Settling the complexity of computing two-player Nash equilibria

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.

Algorithmic Game Theory

“…There has been a frenzy of recent work on these questions, for many different fundamental equilibrium concepts. Perhaps the most celebrated results in the area concern the P P AD-completeness of computing mixed-strategy Nash equilibria in finite games with two or more players [8,12]. To briefly convey the spirit of the area with a minimum of technical fuss, we instead discuss the complexity of converging to and computing pure-strategy Nash equilibria in a variant of the routing games studied in Section 3.…”

“…A few years ago, the problem of computing an MNE of a bimatrix game was shown to be P P ADcomplete [8,12]. Thus, if P = P P AD, there is no general-purpose and computationally efficient algorithm for this problem, and in particular there is no general and tractable way for players to reach a Nash equilibrium in a reasonable amount of time.…”

“…This hardness result casts doubt on the predictive power of the Nash equilibrium concept in arbitrary games. See the papers [8,12] for the details of this tour de force result and the recent CACM article [13] for a high-level survey of the proof.…”

Computing equilibria: a computational complexity perspective

“…22 Recall that the definition of an N P problem -a verifier for candidate solutions and an explicit polynomial bound on its worst-case running time -was motivated by the minimal information required to run brute-force search. What are the minimal ingredients for executing a local search procedure analogous to better-response dynamics?…”

“…22 This class was originally defined to study local search problems, not equilibrium computation problems [64]. The connection between local search algorithms and better-response dynamics was made explicit by Fabrikant, Papadimitriou, and Talwar [42].…”