This paper studies multiplayer turn-based games on graphs in which player preferences are modeled as $ω$-automatic relations given by deterministic parity automata. This contrasts with most existing work, which focuses on specific reward functions. We conduct a computational analysis of these games, starting with the threshold problem in the antagonistic zero-sum case. As in classical games, we introduce the concept of value, defined here as the set of plays a player can guarantee to improve upon, relative to their preference relation. We show that this set is recognized by an alternating parity automaton APW of polynomial size. We also establish the computational complexity of several problems related to the concepts of value and optimal strategy, taking advantage of the $ω$-automatic characterization of value. Next, we shift to multiplayer games and Nash equilibria, and revisit the threshold problem in this context. Based on an APW construction again, we close complexity gaps left open in the literature, and additionally show that cooperative rational synthesis is $\mathsf{PSPACE}$-complete, while it becomes undecidable in the non-cooperative case.

翻译：本文研究图上的多人回合制博弈，其中玩家的偏好被建模为由确定性奇偶自动机给出的ω-自动关系。这与现有大多数工作聚焦于特定奖励函数形成对比。我们对这类博弈进行了计算分析，从对抗性零和情况下的阈值问题入手。与经典博弈类似，我们引入了价值的概念，此处定义为玩家能保证相对于其偏好关系有所改进的玩法集合。我们证明该集合可由一个多项式规模的交替奇偶自动机（APW）识别。利用价值的ω-自动刻画，我们还确立了与价值和最优策略概念相关的若干问题的计算复杂度。随后，我们将研究转向多人博弈与纳什均衡，并在此背景下重新审视阈值问题。基于再次构造的APW，我们填补了文献中遗留的复杂度缺口，并进一步证明合作理性综合是$\mathsf{PSPACE}$-完全的，而在非合作情况下该问题不可判定。