两队多项式矩阵游戏的复杂性
The Complexity of Two-Team Polymatrix Games with Independent Adversaries
AI总结：
研究两队多项式矩阵游戏在独立对手情况下的纳什均衡计算复杂性，证明其CLS难，并提出一种具有1/ε²时间复杂度的近似算法。
摘要：
对抗性多智能体游戏是多智能体学习中的重要研究对象。特别是多项式零和游戏是一种多玩家设置，其中纳什均衡已被证明可以高效计算。为了理解多项式游戏计算极限，我们研究了在每对玩家之间进行零和或协调游戏的计算问题。我们特别关注玩家可以分为少量相同利益团队的情况。虽然三队版本已知为PPAD难，但两队版本的复杂性仍不清楚。我们的主要贡献是证明两队版本仍然困难，即为CLS难。此外，我们证明在其中一个团队由多个独立对手组成的情况下，该下界是紧的（即CLS成员）。通过利用这一结果，我们还获得了一种找到ε-纳什均衡的简单算法，并且其运行时间仅与ε²成反比。在获得主要结果的过程中，我们证明了在最简单的非凸-凹min-max约束优化问题中寻找任何平稳点的难度，即对于一类双线性多项式目标函数。
英文摘要：
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.
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