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Correlated equilibrium
 
 
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In game theory
Game theory

Game theory is a branch of applied mathematics that is used in the social sciences , biology, engineering, political science, international relations, computer science , and philosophy....
, a correlated equilibrium is a solution concept
Solution concept

In game theory, a solution concept is a formal rule for predicting how the game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players, therefore predicting the result of the game....
 that is more general than the well known Nash equilibrium
Nash equilibrium

In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally....
. It was first discussed by mathematician Robert Aumann
Robert Aumann

Robert John Aumann is an Israeli mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel....
 (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.
Formal definition
be a countable probability space
Probability space

A probability space, in probability theory, is the conventional mathematical model of randomness. This mathematical object, sometimes called also probability triple, formalizes three interrelated ideas by three mathematical notions....
.






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In game theory
Game theory

Game theory is a branch of applied mathematics that is used in the social sciences , biology, engineering, political science, international relations, computer science , and philosophy....
, a correlated equilibrium is a solution concept
Solution concept

In game theory, a solution concept is a formal rule for predicting how the game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players, therefore predicting the result of the game....
 that is more general than the well known Nash equilibrium
Nash equilibrium

In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally....
. It was first discussed by mathematician Robert Aumann
Robert Aumann

Robert John Aumann is an Israeli mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel....
 (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.

Formal definition


be a countable probability space
Probability space

A probability space, in probability theory, is the conventional mathematical model of randomness. This mathematical object, sometimes called also probability triple, formalizes three interrelated ideas by three mathematical notions....
. For each player , let be his information partition, be 's posterior and let , assigning the same value to states in the same cell of 's information partition.

A pair is a correlated equilibrium of the strategic game if for every player and for every strategies we have:

where is 's utility function and is the set of all possible profile of strategies that the other players might take.

An example



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.