Stationary ɛ-optimal strategies in stochastic games

We deal with stochastic games with finite state and action spaces for which we examine players' possibilities for playing limiting average (epsilon-)optimal by means of stationary strategies (epsilon > 0). It is well-known that stationary limiting average epsilon-optimal strategies need not exist for all initial states, hence we focus on those particular initial states for which the limiting average value is either maximal or minimal over the set of states.