  Recent results of Ye and Hansen, Miltersen and Zwick show that policy
iteration for one or two player (perfect information) zero-sum stochastic
games, restricted to instances with a fixed discount rate, is strongly
polynomial. We show that policy iteration for mean-payoff zero-sum stochastic
games is also strongly polynomial when restricted to instances with bounded
first mean return time to a given state. The proof is based on methods of
nonlinear Perron-Frobenius theory, allowing us to reduce the mean-payoff
problem to a discounted problem with state dependent discount rate. Our
analysis also shows that policy iteration remains strongly polynomial for
discounted problems in which the discount rate can be state dependent (and even
negative) at certain states, provided that the spectral radii of the
nonnegative matrices associated to all strategies are bounded from above by a
fixed constant strictly less than 1.
