  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.
