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Concept
Correlated equilibrium
Summary
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 in 1974. The idea is that each player chooses their action according to their private 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 their strategy (assuming the others also don't deviate), the distribution from which the signals are drawn is called a correlated equilibrium.
An -player strategic game is characterized by an action set and utility function for each player . When player chooses strategy and the remaining players choose a strategy profile described by the -tuple , then player 's utility is .
A strategy modification for player is a function . That is, tells player to modify his behavior by playing action when instructed to play .
Let be a countable probability space. 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. Then is a correlated equilibrium of the strategic game if for every player and for every strategy modification :
In other words, is a correlated equilibrium if no player can improve his or her expected utility via a strategy modification.
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 both players chicken out with probability 2/3.
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