Motivated by recent advances in both theoretical and applied aspects of multiplayer games, spanning from e-sports to multi-agent generative adversarial networks, we focus on min-max optimization in team zero-sum games. In this class of games, players are split into two teams with payoffs equal within the same team and of opposite sign across the opponent team. Unlike the textbook two-player zero-sum games, finding a Nash equilibrium in our class can be shown to be CLS-hard, i.e., it is unlikely to have a polynomial-time algorithm for computing Nash equilibria. Moreover, in this generalized framework, we establish that even asymptotic last iterate or time average convergence to a Nash Equilibrium is not possible using Gradient Descent Ascent (GDA), its optimistic variant, and extra gradient. Specifically, we present a family of team games whose induced utility is \emph{non} multi-linear with \emph{non} attractive \emph{per-se} mixed Nash Equilibria, as strict saddle points of the underlying optimization landscape. Leveraging techniques from control theory, we complement these negative results by designing a modified GDA that converges locally to Nash equilibria. Finally, we discuss connections of our framework with AI architectures with team competition structures like multi-agent generative adversarial networks.
翻译:在多玩游戏的理论和应用方面,从电子体育到多试剂基因对抗网络,最近出现了一些进步,从电子体育到多试剂基因对抗网络,我们注重在零和球队零和球队中实现微麦优化。在这一类游戏中,球员分成两个球队,球员在球队中报酬相等,对手队之间则有相反的标志。与教科书中的双球运动员零和球队不同,在我们的球队中找到一个纳什平衡,可以证明是CLS-硬的,也就是说,在计算纳什平衡时,我们不可能有一个从电子体育到多试调的多时空算法。此外,在这个普遍的框架内,我们确定即使最后的重复性或平均趋同纳什平衡是不可能使用同一球队中同等报酬的球队。具体地说,我们展示了一组球队游戏的组合,其诱发的效用是cemph{nonnon-nortial, 也就是说,它是一个具有吸引力的耐姆-per-eme-me 混合的纳什-equiliblibrial 网络,这是我们最基础的理论化结构结构的最后讨论。我们从一个严格的基调的基调结构结构,我们最后讨论。