Nonzero-Sum Stochastic Games with Probability Criteria

In this paper, we consider two-person nonzero-sum discrete-time stochastic games under the probability criterion. Taking lambda for player 1 and mu for player 2 as their profit goal, the two players are concerned with the probabilities that the rewards they earn before the first passage to some target state set are more than lambda and mu respectively. We firstly give a characterization of the probabilities, and then, under a mild condition, we show that the optimal value function for each player is the unique solution to the corresponding optimality equation by an iterative approximation, and then establish the existence of Nash equilibria. Finally, a queueing system is provided to show the application of our main result.