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.

翻译：极小化极大定理的一个著名推论是，两人零和博弈具有易于处理的均衡刻画。然而，在许多核心应用中，每一方由多个独立主体组成，他们共享共同目标但无法完美协调行动。这类场景可建模为*团队零和博弈*——它是算法博弈论中研究最充分的两类博弈，即两人零和博弈与势博弈的自然推广。本文通过证明计算纳什均衡是\PPAD完全的，从而确定了团队零和博弈的复杂性。结果表明，尽管存在全局对抗结构，团队零和博弈的难度与一般和博弈相当。我们的困难性结果即使在以下条件下依然成立：(i) 精度为多项式倒数，从而排除了完全多项式时间近似方案（除非$¶= \PPAD$）；(ii) 每个团队仅由两名玩家组成；(iii) 底层博弈类为多矩阵博弈。作为副产品，我们解决了群组零和多项式矩阵博弈的复杂性，这类博弈由Cai和Daskalakis (SODA '11) 的开创性工作引入并研究，近期由Hollender、Maystre和Nagarajan (ICLR '25) 再次强调。此外，我们证明即使对于二次（多线性）目标函数，计算极小-极大优化中的一阶稳定点也是\PPAD完全的。从技术角度，我们开发了一系列团队零和博弈构件，使得能够模拟Bernasconi和Castiglioni (STOC '26) 的突破性归约。同时，为获得二次目标函数的困难性结果，我们利用了一种基于线性局部逼近的通用技术，该技术具有独立的研究价值。