Bayes correlated equilibrium

Let ${\displaystyle I}$ be a set of players, and ${\displaystyle \Theta }$ a set of possible states of the world. A game is defined as a tuple ${\displaystyle G=\langle (A_{i},u_{i})_{i\in I},\Theta ,\psi \rangle }$, where ${\displaystyle A_{i}}$ is the set of possible actions (with ${\displaystyle A=\prod _{i\in I}A_{i}}$) and ${\displaystyle u_{i}:A\times \Theta \rightarrow \mathbb {R} }$ is the utility function for each player, and ${\displaystyle \psi \in \Delta _{++}(\Theta )}$ is a full support common prior over the states of the world.

An information structure is defined as a tuple ${\displaystyle S=\langle (T_{i})_{i\in I},\pi \rangle }$, 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 ${\displaystyle \pi$ :\Theta \rightarrow \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 ${\displaystyle \theta \in \Theta }$.

By joining those two definitions, one can define ${\displaystyle \Gamma =(G,S)}$ as an incomplete information game.^[3] A decision rule for the incomplete information game ${\displaystyle \Gamma =(G,S)}$ is a mapping ${\displaystyle \sigma :T\times \Theta \rightarrow \Delta (A)}$. 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 ${\displaystyle \sigma (\cdot \mid t,\theta )\in \Delta (A)}$ when the joint signal received is ${\displaystyle t\in T}$ and the state of the world is ${\displaystyle \theta \in \Theta }$.

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 ${\displaystyle \Gamma =(G,S)}$ 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

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

${\displaystyle \geq \sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_{i},t_{-i}|\theta )\sigma (a_{i},a_{-i}|t_{i},t_{-i},\theta )u_{i}(a'_{i},a_{-i},\theta )}$

for all ${\displaystyle a'_{i}\in A_{i}}$.

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.

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.^[2]

Formally, let ${\displaystyle \Gamma =(G,S)}$ be an incomplete information game, and let ${\displaystyle s:T\rightarrow \Delta (A)}$ be an equilibrium joint strategy, with each player ${\displaystyle i}$ playing ${\displaystyle s_{i}(a_{i}|t_{i})\in \Delta (A_{i})}$. Therefore, the definition of BNE implies that, for every ${\displaystyle i\in I}$, ${\displaystyle t_{i}\in T_{i}}$ and ${\displaystyle a_{i}\in A_{i}}$ such that ${\displaystyle s_{i}(a_{i}|t_{i})>0}$, we have

${\displaystyle \sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_{i},t_{-i}|\theta )\left(\prod _{j\neq i}s_{j}(a_{j}|t_{j})\right)u_{i}(a_{i},a_{-i},\theta )}$

${\displaystyle \geq \sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_{i},t_{-i}|\theta )\left(\prod _{j\neq i}s_{j}(a_{j}|t_{j})\right)u_{i}(a'_{i},a_{-i},\theta )}$

for every ${\displaystyle a'_{i}\in A_{i}}$.

If we define the decision rule ${\displaystyle \sigma }$ on ${\displaystyle \Gamma }$ as ${\displaystyle \sigma (a|t,\theta )=s(a|t)=\prod _{i}s_{i}(a_{i}|t_{i})}$ for all ${\displaystyle t\in T}$ and ${\displaystyle \theta \in \Theta }$, we directly get a BCE.

If there is no uncertainty about the state of the world (e.g., if ${\displaystyle \Theta }$ is a singleton), then the definition collapses to Aumann's correlated equilibrium solution.^[4] In this case, ${\displaystyle \sigma \in \Delta (A)}$ is a BCE if, for every ${\displaystyle i\in I}$, we have^[1]

${\displaystyle \sum _{a_{-i}\in A{-i}}\sigma (a_{i},a_{-i})u_{i}(a_{i},a_{-i})\geq \sum _{a_{-i}\in A{-i}}\sigma (a_{i},a_{-i})u_{i}(a'_{i},a_{-i})}$

for every ${\displaystyle a'_{i}\in A_{i}}$, which is equivalent to the definition of a correlated equilibrium for such a setting.

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.^[5] More specifically, let ${\displaystyle v:A\times \Theta \rightarrow \mathbb {R} }$ be the information designer's objective function. Then her ex-ante expected utility from a BCE decision rule ${\displaystyle \sigma }$ is given by:^[1]

${\displaystyle V(\sigma )=\sum _{a,t,\theta }\psi (\theta )\pi (t|\theta )\sigma (a|t,\theta )v(a,\theta )}$

If the set of players ${\displaystyle I}$ is a singleton, then choosing an information structure to maximize ${\displaystyle V(\sigma )}$ is equivalent to a Bayesian persuasion problem, where the information designer is called a Sender and the player is called a Receiver.