Smoothed Analysis of Deterministic Discounted and Mean-Payoff Games

Authors Bruno Loff , Mateusz Skomra

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Author Details

Bruno Loff
  • LASIGE, Faculdade de Ciências, Universidade de Lisboa, Portugal
Mateusz Skomra
  • LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, France


MS would like to thank Xavier Allamigeon, Stéphane Gaubert, and Ricardo D. Katz for many useful discussions on mean-payoff games, policy iteration, and the operator approach, for exchanging ideas about the problem of smoothed analysis, for their remarks on a preliminary version of this paper, and for being a perpetual source of friendship and inspiration.

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Bruno Loff and Mateusz Skomra. Smoothed Analysis of Deterministic Discounted and Mean-Payoff Games. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 147:1-147:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We devise a policy-iteration algorithm for deterministic two-player discounted and mean-payoff games, that runs in polynomial time with high probability, on any input where each payoff is chosen independently from a sufficiently random distribution and the underlying graph of the game is ergodic. This includes the case where an arbitrary set of payoffs has been perturbed by a Gaussian, showing for the first time that deterministic two-player games can be solved efficiently, in the sense of smoothed analysis. More generally, we devise a condition number for deterministic discounted and mean-payoff games played on ergodic graphs, and show that our algorithm runs in time polynomial in this condition number. Our result confirms a previous conjecture of Boros et al., which was claimed as a theorem [Boros et al., 2011] and later retracted [Boros et al., 2018]. It stands in contrast with a recent counter-example by Christ and Yannakakis [Christ and Yannakakis, 2023], showing that Howard’s policy-iteration algorithm does not run in smoothed polynomial time on stochastic single-player mean-payoff games. Our approach is inspired by the analysis of random optimal assignment instances by Frieze and Sorkin [Frieze and Sorkin, 2007], and the analysis of bias-induced policies for mean-payoff games by Akian, Gaubert and Hochart [Akian et al., 2018].

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • Mean-payoff games
  • discounted games
  • policy iteration
  • smoothed analysis


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