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Correlated equilibrium
A solution concept in game theory
Relationships
Superset of:	Nash equilibrium
Significance
Proposed by:	Robert Aumann
Example:	Chicken
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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 a strategy profile is chosen at random according to some distribution. If no player would want to deviate from the recommended strategy (assuming the others don't deviate), the distribution is called a correlated equilibrium.

Chicken

	D	C
D	0, 0	7, 2
C	2, 7	6, 6

Consider the game of chicken (pictured to the right). 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). 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. Interestingly, 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.

A probability distribution, is a correlated equilibrium if for all strategies s_i such that p(s_i) > 0 and every alternative strategy s_i'

where is i's utility function and S _{− i} is the set of all possible strategies that i's opponents might take.

• Aumann, Robert (1974) Subjectivity and correlation in randomized strategies. Journal of Mathematical Economics 1:67-96.
• Fudenberg, Drew and Jean Tirole (1991) Game Theory, MIT Press, 1991, ISBN 0-262-06141-4
• Tardos, Eva (2004) Class notes from Algorithmic game theory (note an important typo) [1]

view	Topics in game theory
Definitions	Normal form game · Extensive form game · Cooperative game · Information set · Preference
Equilibrium concepts	Nash equilibrium · Subgame perfection · Bayes-Nash · Trembling hand · Proper equilibrium · Epsilon-equilibrium · Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · ESS · Risk dominance
Strategies	Dominant strategies · Mixed strategy · Tit for tat · Grim trigger
Classes of games	Symmetric game · Perfect information · Dynamic game · Repeated game · Signaling game · Cheap talk · Zero-sum game · Mechanism design
Games	Prisoner's dilemma · Coordination game · Chicken · Battle of the sexes · Stag hunt · Matching pennies · Ultimatum game · Minority game · Rock, Paper, Scissors · Pirate game · Dictator game · Public goods game · Nash bargaining game
Theorems	Minimax theorem · Purification theorems · Folk theorem · Revelation principle · Arrow's Theorem
Related topics	Mathematics · Economics · Behavioral economics · Evolutionary game theory · Population genetics · Behavioral ecology · Adaptive dynamics · List of game theorists