Solving irreducible stochastic mean-payoff games and entropy games by relative Krasnoselskii-Mann iteration

TL;DR

This paper introduces an algorithm combining relative value iteration and Krasnoselskii-Mann damping to efficiently approximate values in irreducible stochastic mean-payoff and entropy games, with explicit complexity bounds.

Contribution

It provides the first parameterized complexity bounds for irreducible stochastic mean-payoff and entropy games using a novel iteration method.

Findings

Approximate game values in $O(| ext{log} \\epsilon|)$ iterations for irreducible stochastic games.

Derived explicit constants depending on minimal non-zero transition probabilities.

Established contraction properties of the iteration method via variational analysis.

Abstract

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.