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A Generic Strategy Improvement Method for Simple Stochastic Games

Authors David Auger, Xavier Badin de Montjoye, Yann Strozecki

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Author Details

David Auger
  • Université Paris Saclay, UVSQ, DAVID, France
Xavier Badin de Montjoye
  • Université Paris Saclay, UVSQ, DAVID, France
Yann Strozecki
  • Université Paris Saclay, UVSQ, DAVID, France

Acknowledgements

The authors want to thank Pierre Coucheney for many interesting discussions on SSGs.

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David Auger, Xavier Badin de Montjoye, and Yann Strozecki. A Generic Strategy Improvement Method for Simple Stochastic Games. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.MFCS.2021.12

Abstract

We present a generic strategy improvement algorithm (GSIA) to find an optimal strategy of simple stochastic games (SSG). We prove the correctness of GSIA, and derive a general complexity bound, which implies and improves on the results of several articles. First, we remove the assumption that the SSG is stopping, which is usually obtained by a polynomial blowup of the game. Second, we prove a tight bound on the denominator of the values associated to a strategy, and use it to prove that all strategy improvement algorithms are in fact fixed parameter tractable in the number r of random vertices. All known strategy improvement algorithms can be seen as instances of GSIA, which allows to analyze the complexity of converge from below by Condon [Condon, 1993] and to propose a class of algorithms generalising Gimbert and Horn’s algorithm [Gimbert and Horn, 2008; Gimbert and Horn, 2009]. These algorithms terminate in at most r! iterations, and for binary SSGs, they do less iterations than the current best deterministic algorithm given by Ibsen-Jensen and Miltersen [Ibsen-Jensen and Miltersen, 2012].

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
Keywords
  • Simple Stochastic Games
  • Strategy Improvement
  • Parametrized Complexity
  • Stopping
  • Meta Algorithm
  • f-strategy

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