On Completely Mixed Stochastic Games

In this paper, we consider a zero-sum undiscounted stochastic game which has finite state space and finitely many pure actions. Also, assume the transition probability of is controlled by one player all optimal strategies are strictly positive. Under above assumptions, show that $\beta$-discounted games with same payoff matrices $\beta$ sufficiently close to 1 also completely mixed. We give counterexample converse result in not true. that, if have non-zero value some for then possess nonzero state.