Bayes correlated equilibrium
Bayes correlated equilibrium | |
---|---|
A solution concept in game theory | |
Relationship | |
Superset of | Correlated equilibrium, Bayesian Nash equilibrium |
Significance | |
Proposed by | Dirk Bergemann, Stephen Morris |
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.[2]
Formal definition
Preliminaries
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
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
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.[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.
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
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
- ^ 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.
- ^ 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.
- ^ Gossner, Olivier (2000). "Comparison of Information Structures". Games and Economic Behavior. 30 (1): 44–63. doi:10.1006/game.1998.0706. hdl:10230/596.
- ^ Aumann, Robert J. (1987). "Correlated Equilibrium as an Expression of Bayesian Rationality". Econometrica. 55 (1): 1–18. doi:10.2307/1911154.
- ^ 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.
- v
- t
- e
- Congestion game
- Cooperative game
- Determinacy
- Escalation of commitment
- Extensive-form game
- First-player and second-player win
- Game complexity
- Graphical game
- Hierarchy of beliefs
- Information set
- Normal-form game
- Preference
- Sequential game
- Simultaneous game
- Simultaneous action selection
- Solved game
- Succinct game
concepts
- Bayes correlated equilibrium
- Bayesian Nash equilibrium
- Berge equilibrium
- Core
- Correlated equilibrium
- Epsilon-equilibrium
- Evolutionarily stable strategy
- Gibbs equilibrium
- Mertens-stable equilibrium
- Markov perfect equilibrium
- Nash equilibrium
- Pareto efficiency
- Perfect Bayesian equilibrium
- Proper equilibrium
- Quantal response equilibrium
- Quasi-perfect equilibrium
- Risk dominance
- Satisfaction equilibrium
- Self-confirming equilibrium
- Sequential equilibrium
- Shapley value
- Strong Nash equilibrium
- Subgame perfection
- Trembling hand
of games
- Go
- Chess
- Infinite chess
- Checkers
- Tic-tac-toe
- Prisoner's dilemma
- Gift-exchange game
- Optional prisoner's dilemma
- Traveler's dilemma
- Coordination game
- Chicken
- Centipede game
- Lewis signaling game
- Volunteer's dilemma
- Dollar auction
- Battle of the sexes
- Stag hunt
- Matching pennies
- Ultimatum game
- Rock paper scissors
- Pirate game
- Dictator game
- Public goods game
- Blotto game
- War of attrition
- El Farol Bar problem
- Fair division
- Fair cake-cutting
- Cournot game
- Deadlock
- Diner's dilemma
- Guess 2/3 of the average
- Kuhn poker
- Nash bargaining game
- Induction puzzles
- Trust game
- Princess and monster game
- Rendezvous problem
figures
- Albert W. Tucker
- Amos Tversky
- Antoine Augustin Cournot
- Ariel Rubinstein
- Claude Shannon
- Daniel Kahneman
- David K. Levine
- David M. Kreps
- Donald B. Gillies
- Drew Fudenberg
- Eric Maskin
- Harold W. Kuhn
- Herbert Simon
- Hervé Moulin
- John Conway
- Jean Tirole
- Jean-François Mertens
- Jennifer Tour Chayes
- John Harsanyi
- John Maynard Smith
- John Nash
- John von Neumann
- Kenneth Arrow
- Kenneth Binmore
- Leonid Hurwicz
- Lloyd Shapley
- Melvin Dresher
- Merrill M. Flood
- Olga Bondareva
- Oskar Morgenstern
- Paul Milgrom
- Peyton Young
- Reinhard Selten
- Robert Axelrod
- Robert Aumann
- Robert B. Wilson
- Roger Myerson
- Samuel Bowles
- Suzanne Scotchmer
- Thomas Schelling
- William Vickrey
- All-pay auction
- Alpha–beta pruning
- Bertrand paradox
- Bounded rationality
- Combinatorial game theory
- Confrontation analysis
- Coopetition
- Evolutionary game theory
- First-move advantage in chess
- Glossary of game theory
- List of game theorists
- List of games in game theory
- No-win situation
- Paradox of tolerance
- Solving chess
- Topological game
- Tragedy of the commons
- Tyranny of small decisions