Chapter summary

This chapter introduces the concept of correlated equilibrium in strategic-form games. The motivation for this concept is that players' choices of pure strategies may be correlated due to the fact that they use the same random events in deciding which pure strategy to play. Consider an extended game that includes an observer who recommends to each player a pure strategy that he should play. The vector of recommended strategies is chosen by the observer according to a probability distribution over the set of pure strategy vectors, which is commonly known among the players. This probability distribution is called a correlated equilibrium if the strategy vector in which all players follow the observer's recommendations is a Nash equilibrium of the extended game.

The probability distribution over the set of strategy vectors induced by any Nash equilibrium is a correlated equilibrium. The set of correlated equilibria is a polytope that can be calculated as a solution of a set of linear equations.

In Chapters 4, 5, and 7 we considered strategic-form games and studied the concept of equilibrium. One of the underlying assumptions of those chapters was that the choices made by the players were independent. In practice, however, the choices of players may well depend on factors outside the game, and therefore these choices may be correlated. Players can even coordinate their actions among themselves.

A good example of such correlation is the invention of the traffic light: when a motorist arrives at an intersection, he needs to decide whether to cross it, or alternatively to give right of way to motorists approaching the intersection from different directions.