  We analyse an algorithm solving stochastic mean-payoff games, combining the
ideas of relative value iteration and of Krasnoselskii-Mann damping. We derive
parameterized complexity bounds for several classes of games satisfying
irreducibility conditions. We show in particular that an
$\epsilon$-approximation of the value of an irreducible concurrent stochastic
game can be computed in a number of iterations in $O(|\log\epsilon|)$ where the
constant in the $O(\cdot)$ is explicit, depending on the smallest non-zero
transition probabilities. This should be compared with a bound in
$O(|\epsilon|^{-1}|\log(\epsilon)|)$ obtained by Chatterjee and Ibsen-Jensen
(ICALP 2014) for the same class of games, and to a $O(|\epsilon|^{-1})$ bound
by Allamigeon, Gaubert, Katz and Skomra (ICALP 2022) for turn-based games. We
also establish parameterized complexity bounds for entropy games, a class of
matrix multiplication games introduced by Asarin, Cervelle, Degorre, Dima, Horn
and Kozyakin. We derive these results by methods of variational analysis,
establishing contraction properties of the relative Krasnoselskii-Mann
iteration with respect to Hilbert's semi-norm.
