Title: Nash solutions for games with bounded memory
Abstract: In [53 Smale considers a repeated game with bounded memory. In his model, the players only keep some kind of average of the past outcomes in their memory, and decisions are based on this memory. Thus, one has a dynamics on the outcome space of this game, depending on players’ memory strategies. A stationary point of this dynamics is called a solution of this game. Stability considerations about these solutions arise naturally in this setting. Nash solutions for a game with memory are defined in [5], which include the ordinary Nash solutions for the same game. It is proved that they are also Nash equilibria considered as an extended solution for the associated supergame. The first order necessary condition for Nash solutions is also obtained via the use of a kind of calculus of variations. In this paper, we investigate the following three general questions and a concrete model example: (1) Study first necessary conditions and second sufficient conditions for (local) Nash solutions. (2) Describe the game with memory strategies from a generic viewpoint. (3) Determine the payoff vectors associated to those Nash solutions. (4) Analyze the continuous prisoner’s dilemma in detail. The results of this paper provide satisfactory or nice answers to the above problems. Roughly speaking, we obtain the following: (1)