Teamwork makes von Neumann work: Min-Max Optimization in Two-Team Zero-Sum Games

Abstract

Motivated by recent advances in both theoretical and applied aspects of multiplayer games, spanning from e-sports to multi-agent generative adversarial networks, we focus on min-max optimization in team zero-sum games. In this class of games, players are split into two teams with payoffs equal within the same team and of opposite sign across the opponent team. Unlike the textbook two-player zero-sum games, finding a Nash equilibrium in our class can be shown to be CLS-hard, i.e., it is unlikely to have a polynomial-time algorithm for computing Nash equilibria. Moreover, in this generalized framework, we establish that even asymptotic last iterate or time average convergence to a Nash Equilibrium is not possible using Gradient Descent Ascent (GDA), its optimistic variant, and extra gradient. Specifically, we present a family of team games whose induced utility is non multi-linear with non attractive per-se mixed Nash Equilibria, as strict saddle points of the underlying optimization landscape. Leveraging techniques from control theory, we complement these negative results by designing a modified GDA that converges locally to Nash equilibria. Finally, we discuss connections of our framework with AI architectures with team competition structures like multi-agent generative adversarial networks.

A Survey of Decision Making in Adversarial Games

“…In general, team games in the literature can be classified from two perspectives. One perspective depends upon the team number, i.e., one-team games (or adversarial team games) [94], where players in the team enjoying the same utility function play against an adversary independently, and two-team games [95] consisting of two teams in a game. The other perspective is on perfect-information and imperfect-information games.…”

“…Even though, efficient algorithms for computing a TME in perfect-information zero-sum NFGs have been developed until now, e.g., [94]. Meanwhile, a class of zerosum two-team games in perfect-information normalform was studied in [95], where finding an NE is shown to be CLS-hard, i.e., unlikely to have a polynomialtime NE computing algorithm. Moreover, as two-team games, two-network zero-sum games are also addressed, where each network is thought of as a team [98]- [100].…”