The Computational Complexity of Team Zero-Sum Games

2026-06-14

AI Summary

This study addresses the computational complexity of computing Nash equilibria in team zero-sum games, where multiple players on each side collaborate against an opposing team but lack full coordination capabilities. By introducing a novel game reduction, employing linear local approximation techniques, and simulating the reduction framework of Bernasconi and Castiglioni, the authors establish for the first time that the problem remains PPAD-complete even in the simplest nontrivial setting—two players per team—with polynomial precision and multi-matrix representations. This result resolves an open question posed by Cai and Daskalakis regarding the complexity of team zero-sum games, demonstrating that their computational hardness matches that of general-sum games. Consequently, a fully polynomial-time approximation scheme cannot exist unless P = PPAD. The work further shows that computing first-order stationary points in minimax optimization is also PPAD-complete.

Abstract

A celebrated consequence of the minimax theorem is that two-player zero-sum games admit a tractable equilibrium characterization. In many central applications, however, each side comprises multiple independent agents who share a common objective but cannot perfectly coordinate their actions. Such settings can be modeled as \emph{team zero-sum games}, a natural generalization of both two-player zero-sum games and potential games -- the two most well-studied classes of games in algorithmic game theory. In this paper, we settle the complexity of team zero-sum games by establishing that computing Nash equilibria is \PPAD-complete. As a result, despite the global adversarial structure, team zero-sum games are as hard as general-sum games. Our hardness result holds even when i) the precision is inverse polynomial, thereby ruling out a fully polynomial-time approximation scheme (unless $¶= \PPAD$); ii) each team consists of only two players; and iii) the underlying class of games is polymatrix. As a byproduct, we resolve the complexity of group-wise zero-sum polymatrix games, a class introduced and examined in the seminal work of Cai and Daskalakis (SODA '11), and more recently highlighted by Hollender, Maystre, and Nagarajan (ICLR '25). Moreover, we show that computing a first-order stationary point in min-max optimization is \PPAD-complete even for quadratic (multilinear) objectives. From a technical standpoint, we develop a series of team zero-sum game gadgets that allow us to simulate the breakthrough reduction of Bernasconi and Castiglioni (STOC '26). Moreover, to obtain hardness results for quadratic objectives, we make use of a general technique based on linear local approximation, which is of independent interest.

Problem

• team zero-sum games
• Nash equilibria
• computational complexity
• PPAD-complete
• polymatrix games

Innovation

• team zero-sum games
• PPAD-completeness
• Nash equilibrium computation
• polymatrix games
• min-max optimization