On Nash equilibria for stochastic games and determining the optimal strategies of the players

We consider $n$-person stochastic games in the sense of Shapley. The main results of the paper are related to the existence of Nash equilibria and determining the optimal stationary strategies of the players in the considered games. We show that a Nash equilibrium for the stochastic game with average payoff functions of the players exists if an arbitrary situation induces an ergodic Markov chain. For the stochastic game with discounted payoff functions we show that a Nash equilibrium always exists. Some approaches for determining Nash equilibria in the considered games are proposed.