in block $n$. Denote by $T_{n}$ the first period of block $n$, which is the first period of the corresponding communication phase. The first period of the corresponding adjustment phase is $T_{n}+L_{n}+1$. Denote by freq ${ }_{n}^{1}$ the average report frequencies in this block at the beginning of the adjustment stage:
$$
\operatorname{freq}_{n}^{1}(\hat{\theta}) \triangleq \frac{1}{L_{n}} \sum_{t=T_{n}}^{T_{n}+L_{n}-1} 1\left(\hat{\theta}_{t}=\hat{\theta}\right) .
$$
If $\operatorname{freq}_{n}^{1}=f$, then the adjustment phase is empty, and the mechanism proceeds to the next block. Otherwise, in the adjustment phase, the mechanism generates reports over $N_{n}$ periods to guarantee that at the end of the adjustment phase the expected frequency of reports in this block equals $f$ that is,
$$
\mathbb{E}\left[\text { freq }_{n}^{2} \mid \text { freq }_{n}^{1}\right]=f
$$
where
$$
\operatorname{freq}_{n}^{2}(\hat{\theta}) \triangleq \frac{1}{L_{n}+N_{n}} \sum_{t=T_{n}}^{T_{n}+L_{n}+N_{n}-1} 1\left(\hat{\theta}_{t}=\hat{\theta}\right) .
$$
To do so, denote by $\eta \triangleq \min _{\theta} f(\theta)$ and observe that $f \in \Delta(\Theta)$ can be surrounded by a ball of radius $\eta$ within the simplex $\Delta(\Theta)$. The adjustment phase lasts for $N_{n}$ periods where: ${ }^{52}$
$$
N_{n}=\left\lceil L_{n} \frac{\left\|\mathrm{freq}_{n}^{1}-f\right\|_{\infty}}{\eta}\right\rceil,
$$
and in each period of the adjustment phase the mechanism generates the reports i.i.d. according to $\tilde{f}_{n}^{a}$ :
$$
f_{n}^{a}=f-\left(\text { freq }_{n}^{1}-f\right) \frac{L_{n}}{N_{n}} .
$$
The construction ensures that $f_{n}^{a} \in \Delta(\Theta)$, because $\left\|f_{n}^{a}-f\right\|_{\infty} \leq \eta$, and that (C.41) holds, because
$$
\mathbb{E}\left[\operatorname{freq}_{n}^{2} \mid \operatorname{freq}_{n}^{1}\right]=\frac{1}{L_{n}+N_{n}}\left(L_{n} \operatorname{freq}_{n}^{1}+N_{n} f_{n}^{a}\right)=f
$$
This in turn guarantees that the long-run distribution of reports (generated jointly by the agent and the mechanism) exists and equals $f$ irrespectively of the agent's strategy. Intuitively, the fact that each block becomes negligible relative to past history over time ensures the agent's reports in each block have less and less effect on the long run frequency of reports, whereas the adjustment phase ensures that the frequency of reports converges to $f$. Formally, for any history and $T$ denote by $n^{\text {last }}(T)$ the number of the block to which $T$ belongs and by $T^{\text {last }}(T)$ the first period of that block. Observe that for


Calibrated Mechanism Design

[^34]any agent's strategy:
$$
N_{n} \leq L_{n}\left(\max _{f^{\prime}} \frac{\left\|f^{\prime}-f\right\|_{\infty}}{\eta}+1\right) \triangleq L_{n} \bar{\rho}
$$
Therefore,
$$
\frac{\left|T-T^{\text {last }}(T)\right|}{T^{\text {last }}(T)} \leq \frac{L_{n^{\text {last }}(T)}(1+\bar{\rho})}{\sum_{k<n^{\text {last }}(T)} L_{k}} \xrightarrow[T \rightarrow \infty]{\text { a.s. }} 0,
$$
where the limit result holds because $n^{\text {last }}(T) \xrightarrow[T \rightarrow \infty]{\text { a.s. }} \infty$ and $L_{n} / \sum_{k \leq n} L_{k} \xrightarrow[n \rightarrow \infty]{ } 0$.
Then, for any agent's strategy, for any $\hat{\theta} \in \Theta$,
$$
\begin{aligned}
\lim _{T \rightarrow \infty} \frac{1}{T} \sum_{t=1}^{T} \operatorname{Pr}\left(\hat{\theta}_{t}=\hat{\theta}\right) & =\lim _{T \rightarrow \infty} \mathbb{E}\left[\frac{f(\hat{\theta}) T^{\text {last }}(T)+f^{\text {last }}(T)\left(T-T^{\text {last }}(T)\right)}{T^{\text {last }}(T)+T-T^{\text {last }}(T)}\right] \\
& =\lim _{T \rightarrow \infty} \mathbb{E}\left[\frac{f(\hat{\theta})+f^{\text {last }}(T)\left(T-T^{\text {last }}(T) / T^{\text {last }}(T)\right.}{1+\left(T-T^{\text {last }}(T)\right) / T^{\text {last }}(T)}\right]=f(\hat{\theta}),
\end{aligned}
$$
where $f^{\text {last }}(T) \in \Delta(\Theta)$ is the report frequency in the last block up to period $T$, and the last line follows from Equation C.46.

Since the mechanism chooses allocations in all periods according to $\alpha^{\prime}$, it follows that for any agent's strategy $\sigma^{\prime}$ the induced occupation measure over allocations and type reports satisfies:

$$
\lim _{T \rightarrow \infty} \frac{1}{T} \mathbb{E}_{\sigma^{\prime}}\left[\sum_{t=1}^{T} \mathbb{1}\left[\left(a_{t}, \hat{\theta}_{t}\right)=(a, \hat{\theta})\right]\right]=f(\hat{\theta}) \alpha^{\prime}(a \mid \hat{\theta})=\vartheta^{\prime}(a, \hat{\theta}) .
$$

In other words, for any reporting strategy the occupation measure over allocations and reports exists.
We now show that under truthtelling the occupation measure over types and allocations exists and equals $f(\theta) \alpha^{\prime}(a \mid \theta)=\vartheta(a, \theta)$. To this end, assume that the agent always reports her true type. For any $T$, denote by $\tilde{L}^{\text {total }}(T)$ the total number of periods spent in communication phases before $T$ and by $\tilde{N}^{\text {total }}(T)$ the total number of periods spent in adjustment phases before $T$. Observe that by the strong law of large numbers, because $L_{n} \rightarrow \infty$,

$$
\frac{N_{n}}{L_{n}} \leq \frac{\left\|\operatorname{freq}_{n}^{1}-f\right\|_{\infty}}{\eta}+\frac{1}{L_{n}} \xrightarrow[n \rightarrow \infty]{\text { a.s. }} 0 .
$$

Therefore,

$$
\frac{\tilde{N}^{\text {total }}(T)}{\tilde{N}^{\text {total }}(T)+\tilde{L}^{\text {total }}(T)} \underset{T \rightarrow \infty}{\stackrel{\text { a.s. }}{\longrightarrow}} 0,
$$

because whenever $N_{n} / L_{n} \rightarrow 0, \lim _{T \rightarrow \infty} N^{\text {total }(T)} /\left(N^{\text {total }}(T)+L^{\text {total }}(T)\right)=\lim _{n \rightarrow \infty} N_{n} /\left(L_{n}+N_{n}\right)=0$.

It follows that

$$
\begin{aligned}
& \lim _{T \rightarrow \infty} \frac{1}{T} \sum_{t=1}^{T} \operatorname{Pr}\left(\left(\theta_{t}, \hat{\theta}_{t}, a_{t}\right)=(\theta, \hat{\theta}, a)\right) \\
& =\lim _{T \rightarrow \infty} \mathbb{E}\left[\frac{\tilde{L}^{\text {total }}(T) 1(\theta=\hat{\theta}) f(\hat{\theta}) \alpha^{\prime}(a \mid \hat{\theta})+\tilde{N}^{\text {total }}(T) f^{\text {adj }}(T)(\theta, \hat{\theta}, a)}{\tilde{N}^{\text {total }}(T)+\tilde{L}^{\text {total }}(T)}\right] \\
& =1(\theta=\hat{\theta}) f(\hat{\theta}) \alpha^{\prime}(a \mid \hat{\theta})
\end{aligned}
$$

where $f^{\text {adj }}(T) \in \Delta(\Theta \times \Theta \times A)$ is the average frequency of types, reports, and allocations in the adjustment phases before $T$. Therefore, under truthtelling, the occupation measure over allocations and types equals

$$
\vartheta^{\prime}(a, \theta)=f(\theta) \alpha^{\prime}(a \mid \theta) .
$$

Hence, the agent's payoff in the dynamic mechanism under truthtelling is:

$$
U^{\mathrm{truth}}=\sum_{(a, \theta)} f(\theta) \alpha^{\prime}(a \mid \theta) u^{\prime}(a, \theta) .
$$

It remains to show that the agent cannot achieve more than $U^{\text {truth }}$ under any other strategy. To this end, fix and alternative strategy $\sigma$, and denote by $U(\sigma)=\limsup _{T \rightarrow \infty} U_{T}(\sigma)$ where:

$$
U_{T}(\sigma)=\frac{1}{T} \sum_{t=1}^{T} \sum_{a, \theta} \operatorname{Pr}\left(\left(a_{t}, \theta_{t}\right)=(a, \theta)\right) u^{\prime}(a, \theta) .
$$

Consider any convergent subsequence $\left(U_{T_{n}}\right)_{n=1}^{\infty}$ along times $\left\{T_{n}\right\}_{n=1}^{\infty}$. Because $\Delta(A \times \Theta \times \Theta)$ is compact (Aliprantis and Border, 2006, Theorem 15.11), a convergent (sub)subsequence at times $\left\{T_{k}\right\}_{k=1}^{\infty} \subseteq$ $\left\{T_{n}\right\}_{n=1}^{\infty}$ exists along which the occupation measure induced by $\sigma$

$$
v_{\sigma}^{T_{k}} \xrightarrow{w^{*}} v_{\sigma},
$$

for some $v_{\sigma} \in \Delta(A \times \Theta \times \Theta)$, which by (C.48) satisfies $v_{\sigma}(a, \hat{\theta})=f(\hat{\theta}) \alpha^{\prime}(a \mid \hat{\theta})$. It follows that for some undetectable deviation $v_{\sigma}(\hat{\theta} \mid \theta)$ :

$$
\lim _{n \rightarrow \infty} U_{T_{n}}=\lim _{k \rightarrow \infty} U_{T_{k}}=\sum_{\theta, \hat{\theta}, a} f(\theta) v_{\sigma}(\hat{\theta} \mid \theta) \alpha^{\prime}(a \mid \hat{\theta}) u^{\prime}(a, \theta) \leq U^{\mathrm{truth}},
$$

where the inequality follows because $\alpha^{\prime}(a \mid \hat{\theta})$ lacks profitable undetectable deviations. Because this inequality holds for any convergent subsequence $\left(U_{T_{n}}\right)_{n=1}^{\infty}$,

$$
U(\sigma)=\limsup _{T \rightarrow \infty} U_{T}(\sigma) \leq U^{\text {truth }}
$$

Finally, observe that the construction ensures that after every history, truthtelling from there on delivers the continuation payoff $U^{\text {truth }}$. Since $\alpha^{\prime}$ is ex ante individually rational, $U^{\text {truth }} \geq \sum_{\theta} f(\theta) u^{\prime}\left(a_{\varnothing}, \theta\right)$, and thus the participation constraints are satisfied. This concludes the proof. $\square$


Calibrated Mechanism Design

Step 2 Consider now the outcome distribution $\vartheta \in \Delta(A \times \Theta \times \Omega)$ satisfying Equation C.39. As we argue in the proof of Theorem 2, a finite $K \leq|A||\Theta||\Omega|,\left\{\mu_{1}, \ldots, \mu_{K}\right\} \in \Delta(\Omega)$, and $\tau^{\prime} \in \Delta(\Delta(\Omega))$ exists such that

$$
\vartheta(a, \theta, \omega)=f(\theta) \sum_{k=1}^{K} \tau^{\prime}\left(\mu_{k}\right) \mu_{k}(\omega) \alpha\left(a \mid \theta, \mu_{k}\right) .
$$

Step 3 We now use steps 1 and 2 to complete the proof of Theorem 4, so in what follows we use the finite support representation of $\vartheta$ in the previous step. By Bayes plausibility, a dynamic mechanism can generate the belief split $\tau^{\prime}$ in $T$ periods with $T \leq\left\lceil\log _{|A|}(|\Omega||\Theta||A|)\right\rceil$, by treating each sequence of allocations of length $T$ as a message. This can be achieved by making the mechanism constant on the agent's type reports during the first $T$ periods. Since each $\alpha\left(\cdot \mid \cdot, \mu_{k}\right)$ for $k \in\{1, \ldots, K\}$ lacks profitable undetectable deviations and is individually rational, Proposition C. 1 implies that a dynamic mechanism exists that implements $\vartheta$ by first generating the belief split $\tau^{\prime}$ and then implementing $\alpha\left(\cdot \mid \cdot, \mu_{k}\right)$ in the corresponding continuation play.


Calibrated Mechanism Design

## D Proof of auxiliary results


Calibrated Mechanism Design

## D. 1 Revelation principle for calibrated mechanism design

In the main text, we restricted attention to incentive compatible and individually rational calibrated mechanisms. We show in this appendix that this restriction is without loss of generality by considering mechanisms with arbitrary message spaces and participation and reporting decisions by the agents that constitute an equilibrium of the game induced by the mechanism and its calibrated information structure.

Mechanisms Let $2^{[N]} \backslash \varnothing$ denote the nonempty subsets of agents. Then, we can define a mechanism as a collection $\left\{\left(M_{J}, \phi_{J}\right): J \in 2^{[N]} \backslash \varnothing\right\}$, where

$$
\phi_{J}: M_{J} \times \Omega \times[0,1] \rightarrow \Delta\left(A_{J}\right),
$$

is the mechanism when agents in $J$ participate, where $M_{J}=\times_{i \in J} M_{i}$ and $A_{J}=\times_{i \in J} A_{i}$.

Information Structure Let $\hat{S}_{i}=\Delta\left(A_{i}\right)^{M_{i}}$ denote the collection of menus of lotteries with labels $M_{i}$, and let $\hat{S}=\times_{i \in[N]} \hat{S}_{i}$. An information structure is $(\pi, \hat{S})$, where $\pi: \Omega \times[0,1] \rightarrow \hat{S}$.

Participation and reporting strategies It is notationally convenient to allow each agent to have her own randomization device $\varepsilon_{i} \sim U[0,1]$ and write agents' strategies as mappings $\left(p_{i}, \sigma_{i}\right): \Theta_{i} \times \hat{S}_{i} \times$ $[0,1] \rightarrow\{0,1\} \times M_{i}$, where $p_{i}$ denotes agent $i$ 's participation decision, and $\sigma_{i}$ her reporting strategy, conditional on participating. To distinguish the agents' randomization from that of the original mechanism, we reserve $\varepsilon_{0}$ for the realization of the mechanism's randomization device.

Given $\left(p_{i}, \sigma_{i}\right)_{i \in[N]}$ and a mechanism $(\phi, M)$, fix a profile $(\theta, \hat{s}, \bar{\varepsilon}) \equiv\left(\theta_{i}, \hat{s}, \varepsilon_{i}\right)_{i \in[N]}$. This determines a set of agents that participate,

$$
J(\theta, \hat{s}, \bar{\varepsilon})=\left\{j \in[N]: p_{j}\left(\theta_{j}, \hat{s}_{j}, \varepsilon_{j}\right)=1\right\},
$$

and let $J_{-i}(\theta, \hat{s}, \bar{\varepsilon})$ denote the projection of $J(\theta, \hat{s}, \bar{\varepsilon})$ on $J \backslash\{i\}$. Note that $J_{-i}$ only depends on $\left(\theta_{-i}, \hat{s}_{-i}, \bar{\varepsilon}_{-i}\right)$. Lastly, write $\phi_{J_{-i}\left(\theta_{-i}, \hat{s}_{-i}, \varepsilon_{-i}\right) \cup\{i\}}\left(m_{i}, \sigma_{J_{-i}\left(\theta_{-i}, \hat{s}_{-i}, \varepsilon_{-i}\right)}, \omega, \varepsilon_{0}\right) \in \Delta\left(A_{J_{-i}\left(\theta_{-i}, \hat{s}_{-i}, \varepsilon_{-i}\right) \cup\{i\}}\right)$ for

$$
\sum_{m_{J_{-i}\left(\theta_{-i}, \hat{s}_{-i}, \varepsilon_{-i}\right)}}\left(\prod_{j \in J_{-i}\left(\theta_{-i}, \hat{s}_{-i}, \varepsilon_{-i}\right)} \sigma_{j}\left(\theta_{j}, \hat{s}_{j}\right)\left(m_{j}\right)\right) \phi_{J_{-i}\left(\theta_{-i}, \hat{s}_{-i}, \varepsilon_{-i}\right) \cup\{i\}}\left(m_{i}, m_{J_{-i}\left(\theta_{-i}, \hat{s}_{-i}, \varepsilon_{-i}\right)}, \omega, \varepsilon_{0}\right)
$$

Calibrated information structures Given $\left(p_{i}, \sigma_{i}\right)_{i \in[N]}$ and a mechanism $(\phi, M)$, the information structure $(\pi, \hat{S})$ is calibrated with the mechanism and the agents' strategies if whenever $\pi\left(\omega, \varepsilon_{0}\right)=$ $\left(\hat{s}_{1}, \ldots, \hat{s}_{N}\right)$, then for all $i, m_{i}$

$$
\hat{s}_{i}\left(\cdot \mid m_{i}\right)=\mathbb{E}_{\tilde{\theta}_{-i} \sim f_{-i}(\cdot \mid \omega), \epsilon_{-i}}\left[\sum_{a_{-i} \in A_{-i}} \phi_{J_{-i}\left(\theta_{-i}, \hat{s}_{-i}, \varepsilon_{-i}\right) \cup\{i\}}\left(m_{i}, \sigma_{J_{-i}\left(\theta_{-i}, \hat{s}_{-i}, \varepsilon_{-i}\right)}, \omega, \epsilon_{0}\right)\left(\cdot, a_{-i}\right)\right] .
$$

Below, to keep the presentation simple, we focus on the case in which the calibrated information structure has finite support.

Equilibrium Given $\left(p_{i}, \sigma_{i}\right)_{i \in[N]}$, a mechanism $(\phi, M)$ and an information structure $(\pi, \hat{S})$ calibrated with the mechanism and the agents' strategies, $\left(p_{i}, \sigma_{i}\right)_{i \in[N]}$ is an equilibrium if for all $i \in[N]$, all $\theta_{i} \in \Theta_{i}$, all $\hat{s}_{i} \in \hat{S}_{i}$, and $\varepsilon_{i} \in[0,1]$, the following hold:

$$
\begin{aligned}
& \sigma_{i}\left(\theta_{i}, \hat{s}_{i}, \varepsilon_{i}\right) \in \arg \max _{m_{i} \in M_{i}} \sum_{a_{i} \in A_{i}} \hat{s}_{i}\left(a_{i} \mid m_{i}\right) \mathbb{E}_{\omega \sim \mu_{i}\left(\cdot \mid \theta_{i}, \hat{s}_{i}\right)}\left[u_{i}\left(a_{i}, \theta_{i}, \omega\right)\right], \\
& p_{i}\left(\theta_{i}, \hat{s}_{i}, \varepsilon_{i}\right) \in \arg \max _{p \in\{0,1\}} p \sum_{a_{i} \in A_{i}} \hat{s}_{i}\left(a_{i} \mid \sigma_{i}\left(\theta_{i}, \hat{s}_{i}, \varepsilon_{i}\right)\right) \mathbb{E}_{\omega \sim \mu_{i}\left(\cdot \mid \theta_{i}, \hat{s}_{i}\right)}\left[u_{i}\left(a_{i}, \theta_{i}, \omega\right)\right]+(1-p) \mathbb{E}_{\omega \sim \mu_{i}\left(\cdot \mid \theta_{i}, \hat{s}_{i}\right)}\left[u_{i}\left(a_{i \varnothing}, \theta_{i}, \omega\right)\right],
\end{aligned}
$$

where $\mu_{i}\left(\theta_{i}, \hat{s}_{i}\right) \in \Delta(\Omega)$ denotes agent $i$ 's updated beliefs about the state when her type is $\theta_{i}$ conditional on receiving signal $\hat{s}_{i}$.

Revelation Principle Fix $\left(p_{i}, \sigma_{i}\right)_{i \in[N]}$, a mechanism $(\phi, M)$ and an information structure $(\pi, \hat{S})$ calibrated with the mechanism such that $\left(p_{i}, \sigma_{i}\right)_{i \in[N]}$ is an equilibrium. We construct a direct mechanism $\left(\phi^{*}, \Theta\right)$ and a calibrated information structure $\left(\pi^{*}, S^{*}\right)$ calibrated with the mechanism under truthtelling and full participation such that truthtelling and full participation is an equilibrium.

First, note that we can extend each $\phi_{J}(\cdot) \in \Delta\left(A_{J}\right)$ to a mechanism $\bar{\phi}_{J}(\cdot) \in \Delta(A)$ as follows: for all $m \in M_{J}, \omega \in \Omega, \varepsilon_{0} \in[0,1]$, and $a_{J} \in A_{J}$,

$$
\bar{\phi}_{J}\left(m_{J}, \omega, \varepsilon_{0}\right)(a)=\phi_{J}\left(m_{J}, \omega, \varepsilon_{0}\right)\left(a_{J}\right) \times \delta_{a_{-J, \phi}} .
$$

Define a "pseudo"-mechanism as follows:


Calibrated Mechanism Design

$$
\hat{\phi}_{N}\left(\theta, \omega, \varepsilon_{0}, \bar{\varepsilon}\right)=\bar{\phi}_{J\left(\theta, \pi\left(\omega, \varepsilon_{0}\right), \bar{\varepsilon}\right)}\left(\sigma_{J(\theta, \hat{s}, \bar{\varepsilon})}, \omega, \varepsilon_{0}\right) .
$$

where $\sigma_{J(\theta, \hat{s}, \bar{\varepsilon})}$ is the message vector generated by the strategies. Define the full participation mechanism $\phi_{N}^{*}: \Theta \times \Omega \times[0,1] \mapsto \Delta(A)$ to be

$$
\phi_{N}^{*}\left(\theta, \omega, \varepsilon_{0}\right)(a)=\int_{[0,1]^{N}} \hat{\phi}_{N}\left(\theta, \omega, \varepsilon_{0}, \bar{\varepsilon}\right)(a) \lambda^{N}(d \bar{\varepsilon})
$$

Let $S_{i}^{*}=\Delta\left(A_{i}\right)^{\Theta_{i}}$ and define $\pi^{*}\left(\omega, \varepsilon_{0}\right)=\left(s_{1}^{*}, \ldots, s_{N}^{*}\right) \in \times_{i \in[N]} S_{i}^{*}$, where

$$
s_{i}^{*}\left(\cdot \mid \hat{\theta_{i}}\right)=\mathbb{E}_{\theta_{-i} \sim f_{-i}(\cdot \mid \omega)}\left[\sum_{a_{-i}} \phi_{N}^{*}\left(\hat{\theta_{i}}, \theta_{-i}, \omega, \varepsilon_{0}\right)\left(\cdot, a_{-i}\right)\right] .
$$

By definition, the information structure is calibrated relative to full participation and truthful reporting.
We now show that full participation and truthful reporting is a best response to others participating and truthfully reporting into the mechanism. To this end, consider agent $i$ 's payoff from submitting report $\theta_{i}^{\prime}$ when observing $s_{i}^{*}$. Denoting by $\Sigma\left(\omega, s_{i}^{*}\right)$ the set of $\varepsilon_{0}$ such that $\pi_{i}^{*}=s_{i}^{*}$, this payoff is given by: ${ }^{53}$

$$
\begin{aligned}
& \sum_{\omega \in \Omega} \frac{\mu_{0}(\omega) f_{i}\left(\theta_{i} \mid \omega\right)}{\operatorname{Pr}\left(s_{i}^{*} \mid \theta_{i}\right)} \sum_{\theta_{-i}} f_{-i}\left(\theta_{-i} \mid \omega\right) \int_{\Sigma\left(\omega, s_{i}^{*}\right)} \sum_{a} \phi_{N}^{*}\left(\theta_{i}^{\prime}, \theta_{-i}, \omega, \varepsilon_{0}\right)\left(a_{i}, a_{-i}\right) \lambda\left(d \varepsilon_{0}\right) u_{i}\left(a_{i}, \theta_{i}, \omega\right)= \\
& \sum_{a_{i} \in A_{i}} \sum_{\omega \in \Omega} \frac{\mu_{0}(\omega) f_{i}\left(\theta_{i} \mid \omega\right)}{\operatorname{Pr}\left(s_{i}^{*} \mid \theta_{i}\right)} u_{i}\left(a_{i}, \theta_{i}, \omega\right) \sum_{\theta_{-i}} f_{-i}\left(\theta_{-i} \mid \omega\right) \int_{\Sigma\left(\omega, s_{i}^{*}\right)} \sum_{a_{-i}} \phi_{N}^{*}\left(\theta_{i}^{\prime}, \theta_{-i}, \omega, \varepsilon_{0}\right)\left(a_{i}, a_{-i}\right) \lambda\left(d \varepsilon_{0}\right) \\
& =\int_{0}^{1}\left[\sum_{a_{i}} \sum_{\omega} \frac{\mu_{0}(\omega) f_{i}\left(\theta_{i} \mid \omega\right)}{\operatorname{Pr}\left(s_{i}^{*} \mid \theta_{i}\right)} u_{i}\left(a_{i}, \theta_{i}, \omega\right) \int_{\Sigma\left(\omega, s_{i}^{*}\right)}(\star) \lambda\left(d \varepsilon_{0}\right)\right] \lambda\left(d \varepsilon_{i}\right)
\end{aligned}
$$

where

$$
\begin{aligned}
& \star=\mathbb{E}_{\theta_{-i} \mid \omega, \bar{\varepsilon}_{-i}}\left[\sum_{a_{-i}} \hat{\phi}_{N}\left(\theta_{i}^{\prime}, \theta_{-i}, \varepsilon_{0}, \bar{\varepsilon}_{-i}\right)\left(a_{i}, a_{-i}\right)\right] \\
& =\mathbb{E}_{\theta_{-i} \mid \omega, \bar{\varepsilon}_{-i}}\left[\sum_{a_{-i}} \bar{\phi}_{J\left(\theta_{i}^{\prime}, \theta_{-i}, \hat{s}\left(\omega, \varepsilon_{0}\right), \bar{\varepsilon}\right)}\left(\sigma_{J\left(\theta_{i}^{\prime}, \theta_{-i}, \hat{s}\left(\omega, \varepsilon_{0}\right), \bar{\varepsilon}\right)}, \omega, \varepsilon_{0}\right)\left(a_{i}, a_{-i}\right)\right] \\
& =\mathbb{1}\left[p_{i}\left(\theta_{i}^{\prime}, \hat{s}_{i}\left(\omega, \varepsilon_{0}\right), \varepsilon_{i}\right)=1\right] \hat{s}_{i}\left(a_{i} \mid \sigma_{i}\left(\theta_{i}^{\prime}, \hat{s}_{i}, \varepsilon_{i}\right)\right)+\left(1-\mathbb{1}\left[p_{i}\left(\theta_{i}^{\prime}, \hat{s}_{i}\left(\omega, \varepsilon_{0}\right), \varepsilon_{i}\right)=1\right]\right) \delta_{a_{i, \phi}}\left(a_{i}\right)
\end{aligned}
$$

Because agent $i$ of type $\theta_{i}$ could have imitated type $\theta_{i}^{\prime}$, reporting $\theta_{i}$ dominates. By the same logic, when the agent reports $\theta_{i}$, she obtains at least the payoff from participating in the mechanism.


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

## 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.

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