  Mean-payoff zero-sum stochastic 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 $T(u)=\lambda e + u$, 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, 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 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 (or tropical) algebra
and nonlinear Perron-Frobenius theory. As an application of our results, we
obtain a perturbation scheme allowing one to solve degenerate instances of
stochastic games by policy iteration.
