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DOI: 10.4230/LIPIcs.ESA.2022.41
A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games

Authors: Argyrios Deligkas, Michail Fasoulakis, and Evangelos Markakis

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute ε-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis [Tsaknakis and Spirakis, 2008], with an approximation guarantee of (0.3393+δ), remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a (1/3+δ)-Nash equilibrium, for any constant δ > 0. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of [Tsaknakis and Spirakis, 2008], and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.
Cite as

Argyrios Deligkas, Michail Fasoulakis, and Evangelos Markakis. A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{deligkas_et_al:LIPIcs.ESA.2022.41,
  author =	{Deligkas, Argyrios and Fasoulakis, Michail and Markakis, Evangelos},
  title =	{{A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{41:1--41:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.41},
  URN =		{urn:nbn:de:0030-drops-169790},
  doi =		{10.4230/LIPIcs.ESA.2022.41},
  annote =	{Keywords: bimatrix games, approximate Nash equilibria}
}
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