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The Core of Bayesian Persuasion
Given a primitive payoff structure, information design provides a framework for rationalizing outcomes as the result of non-cooperative play without having to specify the players' information structure. For this reason, the seminal work of bergemann2016bayes has spurred renewed interest among empirical scholars wishing to obtain identification and estimation results under a weaker set of information assumptions (see, for instance, syrgkanis2017inference,magnolfi2019estimation,koh2022stable).
However, the weaker set of assumptions on the information structure comes at the cost of increasing demands on the data set available to the analyst. Indeed, the usual assumption in the literature is that the analyst is given a joint distribution over payoff relevant states and action profiles. For instance, the literatures on rational inattention and stochastic choice usually assume that the analyst observes an agent's choices conditional on the state of the world (e.g., caplin2015revealed,aguiar2018does). Given this data set, Bayes correlated equilibrium provides an easy to test set of conditions that the joint distribution over states and action profiles must satisfy in order to be consistent with the outcome of non-cooperative play under some information structure.
Oftentimes, however, the analyst's data set is more limited. The analyst may observe the distribution over the payoff relevant states of the world and the distribution over action profiles, but not the distribution over action profiles conditional on the state of the world.\footnote{Whereas state-dependent stochastic choice data is useful to guide the design and interpretation of experiments, this data is oftentimes hard to come by outside the experimental setting. dardanoni2020inferring provides an eloquent discussion of the data voracity of stochastic choice.} We can then ask, given the primitive payoff structure, which marginal distributions can be rationalized as the outcome of non-cooperative play under some information structure. We refer to such marginals as BCE consistent because they satisfy that a joint distribution over states and action profiles exists that is consistent with the marginals and is a Bayes correlated equilibrium. Characterizing the set of BCE-consistent marginal distributions can only increase the practical applicability of Bayes correlated equilibrium.
The set of BCE-consistent marginal distributions is of interest for two other reasons. First, the analyst oftentimes is not just interested in the existence of an information structure that rationalizes the (marginal) distribution of play, but one that satisfies certain properties. For instance, the analyst may want to test whether the agents have private information. As we explain below, our characterization result provides us with a test for the existence of a public information structure that rationalizes the observed distribution of play. The second reason is related to reduced-form implementation in mechanism design matthews1984implementability,border1991implementation. Whenever the information designer only cares about the agents' action profiles, but not the state of the world, the information designer's problem can be expressed as the choice out of the set of BCE-consistent marginals.
In this paper, we take the first step towards characterizing the set of BCE-consistent marginals by considering the single-agent case. (ref) provides a characterization of the set of BCE-consistent marginals building on a theorem in strassen1965existence. Furthermore, marrying the obedience approach in information design with the belief approach in kamenica2011bayesian, (ref) characterizes the Bayes plausible distributions over posteriors that implement a given marginal over actions. We provide two network-based proofs of (ref). Relying on recent extensions of Hall's marriage theorem in barseghyan2021heterogeneous and azrieli2022marginal, the first characterization uncovers a connection between BCE-consistency and the core of the game induced by loosely speaking, some (Bayes plausible) posterior distribution (see (ref) and grabisch2016set). The second proof relies on the demand problem of gale1957theorem. We show that one can interpret BCE-consistency problem as a supply-demand problem in a persuasion economy, in which the marginal action distribution describes the demand and a Bayes plausible posterior distribution describes the supply. We then rely on the results in gale1957theorem to determine when the demand is feasible given the supply.
(ref) illustrates how (ref) already allows us to study multi-agent games. (ref) applies (ref) to the first-order Bayesian persuasion setting of arieli2021feasible to characterize the subset of BCE-consistent marginals that are consistent with a public information structure. Instead, (ref) applies (ref) to characterize BCE-consistent marginals in ring-network games as in kneeland2015identifying.
\paragraph{Related literature} The two closest papers to ours are rehbeck2023revealed and azrieli2022marginal. rehbeck2023revealed studies the same question as us, but when the analyst has access to a decision maker's unconditional stochastic choices, possibly out of different menus. For the case of a single menu, the characterization in rehbeck2023revealed is different from that in (ref) and is stated in terms of the non-existence of a possibly mixed deviation. azrieli2022marginal study a similar question to ours in the context of stochastic choice. In their setting, the analyst has access to a marginal distribution over a decision maker's choices and a marginal distribution over the menus out of which the decision maker made her choices. azrieli2022marginal show that the marginal distributions are consistent if and only if the marginal over choices is in the core of the game induced by the marginal over menus.
A literature in decision theory and experimental economics studies when choices can be rationalized via costly information acquisition and whether the choices can be used to identify the information acquisition costs (see, e.g., caplin2015revealed, caplin2017rationally, chambers2020costly, dewan2020estimating, denti2022posterior). Like we do, many of these papers assume that the decision maker's utility is known. More recently, assuming that the analyst has access to state-dependent stochastic choice data, caplin2023rationalizable study when choices can be rationalized as if the agent has access to some information before choosing her actions. Whereas their analyst has access to a richer data set, they require consistency of the information structure across a family of decision problems.
arieli2021feasible and morris2020no characterize joint distributions over posterior beliefs that are consistent with some information structure.\footnote{Whereas arieli2021feasible study the binary-state case, the characterization in morris2020no requires no such assumption.} Both papers cast the problem as one of distributions with given marginals: they take as given a profile of marginal distributions over posterior beliefs with the same mean and characterize when a joint distribution with the given marginals exists that is consistent with information.
Finally, toikka2022bayesian study reduced-form implementation in a Bayesian persuasion in which the sender and the receiver care only about the posterior mean of the states. They leverage the mean preserving spread property to write a linear programming problem for the sender that only depends on the marginal distribution over actions. Beyond the posterior mean setting, they do not provide a characterization of the set of implementable marginal action distributions.
Anticipating our multi-agent results in (ref), our notation below presumes multiple agents. We then specialize it to the single-agent case in (ref):
\paragraph{Base game:} An incomplete information base game, \ensuremath{G}, is defined as follows. We are given a set of \ensuremath{N}\ players, \ensuremath{[N]}=$\{1,\dots,\ensuremath{N}\}$. Each player $\ensuremath{i}\in\ensuremath{[N]}$ chooses an action from the finite set $\ensuremath{\ensuremath{A}_\ensuremath{i}}$. Payoffs $\ensuremath{\ensuremath{u}_\ensuremath{i}}(\ensuremath{a},\ensuremath{\theta})$ depend on the action profiles $\ensuremath{a}\in\ensuremath{A}\equiv\times_{\ensuremath{i}\in\ensuremath{[N]}}\ensuremath{\ensuremath{A}_\ensuremath{i}}$ and the state of the world, \ensuremath{\theta}, an element of the finite set \ensuremath{\Theta}.\footnote{As we explain in (ref) our single-agent characterization extends to the case in which \ensuremath{\Theta}\ and \ensuremath{A}\ are infinite (see (ref)). However, the set of finitely many states and actions allows us to provide a sharper characterization.} The players share a common prior $\ensuremath{\ensuremath{\mu}_0}\in\Delta(\ensuremath{\Theta})$ over the state of the world. That is, $\ensuremath{G}=\langle\ensuremath{\Theta},(\ensuremath{\ensuremath{A}_\ensuremath{i}},\ensuremath{\ensuremath{u}_\ensuremath{i}})_{\ensuremath{i}\in\ensuremath{[N]}},\ensuremath{\ensuremath{\mu}_0}\rangle$.
\paragraph{Bayes correlated equilibrium:} An outcome is a joint distribution over action profiles and states of the world, $\ensuremath{\pi}\in\Delta(\ensuremath{A}\times\ensuremath{\Theta})$. We are concerned with those outcomes that are consistent with non-cooperative play of the base game, where the solution concept is Bayes Nash equilibrium. The notion of Bayes correlated equilibrium in bergemann2016bayes captures the set of outcomes that are consistent with (Bayes Nash) equilibrium of the base game under some information structure:
In words, a Bayes correlated equilibrium is an outcome distribution that satisfies a series of obedience constraints (ref) and a martingale condition (ref). The first ensures each player's best response condition under some information structure, whereas the second ensures the existence of an information structure that is consistent with the players' prior information. Note that any Bayes correlated equilibrium $\ensuremath{\pi}\in\Delta(\ensuremath{A}\times\ensuremath{\Theta})$ induces two marginal distributions, $(\ensuremath{\pi}_\ensuremath{\Theta},\ensuremath{\pi}_\ensuremath{A})\in\ensuremath{\Delta(\ensuremath{\Theta})}\times\Delta(\ensuremath{A})$. The definition of Bayes correlated equilibrium implies that the primitive base game \ensuremath{G}\ pins down $\ensuremath{\pi}_\ensuremath{\Theta}$, but not necessarily $\ensuremath{\pi}_\ensuremath{A}$.
\paragraph{Information Design with Given Marginals:} We take the point of view of an analyst who knows the base game, but not the information structure under which the base game is played. The analyst is also endowed with information about the actions taken by the players. The analyst's goal is to determine whether this information is consistent with non-cooperative play of the base game under some information structure.
We consider two kinds of information the analyst may have about the players' actions, which are equivalent in the single-agent setting. In the first case, the analyst is endowed with a distribution over action profiles, $\ensuremath{\ensuremath{\nu}_0}\in\Delta(\ensuremath{A})$. In the second case, the analyst is endowed with a profile of action distributions, one for each player, that is, $\ensuremath{\overline{\ensuremath{\nu}}_0}=(\ensuremath{\nu}_{0,1},\dots,\ensuremath{\nu}_{0,N})\in\times_{\ensuremath{i}\in\ensuremath{[N]}}\Delta(\ensuremath{\ensuremath{A}_\ensuremath{i}})$.
In each of these cases, the analyst wants to ascertain whether a Bayes correlated equilibrium $\ensuremath{\pi}\in\ensuremath{\ensuremath{\mathrm{BCE}}\left(\ensuremath{\ensuremath{\mu}_0}\right)}$ exists such that $\ensuremath{\pi}_\ensuremath{A}$ coincides with the analyst's information about the players' actions (i.e., $\ensuremath{\pi}_\ensuremath{A}=\ensuremath{\ensuremath{\nu}_0}$ or $\times_{\ensuremath{i}\in\ensuremath{[N]}}\ensuremath{\pi}_{\ensuremath{\ensuremath{A}_\ensuremath{i}}}=\ensuremath{\overline{\ensuremath{\nu}}_0}$). In this case, we say that the marginals $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ are BCE-consistent or that the profile of marginal distributions $(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\overline{\ensuremath{\nu}}_0})$ are M-BCE-consistent. (ref) records this for future reference:
Note that if $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ are BCE-consistent, then letting $\ensuremath{\nu}_{0,\ensuremath{i}}$ denote the marginal of \ensuremath{\ensuremath{\nu}_0}\ over $\ensuremath{\ensuremath{A}_\ensuremath{i}}$, we have that $(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\nu}_{0,1},\dots,\ensuremath{\nu}_{0,\ensuremath{N}})$ are M-BCE-consistent.
\paragraph{Constrained Optimal Transport} We close this section by noting a connection with optimal transport. Given $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$, let $\ensuremath{\Pi}\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ denote the set of joint distributions $\ensuremath{\pi}\in\Delta(\ensuremath{A}\times\ensuremath{\Theta})$ withmarginals $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$, i.e., $(\ensuremath{\pi}_\ensuremath{\Theta},\ensuremath{\pi}_\ensuremath{A})=\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$. Note that $\ensuremath{\Pi}\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ is always nonempty, e.g., the joint distribution $\ensuremath{\pi}(\ensuremath{a},\ensuremath{\theta})=\ensuremath{\ensuremath{\nu}_0}(\ensuremath{a})\ensuremath{\ensuremath{\mu}_0}(\ensuremath{\theta})$ satisfies the marginal constraints. Instead, the subset $\ensuremath{\Pi}_\mathrm{O}\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ of $\ensuremath{\Pi}\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ that satisfies the obedience constraints (ref) may be empty. Thus, the characterization of the set of BCE-consistent marginals $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ is equivalent to the characterization of when the feasible set of a constrained optimal transport problem--in this case $\ensuremath{\Pi}_\mathrm{O}\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$--is nonempty.\footnote{In their study of credible Bayesian persuasion, lin2022credible characterize the set of credible outcome distributions by noting a connection with optimal transport. In their case, to check whether a given message distribution $\lambda_M$ is implementable, it must be that no other joint distribution over states and messages that respects the given marginals exists and is preferred by the sender to $\lambda_M$.}
In this section we characterize the set of BCE-consistent marginals in the case of a single agent, that is, $\ensuremath{N}=1$. For this reason, in what follows we remove the index $i=1$ from the action set and the utility function.
\paragraph{Distributions over posteriors and stochastic choice} An outcome distribution $\ensuremath{\pi}\in\Delta(\ensuremath{A}\times\ensuremath{\Theta})$ with marginals $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ induces two conditional probability systems: The first, $\{\ensuremath{\mu}(\cdot|\ensuremath{a})\in\ensuremath{\Delta(\ensuremath{\Theta})}:\ensuremath{a}\in\ensuremath{A}\}$, describes the agent's beliefs conditional on action \ensuremath{a}\ and satisfies for all actions $\ensuremath{a}\in\ensuremath{A}$, \[\ensuremath{\ensuremath{\nu}_0}(\ensuremath{a})\ensuremath{\mu}(\ensuremath{\theta}|\ensuremath{a})=\ensuremath{\pi}(\ensuremath{a},\ensuremath{\theta}).\] In this case, one can view \ensuremath{\ensuremath{\nu}_0}\ as a distribution over posteriors and the belief system $\left(\ensuremath{\mu}(\cdot|\ensuremath{a})\right)_{\ensuremath{a}\in\ensuremath{A}}$ as its support.
The second, $\{\ensuremath{\sigma}(\cdot|\ensuremath{\theta})\in\Delta(\ensuremath{A}):\ensuremath{\theta}\in\ensuremath{\Theta}\}$, describes the agent's actions conditional on state \ensuremath{\theta}\ and satisfies for all states $\ensuremath{\theta}\in\ensuremath{\Theta}$, \[\ensuremath{\ensuremath{\mu}_0}(\ensuremath{\theta})\ensuremath{\sigma}(\ensuremath{a}|\ensuremath{\theta})=\ensuremath{\pi}(\ensuremath{a},\ensuremath{\theta}).\] The collection $\{\ensuremath{\sigma}(\cdot|\ensuremath{\theta}):\ensuremath{\theta}\in\ensuremath{\Theta}\}$ is what the stochastic choice literature dubs the agent's stochastic choice rule.
The analysis that follows characterizes the set of BCE-consistent marginals relying on the belief system, $\{\ensuremath{\mu}(\cdot|\ensuremath{a})\in\ensuremath{\Delta(\ensuremath{\Theta})}:\ensuremath{a}\in\ensuremath{A}\}$. Instead, the stochastic choice rule $\ensuremath{\sigma}(\cdot|\ensuremath{\theta})$ is the focus of the analysis in (ref).
\paragraph{The action marginal as a distribution over posteriors} Given marginals $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$, the goal is to determine whether a belief system $\{\ensuremath{\mu}(\cdot|\ensuremath{a}):\ensuremath{a}\in\ensuremath{A}\}$ exists that satisfies for all states $\ensuremath{\theta}\in\ensuremath{\Theta}$
and for all $\ensuremath{a},\ensuremath{\ensuremath{a}^\prime}\in\ensuremath{A}$,
For an action \ensuremath{a}, let $\ensuremath{\ensuremath{\ensuremath{\Delta}^*}(\ensuremath{a})}$ denote the set of beliefs under which \ensuremath{a}\ is optimal.\footnote{Formally, $\ensuremath{\ensuremath{\ensuremath{\Delta}^*}(\ensuremath{a})}=\{\ensuremath{\mu}\in\ensuremath{\Delta(\ensuremath{\Theta})}:(\forall\ensuremath{\ensuremath{a}^\prime}\in\ensuremath{A})\sum_{\ensuremath{\theta}\in\ensuremath{\Theta}}\ensuremath{\mu}(\ensuremath{\theta})\left(\ensuremath{u}(\ensuremath{a},\ensuremath{\theta})-\ensuremath{u}(\ensuremath{\ensuremath{a}^\prime},\ensuremath{\theta})\right)\geq0\}$.} Then, Equations (ref) and (ref) require that (i) \ensuremath{\ensuremath{\nu}_0}\ induces a Bayes plausible distribution over posteriors and (ii) for all actions \ensuremath{a}, the posterior belief $\ensuremath{\mu}(\cdot|\ensuremath{a})$ is an element of $\ensuremath{\ensuremath{\ensuremath{\Delta}^*}(\ensuremath{a})}$. Under this interpretation, the action distribution \ensuremath{\ensuremath{\nu}_0}\ describes the frequency with which inducing beliefs in $\ensuremath{\ensuremath{\ensuremath{\Delta}^*}(\ensuremath{a})}$ is necessary. Unsurprisingly, some of the conditions in (ref) below also check that \ensuremath{\ensuremath{\nu}_0}\ satisfies a version of the martingale condition aumann1995repeated,kamenica2011bayesian.
(ref) characterizes the set of BCE-consistent marginals:
The proof is in (ref). In what follows, we provide intuition for the statement in (ref) and review the main steps of its proof.
(ref) can be interpreted through the lens of the martingale property of beliefs. As discussed before (ref), the action distribution \ensuremath{\ensuremath{\nu}_0}\ describes the frequency with which beliefs in $\ensuremath{\ensuremath{\ensuremath{\Delta}^*}(\ensuremath{a})}$ must be induced to satisfy (ref). For a given state $\ensuremath{\theta}\in\ensuremath{\Theta}$, the term \[\underline{\ensuremath{\mu}}_\ensuremath{a}(\ensuremath{\theta})\equiv\min_{\ensuremath{\mu}\in\ensuremath{\ensuremath{\ensuremath{\Delta}^*}(\ensuremath{a})}}\ensuremath{\mu}(\ensuremath{\theta}),\] describes the smallest probability that the agent can assign to state \ensuremath{\theta}\ and action \ensuremath{a}\ be optimal. Thus, (ref) states that for $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ to be BCE-consistent, it must be that the average under \ensuremath{\ensuremath{\nu}_0}\ of these minimum probabilities, $\underline{\ensuremath{\mu}}_\ensuremath{a}(\ensuremath{\theta})$, are below the prior probability of $\ensuremath{\theta}$, \ensuremath{\ensuremath{\mu}_0}(\ensuremath{\theta}). It is immediate that if for some state \ensuremath{\theta}, (ref) does not hold, then $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ cannot be BCE-consistent.
As we argue next, (ref) can be interpreted through the lens of a martingale property for the utility differences, $\ensuremath{u}(\ensuremath{\ensuremath{a}^\prime},\ensuremath{\theta})-\ensuremath{u}(\ensuremath{\ensuremath{a}^{\prime\prime}},\ensuremath{\theta})$. That is, for all pairs of actions, $\ensuremath{\ensuremath{a}^\prime},\ensuremath{\ensuremath{a}^{\prime\prime}}$, the agent's expected ranking over \ensuremath{\ensuremath{a}^\prime}\ and \ensuremath{\ensuremath{a}^{\prime\prime}}\ under the experiment that rationalizes $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ has to coincide with the agent's ex ante ranking over these actions, which is the right-hand side of (ref). Indeed, because (ref) must hold when we exchange the roles of \ensuremath{\ensuremath{a}^\prime}\ and \ensuremath{\ensuremath{a}^{\prime\prime}}, we obtain that $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ must also satisfy that
That is, the ranking at the prior between \ensuremath{\ensuremath{a}^\prime}\ and \ensuremath{\ensuremath{a}^{\prime\prime}}\ must be in between the worst and best rankings under the “distribution over posteriors” \ensuremath{\ensuremath{\nu}_0}.
This is most easily seen in the simple case that \ensuremath{\ensuremath{a}^{\prime\prime}}\ is strictly optimal at the prior and $\{\ensuremath{\ensuremath{a}^\prime},\ensuremath{\ensuremath{a}^{\prime\prime}}\}$ are the only actions in the support of \ensuremath{\ensuremath{\nu}_0}. Because \ensuremath{\ensuremath{a}^\prime}\ is in the support of \ensuremath{\ensuremath{\nu}_0}, under a BCE \ensuremath{\pi}\ that satisfies the marginal constraints the agent must sometimes find it optimal to take action \ensuremath{\ensuremath{a}^\prime}\ instead of \ensuremath{\ensuremath{a}^{\prime\prime}}. Note, however, that on average it must be the case that the agent finds action \ensuremath{\ensuremath{a}^{\prime\prime}}\ better than \ensuremath{\ensuremath{a}^\prime}. Consequently, under \ensuremath{\pi}, when the agent takes \ensuremath{\ensuremath{a}^{\prime\prime}}, the agent must prefer \ensuremath{\ensuremath{a}^{\prime\prime}}\ over \ensuremath{\ensuremath{a}^\prime}\ (weakly) more than at the prior. Because the left-hand side of (ref) selects beliefs in favor of \ensuremath{\ensuremath{a}^\prime}, it is immediate that if (ref) fails one cannot find an experiment in which the agent would take action \ensuremath{\ensuremath{a}^\prime}\ with sufficiently high probability so as to match \ensuremath{\ensuremath{\nu}_0}.
So far, we have argued that the conditions in (ref) are necessary for $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ to be BCE-consistent. To explain why they are also sufficient, it is useful to review the main steps in the proof of (ref). Key to our proof is the following result from strassen1965existence, which we record in present notation:
Whereas Theorem 3 in strassen1965existence requires that (ref) holds for all directions in $\ensuremath{\mathbb{R}}^{\ensuremath{|\ensuremath{\Theta}|}}$, (ref) states that verifying (ref) holds for finitely many directions is enough to conclude that $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ are BCE-consistent. To see this, note that Equations (ref) and (ref) correspond to (ref) for specific directions $\ensuremath{c}\in\ensuremath{\mathbb{R}}^{\ensuremath{|\ensuremath{\Theta}|}}$. Indeed, (ref) corresponds to $\ensuremath{c}=-e_\ensuremath{\theta}\in\ensuremath{\mathbb{R}}^{\ensuremath{|\ensuremath{\Theta}|}}$, where $e_\ensuremath{\theta}$ is the vector with a $1$ in the \ensuremath{\theta}-coordinate and $0$ otherwise. Instead, (ref) corresponds to the direction $\ensuremath{c}=-\ensuremath{\ensuremath{d}_{\ensuremath{\ensuremath{a}^\prime},\ensuremath{\ensuremath{a}^{\prime\prime}}}}$, where \ensuremath{\ensuremath{d}_{\ensuremath{\ensuremath{a}^\prime},\ensuremath{\ensuremath{a}^{\prime\prime}}}}\ is the vector with \ensuremath{\theta}-coordinate $\ensuremath{\ensuremath{d}_{\ensuremath{\ensuremath{a}^\prime},\ensuremath{\ensuremath{a}^{\prime\prime}}}} (\ensuremath{\theta})=\ensuremath{u}(\ensuremath{\ensuremath{a}^\prime},\ensuremath{\theta})-\ensuremath{u}(\ensuremath{\ensuremath{a}^{\prime\prime}},\ensuremath{\theta})$.
To see why verifying that (ref) holds for directions $\{(-e_\ensuremath{\theta})_{\ensuremath{\theta}\in\ensuremath{\Theta}},(-\ensuremath{\ensuremath{d}_{\ensuremath{\ensuremath{a}^\prime},\ensuremath{\ensuremath{a}^{\prime\prime}}}})_{\ensuremath{\ensuremath{a}^\prime},\ensuremath{\ensuremath{a}^{\prime\prime}}\in\ensuremath{A}}\}$ is enough to determine that (ref) holds for all directions $\ensuremath{c}\in\ensuremath{\mathbb{R}}^{\ensuremath{|\ensuremath{\Theta}|}}$, note the following. First, for a fixed action \ensuremath{\ensuremath{a}^\prime}, the directions $\{(-e_\ensuremath{\theta})_{\ensuremath{\theta}\in\ensuremath{\Theta}},(-\ensuremath{\ensuremath{d}_{\ensuremath{\ensuremath{a}^\prime},\ensuremath{\ensuremath{a}^{\prime\prime}}}})_{\ensuremath{\ensuremath{a}^{\prime\prime}}\in\ensuremath{A}}\}$ are the normal vectors that define the polyhedron \ensuremath{\ensuremath{\ensuremath{\Delta}^*}(\ensuremath{\ensuremath{a}^\prime})}. Indeed, the directions $(-e_\ensuremath{\theta})_{\ensuremath{\theta}\in\ensuremath{\Theta}}$ correspond to the condition that the elements of \ensuremath{\ensuremath{\ensuremath{\Delta}^*}(\ensuremath{\ensuremath{a}^\prime})}\ are non-negative, whereas the directions $(-\ensuremath{\ensuremath{d}_{\ensuremath{\ensuremath{a}^\prime},\ensuremath{\ensuremath{a}^{\prime\prime}}}})_{\ensuremath{\ensuremath{a}^{\prime\prime}}\in\ensuremath{A}}$ correspond to the condition that action \ensuremath{\ensuremath{a}^\prime}\ is optimal for all beliefs in \ensuremath{\ensuremath{\ensuremath{\Delta}^*}(\ensuremath{\ensuremath{a}^\prime})}. Second, it is immediate that in each of the maximization problems on the left hand side of (ref), the maximum is attained at an extreme point of \ensuremath{\ensuremath{\ensuremath{\Delta}^*}(\ensuremath{a})}. Standard results in convex analysis then imply that if (ref) holds at all normal directions defining the polyhedra $\{\ensuremath{\ensuremath{\ensuremath{\Delta}^*}(\ensuremath{a})}:\ensuremath{a}\in\ensuremath{A}\}$, then it holds for all directions (cf. hiriart2004fundamentals).
We close (ref) with a remark on the generality of the results in strassen1965existence. It can be skipped with no loss of continuity.
In this section we provide a different perspective on (ref). Together with the marginal distributions, $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$, we are given a distribution over posteriors $\ensuremath{\tau}\in\Delta(\ensuremath{\Delta(\ensuremath{\Theta})})$ with mean equal to the prior \ensuremath{\ensuremath{\mu}_0}. (ref) below characterizes the set of such distributions over posteriors that can implement the marginal \ensuremath{\ensuremath{\nu}_0}. Whereas this characterization does not substitute that in (ref), it allows us to illustrate how one would go about constructing an information structure that implements \ensuremath{\ensuremath{\nu}_0}. Along the way we also establish formal connections with the literature on stochastic choice. For this reason, we work with the agent's stochastic choice rule $\{\ensuremath{\sigma}(\cdot|\ensuremath{\theta}):\ensuremath{\theta}\in\ensuremath{\Theta}\}$ instead of the belief system $\{\ensuremath{\mu}(\cdot|\ensuremath{a}):\ensuremath{a}\in\ensuremath{A}\}$.
\paragraph{Obedient stochastic choice} To understand the results that follow, it is useful to state the obedience and marginal conditions in terms of the stochastic choice rule: Given $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$, we want a stochastic choice rule that satisfies for all actions $\ensuremath{a}\in\ensuremath{A}$
and for all $\ensuremath{a},\ensuremath{\ensuremath{a}^\prime}\in\ensuremath{A}$,
\paragraph{Distributions over posteriors and stochastic choice} Given a Bayes plausible distribution over posteriors $\ensuremath{\tau}\in\Delta(\ensuremath{\Delta(\ensuremath{\Theta})})$, constructing a state-dependent stochastic choice rule is almost at hand. Almost because a Bayes plausible distribution over posteriors does not specify how the agent breaks ties when indifferent. Indeed, to a Bayes plausible distribution over posteriors, $\ensuremath{\tau}(\ensuremath{\mu})$, we can associate a decision rule $\ensuremath{\alpha}:\ensuremath{\Delta(\ensuremath{\Theta})}\mapsto\Delta(\ensuremath{A})$, describing the probability $\ensuremath{\alpha}(\ensuremath{a}|\ensuremath{\mu})$ with which the agent takes action \ensuremath{a}\ when her belief is \ensuremath{\mu}. The pair $(\ensuremath{\tau},\ensuremath{\alpha})$ determines a stochastic choice rule $\{\ensuremath{\sigma}(\cdot|\ensuremath{\theta}):\ensuremath{\theta}\in\ensuremath{\Theta}\}$ as follows:
(ref) suggests that conditions under which a stochastic choice rule that satisfies (ref) and (ref) are intimately related to the existence of a Bayes plausible distribution \ensuremath{\tau}\ and a decision rule \ensuremath{\alpha}\ that satisfy certain properties. In fact, the analysis that follows identifies conditions on Bayes plausible distributions over posteriors under which a decision rule exists that induces a stochastic choice rule--and hence a joint distribution $\ensuremath{\pi}\in\Delta(\ensuremath{A}\times\ensuremath{\Theta})$--that satisfies all the constraints.
\paragraph{Distributions over posteriors as distributions over menus} Given a Bayes plausible $\ensuremath{\tau}\in\Delta(\ensuremath{\Delta(\ensuremath{\Theta})})$, one can construct a measure over subsets \ensuremath{B}\ of the set of actions \ensuremath{A}\ as follows. For each $\ensuremath{\mu}\in\ensuremath{\Delta(\ensuremath{\Theta})}$, let $\ensuremath{a}^*(\ensuremath{\mu})$ denote the agent's best response when her belief is \ensuremath{\mu}. That is, $\ensuremath{a}^*(\ensuremath{\mu})=\arg\max_{\ensuremath{a}\in\ensuremath{A}}\mathbb{E}_{\ensuremath{\theta}\sim\ensuremath{\mu}}\left[\ensuremath{u}(\ensuremath{a},\ensuremath{\theta})\right]$. For each $\ensuremath{B}\subseteq\ensuremath{A}$, define $\ensuremath{\tau}_\ensuremath{A}(\ensuremath{B})$ as
In words, each action subset \ensuremath{B}\ has mass equal to the probability that \ensuremath{\tau}\ induces a belief under which \ensuremath{B}\ is optimal.
(ref) characterizes when the distribution over posteriors \ensuremath{\tau}\ implements \ensuremath{\ensuremath{\nu}_0}:
To interpret (ref), note the following. The left-hand side of (ref) is the probability under which the agent takes some action \ensuremath{a}\ in the set \ensuremath{B}. Instead, the right-hand side of (ref) is the probability under which the agent finds optimal some action in the set \ensuremath{B}\ (but no action that is not in \ensuremath{B}). (ref) then says that the frequency with which the agent takes actions in \ensuremath{B}\ has to be at least the frequency with which an action in \ensuremath{B}\ is optimal.
The proof of (ref) is based on the following graphical representation of the BCE-consistency problem depicted in (ref). Consider the following graph. Nodes are (i) the actions $\ensuremath{a}\in\ensuremath{A}$, (ii) the (non-empty) action subsets $\ensuremath{B}\subseteq\ensuremath{A}$ (i.e., the elements of $2^\ensuremath{A}\setminus\{\emptyset\}$), (iii) a source node $s$, and (iv) a sink node $t$. Edges are as follows. There is an edge of weight one between $\ensuremath{a}\in\ensuremath{A}$ and $\ensuremath{B}\subseteq\ensuremath{A}$ if and only if $\ensuremath{a}\in\ensuremath{B}$. There is an edge with weight $\ensuremath{\ensuremath{\nu}_0}(\ensuremath{a})$ between the source $s$ and \ensuremath{a}. Finally, there is an edge between $\ensuremath{B}\subseteq\ensuremath{A}$ and the sink $t$ with weight $\ensuremath{\tau}_\ensuremath{A}(\ensuremath{B})$. The condition in (ref) ensures that a feasible flow exists throughout the network.\footnote{That is, a flow $f$ such that $\sum_{\ensuremath{a}\in\ensuremath{A}}f(s,\ensuremath{a})=\sum_{\ensuremath{B}\in 2^\ensuremath{A}}f(\ensuremath{B},t)=1$.}
(ref) provides an alternative proof of (ref) using gale1957theorem's network flow theorem. \paragraph{Connection to stochastic choice:} The proof of (ref) connects two sets of conditional distribution over choices that arise in the stochastic choice literature: stochastic choices conditional on a state of the world--denoted by \ensuremath{\sigma}\ in the proof--and stochastic choices out of a menu--denoted by $\ensuremath{\alpha}^\prime$ in the proof. Indeed, the measure $\ensuremath{\tau}_\ensuremath{A}$ can be interpreted as the frequency with which the agent faces different menus--action subsets in this case--whereas the measure \ensuremath{\ensuremath{\nu}_0}\ represents the frequency with which the agent makes different choices. In other words, the pair $(\ensuremath{\tau}_\ensuremath{A},\ensuremath{\ensuremath{\nu}_0})$ is analogous to the data set in azrieli2022marginal. Our ultimately goal, however, is to obtain the agent's stochastic choice rule, which we obtain relying on the Bayes' plausibility of $\ensuremath{\tau}$.
We consider in this section two applications of (ref) to simple multi-agent settings. (ref) studies under what conditions a pair of marginal distributions $\ensuremath{(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\ensuremath{\nu}_0})}$ can be rationalized by a public information structure. (ref) shows that (ref) characterizes the set of M-BCE-consistent marginals.
We consider in this section the following multiplayer game. We assume $\ensuremath{N}\geq 1$ and that each player's utility function depends only her own action and the state of the world.\footnote{arieli2021feasible dub this setting first-order Bayesian persuasion.} That is, for all players $\ensuremath{i}\in\{1,\dots,N\}$, all action profiles $(\ensuremath{\ensuremath{a}_\ensuremath{i}},\ensuremath{\ensuremath{a}_{-\ensuremath{i}}})\in\ensuremath{A}$, and states of the world $\ensuremath{\theta}\in\ensuremath{\Theta}$, \[\ensuremath{\ensuremath{u}_\ensuremath{i}}\left( \ensuremath{\ensuremath{a}_\ensuremath{i}},\ensuremath{\ensuremath{a}_{-\ensuremath{i}}}, \ensuremath{\theta} \right) =\ensuremath{\ensuremath{u}_\ensuremath{i}}\left( \ensuremath{\ensuremath{a}_\ensuremath{i}}, \ensuremath{\theta} \right). \]
The analyst, who knows the base game \ensuremath{G}\ and the marginal distribution of play $\ensuremath{\ensuremath{\nu}_0}\in\Delta(\ensuremath{A})$, wants to ascertain whether the distribution of play \ensuremath{\ensuremath{\nu}_0}\ can be rationalized by a public information structure (i.e., the players publicly observe the realization of a common signal structure before play).
As we show next, (ref) can be applied to address this question. In what follows, we rely on the following definition:
Consider now an auxiliary single-agent base game $\bar\ensuremath{G}=\langle\ensuremath{\Theta}, (\ensuremath{A},\bar u),\ensuremath{\ensuremath{\mu}_0}\rangle$. In this game, a player with payoff $\bar u(\ensuremath{a})=\sum_{\ensuremath{i}=1}^\ensuremath{N}\ensuremath{\ensuremath{u}_\ensuremath{i}}(\ensuremath{\ensuremath{a}_\ensuremath{i}},\ensuremath{\theta})$ chooses an action $\ensuremath{a}\in\ensuremath{A}=\times_{\ensuremath{i}\in\ensuremath{N}}\ensuremath{\ensuremath{A}_\ensuremath{i}}$ under incomplete information about \ensuremath{\theta}.
The following result is an immediate corollary of (ref) and the focus on public signals:
Because of the focus on public signal structures, the analysis of the multi-agent game reduces to the analysis of a single-agent problem. To see this, in a slight abuse of notation, let $\ensuremath{A}^*(\ensuremath{\mu})$ denote the set of actions that the agent with payoff $\bar\ensuremath{u}$ finds optimal when their belief is \ensuremath{\mu}. It is immediate that $\ensuremath{A}^*(\ensuremath{\mu})=\times_{\ensuremath{i}\in\ensuremath{N}}\ensuremath{a}_i^*(\ensuremath{\mu})$, where for each player \ensuremath{i}, $\ensuremath{a}_i^*(\ensuremath{\mu})$ denotes the set of actions player \ensuremath{i}\ finds optimal when her belief is \ensuremath{\mu}. That is, the profile $\ensuremath{a}=\left( \ensuremath{a}_{1},\dots ,\ensuremath{a}_\ensuremath{N}\right) \in \ensuremath{A}$ is optimal for the agent with payoff $\bar\ensuremath{u}$ if and only if action $\ensuremath{\ensuremath{a}_\ensuremath{i}}$ is optimal for agent $i$, for all $\ensuremath{i}\in \ensuremath{[N]}$. And, given a posterior belief $\ensuremath{\mu}$, any distribution of (optimal) action profiles that the agent with payoff $\bar\ensuremath{u}$ can generate, can also be generated by the players using a public correlation device or by duplicating signal realization, and vice versa.\footnote{For example, suppose that the signal realization $s$ induces the posterior belief $\ensuremath{\mu}$. Suppose also that under $\ensuremath{\mu}$, the agent with payoff $\bar\ensuremath{u}$ selects the two optimal action profiles $\ensuremath{a},\ensuremath{\ensuremath{a}^\prime}\in \ensuremath{A}^*(\ensuremath{\mu})$ with equal probability. The same distribution of actions can be generated by the players: Indeed, one can \textquotedblleft split\textquotedblright\ the signal $s$ into two new signals, $s^{\prime }$ and $s^{\prime \prime }$, such that both new signals induce the same posterior belief $\ensuremath{\mu}$, and each of them is sent with half the probability of the original signal $s$. If whenever $s^{\prime }$ and $s^{\prime \prime }$ are realized, each agent acts according to her corresponding optimal action in the profiles $\ensuremath{a}$ and $\ensuremath{\ensuremath{a}^\prime}$, respectively, the distribution over actions will coincide with that of the agent with payoff $\bar\ensuremath{u}$.} Notice that this equivalence no longer holds if either information is not public, or the players' utilities are interdependent.
We consider here ring-network games as in kneeland2015identifying, extended to account for incomplete information. A ring-network game is a base game \ensuremath{G}\ in which player's payoffs satisfy the following:
In words, player $1$ cares about their action and the state of the world, whereas for $i\geq2$ player $i$ cares about their action and that of player $i-1$. Ring-network games are used in the experimental literature that measures players' higher order beliefs to identify departures from Nash equilibrium.
The analyst knows the ring-network base game and for each player $i$, player $i$'s action distribution, $\ensuremath{\nu}_{0,\ensuremath{i}}\in\Delta(\ensuremath{\ensuremath{A}_\ensuremath{i}})$. The analyst wants to ascertain whether $(\ensuremath{\ensuremath{\mu}_0},\ensuremath{\overline{\ensuremath{\nu}}_0})$ is M-BCE-consistent. Relying on (ref) and the ring-network structure, (ref) characterizes the set of M-BCE-consistent marginals:
Similar to (ref), (ref) exploits the structure of the ring-network game to reduce it to a series of single-agent problems in which except for player $1$, the states are given by the actions of the preceding player and the prior distribution over this state space by the marginal over actions of the preceding player. Indeed, for $\ensuremath{i}\geq2$, BCE-consistency of $(\ensuremath{\nu}_{0,\ensuremath{i}-1},\ensuremath{\nu}_{0,\ensuremath{i}})$ implies that an information structure exists that rationalizes player $\ensuremath{i}$'s choices as the outcome of some information structure under “prior” $\ensuremath{\nu}_{0,\ensuremath{i}-1}$, whereas BCE-consistency of $(\ensuremath{\nu}_{0,\ensuremath{i}-2},\ensuremath{\nu}_{0,\ensuremath{i}-1})$\footnote{With the understanding that $\ensuremath{\nu}_{0,0}=\ensuremath{\ensuremath{\mu}_0}$.} guarantees that the “prior” $\ensuremath{\nu}_{0,\ensuremath{i}-1}$ is consistent with player $\ensuremath{i}-1$ observing the outcome of some information structure given their belief $\ensuremath{\nu}_{0,\ensuremath{i}-2}$.