Game Theory Questions Medium
In game theory, correlated equilibrium is a solution concept that extends the notion of Nash equilibrium by allowing players to use randomization or correlation devices to coordinate their actions.
In a correlated equilibrium, players receive a recommendation or a signal from a trusted third party, such as a mediator or a random device, which suggests a specific action for each player to take. These recommendations are based on the players' private information and are designed to ensure that each player's recommended action is in their best interest, given their beliefs about the other players' actions.
Unlike in a Nash equilibrium, where players independently choose their actions without any external coordination, in a correlated equilibrium, players can coordinate their actions based on the recommendations they receive. However, it is important to note that the recommendations do not guarantee any specific outcome, but rather provide a probabilistic distribution of actions that players should follow.
To illustrate this concept, let's consider a simple example of a two-player game: the Prisoner's Dilemma. In this game, two prisoners are arrested for a crime and are given the option to either cooperate with each other or betray each other. If both players cooperate, they receive a moderate sentence. If both players betray each other, they receive a harsh sentence. However, if one player cooperates while the other betrays, the betrayer receives a lenient sentence while the cooperator receives a severe sentence.
In a correlated equilibrium, a trusted third party could provide each player with a signal that suggests cooperating with a certain probability, say 70%. Based on this signal, each player would then independently decide whether to cooperate or betray, taking into account their own beliefs and the signal they received. By following the recommended actions, the players can achieve a correlated equilibrium where the probability of both players cooperating is higher than the probability of both players betraying.
Overall, correlated equilibrium allows for the possibility of coordination and cooperation among players in a game, even in situations where a Nash equilibrium may not exist. It provides a framework for analyzing strategic interactions when players have access to external information or coordination devices.