The Complexity of Symmetric Bimatrix Games with Common Payoffs

TL;DR

This paper investigates the computational complexity of finding symmetric Nash equilibria in symmetric bimatrix games with common payoffs, establishing that the problem is CLS-complete, which clarifies its intractability status.

Contribution

It proves that computing symmetric equilibria in common-payoff symmetric bimatrix games is CLS-complete, extending complexity results and showing intractability within this specific game class.

Findings

The problem is CLS-complete, not PPAD-hard.

Computing a KKT point of a quadratic program is CLS-hard even on a simplex.

Symmetric common-payoff games have intractable equilibrium computation.

Abstract

We study symmetric bimatrix games that also have the common-payoff property, i.e., the two players receive the same payoff at any outcome of the game. Due to the symmetry property, these games are guaranteed to have symmetric Nash equilibria, where the two players play the same (mixed) strategy. While the problem of computing such symmetric equilibria in general symmetric bimatrix games is known to be intractable, namely PPAD-complete, this result does not extend to our setting. Indeed, due to the common-payoff property, the problem lies in the lower class CLS, ruling out PPAD-hardness. In this paper, we show that the problem remains intractable, namely it is CLS-complete. On the way to proving this result, as our main technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a quadratic program remains CLS-hard, even when the feasible domain is a simplex.