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.
[1] 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.
a set of possible states of the world.
A game is defined as a tuple
is a full support common prior over the states of the world.
An information structure is defined as a tuple
is a set of possible signals (or types) each player can receive (with
is a signal distribution function, informing the probability
of observing the joint signal
By joining those two definitions, one can define
[3] A decision rule for the incomplete information game
can be thought of as a joint recommendation for players to play the joint mixed strategy
when the joint signal received is
A Bayes correlated equilibrium (BCE) is defined to be a decision rule
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.
is obedient (and a Bayes correlated equilibrium) for game
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.
be an equilibrium joint strategy, with each player
If there is no uncertainty about the state of the world (e.g., if
is a singleton), then the definition collapses to Aumann's correlated equilibrium solution.
, 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.
be the information designer's objective function.
Then her ex-ante expected utility from a BCE decision rule
is a singleton, then choosing an information structure to maximize
is equivalent to a Bayesian persuasion problem, where the information designer is called a Sender and the player is called a Receiver.