Communication Equilibria in Repeated Games with Incomplete Information

Correlated equilibria (in the sense of Aumann, i.e., normal form correlated equilibria) are studied in two-person infinitely repeated games with lack of information on one side. Extensions of the solution concept are introduced, several of which are shown to be payoff equivalent in the model under consideration.

A normal form correlated equilibrium is a Nash equilibrium of the game where each player gets a private output from a correlation device before the beginning of the original game. An r-device (r = 0, 1, …, ∞) is a communication device, receiving an input from each player before every stage t = 1, …, r and selecting a pair of outputs, one for each player, at every stage t = 1, 2, …, according to a memory dependent transition probability. An r-communication equilibrium is a Nash equilibrium of the game extended by means of an r-device. Let C (resp. D_r, r = 0, 1, …, ∞) be the set of normal form correlated (resp. r-communication) equilibrium payoffs. D₀ is the set of “extensive form correlated equilibrium payoffs,” achieved by means of an “autonomous device,” acting only by sending a stream of outputs to each player. And D_∞ is the set of all communication equilibrium payoffs, corresponding to communication devices which can receive inputs at every stage (even arbitrarily large). The paper characterizes the different sets and investigate whether they coincide.