Repeated Games with Incomplete Information over Predictable Systems

Abstract

Consider a stationary process taking values in a finite state space. Each state is associated with a finite one-shot zero-sum game. We investigate the infinitely repeated zero-sum game with incomplete information on one side in which the state of the game evolves according to the stationary process. Two players, named the observer and the adversary, play the following game. At the beginning of any stage, only the observer is informed of the state ξn and is therefore the only one who knows the identity of the forthcoming one-shot game. Then, both players take actions, which become publicly known. The paper shows the existence of a uniform value in a new class of stationary processes: ergodic Kronecker systems. Techniques from ergodic theory, probability theory, and game theory are employed to describe the optimal strategies of the two players. Funding: This work was supported by the Israel Science Foundation [Grant 591/21] and Deutsche Forschungsgemeinschaft [Grant KA 5609/1-1].