The Complexity of Sparse Win-Lose Bimatrix Games

TL;DR

This paper proves that finding approximate Nash equilibria in sparse win-lose bimatrix games is computationally hard, specifically PPAD-hard for certain sparsity levels, highlighting the complexity boundary at 3-sparse games.

Contribution

It establishes the PPAD-hardness of computing approximate Nash equilibria in 3-sparse win-lose bimatrix games, showing the problem's computational difficulty.

Findings

PPAD-hardness for 3-sparse games

Polynomial-time solvability for 2-sparse games

Complexity boundary at sparsity level 3

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.