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Correlated equilibrium
	A solution concept in game theory
Relationships
Superset of:	Nash equilibrium
Significance
Proposed by:	Robert Aumann
Example:	Chicken
	v • d • e

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 her action according to 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.

[edit] Formal definition

$\displaystyle(\Omega, \pi)$ be a countable probability space. For each player $\displaystyle i$ , let $\displaystyle P_i$ be his information partition, $\displaystyle q_i$ be $\displaystyle i$ 's posterior and let $\displaystyle s_i:\Omega\rightarrow A_i$ , assigning the same value to states in the same cell of $\displaystyle i$ 's information partition.

A pair $\displaystyle((\Omega, \pi),P_i)$ is a correlated equilibrium of the strategic game $\displaystyle (N,A_i,u_i)$ if for every player $\displaystyle i$ and for every strategies $\displaystyle s'_i$ we have:

$\displaystyle\sum_{\omega \in \Omega} q(\omega)u_i(s_{-i},s_i) \geq \sum_{\omega \in \Omega} q(\omega)u_i(s_{-i},s'_i)$

where $\displaystyle u_i(\cdot)$ is $\displaystyle i$ 's utility function and $\displaystyle S_{-i}$ is the set of all possible profile of strategies that the other players might take.

[edit] An example

*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.

[edit] References

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 978-1-598-29593-1, http://www.gtessentials.org . 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, http://www.masfoundations.org . A comprehensive reference from a computational perspective; see Sections 3.4.5 and 4.6. Downloadable free online.
Tardos, Eva (2004) Class notes from Algorithmic game theory (note an important typo) [1]
Iskander Karibzhanov. MATLAB code to plot the set of correlated equilibria in a two player normal form game