Generic uniqueness of the bias vector of mean payoff zero-sum games

Abstract

Zero-sum mean payoff games can be studied by means of a nonlinear spectral problem. When the state space is finite, the latter consists in finding an eigenpair $(u,\lambda)$ solution of $T(u)=\lambda \mathbf{1} + u$ where $T:\mathbb{R}^n \to \mathbb{R}^n$ is the Shapley (dynamic programming) operator, $\lambda$ is a scalar, $\mathbf{1}$ is the unit vector, and $u \in \mathbb{R}^n$. The scalar $\lambda$ yields the mean payoff per time unit, and the vector $u$, called the bias, allows one to determine optimal stationary strategies. The existence of the eigenpair $(u,\lambda)$ is generally related to ergodicity conditions. A basic issue is to understand for which classes of games the bias vector is unique (up to an additive constant). In this paper, we consider perfect information zero-sum stochastic games with finite state and action spaces, thinking of the transition payments as variable parameters, transition probabilities being fixed. We identify structural conditions on the support of the transition probabilities which guarantee that the spectral problem is solvable for all values of the transition payments. Then, we show that the bias vector, thought of as a function of the transition payments, is generically unique (up to an additive constant). The proof uses techniques of max-plus (tropical) algebra and nonlinear Perron-Frobenius theory.