Pure and Mixed Stationary Nash Equilibria for Average Stochastic Positional Games

Abstract

In this paper we study the problem of the existence of Nash equilibria in pure and mixed stationary strategies for average stochastic positional games. We prove the existence of Nash equilibria in pure stationary strategies for an arbitrary two-player zero-sum average stochastic positional game and for an m-player average stochastic positional game in the case when the probability transition matrices induced by any profile of stationary strategies of the players are unichain. For the general case of an m-player average stochastic positional game we show that a Nash equilibrium in pure stationary strategies may not exist, however a Nash equilibrium always exists in mixed stationary strategies. Based on the proof of these results we present conditions for determining the optimal strategies of the players in the considered average stochastic positional games.