The Complexity of Sparse Win-Lose Bimatrix Games

2026-02-20

AI Summary

This study investigates the computational complexity of computing an ε-approximate Nash equilibrium in constant-sparse win-lose bimatrix games. By constructing a PPAD-completeness reduction, it establishes for the first time that the problem remains PPAD-hard even in the 3-sparse case for inverse-polynomial accuracy ε. This hardness result is tight with respect to sparsity, as the 2-sparse case is known to be solvable in polynomial time. The work thus precisely delineates the boundary between tractability and intractability for sparse win-lose games and contributes a refined understanding of the complexity landscape of approximate Nash equilibrium computation.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Privacy-enhancing technologies

Abstract

We prove that computing an $\epsilon$-approximate Nash equilibrium of a win-lose bimatrix game with constant sparsity is PPAD-hard for inverse-polynomial $\epsilon$. Our result holds for 3-sparse games, which is tight given that 2-sparse win-lose bimatrix games can be solved in polynomial time.

Problem

• Sparse bimatrix games
• Win-lose games
• Nash equilibrium
• PPAD-hardness
• Computational complexity

Innovation

• sparse bimatrix games
• win-lose games
• approximate Nash equilibrium
• PPAD-hardness
• computational complexity