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：对抗性团队博弈模拟了多方战略互动场景，其中一群利益完全一致的玩家组成团队，与对抗性玩家进行零和博弈。这类博弈不仅涵盖博弈论中诸多经典研究设定（如拥塞博弈），更能延伸至合作方团队在缺乏显式通信时受到竞争实体阻碍的环境——尽管此类应用广泛，但其底层机制至今仍认知不足。自Von Stengel与Koller（GEB `97）的开创性工作以来，不同解概念已从算法角度受到关注，然而标准纳什均衡的复杂性始终悬而未决。本文通过证明对抗性团队博弈中纳什均衡的计算属于连续局部搜索类（CLS），并借助Rubinstein与Babichenko（STOC `21）在势博弈中最新证明的CLS-困难性结论，最终确立其CLS-完全性。为此，我们利用线性规划对偶性证明：任意ε-近似平稳策略可在多项式时间内扩展为O(ε)-近似纳什均衡（其中O(·)符号隐藏了博弈描述中的多项式因子）。作为推论，我们揭示特定最佳响应函数的莫罗包络在自然梯度动力学下可充当势函数。