Correlated equilibrium

Correlated equilibrium
A solution concept in game theory
Relationships
Superset of	Nash equilibrium
Significance
Proposed by	Robert Aumann
Example	Chicken

In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann (1974). The idea is that each player chooses his/her action according to his/her observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from the recommended strategy (assuming the others don't deviate), the distribution is called a correlated equilibrium.

Formal definition

An $N$ -player strategic game $\displaystyle (N,A_{i},u_{i})$ is characterized by an action set $A_{i}$ and utility function $u_{i}$ for each player $i$ . When player $i$ chooses strategy $a_i \in A_i$ and the remaining players choose a strategy profile described by the $N-1$ -tuple $a_{{-i}}$ , then player $i$ 's utility is $\displaystyle u_{i}(a_{i},a_{{-i}})$ .

A strategy modification for player $i$ is a function $\phi \colon A_{i}\to A_{i}$ . That is, $\phi$ tells player $i$ to modify his behavior by playing action $\phi (a_{i})$ when instructed to play $a_{i}$ .

Let $(\Omega ,\pi )$ be a countable probability space. For each player $i$ , let $P_{i}$ be his information partition, $q_{i}$ be $i$ 's posterior and let $s_{i}\colon \Omega \rightarrow A_{i}$ , assigning the same value to states in the same cell of $i$ 's information partition. Then $((\Omega ,\pi ),P_{i})$ is a correlated equilibrium of the strategic game $(N,A_{i},u_{i})$ if for every player $i$ and for every strategy modification $\phi$ :

\sum _{{\omega \in \Omega }}q_{i}(\omega )u_{i}(s_{i},s_{{-i}})\geq \sum _{{\omega \in \Omega }}q_{i}(\omega )u_{i}(\phi (s_{i}),s_{{-i}})

In other words, $((\Omega ,\pi ),P_{i})$ is a correlated equilibrium if no player can improve his or her expected utility via a strategy modification.

An example

	Dare	Chicken out
Dare	0, 0	7, 2
Chicken out	2, 7	6, 6
A game of Chicken

Consider the game of chicken pictured. In this game two individuals are challenging each other to a contest where each can either dare or chicken out. If one is going to Dare, it is better for the other to chicken out. But if one is going to chicken out it is better for the other to Dare. This leads to an interesting situation where each wants to dare, but only if the other might chicken out.

In this game, there are three Nash equilibria. The two pure strategy Nash equilibria are (D, C) and (C, D). There is also a mixed strategy equilibrium where each player Dares with probability 1/3.

Now consider a third party (or some natural event) that draws one of three cards labeled: (C, C), (D, C), and (C, D), with the same probability, i.e. probability 1/3 for each card. After drawing the card the third party informs the players of the strategy assigned to them on the card (but not the strategy assigned to their opponent). Suppose a player is assigned D, he would not want to deviate supposing the other player played their assigned strategy since he will get 7 (the highest payoff possible). Suppose a player is assigned C. Then the other player will play C with probability 1/2 and D with probability 1/2. The expected utility of Daring is 0(1/2) + 7(1/2) = 3.5 and the expected utility of chickening out is 2(1/2) + 6(1/2) = 4. So, the player would prefer to Chicken out.

Since neither player has an incentive to deviate, this is a correlated equilibrium. The expected payoff for this equilibrium is 7(1/3) + 2(1/3) + 6(1/3) = 5 which is higher than the expected payoff of the mixed strategy Nash equilibrium.

The following correlated equilibrium has an even higher payoff to both players: Recommend (C, C) with probability 1/2, and (D, C) and (C, D) with probability 1/4 each. Then when a player is recommended to play C, she knows that the other player will play D with (conditional) probability 1/3 and C with probability 2/3, and gets expected payoff 14/3, which is equal (not less than) to the expected payoff when she plays D. In this correlated equilibrium, both players get 5.25 in expectation. It can be shown that this is the correlated equilibrium with maximal sum of expected payoffs to the two players.

Learning correlated equilibria

One of the advantages of correlated equilibria is that they are computationally less expensive than Nash equilibria. This can be captured by the fact that computing a correlated equilibrium only requires solving a linear program whereas solving a Nash equilibrium requires finding its fixed point completely.^[1] Another way of seeing this is that it is possible for two players to respond to each other's historical plays of a game and end up converging to a correlated equilibrium.^[2]

References

↑ Paul W. Goldberg and Christos H. Papadimitriou, "Reducibility Among Equilibrium Problems", ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, 2005.
↑ Foster, Dean P and Rakesh V. Vohra, "Calibrated Learning and Correlated Equilibrium" Games and Economic Behaviour (1996)

Sources

Aumann, Robert (1974) Subjectivity and correlation in randomized strategies. Journal of Mathematical Economics 1:67-96.
Aumann, Robert (1987) Correlated Equilibrium as an Expression of Bayesian Rationality. Econometrica 55(1):1-18.
Fudenberg, Drew and Jean Tirole (1991) Game Theory, MIT Press, 1991, ISBN 0-262-06141-4
Leyton-Brown, Kevin; Shoham, Yoav (2008), Essentials of Game Theory: A Concise, Multidisciplinary Introduction, San Rafael, CA: Morgan & Claypool Publishers, ISBN 978-1-59829-593-1 . An 88-page mathematical introduction; see Section 3.5. Free online at many universities.
Osborne, Martin J. and Ariel Rubinstein (1994). A Course in Game Theory, MIT Press. ISBN 0-262-65040-1 (a modern introduction at the graduate level)
Shoham, Yoav; Leyton-Brown, Kevin (2009), Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations, New York: Cambridge University Press, ISBN 978-0-521-89943-7 . A comprehensive reference from a computational perspective; see Sections 3.4.5 and 4.6. Downloadable free online.
Éva Tardos (2004) Class notes from Algorithmic game theory (note an important typo)
Iskander Karibzhanov. MATLAB code to plot the set of correlated equilibria in a two player normal form game
Noam Nisan (2005) Lecture notes from the course Topics on the border of Economics and Computation (lowercase u should be replaced by u_i)

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game Potential game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede 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 games 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 Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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