The problem of the existence and determining stationary Nash equilibria for stochastic positional games with discounted and average payoffs is considered. We show that, for a stochastic positional game with discounted payoffs, there exists a Nash equilibrium in pure stationary strategies and, for a stochastic positional game with average payoffs, there exists a Nash equilibrium in mixed stationary strategies. Some approaches for determining pure and mixed stationary equilibria in such games are proposed.
