Abstract.

Since the seminal PPAD-completeness result for computing a Nash equilibrium even in two-player games, an important line of research has focused on relaxations achievable in polynomial time. In this paper, we consider the notion of an \(\varepsilon\) -well-supported Nash equilibrium, where \(\varepsilon \in [0,1]\) corresponds to the approximation guarantee. Put simply, in an \(\varepsilon\) -well-supported equilibrium, every player chooses with positive probability actions that are within \(\varepsilon\) of the maximum achievable payoff against the other player’s strategy. Ever since the initial approximation guarantee of 2/3 for well-supported equilibria, which was established more than a decade ago, the progress on this problem has been extremely slow and incremental. Notably, the small improvements to 0.6608, and finally to 0.6528, were achieved by algorithms of growing complexity. Our main result is a simple and intuitive algorithm that improves the approximation guarantee to 1/2. Our algorithm is based on linear programming and in particular on exploiting suitably defined zero-sum games that arise from the payoff matrices of the two players. As a byproduct, we show how to achieve the same approximation guarantee in a query-efficient way.

Keywords

  1. Nash equilibria
  2. well-supported Nash equilibria
  3. query complexity
  4. bimatrix games

MSC codes

  1. 68Q25
  2. 91A05

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Information & Authors

Information

Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 1083 - 1096
ISSN (online): 1095-7111

History

Submitted: 24 January 2023
Accepted: 16 May 2023
Published online: 12 September 2023

Keywords

  1. Nash equilibria
  2. well-supported Nash equilibria
  3. query complexity
  4. bimatrix games

MSC codes

  1. 68Q25
  2. 91A05

Authors

Affiliations

Royal Holloway, University of London - Egham Hill, Egham Surrey TW20 0EX, United Kingdom.
Michail Fasoulakis
Foundation for Research and Technology-Hellas FORTH, GR - 700 13, Heraklion, Crete.
Athens University of Economics and Business, Athina 104 34, Greece.
Evangelos Markakis
Athens University of Economics and Business, Athina 104 34, Greece.
Input Output Global (IOG).

Funding Information

Funding: The research of the second and third authors was 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 HFRI-FM17-3512).

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