Adversarial multiplayer games are an important object of study in multiagent learning. In particular, polymatrix zero-sum games are a multiplayer setting where Nash equilibria are known to be efficiently computable. Towards understanding the limits of tractability in polymatrix games, we study the computation of Nash equilibria in such games where each pair of players plays either a zero-sum or a coordination game. We are particularly interested in the setting where players can be grouped into a small number of teams of identical interest. While the three-team version of the problem is known to be PPAD-complete, the complexity for two teams has remained open. Our main contribution is to prove that the two-team version remains hard, namely it is CLS-hard. Furthermore, we show that this lower bound is tight (i.e., CLS-membership) for the setting where one of the teams consists of multiple independent adversaries. By leveraging this result we also obtain a simple algorithm that finds an $\varepsilon$-Nash equilibrium and only has a $1/\varepsilon^2$ dependence in $\varepsilon$ in its running time. On the way to obtaining our main result, we prove hardness of finding any stationary point in the simplest type of non-convex-concave min-max constrained optimization problem, namely for a class of bilinear polynomial objective functions.

翻译：对抗性多人博弈是多智能体学习中的重要研究对象。特别地，多矩阵零和博弈是一种已知可有效计算纳什均衡的多人博弈框架。为探索多矩阵博弈的可处理性边界，我们研究此类博弈中每对玩家进行零和博弈或协调博弈时纳什均衡的计算问题。我们尤其关注玩家可划分为少量同质利益团队的场景。尽管该问题的三团队版本已知为PPAD完全问题，但两团队情形的复杂度仍悬而未决。我们的主要贡献在于证明两团队版本仍然困难，即属于CLS困难问题。进一步研究表明，当其中一个团队由多个独立对抗者构成时，该下界是紧的（即具有CLS成员性）。基于此结果，我们还获得一种简单算法，可找到$\varepsilon$-纳什均衡，其运行时间对$\varepsilon$的依赖仅为$1/\varepsilon^2$。在得出主要结论的过程中，我们证明了最简非凸-非凹极小极大约束优化问题中寻找驻点的困难性，即针对一类双线性多项式目标函数。