Infinite Horizon Potential Difference Games

Abstract This paper concerns a class of deterministic infinite-horizon noncooperative difference (or discrete-time) games with discounted payoffs. Our main objective is to give conditions under which these are potential games . Hence, for each game in this class, there is an optimal control problem (OCP) whose optimal solutions are Nash equilibria for the given game. To this end, we propose an algorithm that provides first-order and separability conditions on the payoff’s stage reward functions, guaranteeing that a given function, which determines the corresponding OCP, is a potential function for the game. We show that for some difference games, these first-order conditions guarantee that the open-loop structure of the game’s Nash equilibria is contained in the open-loop structure of optimal solutions of the OCP determined by the potential. We also show that considering feedback controls instead of open-loop policies significantly reduces the classes of games that can be shown to be potential.