research-article

Settling the complexity of computing two-player Nash equilibria

Authors:

Shang-Hua TengAuthors Info & Claims

Journal of the ACM (JACM), Volume 56, Issue 3

Article No.: 14, Pages 1 - 57

https://doi.org/10.1145/1516512.1516516

Published: 19 May 2009 Publication History

Abstract

We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991.

Our result, building upon the work of Daskalakis et al. [2006a] on the complexity of four-player Nash equilibria, settles a long standing open problem in algorithmic game theory. It also serves as a starting point for a series of results concerning the complexity of two-player Nash equilibria. In particular, we prove the following theorems:

—Bimatrix does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time.

—The smoothed complexity of the classic Lemke-Howson algorithm and, in fact, of any algorithm for Bimatrix is not polynomial unless every problem in PPAD is solvable in randomized polynomial time.

Our results also have a complexity implication in mathematical economics:

—Arrow-Debreu market equilibria are PPAD-hard to compute.

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Index Terms

Settling the complexity of computing two-player Nash equilibria
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image Journal of the ACM

Journal of the ACM Volume 56, Issue 3

May 2009

328 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/1516512

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Copyright © 2009 ACM.

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Publication History

Published: 19 May 2009

Accepted: 01 January 2009

Revised: 01 September 2008

Received: 01 April 2007

Published in JACM Volume 56, Issue 3

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Kapron BSamieefar KDastani MSichman JAlechina NDignum V(2024)On the Computational Complexity of Quasi-Variational Inequalities and Multi-Leader-Follower GamesProceedings of the 23rd International Conference on Autonomous Agents and Multiagent Systems10.5555/3635637.3663148(2324-2326)Online publication date: 6-May-2024
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Yardim BGoldman AHe NDastani MSichman JAlechina NDignum V(2024)When is Mean-Field Reinforcement Learning Tractable and Relevant?Proceedings of the 23rd International Conference on Autonomous Agents and Multiagent Systems10.5555/3635637.3663068(2038-2046)Online publication date: 6-May-2024
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