The Complexity of Two-Team Polymatrix Games with Independent Adversaries

TL;DR

This paper investigates the computational complexity of Nash equilibria in two-team polymatrix games with independent adversaries, establishing new hardness results and providing an efficient approximation algorithm.

Contribution

It proves that computing Nash equilibria in two-team polymatrix games is CLS-hard, and offers a simple algorithm with polynomial dependence on the approximation parameter.

Findings

Two-team polymatrix games are CLS-hard to solve.

An efficient algorithm approximates Nash equilibria with a $1/\varepsilon^2$ dependence.

Hardness results extend to non-convex min-max optimization problems.

Abstract

Adversarial multiplayer games are an important object of study in multiagent learning. In particular, polymatrix zero-sum games are a multiplayer setting where Nash equilibria are known to be efficiently computable. Towards understanding the limits of tractability in polymatrix games, we study the computation of Nash equilibria in such games where each pair of players plays either a zero-sum or a coordination game. We are particularly interested in the setting where players can be grouped into a small number of teams of identical interest. While the three-team version of the problem is known to be PPAD-complete, the complexity for two teams has remained open. Our main contribution is to prove that the two-team version remains hard, namely it is CLS-hard. Furthermore, we show that this lower bound is tight (i.e., CLS-membership) for the setting where one of the teams consists of multiple independent adversaries. By leveraging this result we also obtain a simple algorithm that finds an $\varepsilon$-Nash equilibrium and only has a $1/\varepsilon^2$ dependence in $\varepsilon$ in its running time. On the way to obtaining our main result, we prove hardness of finding any stationary point in the simplest type of non-convex-concave min-max constrained optimization problem, namely for a class of bilinear polynomial objective functions.