A mathematical programming based characterization of Nash equilibria of some constrained stochastic games

We consider two classes of constrained finite state-action stochas- tic games. First, we consider a two player nonzero sum single controller con- strained stochastic game with both average and discounted cost criterion. We consider the same type of constraints as in (1), i.e., player 1 has subscrip- tion based constraints and player 2, who controls the transition probabilities, has realization based constraints which can also depend on the strategies of player 1. Next, we consider a N-player nonzero sum constrained stochastic game with independent state processes where each player has average cost cri- terion as discussed in (2). We show that the stationary Nash equilibria of both classes of constrained games, which exists under strong Slater and irreducibil- ity conditions (3), (2), has one to one correspondence with global minima of certain mathematical programs. In the single controller game if the constraints of player 2 do not depend on the strategies of the player 1, then the mathe- matical program reduces to the non-convex quadratic program. In two player independent state processes stochastic game if the constraints of a player do not depend on the strategies of another player, then the mathematical pro- gram reduces to a non-convex quadratic program. Computational algorithms for finding global minima of non-convex quadratic program exist (4), (5) and hence, one can compute Nash equilibria of these constrained stochastic games. Our results generalize some existing results for zero sum games (1), (6), (7). ⋆ A portion of Section 3 (two player case) has been presented in the 8th International