This paper investigates Nash equilibria (NEs) in multi-player turn-based games on graphs, where player preferences are modeled as $ω$-automatic relations via deterministic parity automata. Unlike much of the existing literature, which focuses on specific reward functions, our results apply to any preference relation definable by an $ω$-automatic relation. We analyze the computational complexity of determining the existence of an NE (possibly under some constraints), verifying whether a given strategy profile forms an NE, and checking whether a specific outcome can be realized by an NE. When a (constrained) NE exists, we show that there always exists one with finite-memory strategies. Finally, we explore fundamental properties of $ω$-automatic relations and their implications for the existence of equilibria.

翻译：本文研究基于图的多玩家回合制博弈中的纳什均衡，其中玩家偏好通过确定性奇偶自动机建模为ω-自动关系。与现有文献大多关注特定奖励函数不同，我们的结果适用于任何可由ω-自动关系定义的偏好关系。我们分析了判定纳什均衡存在性（可能在某些约束条件下）、验证给定策略组合是否构成纳什均衡，以及检验特定结果能否由纳什均衡实现的计算复杂度。当（受约束的）纳什均衡存在时，我们证明总存在一个采用有限记忆策略的均衡。最后，我们探讨了ω-自动关系的基本性质及其对均衡存在性的影响。