The main objective of this paper is to find structural conditions under which a stochastic game between two players with total reward functions has an $\epsilon$-equilibrium. To reach this goal, the results of Markov decision processes are used to find $\epsilon$-optimal strategies for each player and then the correspondence of a better answer as well as a more general version of Kakutani's Fixed Point Theorem to obtain the $\epsilon$-equilibrium mentioned. Moreover, two examples to illustrate the theory developed are presented.
Nash equilibrium, Markov decision processes, stochastic games, total rewards
91A15, 91A50, 90C40