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DOI: 10.4230/LIPIcs.MFCS.2021.12

A Generic Strategy Improvement Method for Simple Stochastic Games

Authors: David Auger, Xavier Badin de Montjoye, and Yann Strozecki

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

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].

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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)

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@InProceedings{auger_et_al:LIPIcs.MFCS.2021.12,
  author =	{Auger, David and Badin de Montjoye, Xavier and Strozecki, Yann},
  title =	{{A Generic Strategy Improvement Method for Simple Stochastic Games}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{12:1--12:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.12},
  URN =		{urn:nbn:de:0030-drops-144524},
  doi =		{10.4230/LIPIcs.MFCS.2021.12},
  annote =	{Keywords: Simple Stochastic Games, Strategy Improvement, Parametrized Complexity, Stopping, Meta Algorithm, f-strategy}
}

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A Generic Strategy Improvement Method for Simple Stochastic Games

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