A Polynomial-Time Algorithm for 1/2-Well-Supported Nash Equilibria in Bimatrix Games
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.
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© 2023 Society for Industrial and Applied Mathematics.
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Submitted: 24 January 2023
Accepted: 16 May 2023
Published online: 12 September 2023
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Hellenic Foundation for Research and Innovation (HFRI): HFRI-FM17-3512
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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