[Mechanism Design Theme invited talk] The Complexity of Symmetric Equilibria in Min-Max Optimization and Team Zero-Sum Games

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

Speaker: Ioannis Panageas (UC Irvine)

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 adversarial team games -- wherein a team of 2 players competes against a single adversary -- is CLS-complete, resolving the complexity of Nash equilibria in such settings. Our proof proceeds by reducing from symmetric ε-Nash equilibria in 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 symmetric (first-order) equilibria in \emph{symmetric} min-max optimization is 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 PPAD-complete, even for ϵ=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 non-symmetric poly(1/n)-equilibrium in symmetric min-max optimization is FNP-hard.

Short Bio: Ioannis Panageas is an Assistant Professor of Computer Science at UC Irvine and a researcher at Archimedes Research unit. He is interested in the theory of computation, machine learning and its interface with non-convex optimization, dynamical systems, learning in games, statistics and multi-agent reinforcement learning. Before joining UCI, he was an Assistant Professor at Singapore University of Technology and Design. Prior to that he was a MIT postdoctoral fellow. He received his PhD in Algorithms, Combinatorics and Optimization from Georgia Tech in 2016, a Diploma in EECS from National Technical University of Athens, and a MS in Mathematics from Georgia Tech. He is the recipient of the 2019 NRF fellowship for AI.

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