# Bayes correlated equilibrium

In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information. It is both a generalization of the correlated equilibrium perfect information solution concept to bayesian games, and also a broader solution concept than the usual Bayesian Nash equilibrium thereof. Additionally, it can be seen as a generalized multi-player solution of the Bayesian persuasion information design problem.
Intuitively, a Bayes correlated equilibrium allows for players to correlate their actions in a way such that no player has an incentive to deviate for every possible type they may have. It was first proposed by Dirk Bergemann and Stephen Morris.

## Formal definition
### Preliminaries

Let $I$ be a set of players, and $\Theta$ a set of possible states of the world. A game is defined as a tuple $G=\langle (A_{i},u_{i})_{i\in I},\Theta ,\psi \rangle$, where $A_{i}$ is the set of possible actions (with $A=\prod _{i\in I}A_{i}$) and $u_{i}:A\times \Theta \rightarrow \mathbb {R}$ is the utility function for each player, and $\psi \in \Delta _{++}(\Theta )$ is a full support common prior over the states of the world.
An information structure is defined as a tuple $S=\langle (T_{i})_{i\in I},\pi \rangle$, where $T_{i}$ is a set of possible signals (or types) each player can receive (with $T=\prod _{i\in I}T_{i}$), and $\pi :\Theta \rightarrow \Delta (T)$ is a signal distribution function, informing the probability $\pi (t|\theta )$ of observing the joint signal $t\in T$ when the state of the world is $\theta \in \Theta$.
By joining those two definitions, one can define $\Gamma =(G,S)$ as an incomplete information game. A decision rule for the incomplete information game $\Gamma =(G,S)$ is a mapping $\sigma :T\times \Theta \rightarrow \Delta (A)$. Intuitively, the value of decision rule $\sigma (a|t,\theta )$ can be thought of as a joint recommendation for players to play the joint mixed strategy $\sigma (a)\in \Delta (A)$ when the joint signal received is $t\in T$ and the state of the world is $\theta \in \Theta$.

### Definition

A Bayes correlated equilibrium (BCE) is defined to be a decision rule $\sigma$ which is obedient: that is, one where no player has an incentive to unilaterally deviate from the recommended joint strategy, for any possible type they may be. Formally, decision rule $\sigma$ is obedient (and a Bayes correlated equilibrium) for game $\Gamma =(G,S)$ if, for every player $i\in I$, every signal $t_{i}\in T_{i}$ and every action $a_{i}\in A_{i}$, we have
    $\sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_{i},t_{-i}|\theta )\sigma (a_{i},a_{-i}|t_{i},t_{-i},\theta )u_{i}(a_{i},a_{-i},\theta )$
    $\geq \sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_{i},t_{-i}|\theta )\sigma (a_{i},a_{-i}|t_{i},t_{-i},\theta )u_{i}(a'_{i},a_{-i},\theta )$
for all $a'_{i}\in A_{i}$.
That is, every player obtains a higher expected payoff by following the recommendation from the decision rule than by deviating to any other possible action.

## Relation to other concepts
### Bayesian Nash equilibrium

Every Bayesian Nash equilibrium (BNE) of an incomplete information game can be thought of a as BCE, where the recommended joint strategy is simply the equilibrium joint strategy.
Formally, let $\Gamma =(G,S)$ be an incomplete information game, and let $s:T\rightarrow \Delta (A)$ be an equilibrium joint strategy, with each player $i$ playing $s_{i}(a_{i}|t_{i})\in \Delta (A_{i})$. Therefore, the definition of BNE implies that, for every $i\in I$, $t_{i}\in T_{i}$ and $a_{i}\in A_{i}$ such that $s_{i}(a_{i}|t_{i})>0$, we have
    $\sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_{i},t_{-i}|\theta )\left(\prod _{j\neq i}s_{j}(a_{j}|t_{j})\right)u_{i}(a_{i},a_{-i},\theta )$
    $\geq \sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_{i},t_{-i}|\theta )\left(\prod _{j\neq i}s_{j}(a_{j}|t_{j})\right)u_{i}(a'_{i},a_{-i},\theta )$
for every $a'_{i}\in A_{i}$.
If we define the decision rule $\sigma$ on $\Gamma$ as $\sigma (a|t,\theta )=s(a|t)=\prod _{i}s_{i}(a_{i}|t_{i})$ for all $t\in T$ and $\theta \in \Theta$, we directly get a BCE.

### Correlated equilibrium

If there is no uncertainty about the state of the world (e.g., if $\Theta$ is a singleton), then the definition collapses to Aumann's correlated equilibrium solution. In this case, $\sigma \in \Delta (A)$ is a BCE if, for every $i\in I$, we have
    $\sum _{a_{-i}\in A{-i}}\sigma (a_{i},a_{-i})u_{i}(a_{i},a_{-i})\geq \sum _{a_{-i}\in A{-i}}\sigma (a_{i},a_{-i})u_{i}(a'_{i},a_{-i})$
for every $a'_{i}\in A_{i}$, which is equivalent to the definition of a correlated equilibrium for such a setting.

### Bayesian persuasion

Additionally, the problem of designing a BCE can be thought of as a multi-player generalization of the Bayesian persuasion problem from Emir Kamenica and Matthew Gentzkow. More specifically, let $v:A\times \Theta \rightarrow \mathbb {R}$ be the information designer's objective function. Then her ex-ante expected utility from a BCE decision rule $\sigma$ is given by:
    $V(\sigma )=\sum _{a,t,\theta }\psi (\theta )\pi (t|\theta )\sigma (a|t,\theta )v(a,\theta )$
If the set of players $I$ is a singleton, then choosing an information structure to maximize $V(\sigma )$ is equivalent to a Bayesian persuasion problem, where the information designer is called a Sender and the player is called a Receiver.