A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games

Authors Argyrios Deligkas, Michail Fasoulakis, Evangelos Markakis

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Author Details

Argyrios Deligkas
  • Royal Holloway, University of London, Egham, UK
Michail Fasoulakis
  • Foundation for Research and Technology-Hellas (FORTH), Heraklion, Greece
  • Athens University of Economics and Business, Greece
Evangelos Markakis
  • Athens University of Economics and Business, Greece


The authors would like to thank Marcin Jurdzi{ń}ski for some initial discussions on the Tsaknakis-Spirakis algorithm.

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Exact and approximate computation of equilibria
  • bimatrix games
  • approximate Nash equilibria


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