Algorithms and Complexity for Computing Nash Equilibria in Adversarial Team Games

TL;DR

This paper proves that computing Nash equilibria in adversarial team games is computationally hard, specifically CLS-complete, and introduces methods to approximate equilibria using linear programming and potential functions.

Contribution

It establishes the complexity of finding Nash equilibria in adversarial team games as CLS-complete and provides polynomial-time approximation techniques leveraging linear programming duality.

Findings

Nash equilibrium computation in adversarial team games is CLS-complete.

Any epsilon-approximate stationary strategy can be extended to an approximate Nash equilibrium efficiently.

Potential functions based on the Moreau envelope facilitate gradient-based dynamics for equilibrium approximation.

Abstract

Adversarial team games model multiplayer strategic interactions in which a team of identically-interested players is competing against an adversarial player in a zero-sum game. Such games capture many well-studied settings in game theory, such as congestion games, but go well-beyond to environments wherein the cooperation of one team -- in the absence of explicit communication -- is obstructed by competing entities; the latter setting remains poorly understood despite its numerous applications. Since the seminal work of Von Stengel and Koller (GEB `97), different solution concepts have received attention from an algorithmic standpoint. Yet, the complexity of the standard Nash equilibrium has remained open. In this paper, we settle this question by showing that computing a Nash equilibrium in adversarial team games belongs to the class continuous local search (CLS), thereby establishing CLS-completeness by virtue of the recent CLS-hardness result of Rubinstein and Babichenko (STOC `21) in potential games. To do so, we leverage linear programming duality to prove that any $\epsilon$-approximate stationary strategy for the team can be extended in polynomial time to an $O(\epsilon)$-approximate Nash equilibrium, where the $O(\cdot)$ notation suppresses polynomial factors in the description of the game. As a consequence, we show that the Moreau envelop of a suitable best response function acts as a potential under certain natural gradient-based dynamics.