[Paper Review] Settling the Complexity of Computing Two-Player Nash Equilibria

Xi Chen, Xiaotie Deng+1 more|ArXiv.org|Apr 12, 2007

Game Theory and ApplicationsDecision Sciences35 references3 citations

TL;DR

This paper establishes that computing a Nash equilibrium in a two-player game (Bimatrix) is PPAD-complete, resolving a long-standing open problem in algorithmic game theory. The authors prove that no fully polynomial-time approximation scheme exists for Bimatrix unless PPAD is contained in P, and that the smoothed complexity of the Lemke-Howson algorithm is not polynomial unless PPAD problems are solvable in randomized polynomial time.

ABSTRACT

We settle a long-standing open question in algorithmic game theory. 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. This is the first of a series of results concerning the complexity of 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 demonstrate that, even in the simplest form of non-cooperative games, equilibrium computation and approximation are polynomial-time equivalent to fixed point computation. Our results also have two broad complexity implications in mathematical economics and operations research: Arrow-Debreu market equilibria are PPAD-hard to compute. The P-Matrix Linear Complementary Problem is computationally harder than convex programming unless every problem in PPAD is solvable in polynomial time.

Motivation & Objective

• To resolve the long-standing open question regarding the computational complexity of finding a Nash equilibrium in a two-player game.
• To establish that the Bimatrix problem is complete for the complexity class PPAD, which captures total search problems with a parity-based existence proof.
• To demonstrate that approximating Nash equilibria in two-player games is as hard as solving any problem in PPAD.
• To analyze the smoothed complexity of the Lemke-Howson algorithm and show it is not polynomial unless PPAD is contained in randomized polynomial time.
• To extend the complexity results to broader problems in mathematical economics, such as Arrow-Debreu market equilibria and the P-matrix linear complementarity problem.

Proposed method

• Reduction of the Bimatrix problem to the computational version of Brouwer’s fixed point theorem via a discrete fixed point problem.
• Construction of a reduction from the End-of-Line problem to Bimatrix, establishing PPAD-completeness.
• Use of the PPAD-completeness framework to show that any algorithm solving Bimatrix must be as hard as solving any problem in PPAD.
• Application of smoothed analysis to the Lemke-Howson algorithm, showing that its expected runtime remains superpolynomial under small random perturbations.
• Leveraging the fact that PPAD captures problems with a parity-based existence proof, and showing that Bimatrix is complete for this class.
• Use of the fact that PPAD-completeness implies that no fully polynomial-time approximation scheme (FPTAS) exists unless PPAD ⊆ P.

Experimental results

Research questions

• RQ1 Is the problem of computing a Nash equilibrium in a two-player game complete for the complexity class PPAD?
• RQ2 Does the existence of a fully polynomial-time approximation scheme (FPTAS) for Bimatrix imply that all PPAD problems can be solved in polynomial time?
• RQ3 Is the smoothed complexity of the Lemke-Howson algorithm polynomial, or is it superpolynomial under small random perturbations?
• RQ4 Are Arrow-Debreu market equilibria PPAD-hard to compute?
• RQ5 Is the P-matrix linear complementarity problem computationally harder than convex programming unless PPAD is contained in P?

Key findings

• The problem of computing a Nash equilibrium in a two-player game (Bimatrix) is PPAD-complete.
• There is no fully polynomial-time approximation scheme (FPTAS) for Bimatrix unless every problem in PPAD is solvable in polynomial time.
• The smoothed complexity of the Lemke-Howson algorithm is not polynomial unless every problem in PPAD is solvable in randomized polynomial time.
• Arrow-Debreu market equilibria are PPAD-hard to compute.
• The P-matrix linear complementarity problem is computationally harder than convex programming unless PPAD is contained in P.
• The complexity of computing r-player Nash equilibria can be reduced in polynomial time to computing two-player Nash equilibria, though the reduction is not direct.