For decades, two-player (antagonistic) games on graphs have been a framework of choice for many important problems in theoretical computer science. A notorious one is controller synthesis, which can be rephrased through the game-theoretic metaphor as the quest for a winning strategy of the system in a game against its antagonistic environment. Depending on the specification, optimal strategies might be simple or quite complex, for example having to use (possibly infinite) memory. Hence, research strives to understand which settings allow for simple strategies. In 2005, Gimbert and Zielonka provided a complete characterization of preference relations (a formal framework to model specifications and game objectives) that admit memoryless optimal strategies for both players. In the last fifteen years however, practical applications have driven the community toward games with complex or multiple objectives, where memory -- finite or infinite -- is almost always required. Despite much effort, the exact frontiers of the class of preference relations that admit finite-memory optimal strategies still elude us. In this work, we establish a complete characterization of preference relations that admit optimal strategies using arena-independent finite memory, generalizing the work of Gimbert and Zielonka to the finite-memory case. We also prove an equivalent to their celebrated corollary of great practical interest: if both players have optimal (arena-independent-)finite-memory strategies in all one-player games, then it is also the case in all two-player games. Finally, we pinpoint the boundaries of our results with regard to the literature: our work completely covers the case of arena-independent memory (e.g., multiple parity objectives, lower- and upper-bounded energy objectives), and paves the way to the arena-dependent case (e.g., multiple lower-bounded energy objectives).

中文翻译：

---

使用独立于竞技场的有限内存，您可以最佳地玩游戏

几十年来，图上的两人（对抗）博弈一直是理论计算机科学中许多重要问题的首选框架。一个臭名昭著的是控制器综合，它可以通过博弈论的比喻重新表述为在对抗其对抗环境的游戏中寻求系统的制胜策略。根据规范，最佳策略可能很简单，也可能非常复杂，例如必须使用（可能是无限的）内存。因此，研究力求了解哪些设置允许使用简单的策略。2005 年，Gimbert 和 Zielonka 提供了偏好关系的完整特征（一个对规范和博弈目标建模的正式框架），承认双方的无记忆最优策略。然而在过去的十五年里，实际应用已将社区推向具有复杂或多个目标的游戏，在这些游戏中，几乎总是需要有限或无限的内存。尽管付出了很多努力，但我们仍然无法了解允许有限记忆最优策略的偏好关系类别的确切边界。在这项工作中，我们建立了一个完整的偏好关系表征，这些偏好关系允许使用与竞技场无关的有限记忆的最优策略，将 Gimbert 和 Zielonka 的工作推广到有限记忆的情况。我们还证明了他们著名的具有重大实际意义的推论的等价物：如果两个玩家在所有单人游戏中都有最佳的（独立于竞技场的）有限记忆策略，那么在所有双人游戏中也是如此。最后，我们根据文献确定了结果的界限：