Game theory solution
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 formula_1 be a set of players, and formula_2 a set of possible states of the world. A "game" is defined as a tuple formula_3, where formula_4 is the set of possible actions (with formula_5) and formula_6 is the utility function for each player, and formula_7 is a full support common prior over the states of the world.
An "information structure" is defined as a tuple formula_8, where formula_9 is a set of possible signals (or types) each player can receive (with formula_10), and formula_11 is a signal distribution function, informing the probability formula_12 of observing the joint signal formula_13 when the state of the world is formula_14.
By joining those two definitions, one can define formula_15 as an incomplete information game. A "decision rule" for the incomplete information game formula_15 is a mapping formula_17. Intuitively, the value of decision rule formula_18 can be thought of as a joint recommendation for players to play the joint mixed strategy formula_19 when the joint signal received is formula_13 and the state of the world is formula_14.
Definition.
A Bayes correlated equilibrium (BCE) is defined to be a decision rule formula_22 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 formula_22 is obedient (and a Bayes correlated equilibrium) for game formula_24 if, for every player formula_25, every signal formula_26 and every action formula_27, we have
formula_28
formula_29
for all formula_30.
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 formula_15 be an incomplete information game, and let formula_32 be an equilibrium joint strategy, with each player formula_33 playing formula_34. Therefore, the definition of BNE implies that, for every formula_25, formula_26 and formula_27 such that formula_38, we have
formula_39
formula_40
for every formula_30.
If we define the decision rule formula_22 on formula_43 as formula_44 for all formula_13 and formula_14, we directly get a BCE. 
Correlated equilibrium.
If there is no uncertainty about the state of the world (e.g., if formula_2 is a singleton), then the definition collapses to Aumann's correlated equilibrium solution. In this case, formula_48 is a BCE if, for every formula_25, we have
formula_50
for every formula_30, 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 formula_52 be the information designer's objective function. Then her "ex-ante" expected utility from a BCE decision rule formula_22 is given by:
formula_54
If the set of players formula_1 is a singleton, then choosing an information structure to maximize formula_56 is equivalent to a Bayesian persuasion problem, where the information designer is called a Sender and the player is called a Receiver.
References.
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