Noncooperative Game in Cooperation : Reformulation of Correlated Equilibria

Abstract

In the usual Nash equilibrium of strategic noncooperative games, the mixed strategies of all the players are assumed to be stochastically independent. In order to relax this stochastic independence, two essentially different formulations are proposed. In this paper, by using random variables defined on an abstract and universal probability space, I introduce two equilibrium concepts, one is for the framework of a noncooperative game in which a third person, a mediator, influences each player's strategy independently or plays the role of a publicly observable random device. The other concept is for the framework of a noncooperative game in which the restriction of stochastic independence is removed, which could imply possible communications between players. Consequently, we have two different theorems that characterize all the strategies in the correlated equilibrium for each framework. These two theorems show that the two equilibrium concepts, which we shall call “exogenous correlated equilibrium” and “endogenous correlated equilibrium” are entirely different concepts. Finally, we provide some comments on previous related studies 2).