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.[1]

Bayes correlated equilibrium
Solution concept in game theory
Relationship
Superset ofCorrelated equilibrium, Bayesian Nash equilibrium
Significance
Proposed byDirk Bergemann, Stephen Morris

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.[2]

Formal definition

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Preliminaries

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Let   be a set of players, and   a set of possible states of the world. A game is defined as a tuple  , where   is the set of possible actions (with  ) and   is the utility function for each player, and   is a full support common prior over the states of the world.

An information structure is defined as a tuple  , where   is a set of possible signals (or types) each player can receive (with  ), and   is a signal distribution function, informing the probability   of observing the joint signal   when the state of the world is  .

By joining those two definitions, one can define   as an incomplete information game.[3] A decision rule for the incomplete information game   is a mapping  . Intuitively, the value of decision rule   can be thought of as a joint recommendation for players to play the joint mixed strategy   when the joint signal received is   and the state of the world is  .

Definition

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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. Formally, decision rule   is obedient (and a Bayes correlated equilibrium) for game   if, for every player  , every signal   and every action  , we have

 


 

for all  .

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

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Bayesian Nash equilibrium

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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   be an incomplete information game, and let   be an equilibrium joint strategy, with each player   playing  . Therefore, the definition of BNE implies that, for every  ,   and   such that  , we have

 


 

for every  .

If we define the decision rule   on   as   for all   and  , we directly get a BCE.

Correlated equilibrium

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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.[4] In this case,   is a BCE if, for every  , we have[1]

 

for every  , which is equivalent to the definition of a correlated equilibrium for such a setting.

Bayesian persuasion

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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   be the information designer's objective function. Then her ex-ante expected utility from a BCE decision rule   is given by:[1]

 

If the set of players   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.

References

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  1. ^ a b c Bergemann, Dirk; Morris, Stephen (2019). "Information Design: A Unified Perspective". Journal of Economic Literature. 57 (1): 44–95. doi:10.1257/jel.20181489.
  2. ^ a b Bergemann, Dirk; Morris, Stephen (2016). "Bayes correlated equilibrium and the comparison of information structures in games". Theoretical Economics. 11 (2): 487–522. doi:10.3982/TE1808. hdl:10419/150284.
  3. ^ Gossner, Olivier (2000). "Comparison of Information Structures". Games and Economic Behavior. 30 (1): 44–63. doi:10.1006/game.1998.0706. hdl:10230/596.
  4. ^ Aumann, Robert J. (1987). "Correlated Equilibrium as an Expression of Bayesian Rationality". Econometrica. 55 (1): 1–18. doi:10.2307/1911154.
  5. ^ Kamenica, Emir; Gentzkow, Matthew (2011-10-01). "Bayesian Persuasion". American Economic Review. 101 (6): 2590–2615. doi:10.1257/aer.101.6.2590. ISSN 0002-8282.