An Accretive Operator Approach to Ergodic Problems for Zero-Sum Games

Abstract: Mean payoff stochastic games can be studied by means of a nonlinear spectral problem involving the Shapley operator: the ergodic equation. A solution consists in a scalar, called the ergodic constant, and a vector, called bias. The existence of such a pair entails that the mean payoff per time unit is equal to the ergodic constant for any initial state, and the bias gives stationary strategies. By exploiting two fundamental properties of Shapley operators, monotonicity and additive homogeneity, we give a necessary and sufficient condition for the solvability of the ergodic equation for all the Shapley operators obtained by perturbation of the transition payments of a given stochastic game with finite state space. If the latter condition is satisfied, we establish that the bias is unique (up to an additive constant) for a generic perturbation of the transition payments. To show these results, we use the theory of accretive operators, and prove in particular some surjectivity condition.