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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

Funding

Funded by the European Union (ERC, HOFGA, 101041696). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Also supported by FCT through the LASIGE Research Unit, ref. UIDB/00408/2020 and ref. UIDP/00408/2020.

Acknowledgements

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) https://doi.org/10.4230/LIPIcs.ICALP.2024.147

Abstract

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
Keywords
  • Mean-payoff games
  • discounted games
  • policy iteration
  • smoothed analysis

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