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

TL;DR

The paper addresses the computational complexity of stationary points in nonconvex–nonconcave min–max optimization, with a focus on symmetric equilibria and team zero‑sum games. It develops CLS and PPAD hardness results via novel gadget reductions that enforce symmetry and approximate fixed‑point behavior, including a 3‑player adversarial team game and a 6‑player symmetric team zero‑sum polymatrix setting. It shows that even in constrained, structured instances, computing symmetric first‑order Nash equilibria is PPAD‑complete, while non‑symmetric fixed points can be NFN‑hard, and it precludes broad classes of symmetric dynamics from converging to stationary points unless major complexity collapses occur. These results advance the understanding of when efficient equilibrium computation is plausible and highlight fundamental limits for algorithmic approaches in min–max optimization and team games, with implications for learning dynamics and robustness in multi-agent systems.

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 $ε$-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} $ε$-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 $ε$-Nash equilibria in symmetric, $6$-player ($3$ vs. $3$) team zero-sum games is also \textsf{PPAD}-complete, even for $ε= \text{poly}(1/n)$. 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.

Figures (1)

• Figure 1: An example of matrix $\mathbf{A} = \mathbf{A}(G)$ (right) for graph $G$ (left).

Theorems & Definitions (62)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4: Symmetric min-max optimization
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Theorem 2.2: Chen09:Settling
• Theorem 3.1: ghosh2024complexitysymmetricbimatrixgames