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.

翻译：我们分析了一种求解随机平均收益博弈的算法，该算法融合了相对值迭代与Krasnoselskii-Mann阻尼的思想。针对满足不可约性条件的若干博弈类别，我们推导了参数化复杂度界。特别地，我们证明：对于不可约并发随机博弈，其值的$\epsilon$近似解可在$O(|\log\epsilon|)$次迭代内计算得出，其中$O(\cdot)$中的常数为显式表达式，依赖于最小非零转移概率。这一结果需与Chatterjee和Ibsen-Jensen（ICALP 2014）针对同一类博弈得到的$O(|\epsilon|^{-1}|\log(\epsilon)|)$界，以及Allamigeon、Gaubert、Katz和Skomra（ICALP 2022）针对回合制博弈得到的$O(|\epsilon|^{-1})$界进行比较。我们进一步为熵博弈建立了参数化复杂度界——该博弈是一类由Asarin、Cervelle、Degorre、Dima、Horn和Kozyakin提出的矩阵乘法博弈。通过变分分析方法，我们在Hilbert半范数意义下证明了相对Krasnoselskii-Mann迭代的压缩性，从而推导出上述结果。