## Settling the Complexity of Computing Two-Player Nash Equilibria

Citations: | 45 - 3 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 = {}

}

### Years of Citing Articles

### OpenURL

### 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.