Weakly acyclic games generalize potential games and are fundamental to the study of game theoretic control. In this paper, we present a generalization of weakly acyclic games, and we observe its importance in multi-agent learning when agents employ experimental strategy updates in periods where they fail to best respond. While weak acyclicity is defined in terms of path connectivity properties of a game's better response graph, our generalization is defined using a generalized better response graph. We provide sufficient conditions for this notion of generalized weak acyclicity in both two-player games and $n$-player games. To demonstrate that our generalization is not trivial, we provide examples of games admitting a pure Nash equilibrium that are not generalized weakly acyclic. The generalization presented in this work is closely related to the recent theory of satisficing paths, and the counterexamples presented here constitute the first negative results in that theory.

翻译：弱非循环博弈是势博弈的推广，对博弈论控制研究具有基础性意义。本文提出弱非循环博弈的一种推广形式，并指出当智能体在未能进行最优响应时采用实验性策略更新时，该推广对多智能体学习具有重要价值。原始弱非循环性通过博弈的"更优响应图"路径连通性定义，而本文的推广则基于"广义更优响应图"进行定义。我们给出了两人博弈与$n$人博弈中这种广义弱非循环性的充分条件。为证明该推广并非平凡，我们提供了存在纯纳什均衡但不满足广义弱非循环性的博弈实例。本文提出的推广与近期提出的满意度路径理论密切相关，文中所举的反例构成了该理论的首批否定性结果。