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.

Learning correlated equilibria

One of the advantages of correlated equilibria is that they are computationally less expensive than are 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

1. Paul W. Goldberg and Christos H. Papadimitriou, "Reducibility Among Equilibrium Problems", ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, 2005.
2. 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)