Calibrated Mechanism Design

## Mathematical conventions

Throughout the appendix, we take all sets to be Polish spaces, that is, completely metrizable, separable, topological spaces, and endow them with their Borel $\sigma$-algebra. We endow product spaces with their product $\sigma$-algebra. For a Polish space $X$, we let $\mathcal{B}_{X}$ denote its Borel $\sigma$-algebra and $\Delta(X)$ the set of all Borel probability measures on $X$, endowed with the weak* topology. Thus, $\Delta(X)$ is also a Polish space (Aliprantis and Border, 2006), and it is compact, whenever $X$ is compact (Aliprantis and Border, 2006, Theorem 15.11 and Theorem 15.15).

Notational conventions If $X$ is a Polish space, $\tilde{X}$ denotes a measurable subset of $X$, i.e., an element of the Borel $\sigma$-algebra on $X$, and $C_{b}(X)$ denotes the set of continuous and bounded functions on $X$. Given a measure $v \in \Delta\left(\times_{i=1}^{N} Y_{i}\right)$, we denote by $v_{Y_{j} Y_{k} \ldots Y_{l}}$ the marginal of $v$ on $Y_{j} Y_{k} \ldots Y_{l}$. When one of the $Y_{i}=\Delta\left(X_{i}\right)$, we write $\Delta$ instead of $Y_{i}$ in the subscript, when it is unlikely to generate confusion.

Throughout the appendix, we define different distributions that arise in our proofs. Because we endow product spaces with their product topology and their product Borel $\sigma$-algebra, it is enough to define these new measures on the measurable rectangles and we follow this convention throughout.

Disintegration We rely on the notion of disintegration in many of our proofs (Bogachev, 2007, Chapter 10.6). We define disintegration in the context of product sets $X \times Y$, as this is the one that shows up in the proof, but it is more general than this. Given a measure $v \in \Delta(X \times Y), \lambda: X \times \mathcal{B}_{Y} \rightarrow[0,1]$ is the disintegration of $v$ along $X$ if the following holds

1. For all $\tilde{Y} \in \mathcal{B}_{Y}, x \mapsto \lambda_{x}(\tilde{Y})$ is measurable,
2. For $v_{X}$-almost everywhere $x \in X, \tilde{Y} \mapsto \lambda_{x}(\tilde{Y})$ is a probability measure, and
3. For every bounded measurable function $g: X \times Y \rightarrow \mathbb{R}$,
$$
\int_{X \times Y} g(x, y) v(d(x, y))=\int_{X} \int_{Y} g(x, y) \lambda_{x}(d y) v_{X}(d x)
$$

Kallenberg (2017, Theorem 1.23) ensures that $\left\{\lambda_{x}: x \in X\right\}$ exists and is unique $v_{X}$-almost everywhere.


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.

## A Omitted proofs from Section 2

Proof of Theorem 1. Suppose the agents' payoffs are state independent and in a slight abuse of notation let $u_{i}\left(a_{i}, \theta_{i}\right)$ denote agent $i$ 's utility.

The calibrated mechanism design problem is

$$
\max _{\phi: \Theta \times \Omega \times[0,1] \rightarrow \Delta(A)} \sum_{\omega \in \Omega} \mu_{0}(\omega) \sum_{\theta \in \Theta} f(\theta \mid \omega) \int_{0}^{1} w(\phi(\theta, \omega, \varepsilon), \theta, \omega) \lambda(d \varepsilon),
$$

subject to the following constraints holding for all $(\omega, \varepsilon) \in \Omega \times[0,1], i \in[N], \theta_{i} \in \Theta$ and $\theta_{i}^{\prime} \in \Theta$ :

$$
\begin{aligned}
& \sum_{a_{i} \in A_{i}} \mathbb{E}_{f_{-i}(\cdot \mid \omega)}\left[\sum_{a_{-i} \in A_{-i}} \phi\left(\theta_{i}, \theta_{-i}, \omega, \varepsilon\right)\left(a_{i}, a_{-i}\right)\right] u_{i}\left(a_{i}, \theta_{i}\right) \geq \sum_{\left(a_{i} \in A_{i}\right)} \mathbb{E}_{f_{-i}(\cdot \mid \omega)}\left[\sum_{a_{-i} \in A_{-i}} \phi\left(\theta_{i}^{\prime}, \theta_{-i}, \omega, \varepsilon\right)\left(a_{i}, a_{-i}\right)\right] u_{i}\left(a_{i}, \theta_{i}\right) \\
& \sum_{a_{i} \in A_{i}} \mathbb{E}_{f_{-i}(\cdot \mid \omega)}\left[\sum_{a_{-i} \in A_{-i}} \phi\left(\theta_{i}, \theta_{-i}, \omega, \varepsilon\right)\left(a_{i}, a_{-i}\right)\right] u_{i}\left(a_{i}, \theta_{i}\right) \geq u_{i}\left(a_{i \varnothing}, \theta_{i}\right) .
\end{aligned}
$$

In other words, for each agent $i$, her interim allocation rule $\pi_{\phi, i}(\omega, \varepsilon)$ must be an element of $S_{I C / I R, i}^{*}$, where the latter is the set of interim allocation rules $S_{i}^{*}: \Theta_{i} \rightarrow \Delta\left(A_{i}\right)$ that satisfy the following incentive compatibility and individual rationality constraints:

$$
\begin{gathered}
\left(\forall \theta_{i}, \theta_{i}^{\prime} \in \Theta_{i}\right) \sum_{a_{i} \in A_{i}} s_{i}^{*}\left(a_{i} \mid \theta_{i}\right) u_{i}\left(a_{i}, \theta_{i}\right) \geq \sum_{a_{i} \in A_{i}} s_{i}^{*}\left(a_{i} \mid \theta_{i}^{\prime}\right) u_{i}\left(a_{i}, \theta_{i}\right) \\
\left(\forall \theta_{i} \in \Theta_{i}\right) \sum_{a_{i} \in A_{i}} s_{i}^{*}\left(a_{i} \mid \theta_{i}\right) u_{i}\left(a_{i}, \theta_{i}\right) \geq u_{i}\left(a_{i \varnothing}, \theta_{i}\right)
\end{gathered}
$$

Because the individual rationality and incentive constraints must hold for each pair $(\omega, \varepsilon)$, the designer's problem is separable across variables for different $\omega, \varepsilon$ : the sets of variables $\phi(\cdot, \omega, \varepsilon)$ appear in different sets of constraints and the objective function is additively separable across those variables. Consequently, the designer's problem can be solved as a collection of independent problems, one for each $\omega, \varepsilon$. $\square$


Calibrated Mechanism Design

## B Omitted proofs from Section 3

In this section, we present the proofs of Theorem 2 and Proposition 1. We proceed as follows: We first prove Proposition 1, as when $N=1$ its proof implies the "if" direction of Theorem 2. We then prove the "only if" direction of Theorem 2.

Proof of Proposition 1. We focus on the case in which types and states are independently distributed, and explain how to extend the proof when they are not.

Let $\vartheta \in \Delta(A \times \Theta \times \Omega)$ denote the outcome distribution implemented by an incentive compatible and individually rational calibrated mechanism. We show transition probabilities $\beta: \Omega \rightarrow \Delta\left(\Delta(\Omega)^{N}\right)$, $\bar{\alpha}: \Theta \times \Omega \times \Delta(\Omega)^{N} \rightarrow \Delta(A)$, and $\alpha_{i}: \Theta_{i} \times \Delta(\Omega) \rightarrow \Delta\left(A_{i}\right)$ for $i \in\{1, \ldots, N\}$ exist such that

$$
\vartheta(a, \theta, \omega)=\mu_{0}(\omega) f(\theta) \int_{\Delta(\Omega)^{N}} \bar{\alpha}\left(a \mid \theta, \omega, \mu_{1}, \ldots, \mu_{N}\right) \beta\left(d\left(\mu_{1}, \ldots, \mu_{N}\right) \mid \omega\right),
$$

and for all $i \in\{1, \ldots, N\}$, (i) $\alpha_{i}$ satisfies item 3 of Definition 4, and (ii) on the support of $\mu_{0} \otimes \beta, \alpha_{i}\left(\cdot \mid \cdot, \mu_{i}\right)$ is incentive compatible and individually rational when agent $i$ holds belief $\mu_{i}$.

Let $\pi_{\omega, i}:[0,1] \rightarrow S_{i}^{*}$ denote the mapping $\varepsilon \mapsto \pi_{i}(\omega, \cdot)$. For $\tilde{S}_{i}^{*} \in \Delta\left(A_{i}\right)^{\Theta_{i}}$, define

$$
\operatorname{Pr}_{i}\left(\{\omega\} \times \tilde{S}_{i}^{*}\right)=\mu_{0}(\omega) \lambda\left(\pi_{\omega, i}^{-1}\left(\tilde{S}_{i}^{*}\right)\right)=\int_{\tilde{S}_{i}^{*}} \mu_{i}\left(\omega \mid s_{i}^{*}\right) \tau_{\phi, i}\left(d s_{i}^{*}\right)
$$

where the second equality follows from disintegration of $\operatorname{Pr}_{i} \in \Delta\left(\Omega \times S_{i}^{*}\right)$ along $S_{i}^{*}$, and corresponds to the definition of Bayes rule for agent $i$. Define the measurable mappings, $T_{i}: S_{i}^{*} \rightarrow \Delta(\Omega)$ and
$T: S^{*} \rightarrow \Delta(\Omega)^{N}$ as follows: $T_{i}\left(s_{i}^{*}\right)=\mu_{i}\left(\cdot \mid s_{i}^{*}\right)$ and $T\left(s^{*}\right)=\left(T_{1}\left(s_{1}^{*}\right), \ldots, T_{N}\left(s_{N}^{*}\right)\right)$.
Define a joint distribution $Q \in \Delta\left(A \times \Theta \times \Omega \times \Delta(\Omega)^{N}\right)$ as follows:

$$
Q\left(\{(a, \theta, \omega)\} \times \times_{i=1}^{N} \tilde{\Delta}_{i}\right)=\mu_{0}(\omega) f(\theta) \int_{\pi_{\omega}^{-1}\left(T^{-1}\left(\times \tilde{\Delta}_{i}\right)\right)} \phi(a \mid \theta, \omega, \varepsilon) \lambda(d \varepsilon)
$$

where $\pi_{\omega}^{-1}\left(T^{-1}\left(\times \tilde{\Delta}_{i}\right)\right)=\cap_{i=1}^{N}\left\{\varepsilon: T_{i}\left(\pi_{\omega, i}(\varepsilon)\right) \in \tilde{\Delta}_{i}\right\}$.
We note the following properties of $Q$. First, consider its marginal over $\Theta \times \Omega \times \Delta(\Omega)^{N}$,

$$
Q_{\Theta \Omega \Delta^{N}}\left(\{(\theta, \omega)\} \times \times_{i=1}^{N} \tilde{\Delta}_{i}\right)=\mu_{0}(\omega) f(\theta) \lambda\left(\cap_{i=1}^{N}\left\{\varepsilon: T_{i}\left(\pi_{\omega, i}(\varepsilon)\right) \in \tilde{\Delta}_{i}\right\}\right),
$$

which implies that the disintegration of $Q_{\Theta \Omega \Delta^{N}}$ along $\Theta \times \Omega, \beta: \Theta \times \Omega \rightarrow \Delta\left(\Delta(\Omega)^{N}\right)$ does not depend on $\theta$. This automatically implies that $Q$ admits the following disintegration:

$$
Q\left(\{(a, \theta, \omega)\} \times \times_{i=1}^{N} \tilde{\Delta}_{i}\right)=\mu_{0}(\omega) f(\theta) \int_{\times_{i=1}^{N} \tilde{\Delta}_{i}} \bar{\alpha}\left(a \mid \theta, \omega, \mu_{1}, \ldots, \mu_{N}\right) \beta\left(d\left(\mu_{1}, \ldots, \mu_{N}\right) \mid \omega\right)
$$

which, in turn, delivers Equation B.1. Moreover, note that the marginal of $\beta$ on the beliefs of agent $i$, $\beta_{i}: \Omega \rightarrow \Delta(\Delta(\Omega))$, satisfies

$$
\beta_{i}\left(\tilde{\Delta}_{i} \mid \omega\right)=\lambda\left(\pi_{\omega, i}^{-1}\left(T_{i}^{-1}\left(\tilde{\Delta}_{i}\right)\right)\right) .
$$

Consider now the marginal on $A_{i} \times \Theta_{i} \times \Omega \times \Delta(\Omega)$ of $Q, Q_{A_{i} \Theta_{i} \Omega \Delta_{i}}$, which satisfies:

$$
\begin{aligned}
& Q_{A_{i} \Theta i \Omega \Delta_{i}}\left(\left\{\left(a_{i}, \theta_{i}, \omega\right)\right\} \times \tilde{\Delta}_{i}\right)=\sum_{\theta_{-i} \in \Theta_{-i}} \sum_{a_{-i} \in A_{-i}} Q\left(\{(a, \theta, \omega)\} \times \tilde{\Delta}_{i} \times \Delta(\Omega)^{N-1}\right)= \\
& =\mu_{0}(\omega) f_{i}\left(\theta_{i}\right) \int_{\pi_{\omega, i}^{-1}\left(T_{i}^{-1}\left(\tilde{\Delta}_{i}\right)\right)}\left(\sum_{\theta_{-i} \in \Theta_{-i}} f_{-i}\left(\theta_{-i}\right) \sum_{a_{-i} \in A_{-i}} \phi\left(a_{i}, a_{-i} \mid \theta_{i}, \theta_{-i}, \omega, \varepsilon\right)\right) \lambda(d \varepsilon) \\
& =\mu_{0}(\omega) f_{i}\left(\theta_{i}\right) \int_{\pi_{\omega, i}^{-1}\left(T_{i}^{-1}\left(\tilde{\Delta}_{i}\right)\right)} \pi_{\omega, i}(\varepsilon)\left(a_{i} \mid \theta_{i}\right) \lambda(d \varepsilon)=\mu_{0}(\omega) f_{i}\left(\theta_{i}\right) \int_{T_{i}^{-1}\left(\tilde{\Delta}_{i}\right)} s_{i}^{*}\left(a_{i} \mid \theta_{i}\right)\left(\lambda \circ \pi_{\omega, i}^{-1}\right)\left(d s_{i}^{*}\right)
\end{aligned}
$$

Lastly, $Q_{A_{i} \Theta_{i} \Omega \Delta_{i}}$ admits the following representation via disintegration:

$$
\begin{aligned}
& Q_{A_{i} \Theta_{i} \Omega \Delta_{i}}\left(\left\{\left(a_{i}, \theta_{i}, \omega\right)\right\} \times \tilde{\Delta}_{i}\right)=\mu_{0}(\omega) f_{i}\left(\theta_{i}\right) \int_{\tilde{\Delta}_{i}} \alpha_{i}\left(a_{i} \mid \theta_{i}, \omega, \mu_{i}\right) \beta_{i}\left(d \mu_{i} \mid \omega\right) \\
& =\mu_{0}(\omega) f_{i}\left(\theta_{i}\right) \int_{\tilde{\Delta}_{i}} \alpha_{i}\left(a_{i} \mid \theta_{i}, \omega, \mu_{i}\right)\left(\lambda \circ \pi_{\omega, i}^{-1} \circ T_{i}^{-1}\right)\left(d \mu_{i}\right)
\end{aligned}
$$


Calibrated Mechanism Design

Together with the uniqueness of disintegration and the sufficiency property of beliefs, Equations B. 3 and B. 4 imply that $\alpha_{i}$ does not depend on $\omega$. The incentive compatibility and individual rationality of $\alpha_{i}$ follows from that of the calibrated mechanism.

Finally, consider the case in which $\theta$ and $\omega$ are not independent. Then, the experiment $\beta$ in Equation B. 2 induces a joint distribution over the beliefs of $N$ fictitious agents whose prior over the state is given by $\mu_{0}$. Agent $i$ 's updated beliefs when her type is $\theta_{i}$ obtain from a transformation of $\mu$ (Alonso and Câmara, 2016; Laclau and Renou, 2017). ${ }^{41}$ Thus, up to changing $f(\theta)$ by $f(\theta \mid \omega)$, and interpreting

[^25]the draw from the Blackwell experiment as the posterior of an agent with prior belief $\mu_{0}$, the result follows. $\square$

Proof of Theorem 2.
"Only if" direction Similar to the proof of Proposition 1 , we focus on the case in which $\theta$ and $\omega$ are independent. Suppose $\vartheta \in \Delta(A \times \Theta \times \Omega)$ is implemented by an incentive compatible and individually rational two-stage mechanism. That is,

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

and $\alpha$ is incentive compatible and individually rational on the support of $\mu_{0} \otimes \beta$. We construct an incentive compatible and individually rational calibrated mechanism that implements $\vartheta$.

First, if $\vartheta$ satisfies Equation B.5, Rubin and Wesler (1958) and Carathéodory's theorem (Aliprantis and Border, 2006, Theorem 5.32) imply that a finite support $\beta^{\prime}: \Omega \rightarrow \Delta\left(\left\{\mu_{1}, \ldots, \mu_{K}\right\}\right)$ exists such that for all $(a, \theta, \omega) \in A \times \Theta \times \Omega^{42}$

$$
\vartheta(a, \theta, \omega)=\mu_{0}(\omega) f(\theta) \sum_{k=1}^{K} \alpha\left(a \mid \theta, \mu_{k}\right) \beta^{\prime}\left(\left\{\mu_{k}\right\} \mid \omega\right) .
$$

For each $\omega \in \Omega$, partition $[0,1]=\cup_{k=1}^{K-2}\left[b_{k}^{\omega}, b_{k+1}^{\omega}\right) \cup\left[b_{K-1}^{\omega}, 1\right]$, where $b_{1}=0$, and for all $k \in\{1, \ldots, K-2\}$, $b_{k+1}^{\omega}=\sum_{l=1}^{k} \beta^{\prime}\left(\left\{\mu_{l}\right\} \mid \omega\right)$. Define for $\varepsilon \in\left[b_{k}^{\omega}, b_{k+1}^{\omega}\right)$

$$
\phi(a \mid \theta, \omega, \varepsilon)=\alpha\left(a \mid \theta, \mu_{k}\right) .
$$

The calibrated information structure is $\pi_{\phi}(\omega, \varepsilon)=\phi(\cdot \mid \cdot, \omega, \varepsilon)=\alpha\left(\cdot \mid \theta, \mu_{k}\right)$ for $\varepsilon \in\left[b_{k}^{(w}, b_{k+1}^{(w)}\right.$ if $m \leq K-2$ or $\varepsilon \in\left[b_{K-1}^{\omega}, 1\right]$.

We now show that for all $\theta$ and all $s \in \operatorname{supp} \pi_{\phi}$, the mechanism $\phi$ is incentive compatible and individually rational. Note that $\pi_{\phi}$ has finite support, and let $s \in \operatorname{supp} \pi_{\phi}$ and let $\mu(\cdot \mid s)$ denote the updated posterior. Then, $k$ exists such that the following holds:

$$
\mu(\omega \mid s)=\frac{\mu_{0}(\omega) \lambda(\{\varepsilon: \pi(\omega, \varepsilon)=s\})}{\sum_{\omega^{\prime} \in \Omega} \mu_{0}\left(\omega^{\prime}\right) \lambda\left(\left\{\varepsilon: \pi\left(\omega^{\prime}, \varepsilon\right)=s\right\}\right)}=\frac{\mu_{0}(\omega)\left(b_{k+1}^{\omega}-b_{k}^{\omega}\right)}{\sum_{\omega^{\prime} \in \Omega} \mu_{0}\left(\omega^{\prime}\right)\left(b_{k+1}^{\omega^{\prime}}-b_{k}^{\omega^{\prime}}\right)}=\mu_{k}(\omega) .
$$

Moreover, because $\phi(\cdot \mid \cdot, \omega, \varepsilon)=\alpha\left(\cdot \mid \cdot, \mu_{k}\right)$, then it satisfies the agent's incentive compatibility and individual rationality constraints when she holds belief $\mu_{k} .{ }^{43}$ $\square$

[^26]


Calibrated Mechanism Design

## Supplementary Appendix


Calibrated Mechanism Design

## C Omitted proofs from Section 5


Calibrated Mechanism Design

## C. 1 Repeated Mechanisms

In this section, we present the proof of Theorem 3. To do so, we first complete the formal definition of the game induced by repeating mechanism $\phi: M \times \Omega \times \mathcal{E} \rightarrow \Delta(A)$, by specifying the histories, strategy space, and the distribution over terminal histories induced by the agent's strategy and the mechanism. Having laid this groundwork, we describe the proof strategy, and then provide the formal details of the proof. Throughout this section, we use the shorthand $\tilde{\Omega}=\Omega \times \mathcal{E}$, and denote its elements by $\tilde{\omega}$.

Histories and strategies Histories through period $t \in \mathbb{N}$ are defined as $H^{t} \equiv(\Theta \times M \times A)^{t-1}$. The set of infinite histories from the agent's point of view is $H^{\infty}$. The set of terminal histories is $\mathcal{H}^{\infty} \equiv \tilde{\Omega} \times H^{\infty}$, where recall $\mathcal{E}$ is finite and endowed with some measure $\eta$.

The agent's behavioral strategy is defined as a collection $\sigma \equiv\left(\sigma_{t}\right)_{t \in \mathbb{N}}$ such that for all $t \geq 1$

$$
\sigma_{t}: H^{t} \times \Theta \rightarrow \Delta(M) .
$$

The tuple of distributions $\left(\mu_{0}, \eta, f\right)$ together with the mechanism $\phi$ and the agent's strategy $\sigma$ determine a joint distribution over $\mathcal{H}^{\infty}$ by the Ionescu-Tulcea theorem (Bogachev, 2007, Theorem 10.7.3). We provide more details on this probability distribution below. Denote by $\mathbb{P}_{\left(\mu_{0}, \eta, f, \phi, \sigma\right)}$ and $\mathbb{E}_{\left(\mu_{0}, \eta, f, \phi, \sigma\right)}$ the probability distribution over the terminal histories and the expectation with respect to this distribution, respectively. Whenever it is not likely to lead to confusion, we drop the dependence on $\left(\mu_{0}, \eta, f, \phi, \sigma\right)$, and whenever we want to emphasize the dependence on the agent's strategy we note the dependence on $\sigma$.

The distribution over terminal histories $\mathcal{H}^{\infty}$ For future use, we review the construction of $\mathbb{P}_{\sigma}$. For each $t$, the distributions $\left(\mu_{0}, \eta, f\right)$ together with the mechanism $\phi$ and the agent's strategy $\sigma$ determine a distribution over $\tilde{\Omega} \times H^{t}$, which we denote by $\mathbb{P}_{\sigma}^{t} \in \Delta\left(\tilde{\Omega} \times H^{t}\right)$. Note that for any subset $\tilde{\mathcal{H}^{t}} \subset \tilde{\Omega} \times H^{t}$,

$$
\mathbb{P}_{\sigma}^{t}\left(\tilde{\mathcal{H}}^{t}\right)=\mathbb{P}_{\sigma}^{t+1}\left(\tilde{\mathcal{H}}^{t} \times(\Theta \times M \times A)\right) .
$$

Moreover,

$$
\mathbb{P}_{\sigma}^{t+1}\left(\tilde{\omega}, h^{t}, \theta, m, a\right)=\mathbb{P}_{\sigma}^{t}\left(\tilde{\omega}, h^{t}\right) f(\theta) \sigma_{t}\left(h^{t}, \theta\right)(m) \phi(a \mid m, \tilde{\omega}) .
$$

By the Ionescu-Tulcea theorem, the distribution $\mathbb{P}_{\sigma} \in \Delta\left(\tilde{\Omega} \times H^{\infty}\right)$ is the unique distribution that satisfies that for all $t \in \mathbb{N}, \tilde{\mathcal{H}}^{t} \subset \tilde{\Omega} \times H^{t}$,

$$
\mathbb{P}_{\sigma}\left(\tilde{\mathcal{H}}^{t} \times \prod_{s=t+1}^{\infty}(\Theta \times M \times A)\right)=\mathbb{P}_{\sigma}^{t}\left(\tilde{\mathcal{H}}^{t}\right) .
$$

[^27]Belief system The agent's beliefs over $\tilde{\Omega}$ at the beginning of each $t$ are determined by the belief system, which in a slight abuse of notation we denote by $\mu_{t}: H^{t} \rightarrow \Delta(\tilde{\Omega})$. The belief system satisfies

$$
\mathbb{P}_{\sigma}^{t}\left(h^{t}\right) \mu_{t}\left(\tilde{\omega} \mid h^{t}\right)=\mathbb{P}_{\sigma}^{t}\left(\tilde{\omega}, h^{t}\right) .
$$

That is, whenever $h^{t}$ is such that $\mathbb{P}_{\sigma}\left(\left\{\tilde{h} \in H^{\infty}: \tilde{h}^{t}=h^{t}\right\}\right)>0$,

$$
\mu_{t}\left(\tilde{\omega} \mid h^{t}\right)=\frac{\mathbb{P}_{\sigma}^{t}\left(\tilde{\omega}, h^{t}\right)}{\mathbb{P}_{\sigma}^{t}\left(h^{t}\right)}=\mathbb{P}_{\sigma}^{t}\left(\tilde{\omega} \mid h^{t}\right) .
$$

Given $\mathbb{P}_{\sigma} \in \Delta\left(\mathcal{H}^{\infty}\right)$, define $\mu_{\infty}\left(\tilde{\omega} \mid h^{\infty}\right) \equiv \mathbb{P}_{\sigma}\left(\tilde{\omega} \mid h^{\infty}\right)$ to be the belief system conditional on the whole terminal history $h^{\infty}$.

Remark C. 1 (Belief system and strategies as functions on $\mathcal{H}^{\infty}$ ). Whereas the beliefs and strategies are defined on the finite histories, it is sometimes convenient to write them as functions on $\mathcal{H}^{\infty}$ that are adapted to $H^{t}$.

A property of the belief system We collect here a property of the belief system which we use in our proofs below.

Lemma C. 1 (Martingale property under weak* convergence). $\mu_{t}\left(h^{\infty}\right) \xrightarrow{w^{*}} \mu_{\infty}\left(h^{\infty}\right) \mathbb{P}_{\sigma}$-almost surely. This and the proof of other technical results are in Appendix D.


Calibrated Mechanism Design

## C.1.1 Proof of Theorem 3 (necessity)

We are now ready to present the proof of Theorem 3, starting by the "only if" direction. Let $\vartheta \in \Delta(A \times$ $\Theta \times \Omega$ ) denote the outcome distribution implemented by repeated mechanism $\phi$ under best response $\sigma$, and let $v_{\sigma}$ denote the associated occupation measure, the definition of which we reproduce below for ease of reference:

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

where recall limits are in the weak* sense. We show that $\vartheta$ can be implemented by an incentive compatible and individually rational two-stage mechanism.

To this end, we consider two sequences of extended occupation measures on $A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega})$ :

$$
\begin{array}{r}
\bar{v}_{\sigma}^{T, 1}(\{(a, \theta, m, \tilde{\omega})\} \times \tilde{\Delta})=\frac{1}{T} \mathbb{E}_{\sigma}\left[\sum_{t=1}^{T} \mathbb{1}\left[\left(a_{t}, \theta_{t}, m_{t}, \tilde{\omega}^{\prime}\right)=(a, \theta, m, \tilde{\omega})\right] \mathbb{1}\left[\mu_{t} \in \tilde{\Delta}\right]\right], \\
\bar{v}_{\sigma}^{T, 2}(\{(a, \theta, m, \tilde{\omega})\} \times \tilde{\Delta})=\frac{1}{T} \mathbb{E}_{\sigma}\left[\sum_{t=1}^{T} \mathbb{1}\left[\left(a_{t}, \theta_{t}, m_{t}, \tilde{\omega}^{\prime}\right)=(a, \theta, m, \tilde{\omega})\right] \mathbb{1}\left[\mu_{t+1} \in \tilde{\Delta}\right]\right] .
\end{array}
$$

We note the following. First, Equation C. 5 counts the beliefs at the beginning of period $t$, while Equation C. 6 counts the beliefs at the end of period $t$ (after the realization of $\theta, m$, and $a$.) Equation C. 5 is key to obtain the (limit) independence of the belief and type distributions, while Equation C. 6 allows us to obtain the (limit) independence of the allocation and the state, conditional on the induced belief.

Second, $v_{\sigma}^{T}$ is the marginal of both $\bar{v}_{\sigma}^{T, 1}$ and $\bar{v}_{\sigma}^{T, 2}$. Third, by Lemma C.1, $\mu_{t} \xrightarrow{w^{*}} \mu_{\infty}$, and hence both $\bar{v}_{\sigma}^{T, 1}$ and $\bar{v}_{\sigma}^{T, 2}$ have the same set of subsequential limits, which we record for future reference below (see Appendix D for the proof):

Lemma C.2. The occupation measures $\bar{v}_{\sigma}^{T, 1}$ and $\bar{v}_{\sigma}^{T, 2}$ have the same set of subsequential limits.
The proof of necessity of Theorem 3 proceeds in five steps. First, we show that the marginal of $\bar{v}_{\sigma}^{T, 1}$ on $\Delta(\tilde{\Omega})$, which we denote by $\tau_{\sigma}^{T}$ weak*-converges to $\mathbb{P}_{\sigma} \circ \mu_{\infty}^{-1}$. We denote this limit measure by $\tau_{\sigma}$. By Lemma C.2, $\tau_{\sigma}$ is also the (limit) marginal of $\bar{v}_{\sigma}^{T, 2}$ on $\Delta(\tilde{\Omega})$.

Second, we show that up to a subsequence $\bar{v}_{\sigma}^{T, 1}, \bar{v}_{\sigma}^{T, 2} \xrightarrow{w^{*}} \bar{v}_{\sigma}$. Furthermore, transition probabilities $\tau_{\sigma} \in \Delta(\Delta(\tilde{\Omega})), \rho: \Theta \times \Delta(\tilde{\Omega}) \rightarrow \Delta(M), \alpha^{\prime}: M \times \Delta(\tilde{\Omega}) \rightarrow \Delta(A)$ exist such that

$$
v_{\sigma}(a, \theta, m, \tilde{\omega})=\int_{\Delta(\tilde{\Omega})} \mu(\tilde{\omega}) f(\theta) \rho(m \mid \theta, \mu) \alpha^{\prime}(a \mid m, \mu) \tau_{\sigma}(d \mu)
$$

Hence, the agent's payoff when faced with mechanism $\phi$ and playing strategy $\sigma$ can be written as:

$$
\mathbb{E}_{\bar{v}_{\sigma}}[u(a, \theta, \omega)]=\int_{\Delta(\tilde{\Omega})} \sum_{\theta \in \Theta} f(\theta) \sum_{m \in M} \rho(m \mid \theta, \mu) \mathbb{E}_{\tilde{\omega} \sim \mu}\left[\sum_{a \in A} \alpha^{\prime}(a \mid \mu, m) u(a, \theta, \omega)\right] \tau_{\sigma}(d \mu) .
$$

Third, we show that for all $\theta \in \Theta$

$$
\mathbb{E}_{\tau_{\sigma}}\left\{\sum_{m \in M} \rho(m \mid \theta, \mu) \mathbb{E}_{\tilde{\omega} \sim \mu}\left[\sum_{a \in A} \alpha^{\prime}(a \mid \mu, m) u(a, \theta, \omega)\right]-\max _{m \in M} \mathbb{E}_{\tilde{\omega} \sim \mu}\left[\sum_{a \in A} \alpha^{\prime}(a \mid \mu, m) u(a, \theta, \omega)\right]\right\}=0 .
$$

Equations C. 8 and C. 9 allow us to identify the incentive compatible and individually rational allocation rule of the two-stage mechanism that implements $\vartheta$.

Fourth, whereas the previous steps identify a two-stage mechanism expressed in terms of posterior beliefs over $\tilde{\Omega}$, we show how to obtain a two-stage mechanism expressed in terms of posterior beliefs over $\Omega$. Finally, we show that the agent's payoff in Equation C. 8 coincides with the payoff she would get when best responding to the information structure calibrated to $\phi$.

Step 1 Having defined the extended occupation measure in Equation C.5, we present here a property we use in our proof. Let $\tau_{\sigma}^{T}$ denote the marginal of $\bar{v}_{\sigma}^{T, 1}$ on $\Delta(\tilde{\Omega})$. That is, for any measurable subset $\tilde{\Delta} \subset \Delta(\tilde{\Omega})$, define

$$
\tau_{\sigma}^{T}(\tilde{\Delta})=\frac{1}{T} \mathbb{E}_{\sigma}\left[\sum_{t=1}^{T} \mathbb{1}\left[\mu_{t} \in \tilde{\Delta}\right]\right] .
$$

In Appendix D, we prove the following:
Lemma C.3. The sequence of measures $\left(\tau_{\sigma}^{T}\right)_{T \in \mathbb{N}}$ defined by Equation C. 10 converges in the weak* sense to the push-forward measure $\tau_{\sigma} \equiv \mathbb{P}_{\sigma} \circ \mu_{\infty}^{-1}$, where $\mu_{\infty}\left(h^{\infty}\right)=\mathbb{P}_{\sigma}\left(\cdot \mid h^{\infty}\right)$.

Step 2 To show that Equation C. 7 holds, we show the following properties of $\bar{v}_{\sigma}^{T, 1}$ and $\bar{v}_{\sigma}^{T, 2}$. On the one hand, $\bar{v}_{\sigma}^{T, 1}$ satisfies that for all $g \in C_{b}(A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega}))$,


Calibrated Mechanism Design

$$
\int_{A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega})} g(a, \theta, m, \tilde{\omega}, \mu) d \bar{v}_{\sigma}^{T, 1}=\int_{\Theta \times M \times \Delta(\tilde{\Omega})} \mathbb{E}_{\mu}\left[\mathbb{E}_{\phi(\cdot \mid m, \tilde{\omega})}[g(a, \theta, m, \tilde{\omega}, \mu)]\right] d \bar{v}_{\sigma, \Theta M \Delta^{\prime}}^{T, 1}
$$

and for all $q \in C_{b}(\Theta \times \Delta(\Omega))$,

$$
\int_{\Theta \times \Delta(\tilde{\Omega})} q(\theta, \mu) d \bar{v}_{\sigma, \Theta \Delta}^{T, 1}=\int_{\Delta(\tilde{\Omega})} \int_{\Theta} f(\theta) q(\theta, \mu) d \tau_{\sigma}^{T}
$$

On the other hand, $\bar{v}_{\sigma}^{T, 2}$ satisfies that for all $g \in C_{b}(A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega}))$,

$$
\int_{A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega})} g(a, \theta, m, \tilde{\omega}, \mu) d \bar{v}_{\sigma}^{T, 2}=\int_{\Theta \times M \times A \times \Delta(\tilde{\Omega})} \mathbb{E}_{\tilde{\omega} \sim \mu}[g(a, \theta, m, \tilde{\omega}, \mu)] d \bar{v}_{\sigma, \Theta M A \Delta}^{T, 2}
$$

In the expressions above, the subscripts on $\bar{v}_{\sigma}^{T, k}$ next to $\sigma$ are the spaces over which we take the marginals, and $\Delta$ is shorthand notation for $\Delta(\tilde{\Omega})$. Because $\Delta(A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega}))$ is compact (Aliprantis and Border, 2006, Theorem 15.11), $\bar{v}_{\sigma}^{T, 1}$ has a convergent subsequence $\left(\bar{v}_{\sigma}^{T_{n}, 1}\right)_{n \in \mathbb{N}}$, which by Lemma C. 2 is also a convergent subsequence of $\bar{v}_{\sigma}^{T, 2}$. Let $\bar{v}_{\sigma}$ denote the weak* limit along $T_{n}$. The continuity of the projection implies that $v_{\sigma}$ is the marginal of $\bar{v}_{\sigma}$ on $A \times \Theta \times M \times \tilde{\Omega}$, and $\tau_{\sigma} \equiv \mathbb{P}_{\sigma} \circ \mu_{\infty}^{-1}$ is the marginal on $\Delta(\Omega)$. Equations C. 12 and C. 13 together imply that $\bar{v}_{\sigma}$ admits the decomposition in the right hand side of Equation C.7, and the result follows.

To show Equation C. 11 holds, use that $\bar{v}_{\sigma}^{T, 1}$ has finite support to write it as follows:

$$
\begin{aligned}
\bar{v}_{\sigma}^{T, 1}(a, \theta, m, \tilde{\omega}, \mu) & =\frac{1}{T} \sum_{t=1}^{T} \sum_{h^{t} \in H^{t}} \mathbb{P}_{\sigma}^{t}\left(\tilde{\omega}, h^{t}\right) f(\theta) \sigma_{t}\left(h^{t}, \theta\right)(m) \phi(a \mid m, \tilde{\omega}) \mathbb{1}\left[\mu_{t}=\mu\right] \\
& =\frac{1}{T} \sum_{t=1}^{T} \sum_{h^{t} \in H^{t}} \mathbb{P}_{\sigma}^{t}\left(h^{t}\right) \mu(\tilde{\omega}) f(\theta) \sigma_{t}\left(h^{t}, \theta\right)(m) \phi(a \mid m, \tilde{\omega}) \mathbb{1}\left[\mu_{t}=\mu\right]
\end{aligned}
$$

where the second equality uses that $\mu_{t}\left(h^{t}\right)=\mathbb{P}_{\sigma}^{t}\left(\cdot \mid h^{t}\right)$.
Equation C. 14 implies the following holds for every bounded continuous function $g \in C_{b}(A \times \Theta \times M \times$ $\tilde{\Omega} \times \Delta(\tilde{\Omega})):$

$$
\mathbb{E}_{\bar{v}_{\sigma}^{T, 1}}[g(a, \theta, m, \tilde{\omega}, \mu)]=\mathbb{E}_{\bar{v}_{\sigma, \Theta M \Delta}^{T, 1}}\left[\mathbb{E}_{\mu}\left[\mathbb{E}_{\phi(\cdot \mid m, \tilde{\omega})}[g(a, \theta, m, \tilde{\omega}, \mu)]\right]\right],
$$

This completes the proof that Equation C. 11 holds. Letting $V_{g}(\theta, M, \mu)=\mathbb{E}_{\mu}\left[\mathbb{E}_{\phi(\cdot \mid m, \tilde{\omega})}[g(a, \theta, m, \tilde{\omega}, \mu)]\right]$, we have that

$$
\int_{A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega})} g d \bar{v}_{\sigma}^{T, 1}=\int_{\Theta \times M \times \Delta(\tilde{\Omega})} V_{g} d \bar{v}_{\sigma, \Theta M \Delta}^{T, 1} \Leftrightarrow \int_{A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega})}\left(g-V_{g}\right) d \bar{v}_{\sigma}^{T, 1}=0 .
$$

Because $g-V_{g} \in C_{b}(A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega}))$ and $\bar{v}_{\sigma}^{T_{n}, 1} \xrightarrow{w^{*}} \bar{v}_{\sigma}$, we conclude that

$$
\mathbb{E}_{\bar{v}_{\sigma}}[g(a, \theta, m, \tilde{\omega}, \mu)]=\int_{\Theta \times M \times \Delta(\tilde{\Omega})} \mathbb{E}_{\mu}\left[\mathbb{E}_{\phi(\cdot \mid m, \tilde{\omega})}[g(a, \theta, m, \tilde{\omega}, \mu)]\right] d \bar{v}_{\sigma, \Theta M \Delta} .
$$

To show that Equation C. 12 holds, note that the marginal of $\bar{v}_{\sigma}^{T, 1}$ on $\Theta \times \Delta(\tilde{\Omega})$ equals $\tau_{\sigma}^{T} \otimes f$. Indeed, fix
any continuous function $q \in C_{b}(\Theta \times \Delta(\tilde{\Omega}))$ and note that for all $T$

$$
\mathbb{E}_{\bar{v}_{\sigma, \Theta \Delta}^{T, 1}}[q(\theta, \mu)]=\mathbb{E}_{\tau_{\sigma}^{T}}\left[\sum_{\theta \in \Theta} f(\theta) q(\theta, \mu)\right] .
$$

Letting $V_{q}(\mu)=\sum_{\theta \in \Theta} f(\theta) q(\theta, \mu)$, we have that for all $T$

$$
\int_{\Delta(\tilde{\Omega}) \times \Theta}\left(q(\theta, \mu)-V_{q}(\mu)\right) d \bar{v}_{\sigma, \Theta \Delta}^{T, 1}=0 .
$$

Because $q-V_{q} \in C_{b}(\Theta \times \Delta(\tilde{\Omega}))$ and $\bar{v}_{\sigma}^{T_{n}, 1} \xrightarrow{w^{*}} \bar{v}_{\sigma}$, we conclude that

$$
\int_{\Theta \times \Delta(\tilde{\Omega})} q(\theta, \mu) d \bar{v}_{\sigma, \Theta \Delta}=\int_{\Delta(\tilde{\Omega})} \int_{\Theta} f(\theta) q(\theta, \mu) d \tau_{\sigma} .
$$

Lastly, to show that Equation C. 13 holds, note that we can write $\bar{v}_{\sigma}^{T, 2}$ as follows (once again, we use that for finite $T$, it has finite support):


Calibrated Mechanism Design

$$
\begin{aligned}
& \bar{v}_{\sigma}^{T, 2}(a, \theta, m, \tilde{\omega}, \mu)=\frac{1}{T} \sum_{t=1}^{T} \sum_{h^{t} \in H^{t}} \mathbb{P}_{\sigma}^{t}\left(h^{t}\right) \mathbb{P}_{\sigma}^{t}\left(\tilde{\omega} \mid h^{t}\right) f(\theta) \sigma_{t}\left(h^{t}, \theta\right)(m) \phi(a \mid m, \tilde{\omega}) \mathbb{1}\left[\mu_{t+1}\left(h^{t}, \theta, m, a\right)=\mu\right]= \\
& =\frac{1}{T} \sum_{t=1}^{T} \sum_{h^{t} \in H^{t}: \mu_{t+1}\left(h^{t}, \theta, m, a\right)=\mu} \mathbb{P}_{\sigma}^{t+1}\left(\tilde{\omega}, h^{t}, \theta, m, a\right)=\frac{1}{T} \sum_{t=1}^{T} \sum_{h^{t} \in H^{t}: \mu_{t+1}\left(h^{t}, \theta, m, a\right)=\mu} \mu(\tilde{\omega}) \mathbb{P}_{\sigma}^{t+1}\left(h^{t}, \theta, m, a\right) \\
& =\mu(\tilde{\omega}) \frac{1}{T} \sum_{t=1}^{T} \sum_{h^{t} \in H^{t}: \mu_{t+1}\left(h^{t}, \theta, m, a\right)=\mu} \mathbb{P}_{\sigma}^{t+1}\left(h^{t}, \theta, m, a\right)=\mu(\tilde{\omega}) \bar{v}_{\sigma, A \Theta M \Delta}^{T, 2}(a, \theta, m, \mu),
\end{aligned}
$$

where the last expression follows from noting that the term multiplying $\mu(\omega)$ in the first expression in the third line is $\sum_{\tilde{\omega}} v_{\sigma}^{T, 2}(a, \theta, m, \tilde{\omega}, \mu)$.

Then, for every $g \in C_{b}(A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega}))$, we have that

$$
\mathbb{E}_{\bar{v}_{\sigma}^{T, 2}}[g(a, \theta, m, \tilde{\omega}, \mu)]=\mathbb{E}_{\bar{v}_{\sigma, A \Theta M \Delta(\tilde{\Omega})}^{T, 2}}\left[\mathbb{E}_{\tilde{\omega} \sim \mu}[g(a, \theta, m, \tilde{\omega}, \mu)]\right]
$$

which completes the proof that Equation C. 13 holds. Letting $V_{g}(a, \theta, m, \mu)=\mathbb{E}_{\tilde{\omega} \sim \mu}[g(a, \theta, m, \tilde{\omega}, \mu)]$, we have that

$$
\int\left(g-V_{g}\right) d \bar{v}_{\sigma}^{T, 2}=0
$$

Because $g-V_{g} \in C_{b}(A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega}))$ and $\bar{v}_{\sigma}^{T_{n}, 2} \xrightarrow{w^{*}} \bar{v}_{\sigma}$, we conclude that

$$
\mathbb{E}_{\bar{v}_{\sigma}}[g(a, \theta, m, \tilde{\omega}, \mu)]=\int_{A \times \Theta \times M \times \Delta(\tilde{\Omega})} \mathbb{E}_{\tilde{\omega} \sim \mu}[g(a, \theta, m, \tilde{\omega}, \mu)] \bar{v}_{\sigma, A \Theta M \Delta}(d(a, \theta, m, \mu)) .
$$

Equations C. 16 and C. 17 imply $\bar{v}_{\sigma}$ admits the following disintegration:

$$
\bar{v}_{\sigma}(\{(a, \theta, m, \tilde{\omega})\} \times \tilde{\Delta})=\int_{\tilde{\Delta}} \mu(\tilde{\omega}) f(\theta) \alpha^{\prime}(a \mid \theta, m, \mu) \rho(m \mid \theta, \mu) \tau_{\sigma}(d \mu)
$$

where we disintegrated $\bar{v}_{\sigma, A \Theta M \Delta}$ first along $\Theta \times \Delta(\tilde{\Omega})$-and used Equation C. 16 to obtain the independence
of $\Theta$ and $\Delta(\tilde{\Omega})$-and then further disintegrated the distribution of $A \times M$ conditional on $\Theta \times \Delta(\tilde{\Omega})$. Now, Equation C. 15 implies that the following also holds

$$
\bar{v}_{\sigma}(\{(a, \theta, m, \tilde{\omega})\} \times \tilde{\Delta})=\int_{\tilde{\Delta}} \mu(\tilde{\omega}) f(\theta) \phi(a \mid m, \tilde{\omega}) \rho(m \mid \theta, \mu) \tau_{\sigma}(d \mu)
$$

where once again we use the uniqueness of disintegration. Because Equations C. 18 and C. 19 hold for any tuple $(a, \theta, m, \tilde{\omega})$ and measurable subset $\tilde{\Delta}$ of $\Delta(\tilde{\Omega})$, we conclude that (i) $\alpha^{\prime}(a \mid \theta, m, \mu)$ does not depend on $\theta \tau_{\sigma}$-almost everywhere, and (ii) $\phi(\cdot \mid m, \tilde{\omega})$ is constant on $\tilde{\omega}$ in the support of $\mu \tau_{\sigma}$-almost everywhere. This concludes the proof of Step 2.

Step 3 We now argue that the agent achieves

$$
u^{*}(\mu) \equiv \sum_{\theta \in \Theta} f(\theta) \max _{m \in M} \sum_{\tilde{\omega} \in \tilde{\Omega}} \mu(\tilde{\omega}) \sum_{a \in A} \phi(a \mid m, \tilde{\omega}) u(a, \theta, \omega)=\sum_{\theta \in \Theta} f(\theta) \max _{m \in M} \sum_{\tilde{\omega} \in \tilde{\Omega}} \mu(\tilde{\omega}) \sum_{a \in A} \alpha^{\prime}(a \mid m, \mu) u(a, \theta, \omega),
$$

on the support of $\tau_{\sigma}$, where the second equality follows from Step 2. Toward a contradiction, suppose this is not the case; that is,

$$
\mathbb{E}_{\tau_{\sigma}}\left[\sum_{\theta \in \Theta} f(\theta) \sum_{m \in M} \rho(m \mid \theta, \mu) \sum_{\tilde{\omega} \in \tilde{\Omega}} \mu(\tilde{\omega}) \sum_{a} \phi(a \mid m, \tilde{\omega}) u(a, \theta, \omega)\right]<\mathbb{E}_{\tau_{\sigma}}\left[\sum_{\theta \in \Theta} u^{*}(\mu)\right]=U^{*} .
$$

We show that the agent can achieve a payoff arbitrarily close to $U^{*}$ by playing according to $\sigma$ until some finite $T$ and then best-responding to her beliefs at time $T$ in every period thereafter; a contradiction.

Consider a strategy $\sigma^{\prime}$ which until some period $T$ plays according to $\sigma$ and after period $T$ best responds to $\mu_{T}\left(h^{T}\right) \in \Delta(\tilde{\Omega})$. Because payoffs accumulated on a finite number of periods are irrelevant to longrun payoffs, this strategy results in a payoff:

$$
\begin{aligned}
& \sum_{h^{T} \in H^{T}} \mathbb{P}_{\sigma}^{T}\left(h^{T}\right) \sum_{\theta \in \Theta} f(\theta) \max _{m \in M}\left[\sum_{\tilde{\omega} \in \tilde{\Omega}} \mu_{T}(\tilde{\omega}) \sum_{a \in A} \phi(a \mid m, \tilde{\omega}) u(a, \theta, \omega)\right]=\sum_{h^{T} \in H^{T}} \mathbb{P}_{\sigma}^{T}\left(h^{T}\right) u^{*}\left(\mu_{T}\left(h^{T}\right)\right) \\
& =\mathbb{E}_{\mathbb{P}_{\sigma}^{T} \circ \mu_{T}^{-1}}\left[u^{*}(\mu)\right]=\mathbb{E}_{\mathbb{P}_{\sigma} \circ \mu_{T}^{-1}}\left[u^{*}(\mu)\right],
\end{aligned}
$$

where the last equality follows as $\mu_{T}$ is adapted to the histories through $T$. Similar arguments to Lemma C. 3 imply that $\mathbb{P}_{\sigma} \circ \mu_{T}^{-1} \xrightarrow{w^{*}} \mathbb{P}_{\sigma} \circ \mu_{\infty}^{-1} \equiv \tau_{\sigma}$. Noting that $u^{*}: \Delta(\tilde{\Omega}) \rightarrow \mathbb{R}$ is continuous and bounded (as it is the maximum of linear functions in beliefs), we obtain that as $T \rightarrow \infty$,

$$
\mathbb{E}_{\mathbb{P}_{\sigma} \circ \mu_{T}^{-1}}\left[u^{*}(\mu)\right] \rightarrow \mathbb{E}_{\tau_{\sigma}}\left[u^{*}(\mu)\right] .
$$


Calibrated Mechanism Design

It follows that for every $\delta>0$, we can find $T$ large enough so that $\left|\mathbb{E}_{\mathbb{P}_{\sigma^{\circ}} \mu_{T}^{-1}}\left[u^{*}(\mu)\right]-U^{*}\right|<\delta$, contradicting the optimality of $\sigma$.

We conclude that $\alpha: \Theta \times \Delta(\tilde{\Omega}) \rightarrow \Delta(A)$ defined as follows:

$$
\alpha(a \mid \theta, \mu)=\sum_{m \in M} \rho(m \mid \theta, \mu) \alpha^{\prime}(a \mid m, \mu),
$$

is incentive compatible and individually rational $\tau_{\sigma}$-almost everywhere.

Step 4 We now show how to derive a two-stage mechanism $\beta^{*}: \Omega \rightarrow \Delta(\Delta(\Omega))$ and an allocation rule $\alpha^{*}: \Theta \times \Delta(\Omega) \rightarrow \Delta(A)$ that implement $\vartheta$. First, note that the agent's payoff when her type is $\theta$ and the induced belief is $\mu \in \Delta(\tilde{\Omega})$, can be written as

$$
\sum_{\tilde{\omega} \in \tilde{\Omega}} \mu(\tilde{\omega}) \sum_{a \in A} \alpha(a \mid \theta, \mu) u(a, \theta, \omega)=\sum_{\omega \in \Omega} \mu_{\Omega}(\omega) \sum_{a \in A} \alpha(a \mid \theta, \mu) u(a, \theta, \omega),
$$

where $\mu_{\Omega}$ is the marginal of $\mu$ on $\Omega$ and the equality follows because the realization of $\varepsilon$ is payoffirrelevant. By Step 3, $\alpha(\cdot \mid \theta, \mu)$ is individually rational and incentive compatible when the agent holds belief $\mu_{\Omega}$.

Furthermore, for each $(a, \theta, \omega) \in A \times \Theta \times \Omega$, we have

$$
\vartheta(a, \theta, \omega)=\sum_{\varepsilon \in \mathcal{E}} \int_{\Delta(\tilde{\Omega})} \mu(\omega, \varepsilon) f(\theta) \alpha(a \mid \theta, \mu) \tau_{\sigma}(d \mu)=\int_{\Delta(\tilde{\Omega})} \mu_{\Omega}(\omega) f(\theta) \alpha(a \mid \theta, \mu) \tau_{\sigma}(d \mu)
$$

For each $\theta \in \Theta$, consider the joint distribution $Q_{\theta} \in \Delta(A \times \Delta(\Omega))$ defined as follows:

$$
Q_{\theta}(\{a\} \times \tilde{\Delta})=\int_{\Delta(\tilde{\Omega})} \mathbb{1}\left[\mu_{\Omega} \in \tilde{\Delta}\right] \alpha(a \mid \theta, \mu) \tau_{\sigma}(d \mu)=\int_{\tilde{\Delta}} \alpha^{*}\left(a \mid \theta, \mu_{\Omega}\right) \tau^{*}\left(d \mu_{\Omega}\right)
$$

where the third equality follows from disintegration (note $\alpha^{*}\left(\cdot \mid \cdot, \mu_{\Omega}\right)=\mathbb{E}\left[\alpha(\cdot \mid \cdot, \tilde{\mu}) \mid \tilde{\mu}_{\Omega}=\mu_{\Omega}\right]$ ). By the first argument in Step 4, $\alpha^{*}\left(\cdot \mid \cdot, \mu_{\Omega}\right)$ is individually rational and incentive compatible when the agent holds $\mu_{\Omega}$. We obtain that

$$
\vartheta(a, \theta, \omega)=\int_{\Delta(\Omega)} \mu_{\Omega}(\omega) f(\theta) \alpha^{*}\left(a \mid \theta, \mu_{\Omega}\right) \tau^{*}\left(d \mu_{\Omega}\right)
$$

Defining for all $\omega \in \Omega$ and measurable subsets $\tilde{\Delta} \in \Delta(\Omega)$,

$$
\beta^{*}(\tilde{\Delta} \mid \omega)=\int_{\tilde{\Delta}} \frac{\mu(\omega)}{\mu_{0}(\omega)} \tau^{*}(d \mu)
$$

Steps 2-4 together imply that $\vartheta$ can be implemented by the incentive compatible and individually rational two-stage mechanism $\left(\beta^{*}, \alpha^{*}\right)$.

Step 5: The agent adequately learns Finally, we argue that the agent earns the same payoff as if she had access to the information structure calibrated to $\phi, \pi_{\phi}$.

Lemma C.4. Let $\tau_{\phi}$ denote the belief distribution induced by the calibrated information structure $\pi_{\phi}$. Then, the agent's payoff under $\sigma$ equals

$$
U\left(\tau_{\phi}\right) \equiv \mathbb{E}_{\tau_{\phi}}\left[\sum_{\theta \in \Theta} f(\theta) \max _{m \in M} \sum_{\tilde{\omega} \in \tilde{\Omega}} \mu(\tilde{\omega}) \sum_{a \in A} \phi(a \mid m, \tilde{\omega}) u(a, \theta, \omega)\right] .
$$

The proof of this is standard, and hence we defer it to Appendix D.


Calibrated Mechanism Design

## C.1.2 Proof of Theorem 3 (sufficiency)

Suppose $\vartheta \in \Delta(A \times \Theta \times \Omega)$ is implemented by an incentive compatible and individually rational twostage mechanism. That is,

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

and $\alpha$ is incentive compatible and individually rational on the support of $\mu_{0} \otimes \beta$. As in the proof of Theorem 2, a finite support $\beta^{\prime}: \Omega \rightarrow \Delta\left(\left\{\mu_{1}, \ldots, \mu_{K}\right\}\right)$ exists such that $\left(\beta^{\prime}, \alpha\right)$ implement $\vartheta$. As in Green and Stokey (2022), the experiment $\beta^{\prime}$ can be generated by a finite information structure $\pi: \Omega \times \mathcal{E} \rightarrow$ $\Delta(\Omega)$, where (i) $\mathcal{E}$ is finite, (ii) $\mathcal{E}$ is independent of $\Omega$, and (iii) $\mu=\pi(\omega, \varepsilon)$.

Construct a mechanism $\phi: \Theta \times \Omega \times \mathcal{E} \rightarrow \Delta(A)$ such that $\phi(\cdot \mid \theta, \omega, \varepsilon)=\alpha(\cdot \mid \theta, \pi(\omega, \varepsilon))$. (Note that $\pi$ is information structure calibrated to $\phi$, but expressed in beliefs.) Consider now the extensive form game induced by such a mechanism. ${ }^{44}$

If the agent truthfully reports her type, then the occupation measure induces outcome distribution $\vartheta$. Hence, under truthtelling, the agent's payoff is:

$$
\begin{aligned}
& U\left(\sigma_{\text {truth }}\right)=\sum_{(a, \theta, \omega) \in A \times \Theta \times \Omega} \vartheta(a, \theta, \omega) u(a, \theta, \omega)=\mathbb{E}_{\tau_{\phi}}\left[\sum_{\theta \in \Theta} f(\theta) \sum_{a \in A} \alpha(a \mid \theta, \mu) \sum_{\omega \in \Omega} \mu(\omega) u(a, \theta, \omega)\right] \\
& =\mathbb{E}_{\tau_{\phi}}\left[\sum_{\theta \in \Theta} f(\theta) \max \left\{\max _{\theta^{\prime} \in \Theta} \sum_{a \in A} \alpha\left(a \mid \theta^{\prime}, \mu\right) \sum_{\omega \in \Omega} \mu(\omega) u(a, \theta, \omega), \sum_{\omega \in \Omega} \mu(\omega) u\left(a_{\varnothing}, \theta, \omega\right)\right\}\right]= \\
& =\mathbb{E}_{\mathcal{E}}\left[\sum_{\theta \in \Theta} f(\theta) \max \left\{\max _{\theta^{\prime} \in \Theta} \sum_{\omega \in \Omega} \mu(\omega) \sum_{a \in A} \phi\left(a \mid \theta^{\prime}, \omega, \varepsilon\right) u(a, \theta, \omega), \sum_{\omega \in \Omega} \mu(\omega) u\left(a_{\varnothing}, \theta, \omega\right)\right\}\right],
\end{aligned}
$$

where (i) $\tau_{\phi}$ is the belief distribution induced by the information structure $\pi$, and (ii) the first equality is by definition of the occupation measure, the second is the definition that $\vartheta$ is implemented by the two-stage mechanism, the third follows from incentive compatibility and individual rationality of $\alpha$, and the fourth is definitional.

Moreover, the payoff in the last line of Equation C. 23 is the payoff the agent obtains by using the "learning" strategy in Lemma C.4, which first extracts all the mechanism can teach her about the state and then uses that information to optimize over her participation and reporting strategies. It follows that truthtelling (and participation) are optimal and $\vartheta$ is implemented by repeated mechanism $\phi$.


Calibrated Mechanism Design

## C. 2 Dynamic Mechanisms

In this section, we present the proof of Theorem 4. To do so, we first complete the formal definition of the game, by specifying the histories, strategy space, and the distribution over terminal histories induced by the agent's strategy and the mechanism. Having laid this groundwork, we describe the proof strategy, and then provide the formal details of the proof.

[^28]Mechanisms, histories, and strategies A dynamic mechanism $\left(\varphi_{t}\right)_{t \in \mathbb{N}}$ is a sequence of mappings that condition on the state, the agent's report history, the allocation history, and today's report and output an allocation. By the revelation principle, it is without loss of generality to restrict attention to mechanisms that solicit type reports.

As in the main text, we expand the set of type reports and allocations by the non-participation decision and the outside option, which we denote by $\Theta A_{\varnothing} \equiv A \times \Theta \cup\left\{\left(\varnothing, a_{\varnothing}\right)\right\}$. Then, $\hat{H}^{t}=\left(\Theta A_{\varnothing}\right)^{t-1}$ denotes the histories of reports (inclusive of the non-participation decision) and allocations at the beginning of time $t \in \mathbb{N}$, and let $\hat{\mathcal{H}}^{t}=\Omega \times \hat{H}^{t}$. Similarly, let $\hat{H}^{\infty}=\times_{t \in \mathbb{N}}\left(\Theta A_{\varnothing}\right)$ denote the set of all possible report-allocation outcome paths, and let $\hat{\mathcal{H}}^{\infty}=\Omega \times \hat{H}^{\infty}$. A dynamic mechanism is then a collection of mappings $\left(\varphi_{t}\right)_{t \in \mathbb{N}}$ such that $\varphi_{t}: \hat{\mathcal{H}}^{t} \times \Theta \rightarrow \Delta(A)$.

To define the agent's strategy, let $H^{t}=\Theta^{t-1} \times \hat{H}^{t-1}$, where the coordinates denote the sequence of realized types, reports (inclusive of participation decisions), and allocations through period $t-1$. A behavioral strategy is a mapping $\left(p_{t}, \sigma_{t}\right): H^{t} \times \Theta \rightarrow[0,1] \times \Delta(\Theta)$.

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

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

where the distributions $\left(\mathbb{P}_{(p, \sigma)}^{t}\right)_{t \in \mathbb{N}}$ satisfy (under participation and truthtelling)

$$
\mathbb{P}_{(p, \sigma)}^{t+1}\left(\omega, h^{t}, \theta, \theta^{\prime}, a\right)=\mathbb{P}_{(p, \sigma)}^{t}\left(\omega, h^{t}\right) f(\theta) \mathbb{1}\left[\theta^{\prime}=\theta\right] \varphi_{t}\left(a \mid \omega, \hat{h}^{t}, \theta^{\prime}\right) .
$$

Implementation We focus on incentive-compatible mechanisms $\varphi$ for which (i) a best response, ( $p, \sigma$ ), exists, and (ii) the occupation measure $v_{\sigma} \in \Delta(A \times \Theta \times \Omega)$ exists, where

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

where the limit is in the weak* sense. In contrast to Appendix C.1, we do not keep track of the agent's type reports in the occupation measure, only the agent's types. Under $(p, \sigma)$ only truthtelling histories have positive probability.


Calibrated Mechanism Design

## C.2.1 Proof of Theorem 4 (necessity)

Let $\vartheta \in \Delta(A \times \Theta \times \Omega)$ denote the outcome distribution implemented by an incentive compatible dynamic mechanism $\varphi$, and let $v_{(p, \sigma)}$ denote the occupation measure under the agent's truthtelling strategy. Below, we show that $v_{(p, \sigma)}$, and hence $\vartheta$, can be implemented by a two-stage mechanism which lacks profitable undetectable deviations and is ex ante individually rational.

Analogously to the proof of Theorem 3, we define two sequences of extended occupation measures on $A \times \Theta \times \Omega \times \Delta(\Omega)$ defined as follows. Letting $\tilde{\Delta}$ denote a measurable subset of $\Delta(\Omega)$, define

$$
\begin{aligned}
& \bar{v}_{(p, \sigma)}^{T, 1}(\{(a, \theta, \omega)\} \times \tilde{\Delta})=\frac{1}{T} \sum_{t=1}^{T} \sum_{h^{t} \in H^{t}} \mathbb{P}_{(p, \sigma)}^{t}\left(\omega, h^{t}\right) f(\theta) \sigma_{t}\left(h^{t}, \theta\right)(\theta) \varphi\left(\omega, \hat{h}^{t}, \theta\right)(a) \mathbb{1}\left[\mu_{t}\left(h^{t}\right) \in \tilde{\Delta}\right] \\
& \bar{v}_{(p, \sigma)}^{T, 2}(\{(a, \theta, \omega)\} \times \tilde{\Delta})=\frac{1}{T} \sum_{t=1}^{T} \sum_{h^{t} \in H^{t}} \mathbb{P}_{(p, \sigma)}^{t}\left(\omega, h^{t}\right) f(\theta) \sigma_{t}\left(h^{t}, \theta\right)(\theta) \varphi\left(\omega, \hat{h}^{t}, \theta\right)(a) \mathbb{1}\left[\mu_{t+1}\left(h^{t}, \theta, \theta, a\right) \in \tilde{\Delta}\right]
\end{aligned}
$$

The proof proceeds similarly to that in Appendix C.1. First, we show that the occupation measure $v_{(p, \sigma)} \in \Delta(A \times \Theta \times \Omega)$ admits the following decomposition

$$
v_{(p, \sigma)}(a, \theta, \omega)=\int_{\Delta(\Omega)} f(\theta) \mu(\omega) \alpha(a \mid \theta, \mu) \tau_{(p, \sigma)}(d \mu)
$$

where $\tau_{(p, \sigma)}$ is the distribution over terminal beliefs (cf. Lemma C.3) and the transition probability $\alpha: \Theta \times \Delta(\Omega) \rightarrow \Delta(A)$ is our candidate allocation rule. Consequently, the agent's equilibrium payoff can be written as follows:

$$
\sum_{(a, \theta, \omega) \in A \times \Theta \times \Omega} v_{(p, \sigma)}(a, \theta, \omega) u(a, \theta, \omega)=\int_{\Delta(\Omega)}\left[\sum_{\theta \in \Theta} f(\theta) \sum_{\omega \in \Omega} \mu(\omega) \sum_{a \in A} \alpha(a \mid \theta, \mu) u(a, \theta, \omega)\right] \tau_{(p, \sigma)}(d \mu) .
$$

Second, we show that the allocation rule lacks profitable undetectable deviations and is ex ante individually rational.

The occupation measure satisfies Equation C. 27 To prove that Equation C. 27 holds, we first show that for all $g \in C_{b}(A \times \Theta \times \Omega \times \Delta(\Omega))$ and all $T \in \mathbb{N}$,

$$
\int_{A \times \Theta \times \Omega \times \Delta(\Omega)} g(a, \theta, \omega, \mu) d \bar{v}_{(p, \sigma)}^{T, 2}=\int_{A \times \Theta \times \Delta(\Omega)} \mathbb{E}_{\mu}[g(a, \theta, \omega, \mu)] d \bar{v}_{(p, \sigma), A \Theta \Delta(\Omega)}^{T, 2}
$$

and for all $q \in C_{b}(\Theta \times \Delta(\Omega))$ and all $T \in \mathbb{N}$,

$$
\int_{\Theta \times \Delta(\Omega)} q(\theta, \mu) d \bar{v}_{(p, \sigma), \Theta \Delta}^{T, 1}=\int_{\Delta(\Omega)} \sum_{\theta \in \Theta} f(\theta) q(\theta, \mu) d \bar{v}_{(p, \sigma), \Delta}^{T, 1}
$$

where the subscripts on $\bar{v}$ next to ( $p, \sigma$ ) are the spaces over which we take the marginals, and $\Delta$ is shorthand notation for $\Delta(\Omega)$. We skip the proof of this step as it basically repeats the proof of the analogous step in Appendix C.1.

Because $\Delta(A \times \Theta \times \Omega \times \Delta(\Omega))$ is compact (Aliprantis and Border, 2006, Theorem 15.11), $\bar{v}_{(p, \sigma)}^{T, 1}$ has a convergent subsequence $\left(\bar{v}_{(p, \sigma)}^{T_{n}, 1}\right)_{n \in \mathbb{N}}$, which by Lemma C. 2 is also a convergent subsequence of $\bar{v}_{(p, \sigma)}^{T, 2}$. Let $\bar{v}_{(p, \sigma)}$ denote the weak* limit along $T_{n}$. The continuity of the projection implies that $v_{(p, \sigma)}$ is the marginal of $\bar{v}_{(p, \sigma)}$ on $A \times \Theta \times \Omega$, and $\tau_{(p, \sigma)} \equiv \mathbb{P}_{\sigma} \circ \mu_{\infty}^{-1}$ is the marginal on $\Delta(\Omega)$. Moreover, Equation C. 29 and Equation C. 30 together imply that $v_{(p, \sigma)}$ admits the decomposition on the right hand side of Equation C.27, and the result follows.

The allocation rule lacks profitable undetectable deviations We now show the allocation rule $\alpha$ admits no profitable undetectable deviations. An undetectable deviation is a transition probability $\sigma^{\prime}$ from $\Theta \times \Delta(\Omega)$ to $\Delta(\Theta)$ such that for all $\mu \in \Delta(\Omega)$ and $\theta^{\prime} \in \Theta$

$$
\sum_{\theta \in \Theta} f(\theta) \sigma^{\prime}\left(\theta^{\prime} \mid \theta, \mu\right)=f\left(\theta^{\prime}\right) .
$$

Consider a deviation by the agent to $\left(p, \sigma^{\prime}\right)$ instead of $(p, \sigma)$. That is, when his type is $\theta$ and belief is $\mu$, the agent chooses type $\theta^{\prime}$ with probability $\sigma^{\prime}\left(\theta^{\prime} \mid \theta, \mu\right)$. In what follows, we index the induced distributions over histories only by $\sigma$ and $\sigma^{\prime}$ as we are only changing the agent's reporting strategy. In particular, denote by $\mathbb{P}_{\sigma^{\prime}}$ the induced probability distribution over terminal histories when the agent uses $\left(p, \sigma^{\prime}\right)$ instead of $(p, \sigma)$.

We first claim that for every $t$ the marginal of $\mathbb{P}_{\sigma^{\prime}}^{t}$ over $\Omega \times \hat{H}^{t}$ coincides with that of $\mathbb{P}_{\sigma}^{t}$. Recall that for every $t$ we have that

$$
\mathbb{P}_{\sigma^{\prime}}^{t+1}\left(\omega, h^{t}, \theta, \theta^{\prime}, a\right)=\mathbb{P}_{\sigma^{\prime}}^{t}\left(\omega, h^{t}\right) f(\theta) \sigma^{\prime}\left(\theta^{\prime} \mid \mu_{t}\left(h^{t}\right), \theta\right) \varphi_{t}\left(a \mid \omega, \hat{h}^{t}, \theta^{\prime}\right) .
$$

Adding up over $\theta$ on both sides and using Equation C.31, we get:

$$
\sum_{\theta \in \Theta} \mathbb{P}_{\sigma^{\prime}}^{t+1}\left(\omega, h^{t}, \theta, \theta^{\prime}, a\right)=\mathbb{P}_{\sigma^{\prime}}^{t}\left(\omega, h^{t}\right) f\left(\theta^{\prime}\right) \varphi_{t}\left(a \mid \omega, \hat{h}^{t}, \theta^{\prime}\right) .
$$


Calibrated Mechanism Design

Now, note that $h^{t}=\left(\hat{h}^{t}, \tilde{\theta}^{t-1}\right)$ for some sequence $\tilde{\theta}^{t-1} \in \Theta^{t-1}$. If we add up on both sides over all such sequences we get

$$
\sum_{\theta \in \Theta, \tilde{\theta}^{t-1} \in \Theta^{t-1}} \mathbb{P}_{\sigma^{\prime}}^{t+1}\left(\omega, \hat{h}^{t}, \tilde{\theta}^{t-1}, \theta, \theta^{\prime}, a\right)=\sum_{\tilde{\theta}^{t-1} \in \Theta^{t-1}} \mathbb{P}_{\sigma^{\prime}}^{t}\left(\omega, \hat{h}^{t}, \tilde{\theta}^{t-1}\right) f\left(\theta^{\prime}\right) \varphi_{t}\left(a \mid \omega, \hat{h}^{t}, \theta^{\prime}\right) .
$$

Note that if the distribution over $\Omega \times \hat{H}^{t}$ induced by $\sigma^{\prime}$ up to period $t$ is the same as that induced by $\sigma$, we get that the right-hand side equals:

$$
\mathbb{P}_{\sigma, \hat{\mathcal{H}^{t}}}^{t}\left(\omega, \hat{h}^{t}\right) f\left(\theta^{\prime}\right) \varphi_{t}\left(a \mid \omega, \hat{h}^{t}, \theta^{\prime}\right),
$$

and hence $\mathbb{P}_{\sigma^{\prime}, \hat{\mathcal{H}}^{t+1}}^{t+1}\left(\omega, \hat{h}^{t}, \theta^{\prime}, a\right)=\mathbb{P}_{\sigma, \hat{\mathcal{H}}^{t}}^{t}\left(\omega, \hat{h}^{t}\right) f\left(\theta^{\prime}\right) \varphi_{t}\left(a \mid \omega, \hat{h}^{t}, \theta^{\prime}\right)=\mathbb{P}_{\sigma, \hat{\mathcal{H}}^{t+1}}^{t+1}\left(\omega, \hat{h}^{t}, \theta^{\prime}, a\right)$. By definition of $\mathbb{P}_{\sigma^{\prime}}$, we conclude that $\mathbb{P}_{\sigma^{\prime}, \hat{\mathcal{H}}^{\infty}}=\mathbb{P}_{\sigma, \hat{\mathcal{H}}^{\infty}}$. Hence, the joint distribution over states, reports, and allocations is the same under $\sigma$ and $\sigma^{\prime}$.

Let $\bar{v}_{\sigma^{\prime}}^{T, 1}, \bar{v}_{\sigma^{\prime}}^{T, 2} \in \Delta(A \times \Theta \times \hat{\Theta} \times \Omega \times \Delta(\Omega))$ denote the analogue of the occupation measures in Equations C. 25 and C. 26 corresponding to $\sigma^{\prime}$, extended to account for the agent's reports. Below, the notation $\hat{\Theta}$ signifies those are the agent's reports. In what follows, recalling that the belief system depends only on the reported history and not the type history is useful. Equation C. 31 implies that for all measurable
subsets $\tilde{\Delta}$ of $\Delta(\Omega)$,

$$
\begin{aligned}
& \sum_{\theta \in \Theta} \bar{v}_{\sigma^{\prime}}^{T, 1}\left(\left\{\left(a, \theta, \theta^{\prime}, \omega\right)\right\} \times \tilde{\Delta}\right)=\frac{1}{T} \sum_{t=1}^{T} \sum_{h^{t} \in H^{t}} \mathbb{P}_{\sigma^{\prime}}^{t}\left(\omega, h^{t}\right) f\left(\theta^{\prime}\right) \varphi_{t}\left(\omega, \hat{h}^{t}, \theta^{\prime}\right)(a) \mathbb{1}\left[\mu_{t}\left(h^{t}\right) \in \tilde{\Delta}\right] \\
& =\frac{1}{T} \sum_{t=1}^{T} \sum_{\hat{h}^{t} \in \hat{H}^{t}} \mathbb{P}_{\sigma^{\prime}}^{t}\left(\omega, \hat{h}^{t}\right) f\left(\theta^{\prime}\right) \varphi_{t}\left(\omega, \hat{h}^{t}, \theta^{\prime}\right)(a) \mathbb{1}\left[\mu_{t}\left(\hat{h}^{t}\right) \in \tilde{\Delta}\right] \\
& =\frac{1}{T} \sum_{t=1}^{T} \sum_{\hat{h}^{t} \in \hat{H}^{t}} \mathbb{P}_{\sigma}^{t}\left(\omega, \hat{h}^{t}\right) f\left(\theta^{\prime}\right) \varphi_{t}\left(\omega, \hat{h}^{t}, \theta^{\prime}\right)(a) \mathbb{1}\left[\mu_{t}\left(\hat{h}^{t}\right) \in \tilde{\Delta}\right]=\bar{v}_{\sigma}^{T, 1}\left(\left\{\left(a, \theta^{\prime}, \omega\right)\right\} \times \tilde{\Delta}\right) .
\end{aligned}
$$

The first equality uses the definition of undetectability, the second uses that all the terms depend only on the reported history, the third uses that $\sigma$ and $\sigma^{\prime}$ induce the same distribution over states, reports, and allocations, and the last is the definition of the occupation measure induced by $\sigma$. In words, the marginal of $\bar{v}_{\sigma^{\prime}}^{T, 1}$ over allocations, reports, states, and beliefs, $\bar{v}_{\sigma^{\prime}, A \hat{\Theta} \Omega \Delta}^{T, 1}$ coincides with $\bar{v}_{\sigma}^{T, 1}$.

We now show that $\bar{v}_{\sigma^{\prime}}^{T, 1}$ and $\bar{v}_{\sigma^{\prime}}^{T, 2}$ have a convergent subsequence with limit $\bar{v}_{\sigma^{\prime}} \in \Delta(A \times \Theta \times \hat{\Theta} \times \Omega \times \Delta(\Omega))$ that admits the following decomposition:

$$
\mathbb{E}_{\bar{v}_{\sigma^{\prime}}}[u(a, \theta, \omega)]=\int_{\Delta(\Omega)}\left[\sum_{\theta \in \Theta} f(\theta) \sum_{\theta^{\prime} \in \Theta} \sigma^{\prime}(\theta, \mu)\left(\theta^{\prime}\right) \sum_{a} \alpha\left(a \mid \theta^{\prime}, \mu\right) u(a, \theta, \mu)\right] d \tau_{(p, \sigma)},
$$

where $u(a, \theta, \mu)$ is the linear extension of $u(a, \theta, \cdot)$.
We proceed as follows: First, we show that for each $T$, under $v_{\sigma^{\prime}}^{T, 1}$, the allocation is independent of the true type conditional on the period- $t$ belief and the reported type. Indeed,

$$
\begin{aligned}
\sum_{\omega \in \Omega} v_{\sigma^{\prime}}^{T, 1}\left(a, \theta, \theta^{\prime}, \omega, \mu\right) & =\frac{1}{T} \sum_{t=1}^{T} \sum_{h^{t} \in H^{t}} \mathbb{P}_{\sigma^{\prime}}\left(h^{t}\right)\left(\sum_{\omega \in \Omega} \mathbb{P}_{\sigma^{\prime}}\left(\omega \mid h^{t}\right) \varphi_{t}\left(\omega, \hat{h}^{t}, \theta^{\prime}\right)(a)\right) f(\theta) \sigma^{\prime}\left(\mu_{t}\left(h^{t}\right), \theta\right)\left(\theta^{\prime}\right) \mathbb{1}\left[\mu_{t}\left(h^{t}\right)=\mu\right] \\
& =\frac{f(\theta) \sigma^{\prime}(\theta, \mu)\left(\theta^{\prime}\right)}{f\left(\theta^{\prime}\right)}\left(\frac{1}{T} \sum_{t=1}^{T} \sum_{\hat{h}^{t}: \mu_{t}\left(\hat{h}^{t}\right)=\mu} \mathbb{P}_{\sigma^{\prime}}^{t}\left(\hat{h}^{t}\right) \sum_{\omega \in \Omega} \mathbb{P}_{\sigma^{\prime}}\left(\omega \mid \hat{h}^{t}\right) f\left(\theta^{\prime}\right) \varphi_{t}\left(\hat{h}^{t}, \theta^{\prime}\right)(a)\right) \\
& =\frac{f(\theta) \sigma^{\prime}(\theta, \mu)\left(\theta^{\prime}\right)}{f\left(\theta^{\prime}\right)} v_{\sigma^{\prime}, A \hat{\Theta} \Delta}^{T, 1}\left(a, \theta^{\prime}, \mu\right)=\frac{f(\theta) \sigma^{\prime}(\theta, \mu)\left(\theta^{\prime}\right)}{f\left(\theta^{\prime}\right)} v_{\sigma, A \Theta \Delta}^{T, 1}\left(a, \theta^{\prime}, \mu\right),
\end{aligned}
$$

where the third and fourth equalities use Equation C.25. Moreover, the same analysis as that under $\sigma$ implies the agent's true type is independent of the belief.


Calibrated Mechanism Design

Therefore, $\bar{v}_{\sigma^{\prime}, A \Theta \Theta \leq}^{T, 1}$ admits decomposition:

$$
\bar{v}_{\sigma^{\prime}, A \Theta \hat{\Theta} \Delta}^{T, 1}\left(a, \theta, \theta^{\prime}, \mu\right)=\frac{f(\theta) \sigma^{\prime}\left(\theta^{\prime} \mid \theta, \mu\right)}{f\left(\theta^{\prime}\right)} \bar{v}_{\sigma^{\prime}, A \hat{\Theta} \Delta}^{T, 1}\left(a, \theta^{\prime}, \mu\right)=\frac{f(\theta) \sigma^{\prime}\left(\theta^{\prime} \mid \theta, \mu\right)}{f\left(\theta^{\prime}\right)} \bar{v}_{\sigma, A \Theta \Delta}^{T, 1}\left(a, \theta^{\prime}, \mu\right),
$$

where the second equality follows from Equation C.32.
Second, by the same arguments as in Appendix C.1, $\bar{v}_{\sigma^{\prime}}^{T, 2}\left(a, \theta, \theta^{\prime}, \omega, \mu\right)$ admits decomposition $\mu(\omega) \bar{v}_{\sigma^{\prime}, A \Theta \Theta \leq}^{T, 2}\left(a, \theta, \theta^{\prime}, \mu\right)$ for each $T$.

Third, convergent subsequences $v_{\sigma^{\prime}}^{T_{n_{m}}, 1}$ and $v_{\sigma^{\prime}}^{T_{n_{m}}, 2}$ exist with limit $\bar{v}_{\sigma^{\prime}}$ (cf. Lemma C.2). ${ }^{45}$ We note two

[^29]things. On the one hand, because our previous arguments show that the set of measures admitting the above decompositions is closed, the limit $\bar{v}_{\sigma^{\prime}}$ admits the decomposition. That is,
$$
\int_{A \times \Theta \times \hat{\Theta} \times \Omega \times \Delta(\Omega)} u(a, \theta, \omega) \bar{v}_{\sigma^{\prime}}\left(d\left(a, \theta, \theta^{\prime}, \omega, \mu\right)\right)=\int_{A \times \hat{\Theta} \times \Delta(\Omega)} \sum_{\theta \in \Theta} \frac{f(\theta) \sigma^{\prime}\left(\theta^{\prime} \mid \theta, \mu\right)}{f\left(\theta^{\prime}\right)}\left(\sum_{\omega \in \Omega} \mu(\omega) u(a, \theta, \omega)\right) d \bar{v}_{\sigma^{\prime}, A \hat{\Theta} \Delta}
$$
On the other hand, because $T_{n_{m}}$ is a subsequence of $T_{n}$ and $\bar{v}_{\sigma^{\prime}, A \hat{\Theta} \Omega \Delta}^{T, 1}=\bar{v}_{\sigma, A \Theta \Omega \Delta}^{T, 1}$ and $\bar{v}_{\sigma, A \Theta \Omega \Delta}^{T_{n}, 1} \xrightarrow{w^{*}} \bar{v}_{\sigma}$, we can conclude that $\bar{v}_{\sigma^{\prime}, A \hat{\Theta} \Delta}=\bar{v}_{\sigma, A \Theta \Delta}$ and admits the same decomposition as $\bar{v}_{\sigma}$. We conclude that
$$
\mathbb{E}_{\bar{v}_{\sigma^{\prime}}}[u(a, \theta, \omega)]=\int_{\Delta(\Omega)}\left[\sum_{\theta, \theta^{\prime} \in \Theta} f(\theta) \sigma^{\prime}\left(\theta^{\prime} \mid \theta, \mu\right) \sum_{a} \alpha\left(a \mid \theta^{\prime}, \mu\right)\left(\sum_{\omega \in \Omega} \mu(\omega) u(a, \theta, \omega)\right)\right] d \tau_{(p, \sigma)}
$$
Consequently,
$$
\begin{aligned}
\lim \sup _{T \rightarrow \infty} \mathbb{E}_{\sigma^{\prime}}\left[U_{T}\right] \geq \lim _{m \rightarrow \infty} \mathbb{E}_{\sigma^{\prime}}\left[U_{T_{n_{m}}}\right] & =\mathbb{E}_{\bar{v}_{\sigma^{\prime}}}[u(a, \theta, \omega)] \\
& =\mathbb{E}_{\tau_{(p, \sigma)}}\left[\sum_{\theta, \theta^{\prime}, a} f(\theta) \sigma^{\prime}(\theta, \mu)\left(\theta^{\prime}\right) \alpha\left(a \mid \theta^{\prime}, \mu\right) u(a, \theta, \mu)\right],
\end{aligned}
$$
where $u(a, \theta, \mu)$ is the linear extension of $u(a, \theta, \cdot)$. Because $\sigma$ is a best response, we have that
$$
\mathbb{E}_{\tau_{(p, \sigma)}}\left[\sum_{\theta, a} f(\theta) \alpha(a \mid \theta, \mu) u(a, \theta, \mu)\right] \geq \mathbb{E}_{\tau_{(p, \sigma)}}\left[\sum_{\theta, \theta^{\prime}, a} f(\theta) \sigma^{\prime}(\theta, \mu)\left(\theta^{\prime}\right) \alpha\left(a \mid \theta^{\prime}, \mu\right) u(a, \theta, \mu)\right],
$$
which implies the two-stage mechanism lacks profitable undetectable deviations.

The allocation rule is ex ante individually rational Define

$$
U_{\mathrm{net}}(\mu)=\sum_{\theta \in \Theta} f(\theta) \sum_{\omega \in \Omega} \mu(\omega)\left[\sum_{a \in A} \alpha(a \mid \theta, \mu) u(a, \theta, \omega)-u\left(a_{\varnothing}, \theta, \omega\right)\right],
$$

to be the agent's (ex ante) payoff net of the outside option at belief $\mu$. Ex ante individual rationality of $\alpha$ is equivalent to $U_{\text {net }}(\mu) \geq 0$ for all $\mu$ in the support of $\tau_{(p, \sigma)}$.

Toward a contradiction, assume that $U_{\text {net }}(\mu)<0$ with positive probability under $\tau_{(p, \sigma)}$. By Lemma D. 1

[^30]in Appendix D, a set $B \subset \Delta(\Omega)$ open relative to $\Delta(\Omega)$ exists such that ${ }^{46}$
$$
\int_{B} U_{\text {net }}(\mu) \tau_{(p, \sigma)}(d \mu)<0
$$
Moreover, we can pick $B$ such that $\tau_{(p, \sigma)}(\partial B)=0$, where $\partial B$ denotes the boundary of $B$ relative to $\Delta(\Omega) .{ }^{47}$ Lastly, let $\delta>0$ be such that
$$
\int_{B} U_{\text {net }}(\mu) \tau_{(p, \sigma)}(d \mu) \leq-2 \delta
$$
Let $\left(\mu_{t}\left(h^{t}\right)\right)_{t \in \mathbb{N}, h^{t} \in H^{t}}$ denote the belief process under $(p, \sigma)$. For $L \in \mathbb{N}$, define a strategy $\left(p^{L}, \sigma^{L}\right)$ as follows:

1. $\left(p_{t}^{L}\left(h^{t}, \cdot\right), \sigma_{t}^{L}\left(h^{t}, \cdot\right)\right)=\left(p_{t}\left(h^{t}, \cdot\right), \sigma_{t}\left(h^{t}, \cdot\right)\right)$ if either $t<L$ OR $\left(t \geq L\right.$ and $\left.\mu_{L}\left(h^{L}\right) \notin B\right)$, where $h^{L}$ precedes $h^{t}$,
2. Otherwise, $\left(p_{t}^{L}\left(h^{t}, \cdot\right), \sigma_{t}^{L}\left(h^{t}, \cdot\right)\right)=\left(0, \sigma_{t}\left(h^{t}, \cdot\right)\right)$ (note that when the agent quits the strategy can be specified arbitrarily.)


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

## 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.