Solving Irreducible Stochastic Mean-Payoff Games and Entropy Games by Relative Krasnoselskii-Mann Iteration

Authors Marianne Akian, Stéphane Gaubert, Ulysse Naepels, Basile Terver

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Marianne Akian
  • INRIA and CMAP, École polytechnique, IP Paris, CNRS, France
Stéphane Gaubert
  • INRIA and CMAP, École polytechnique, IP Paris, CNRS, France
Ulysse Naepels
  • École polytechnique, IP Paris, France
Basile Terver
  • École polytechnique, IP Paris, France


We thank the reviewers for helpful comments.

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Marianne Akian, Stéphane Gaubert, Ulysse Naepels, and Basile Terver. Solving Irreducible Stochastic Mean-Payoff Games and Entropy Games by Relative Krasnoselskii-Mann Iteration. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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(|log(ε)|) 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • Stochastic mean-payoff games
  • concurrent games
  • entropy games
  • relative value iteration
  • Krasnoselskii-Mann fixed point algorithm
  • Hilbert projective metric


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