Bayes correlated equilibrium

Intuitively, a Bayes correlated equilibrium allows for players to correlate their actions in such a way 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 ${\displaystyle I}$ be a set of players, and ${\displaystyle \mathrm{\Theta }}$ a set of possible states of the world. A game is defined as a tuple ${\displaystyle G=⟨(A_i,u_i{)}_{i\in I},\mathrm{\Theta },\psi ⟩}$, where ${\displaystyle A_i}$ is the set of possible actions (with ${\displaystyle A=\prod _{i\in I}{A}_i}$) and ${\displaystyle u_i:A\times \mathrm{\Theta }\to \mathbb{R}}$ is the utility function for each player, and ${\displaystyle \psi \in {\mathrm{\Delta }}_{++}(\mathrm{\Theta })}$ is a full support common prior over the states of the world.

An information structure is defined as a tuple ${\displaystyle S=⟨(T_i{)}_{i\in I},\pi ⟩}$, where ${\displaystyle T_i}$ is a set of possible signals (or types) each player can receive (with ${\displaystyle T=\prod _{i\in I}{T}_i}$), and ${\displaystyle \pi :\mathrm{\Theta }\to \mathrm{\Delta }(T)}$ is a signal distribution function, informing the probability ${\displaystyle \pi (t|\theta )}$ of observing the joint signal ${\displaystyle t\in T}$ when the state of the world is ${\displaystyle \theta \in \mathrm{\Theta }}$.

By joining those two definitions, one can define ${\displaystyle \mathrm{\Gamma }=(G,S)}$ as an incomplete information game. A decision rule for the incomplete information game ${\displaystyle \mathrm{\Gamma }=(G,S)}$ is a mapping ${\displaystyle \sigma :T\times \mathrm{\Theta }\to \mathrm{\Delta }(A)}$. Intuitively, the value of decision rule ${\displaystyle \sigma (a|t,\theta )}$ can be thought of as a joint recommendation for players to play the joint mixed strategy ${\displaystyle \sigma (\cdot \mid t,\theta )\in \mathrm{\Delta }(A)}$ when the joint signal received is ${\displaystyle t\in T}$ and the state of the world is ${\displaystyle \theta \in \mathrm{\Theta }}$.

Definition

A Bayes correlated equilibrium (BCE) is defined to be a decision rule ${\displaystyle \sigma }$ 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 ${\displaystyle \sigma }$ is obedient (and a Bayes correlated equilibrium) for game ${\displaystyle \mathrm{\Gamma }=(G,S)}$ if, for every player ${\displaystyle i\in I}$, every signal ${\displaystyle t_i\in T_i}$ and every action ${\displaystyle a_i\in A_i}$, we have

${\displaystyle \sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_i,t_{-i}|\theta )\sigma (a_i,a_{-i}|t_i,t_{-i},\theta )u_i(a_i,a_{-i},\theta )}$

${\displaystyle \ge \sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_i,t_{-i}|\theta )\sigma (a_i,a_{-i}|t_i,t_{-i},\theta )u_i(a_i^{\prime },a_{-i},\theta )}$

for all ${\displaystyle a_i^{\prime }\in A_i}$.

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.