Solving irreducible stochastic mean-payoff games and entropy games by relative Krasnoselskii-Mann iteration
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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 ϵ-approximation of the value of an irreducible concurrent stochastic game can be computed in a number of iterations in O(|logϵ|) where the constant in the O(·) is explicit, depending on the smallest non-zero transition probabilities. This should be compared with a bound in O(|ϵ|^-1|log(ϵ)|) obtained by Chatterjee and Ibsen-Jensen (ICALP 2014) for the same class of games, and to a O(|ϵ|^-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.
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