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

Authors Argyrios Deligkas, Michail Fasoulakis, Evangelos Markakis

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  • Theory of computation → Exact and approximate computation of equilibria
Keywords
  • bimatrix games
  • approximate Nash equilibria

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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.

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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) https://doi.org/10.4230/LIPIcs.ESA.2022.41

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

Funding

  • Markakis, Evangelos: This work has been supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the "First Call for H.F.R.I. Research Projects to support faculty members and researchers and the procurement of high-cost research equipment" grant (Project Number: HFRI-FM17-3512).

Acknowledgements

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

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