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