Calibrated Mechanism Design

[^20]in the next section. For simplicity, let $\tilde{\Omega}=\Omega \times \mathcal{E}$ with elements $\tilde{\omega}$. Suppose repeated mechanism $\phi$ implements $\vartheta$, and let $v_{\sigma} \in \Delta(A \times \Theta \times M \times \tilde{\Omega})$ denote the induced occupation measure. As the analysis so far illustrates, tracking the joint distribution over allocations, types, states, and beliefs is important to show that $\vartheta$ can be implemented via a two-stage mechanism. To this end, we extend the up to period $T$ occupation measures, $v_{\sigma}^{T} \in \Delta(A \times \Theta \times M \times \tilde{\Omega})$, to account for the frequency of beliefs through period $T$. In fact, we define two sequences of extended occupation measures over $A \times \Theta \times M \times \tilde{\Omega} \times \Delta(\tilde{\Omega})$ : the first, $\bar{v}_{\sigma}^{T, 1}$, calculates the frequency of a tuple $(a, \theta, m, \tilde{\omega}, \mu)$ by counting the beliefs at the beginning of period $t$ and the second, $\bar{v}_{\sigma}^{T, 2}$, by counting the beliefs at the end of period $t$. Whereas the martingale property of beliefs implies these two sequences have the same (subsequential) limits, they have different conditional independence properties, which we use to derive the representation of $\vartheta$ via a two-stage mechanism. Suppose for simplicity that $\bar{v}_{\sigma}^{T, 1}$ (and hence, $\bar{v}_{\sigma}^{T, 2}$ ), have limit $\bar{v}_{\sigma}$, though this assumption is not needed for the proof. ${ }^{34}$ A consequence of the martingale property of beliefs is that only the long-run beliefs of the agent are in the support of $\bar{v}_{\sigma}$.

The proof consists of three steps. In the first step, we show that $\bar{v}_{\sigma}$ admits the following decomposition:

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

where (i) $\tau \in \Delta(\Delta(\tilde{\Omega}))$ has mean $\mu_{0} \otimes \eta$, where recall $\eta$ is the measure on $\mathcal{E}$, and (ii) $\rho: \Theta \times \Delta(\tilde{\Omega}) \rightarrow$ $\Delta(M)$ is a "Markovian reporting strategy", and (iii) $\alpha^{\prime}$ is almost the allocation rule in the two-stage mechanism, and hence the prime notation. Moreover, on the support of $\mu, \alpha^{\prime}(\cdot \mid \cdot, \mu)$ coincides with $\phi(\cdot \mid \cdot, \tilde{\omega})$, implying that $\phi(\cdot \mid \cdot, \tilde{\omega})$ is constant in $\tilde{\omega}$ on the support of $\mu$. This is the step which exploits the different conditional independence properties of $\bar{v}_{\sigma}^{T, 1}$ and $\bar{v}_{\sigma}^{T, 2}$. We use $\bar{v}_{\sigma}^{T, 1}$ to show the conditional independence of types and beliefs-all agent types in period $t$ have the same belief at the beginning of period $t$-and $\bar{v}_{\sigma}^{T, 2}$ to show the conditional independence of the allocation and the state-the belief at the end of period $t$ contains all the information about the state contained in the allocation.

In the second step, we show that $\rho: \Theta \times \Delta(\tilde{\Omega}) \rightarrow \Delta(M)$ is indeed a best response for the agent when her type is $\theta$ and her belief is $\mu$. In other words, the support of $\rho(\cdot \mid \theta, \mu)$ is contained in

$$
\arg \max _{m \in M} \sum_{\tilde{\omega} \in \tilde{\Omega}} \mu(\tilde{\omega}) \sum_{a \in A} \phi(a \mid m, \tilde{\omega}) u(a, \theta, \omega),
$$

for beliefs on the support of $\tau$. Hence, we can use $\rho$ and $\phi$ to define a direct mechanism $\alpha(\cdot \mid \cdot, \mu): \Theta \rightarrow$ $\Delta(A)$ that satisfies the agent's participation and incentive constraints when her belief is $\mu$. Together, Steps 1 and 2 allow us to obtain the representation of $\vartheta$ as in Equation $15 .^{35}$

Whereas the above steps are enough to show that $\vartheta$ is implementable by some individually rational and incentive compatible two-stage mechanism, they do not necessarily imply that the distribution over posteriors $\tau$ is the one induced by the information structure calibrated to $\phi, \pi_{\phi}$. The last step of the proof shows that even if this is not the case, the agent adequately learns the information contained in $\pi_{\phi}$ in the sense of Aghion et al. (1991). Indeed, Lemma C. 4 shows a strategy exists that approximately

[^21]delivers the payoff from learning $\pi_{\phi}$, so that the agent's payoff under $\sigma$ is at least the payoff she would obtain if she had learned $\pi_{\phi}$. Because the payoff from learning $\pi_{\phi}$ is the maximal payoff the agent can possibly attain, we conclude that the payoff under $\sigma$ is the payoff the agent would attain when facing the calibrated information structure $\pi_{\phi}$ (and best responding to it).


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.

### 5.2 Dynamic Mechanisms

In this section, we consider the case in which the designer can offer the agent a dynamic mechanism, that is, one that conditions the allocation in each period on the history of past allocations and reports. The analysis herein allows us to describe the limits implied by calibration on the set of implementable outcomes.

Dynamic mechanisms A dynamic mechanism $\varphi=\left(\varphi_{t}\right)_{t \in \mathbb{N}}$ is a sequence of mappings that condition on the state, the history of participation decisions, type reports and allocations, and today's report, and output an allocation. Formally, expand the set of type reports and allocations by a non-participation message and the outside option, which we denote by $\Theta A_{\varnothing}=\Theta \times A \cup\left\{\left(\varnothing, a_{\varnothing}\right)\right\} .{ }^{36}$ For each $t \in \mathbb{N}$, define the mechanism in period $t, \varphi_{t}: \Omega \times\left(\Theta A_{\varnothing}\right)^{t-1} \times \Theta \rightarrow \Delta(A)$. Because the designer can flexibly design the mechanism in each period we no longer rely on the randomization device.

A dynamic mechanism induces an extensive-form game for the agent, in which in each period, the agent decides whether to participate, and conditional on participation what type to report. Whenever the agent chooses not to participate, she obtains her outside option $a_{\varnothing}$. We denote by $p$ the agent's participation strategy and by $\sigma$ the agent's reporting strategy.

Implementation Our notion of implementation continues to be based on the occupation measure over the set of allocations, types, participation decisions, type reports, and states, induced by the distributions $\mu_{0}$ and $f$, the mechanism $\varphi$, and the agent's participation and reporting strategy. However, as we show in Appendix D.3.1, it is without loss to focus on mechanisms such that (i) participation with probability 1 and truthtelling is a best response for the agent, and (ii) the mechanism implements the outside option with probability 1 in all future periods following a non-participation decision by the agent. ${ }^{37}$ Thus, we focus on dynamic mechanisms $\varphi$ such that (i) a best response exists, and (ii) the occupation measure over $A \times \Theta \times \Omega$ is well-defined.

Incentives in dynamic mechanisms Dynamic mechanisms allow the designer to condition the agent's allocation on the history of past participation decisions and reports (and allocations), and hence allow the designer to implement outcomes that satisfy weaker notions of truthtelling and participation, which we explain next.

Because the designer can condition the mechanism on the history of past reports, he can compare the frequency of type reports against the type distribution. So long as the agent is telling the truth, the

[^22]frequency of reports will match the type distribution $f$ over large blocks of time. In fact, any reporting strategy whose expected frequency of reports matches the type distribution will be indistinguishable from truthtelling.

Definition 6 (Undetectable deviations). An undetectable deviation is a reporting strategy $\sigma: \Theta \rightarrow \Delta(\Theta)$ such that for all $\theta^{\prime} \in \Theta$

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

By tracking the empirical distribution of type reports, the designer can dissuade the agent from employing detectable deviations. Thus, in a dynamic mechanism, the designer should be concerned with only discouraging undetectable deviations. This leads to a weaker notion of incentive compatibility for allocation rules:

Definition 7 (Unprofitable undetectable deviations). The allocation rule $\alpha: \Theta \times \Delta(\Omega) \rightarrow \Delta(A)$ lacks profitable undetectable deviations at belief $\mu \in \Delta(\Omega)$ if for all undetectable deviations $\sigma$,

$$
\sum_{\theta \in \Theta} f(\theta) \sum_{a \in A} \alpha(a \mid \theta, \mu) \sum_{\omega \in \Omega} \mu(\omega) u(a, \theta, \omega) \geq \sum_{\theta \in \Theta} f(\theta) \sum_{\theta^{\prime} \in \Theta} \sigma\left(\theta^{\prime} \mid \theta\right) \sum_{a \in A} \alpha\left(a \mid \theta^{\prime}, \mu\right) \sum_{\omega \in \Omega} \mu(\omega) u(a, \theta, \omega) .
$$

A two-stage mechanism $\psi$ with allocation rule $\alpha$ lacks profitable undetectable deviations if $\alpha(\cdot \mid \cdot, \mu)$ lacks profitable undetectable deviations for all beliefs in the support of the mechanism.

To illustrate the difference between the lack of profitable undetectable deviations and incentive compatibility, consider the following example from Ball and Kattwinkel (2023). Suppose the agent types are binary, $\left\{\theta_{1}, \theta_{2}\right\}$, and equally likely. The set of allocations, $q \in\{0,1\}$, describes whether the agent receives a good. Finally, suppose the agent's payoff is $u(q, \theta)=q \theta$ and $\theta_{1}<\theta_{2}$. Consider the mechanism that allocates the good to $\theta_{2}$ : While it is not incentive compatible, it lacks profitable undetectable deviations. The constraint that the deviation must be undetectable implies the gains from $\theta_{1}$ obtaining the good come at the expense of $\theta_{2}$ getting the good.

Consider now the agent's participation incentives in the dynamic mechanism: once the agent rejects the mechanism once, the agent obtains her outside option in all continuation histories independent of her participation decision and her types. In other words, whereas the agent can always ensure her outside option by rejecting the mechanism in a given period, she is effectively quitting the mechanism forever for all her types. The following definition introduces the notion of individual rationality satisfied by the mechanism in the long run.

Definition 8 (Ex ante individual rationality). The allocation rule $\alpha: \Theta \times \Delta(\Omega) \rightarrow \Delta(A)$ is ex ante individually rational at belief $\mu \in \Delta(\Omega)$ if

$$
\sum_{\theta \in \Theta} f(\theta) \sum_{a \in A} \alpha(a \mid \theta, \mu) \sum_{\omega \in \Omega} \mu(\omega) u(a, \theta, \omega) \geq \sum_{\theta \in \Theta} f(\theta) \sum_{\omega \in \Omega} \mu(\omega) u\left(a_{\varnothing}, \theta, \omega\right) .
$$


Calibrated Mechanism Design

A two-stage mechanism $\psi$ with allocation rule $\alpha$ is ex ante individually rational if $\alpha(\cdot \mid \cdot, \mu)$ is ex ante individually rational for all beliefs in the support of the mechanism.

We are now ready to state the main result of this section:
Theorem 4 (Implementable Outcomes via Dynamic Mechanisms). A dynamic mechanism exists that implements outcome $\vartheta \in \Delta(A \times \Theta \times \Omega)$ if and only if $\vartheta$ can be implemented by an ex ante individually rational two-stage mechanism which lacks profitable undetectable deviations. That is, if and only if for all $(a, \theta, \omega) \in A \times \Theta \times \Omega$

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

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

Theorem 4 characterizes the outcome distributions implementable by dynamic mechanisms as those implemented by two-stage mechanisms that satisfy the incentive constraints: unprofitability of undetectable deviations and ex ante individual rationality. Notably, both notions of incentive constraints apply in the aggregate over the type distribution, which reflects the transient nature of the agent's private information.

Comparing Theorem 3 and Theorem 4, we see that dynamic mechanisms allow the designer to weaken the incentive constraints of the agent, but do not allow him to engage in richer-i.e., typedependent-disclosures. Despite dynamic mechanisms implying weaker incentive constraints, we can build on the results of Rochet (1987) and Rahman (2024) to show that in settings with transferable utility, where $a=(q, t)$, dynamic and repeated mechanisms implement the same set of physical allocations $q: \Theta \times \Omega \rightarrow \mathbb{R} .^{38}$ Indeed, Rahman (2024) shows the lack of profitable undetectable deviations is equivalent to cyclical monotonicity in Rochet (1987). Thus, in settings with transferable utility, Theorems 3 and 4 imply that dynamic mechanisms do not allow the designer to expand on the set of implementable distributions over $(q, \theta, \omega)$.

The proof of the only if direction is similar to that of Theorem 3, in that we similarly extend the occupation measure to account for the agent's beliefs and show it satisfies the conditional independence properties implied by a two-stage mechanism. In a dynamic mechanism, however, the agent can ensure the payoff of some, but not all deviations. The latter property is what delivers that the two-stage mechanism must lack profitable undetectable deviations.

The proof of the if direction, instead, harnesses a construction in Margaria and Smolin (2018). The proof proceeds in two steps. In the first step, we analyze a fictitious model without state uncertainty in which a designer faces a privately informed agent, so that implementable outcomes are elements of $\Delta(A \times \Theta)$. We show that if $\vartheta^{\prime}(a, \theta)=f(\theta) \alpha^{\prime}(a \mid \theta) \in \Delta(A \times \Theta)$ is such that $\alpha^{\prime}$ lacks profitable undetectable deviations and is ex ante individually rational, then a dynamic mechanism exists that implements $\vartheta^{\prime} .^{39}$ This is the step that relies on Margaria and Smolin (2018). We construct a dynamic mechanism, which can be split into blocks of random length. Each block consists of two phases: a reporting

[^23]phase and an adjustment phase. In the reporting phase, the mechanism uses the agent's reports to determine the allocation. Instead, in the adjustment phase, the mechanism simulates type reports so that the frequency of type reports matches the type distribution (in expectation) over the length of the block, whenever this is not the case at the end of the reporting phase. These two steps ensure that the expected frequency of type reports and allocations matches $\vartheta^{\prime}$. We then leverage that $\alpha^{\prime}(\cdot \mid \theta)$ lacks profitable undetectable deviations to show the agent cannot do better than by telling the truth. Hence, the induced frequency of types and allocations also matches $\vartheta^{\prime}$. Moreover, the construction ensures that after any history, truthtelling delivers a continuation payoff equal to the ex ante payoff. Because $\alpha^{\prime}$ is ex ante individually rational, we conclude the participation constraints are satisfied.

The second step uses the above result and the representation of the outcome distribution via a twostage mechanism to construct a dynamic mechanism that implements any outcome distribution that satisfies the properties in Theorem 4. Indeed, one can construct a dynamic mechanism which uses a finite number of steps to disclose information to the agent via the realized allocations, ${ }^{40}$ and then continues as in the above construction to implement the allocation rule $\alpha(\cdot \mid \cdot, \mu)$.


Calibrated Mechanism Design

## 6 Conclusions

Many economic institutions-online platforms, lenders, regulators-rely on mechanisms that remain fixed while agents interact with them repeatedly. When the mechanism's operation depends on a state known only to the designer, agents can learn this state from their outcomes, constraining what the mechanism can implement. We introduce calibrated mechanism design, a static solution concept that requires mechanisms to remain incentive compatible given the information they endogenously reveal about the designer's private state through repeated use. In private value environments, the calibration constraint pushes the designer toward full transparency, precluding Crémer-McLean-style schemes under transferable utility. In single agent-settings, calibrated mechanisms are equivalent to two-stage mechanisms. This equivalence yields a practical algorithm for finding optimal calibrated mechanisms, combining tools from information design and mechanism design. We provide a microfoundation by showing calibrated mechanisms characterize exactly what is implementable when an infinitely patient agent repeatedly interacts with the same mechanism, and study the implications on implementable outcomes of allowing the designer to offer fully dynamic mechanisms.

The most important direction for future work is deepening the analysis of multi-agent settings. On the one hand, understanding when generalized two-stage mechanisms coincide with calibrated mechanisms would enable the study of multi-agent applications, while abstracting from the dynamics of experimentation. On the other hand, extending our microfoundation to the multi-agent case would further ground calibrated mechanism design. More broadly, our framework suggests that any institution whose repeated operation leaks information about its designer's knowledge faces a fundamental tradeoff between conditioning the mechanism on this information and the information this leaks to participants, and calibrated mechanism design offers a disciplined way to analyze it.

[^24]


Calibrated Mechanism Design

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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

## 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.)