Adversarial team games model multiplayer strategic interactions in which a team of identically-interested players is competing against an adversarial player in a zero-sum game. Such games capture many well-studied settings in game theory, such as congestion games, but go well-beyond to environments wherein the cooperation of one team -- in the absence of explicit communication -- is obstructed by competing entities; the latter setting remains poorly understood despite its numerous applications. Since the seminal work of Von Stengel and Koller (GEB `97), different solution concepts have received attention from an algorithmic standpoint. Yet, the complexity of the standard Nash equilibrium has remained open. In this paper, we settle this question by showing that computing a Nash equilibrium in adversarial team games belongs to the class continuous local search (CLS), thereby establishing CLS-completeness by virtue of the recent CLS-hardness result of Rubinstein and Babichenko (STOC `21) in potential games. To do so, we leverage linear programming duality to prove that any $\epsilon$-approximate stationary strategy for the team can be extended in polynomial time to an $O(\epsilon)$-approximate Nash equilibrium, where the $O(\cdot)$ notation suppresses polynomial factors in the description of the game. As a consequence, we show that the Moreau envelop of a suitable best response function acts as a potential under certain natural gradient-based dynamics.

翻译：对抗性团队博弈刻画了多智能体战略交互场景：一组利益完全一致的玩家团队在零和博弈中与单个对抗性玩家竞争。这类博弈不仅涵盖博弈论中诸多经典设定（如拥塞博弈），更拓展至缺乏显式通信条件下团队协作受到竞争主体阻碍的环境——尽管该场景具有广泛应用，但其理论理解仍相当有限。自Von Stengel与Koller（GEB `97）开创性工作以来，不同解概念从算法角度受到关注，然而标准纳什均衡的计算复杂度问题始终悬而未决。本文通过证明对抗性团队博弈中纳什均衡计算属于连续局部搜索类（CLS），并结合Rubinstein与Babichenko（STOC `21）在势博弈中提出的CLS-hardness结论，确立了该问题的CLS完全性。为此，我们利用线性规划对偶性证明：团队的任意$\epsilon$-近似平稳策略可在多项式时间内扩展为$O(\epsilon)$-近似纳什均衡，其中$O(\cdot)$符号隐去了博弈描述中的多项式因子。作为推论，我们表明在特定自然梯度动力学下，适当最优响应函数的Moreau包络可充当势函数。