The Complexity of Symmetric Equilibria in Min-Max Optimization and Team Zero-Sum Games
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| Format: | Preprint |
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2025
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| author | Anagnostides, Ioannis Panageas, Ioannis Sandholm, Tuomas Yan, Jingming |
| author_facet | Anagnostides, Ioannis Panageas, Ioannis Sandholm, Tuomas Yan, Jingming |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_08519 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Complexity of Symmetric Equilibria in Min-Max Optimization and Team Zero-Sum Games Anagnostides, Ioannis Panageas, Ioannis Sandholm, Tuomas Yan, Jingming Computer Science and Game Theory Computational Complexity 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. |
| title | The Complexity of Symmetric Equilibria in Min-Max Optimization and Team Zero-Sum Games |
| topic | Computer Science and Game Theory Computational Complexity |
| url | https://arxiv.org/abs/2502.08519 |