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# Settling the Complexity of 2-Player Nash-Equilibrium

Electronic Colloquium on Computational Complexity (ECCC), (2005)

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Abstract

We prove that nding a solution of two player Nash Equilibrium is PPAD-complete.

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Introduction

- In 1944, Morgenstern and von Neumann [43] initiated the study of game theory and its applications to economic behavior.
- In [46], Papadimitriou proved that Bimatrix, the problem of finding a Nash equilibrium in a two-player game with rational payoffs is member of PPAD.
- In a complexity-theoretic breakthrough, Daskalakis, Goldberg and Papadimitriou [18] proved that the problem of computing a Nash equilibrium in a game among four or more players is complete for PPAD.

Highlights

- In 1944, Morgenstern and von Neumann [43] initiated the study of game theory and its applications to economic behavior
- Bimatrix does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time
- The P-Matrix Linear Complementary Problem is computationally harder than convex programming unless every problem in PPAD is solvable in polynomial time
- Because the two-player Nash equilibrium enjoys several structural properties that Nash equilibria with three or more players do not have, our result enables us to answer some other long-standing open questions in mathematical economics and operations research
- If Bimatrix is in smoothed polynomial time under uniform or Gaussian perturbations, for all ǫ > 0, there exists a randomized algorithm to compute an ǫ-approximate Nash equilibrium in a two-player game with expected time O poly(m, n, 1/ǫ) or O poly(m, n, log max(m, n)/ǫ), respectively
- There remains a complexity gap in the approximation of two-player Nash equilibria: Lipton, Markakis and Mehta [40] show that an ǫ-approximate Nash equilibrium can be computed in nO-time, while this paper shows that, for ǫ of order 1/poly(n), no algorithm can find an ǫ-approximate Nash equilibrium in poly(n, 1/ǫ)-time, unless PPAD is contained in P

Results

- The authors show that from each discrete Brouwer function f , the authors can build a two-player game G and a polynomial-time map Π from the Nash equilibria of G to the fixed points of f .
- For any c > 0, the problem of computing an n−c-approximate Nash equilibrium of a two-player game is PPAD-complete.
- For any c > 0, the problem of finding the first (1 + c) log n bits of an exact Nash equilibrium in a two-player game, even when the payoffs are integers of polynomial magnitude, is polynomial-time equivalent to Bimatrix.
- Let Expc-Bimatrix denote the following search problem: Given a rational and positively normalized bimatrix game (A, B), compute a 2−cn-approximate Nash equilibrium of (A, B), if A and B are n × n matrices;
- Let Polyc-Bimatrix denote the following search problem: Given a rational and positively normalized bimatrix game (A, B), compute an n−c-approximate Nash equilibrium of (A, B), if A and B are n × n matrices.
- For a positive integer P , the authors will use P -Bit-Bimatrix to denote the search problem of computing the first P bits of the entries of a Nash equilibrium in a rational bimatrix game.
- The authors define the perturbation models in the smoothed analysis of Bimatrix and show that if the smoothed complexity of Bimatrix is polynomial, the authors can compute an ǫ-approximate Nash equilibrium of a bimatrix game in randomized poly(n, 1/ǫ) time.

Conclusion

- The following lemma shows that if the smoothed complexity of Bimatrix is low, under uniform or Gaussian perturbations, one can quickly find an approximate Nash equilibrium.
- If Bimatrix is in smoothed polynomial time under uniform or Gaussian perturbations, for all ǫ > 0, there exists a randomized algorithm to compute an ǫ-approximate Nash equilibrium in a two-player game with expected time O poly(m, n, 1/ǫ) or O poly(m, n, log max(m, n)/ǫ) , respectively.
- As every two-player game has a Nash equilibrium, this reduction implies that every generalized circuit with K nodes has a 1/K3-approximate solution.

Funding

- We would like to thank everyone who asked about the smoothed complexity of the Lemke-Howson algorithm, especially John Reif for being the first player to ask us this question. Xi Chen’s work was supported by the Chinese National Key Foundation Plan (2003CB317807, 2004CB318108), the National Natural Science Foundation of China Grant 60553001 and the National Basic Research Program of China Grant (2007CB807900, 2007CB807901)
- Xiaotie Deng’s work was supported by City University of Hong Kong for his research
- Shang-Hua Teng’s work was supported by the NSF grants CCR-0311430 and ITR CCR-0325630

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