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Settling the Complexity of Computing Two-Player Nash Equilibria
| Citations: | 88 - 5 self |
BibTeX
@MISC{Chen_settlingthe,
author = {Xi Chen and Xiaotie Deng and Shang-hua Teng},
title = {Settling the Complexity of Computing Two-Player Nash Equilibria },
year = {}
}
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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, Goldberg, and Papadimitriou on the complexity of four-player Nash equilibria [21], 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.
Keyphrases
two-player nash equilibrium long standing open problem nash equilibrium smoothed complexity randomized polynomial time following theorem arrow-debreu market equilibrium starting point polynomial parity argument directed version polynomial time complexity implication mathematical economics algorithmic game theory complexity class ppad four-player nash equilibrium two-player game polynomial-time approximation scheme classic lemke-howson algorithm
