Correlated Equilibrium - Formal Definition

Formal Definition

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 expected utility via a strategy modification.