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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 β-discounted stochastic games with same payoff matrices and β sufficiently close to 1 are also completely mixed. We also provide a necessary condition under which the individual matrix games are completely mixed. We also show that, if we have non-zero value in some state for the undiscounted stochastic game then for β sufficiently close to 1 the β-discounted stochastic game also possess nonzero value in the same state.

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In this paper, we consider a two-person finite state stochastic games with finite number of pure actions for both players in all the states. In particular, for a large number of results we also consider one-player controlled transition probability and show that if all the optimal strategies of the undiscounted stochastic game are completely mixed then for $$\beta$$ β sufficiently close to 1; all the optimal strategies of $$\beta$$ β -discounted stochastic games are also completely mixed. A counterexample is provided to show that the converse is not true. Further, for single-player controlled completely mixed stochastic games if the individual payoff matrices are symmetric in each state, then we show that the individual matrix games are also completely mixed. For the non-zerosum single-player controlled stochastic game under some non-singularity conditions, we show that if the undiscounted game is completely mixed, then the Nash equilibrium is unique. For non-zerosum $$\beta$$ β -discounted stochastic games when Nash equilibrium exists, we provide equalizer rules for corresponding value of the game.