  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.
