Settling the Complexity of Computing Two-Player Nash Equilibria (0704.1678v1)

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.

• The paper proves that computing Nash equilibria in two-player games is PPAD-complete, establishing the Bimatrix problem as hard as any in PPAD.
• It demonstrates that no fully polynomial-time approximation scheme exists and that the smoothed complexity of the problem is non-polynomial.
• Novel encoding methods and geometric lemmas were developed, extending the complexity insights to related equilibrium and optimization challenges.

Complexity of Computing Two-Player Nash Equilibria: An Overview

The paper "Settling the Complexity of Computing Two-Player Nash Equilibria" addresses a significant problem in algorithmic game theory: determining the complexity of finding a Nash equilibrium in two-player games, specifically, the {\sc Bimatrix} problem. The authors establish that this problem is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version), a class introduced by Papadimitriou in 1991. This result resolves a long-standing open question and places the problem of finding Nash equilibria within a definitive complexity framework.

Key Contributions:

1. PPAD-Completeness of {\sc Bimatrix}: The authors prove that {\sc Bimatrix} is PPAD-complete. This establishes that computing a Nash equilibrium in two-player games is as hard as the most challenging problems in PPAD, linking it to the broader class of problems involving fixed-point computations.
2. Implications for Approximation and Smoothed Complexity: The results indicate that unless every problem in PPAD can be solved in polynomial time, there is no fully polynomial-time approximation scheme for {\sc Bimatrix}. Additionally, the smoothed complexity of any algorithm for this problem is not polynomial, assuming PPAD is not contained in RP (randomized polynomial time).
3. Extension to Related Problems: The paper extends its findings to demonstrate that related problems, such as computing Arrow-Debreu market equilibria and the P-Matrix Linear Complementary Problem, are also complex, asserting PPAD-hardness.

Theoretical Framework:

• Fixed Point Computation Equivalence: The results demonstrate that even in simplest non-cooperative games (two-player), equilibrium computation is polynomial-time equivalent to discrete fixed point computation.
• Core Complexity Insights: The PPAD-completeness reflects the inherent difficulty of equilibrium computation, found to be equivalent to classic fixed-point theorems like Brouwer's, indicating the physical necessity of fixed-point approaches for proofs in game theory.

Techniques Employed:

• Novel Encoding Methods: The authors introduce new techniques for encoding variables using mixed strategies, and they simulate boolean and arithmetic operations to reduce fixed point computations to bimatrix games directly, bypassing graphical models.
• Dimensionality Management: Their approach includes a novel geometric lemma useful for reasoning about high-dimensional fixed points, crucial for managing the exponential dependencies in dimensions that complicate fixed-point searches.

Future Directions and Speculation:

The implications of this work are extensive for both mathematical economics and operations research. It suggests a need for further exploration of PPAD and refinement of approximation algorithms. Potential future developments could involve improving approximation schemes or exploring the practical applicability of these theoretical limits in fields like market analysis and economic computations. Furthermore, the barrier between PPAD and other complexity classes remains a stimulating frontier for exploration, possibly influencing cryptographic and computational efficiency understanding.

In summary, this paper provides a critical milestone in understanding the complexity of Nash equilibria in two-player games, linking theoretical computational complexity with practical algorithmic methodologies in game theory and economics.