The Complexity of Symmetric Equilibria in Min-Max Optimization and Team Zero-Sum Games

TL;DR

This paper investigates the computational complexity of finding equilibria in min-max optimization and team zero-sum games, revealing that such problems are generally hard ( extsf{CLS}-complete, extsf{PPAD}-complete, and extsf{FNP}-hard), especially in symmetric settings.

Contribution

It establishes the complexity classifications of equilibrium computation in symmetric and adversarial team games, resolving open questions and highlighting the difficulty of designing convergent algorithms.

Findings

Computing $ ext{ extsf{CLS}}$-complete for 3-player adversarial team games.

Symmetric equilibria in symmetric min-max optimization are extsf{PPAD}-complete.

Non-symmetric $ ext{ extsf{poly}}(1/n)$-equilibria are extsf{FNP}-hard.

Abstract

We consider the problem of computing stationary points in min-max optimization, with a particular focus on the special case of computing Nash equilibria in (two-)team zero-sum games. We first show that computing $\epsilon$-Nash equilibria in $3$-player \emph{adversarial} team games -- wherein a team of $2$ players competes against a \emph{single} adversary -- is \textsf{CLS}-complete, resolving the complexity of Nash equilibria in such settings. Our proof proceeds by reducing from \emph{symmetric} $\epsilon$-Nash equilibria in \emph{symmetric}, identical-payoff, two-player games, by suitably leveraging the adversarial player so as to enforce symmetry -- without disturbing the structure of the game. In particular, the class of instances we construct comprises solely polymatrix games, thereby also settling a question left open by Hollender, Maystre, and Nagarajan (2024). We also provide some further results concerning equilibrium computation in adversarial team games. Moreover, we establish that computing \emph{symmetric} (first-order) equilibria in \emph{symmetric} min-max optimization is \textsf{PPAD}-complete, even for quadratic functions. Building on this reduction, we further show that computing symmetric $\epsilon$-Nash equilibria in symmetric, $6$-player ($3$ vs. $3$) team zero-sum games is also \textsf{PPAD}-complete, even for $\epsilon = \text{poly}(1/n)$. As an immediate corollary, this precludes the existence of symmetric dynamics -- which includes many of the algorithms considered in the literature -- converging to stationary points. Finally, we prove that computing a \emph{non-symmetric} $\text{poly}(1/n)$-equilibrium in symmetric min-max optimization is \textsf{FNP}-hard.