In a recent work, the authors considered a finite state Markov ratio decision process in which the objective was to maximize the ratio of total discounted rewards. In this paper, discounted Markov ratio decision processes are generalized to discounted stochastic ratio games. These may also be viewed as generalizations of ratio games to a stochastic context where the payoff is the ratio of the two total discounted rewards. We show that in the discounted stochastic ratio game the players have stationary optimal strategies with a unique value. The solution may depend on the initial probability distribution. We also provide a convergent algorithm.