Towards convergence to Nash equilibria in two-team zero-sum games

Fivos Kalogiannis, Ioannis Panageas, Emmanouil-Vasileios Vlatakis-Gkaragkounis

Keywords: no-regret-learning, no-regret, optimization, learning-in-games, nash-equilibrium, game-theory, min-max-optimization, min-max

TL;DR: Common no-regret algorithms fail to converge to a Nash equilibrium in two-team zero-sum games but a novel approach does converge locally.

Abstract: Contemporary applications of machine learning raise important and overlooked theoretical questions regarding optimization in two-team games. Formally, two-team zero-sum games are defined as multi-player games where players are split into two competing sets of agents, each experiencing a utility identical to that of their teammates and opposite to that of the opposing team. We focus on the solution concept of Nash equilibria and prove $\textrm{CLS}$-hardness of computing them in this class of games. To further examine the capabilities of online learning algorithms in games with full-information feedback, we propose a benchmark of a simple ---yet nontrivial--- family of such games. These games do not enjoy the properties used to prove convergence for relevant algorithms. In particular, we use a dynamical systems perspective to demonstrate that gradient descent-ascent, its optimistic variant, optimistic multiplicative weights update, and extra gradient fail to converge (even locally) to a Nash equilibrium. On a brighter note, we propose a first-order method that leverages control theory techniques and under some conditions enjoys last-iterate local convergence to a Nash equilibrium. We also believe our proposed method is of independent interest for general min-max optimization.

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