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Mean-payoff zero-sum stochastic games can be studied by means of a nonlinearspectral problem. When the state space is finite, the latter consists infinding an eigenpair $(u,\lambda)$ solution of $T(u)=\lambda e + u$, where$T:\mathbb{R}^n \to \mathbb{R}^n$ is the Shapley (or dynamic programming)operator, $\lambda$ is a scalar, $e$ is the unit vector, and $u \in\mathbb{R}^n$. The scalar $\lambda$ yields the mean payoff per time unit, andthe vector $u$, called the bias, allows one to determine optimal stationarystrategies. The existence of the eigenpair $(u,\lambda)$ is generally relatedto ergodicity conditions. A basic issue is to understand for which classes ofgames the bias vector is unique (up to an additive constant). In this paper, weconsider perfect-information zero-sum stochastic games with finite state andaction spaces, thinking of the transition payments as variable parameters,transition probabilities being fixed. We show that the bias vector, thought ofas a function of the transition payments, is generically unique (up to anadditive constant). The proof uses techniques of max-plus (or tropical) algebraand nonlinear Perron-Frobenius theory. As an application of our results, weobtain a perturbation scheme allowing one to solve degenerate instances ofstochastic games by policy iteration.

Generic uniqueness of the bias vector of finite stochastic games with perfect information