NP-hardness of Computing Uniform Nash Equilibria on Planar Bimatrix Games

TL;DR

This paper proves that determining the existence of uniform Nash equilibria in planar bimatrix games is NP-complete when the game is not restricted to win-lose payoffs, extending known complexity results.

Contribution

It establishes NP-completeness for computing uniform Nash equilibria in planar non-win-lose bimatrix games, resolving an open complexity question.

Findings

Deciding uniform Nash equilibria in planar non-win-lose bimatrix games is NP-complete.

Uniform Nash equilibria always exist in planar win-lose bimatrix games and can be found in polynomial time.

The complexity of finding such equilibria increases significantly when moving from win-lose to general payoffs.

Abstract

We study the complexity of computing a uniform Nash equilibrium on a non-win-lose bimatrix game. It is known that such a problem is NP-complete even if a bimatrix game is win-lose (Bonifaci et al., 2008). Fortunately, if a win-lose bimatrix game is planar, then uniform Nash equilibria always exist. We have a polynomial-time algorithm for finding a uniform Nash equilibrium of a planar win-lose bimatrix game (Addario-Berry et al., 2007). The following question is left: How hard to compute a uniform Nash equilibrium on a planar non-win-lose bimatrix game? This paper resolves this issue. We prove that the problem of deciding whether a non-win-lose planar bimatrix game has uniform Nash equilibrium is also NP-complete.