Two-player (antagonistic) games on (possibly stochastic) graphs are a prevalent model in theoretical computer science, notably as a framework for reactive synthesis. Optimal strategies may require randomisation when dealing with inherently probabilistic goals, balancing multiple objectives, or in contexts of partial information. There is no unique way to define randomised strategies. For instance, one can use so-called mixed strategies or behavioural ones. In the most general settings, these two classes do not share the same expressiveness. A seminal result in game theory - Kuhn's theorem - asserts their equivalence in games of perfect recall. This result crucially relies on the possibility for strategies to use infinite memory, i.e., unlimited knowledge of all past observations. However, computer systems are finite in practice. Hence it is pertinent to restrict our attention to finite-memory strategies, defined as automata with outputs. Randomisation can be implemented in these in different ways: the initialisation, outputs or transitions can be randomised or deterministic respectively. Depending on which aspects are randomised, the expressiveness of the corresponding class of finite-memory strategies differs. In this work, we study two-player turn-based stochastic games and provide a complete taxonomy of the classes of finite-memory strategies obtained by varying which of the three aforementioned components are randomised. Our taxonomy holds both in settings of perfect and imperfect information, and in games with more than two players.
翻译:在(可能是随机的)图形上,双玩者游戏(超自然的)游戏是理论计算机科学中流行的模型,特别是作为被动合成的框架。最佳战略可能需要随机处理内在概率目标、平衡多重目标或部分信息。没有独特的方法来定义随机化战略。例如,可以使用所谓的混合策略或行为策略。在最一般的环境下,这两类游戏不具有相同的表达性。游戏理论的随机性结果 - Kuhn的理论 - 在完全回顾的游戏中显示其等同性。这种结果关键地取决于战略使用无限记忆的可能性,即对过去所有观察的无限知识。然而,计算机系统在实践中是有限的。因此,它关系到将我们的注意力限制在限定的模数战略上,用输出来定义为自动的。随机化可以以不同的方式实施:初始化、输出或转变的任意性分别在库恩的理论中显示其等值。这取决于哪些方面是随机化的,两个方面是如何使用无限的记忆,即对过去所有观测的无限知识。但是,计算机系统是有限的,因此,实际上是有限的。因此,将注意力限制性战略分为三个类的税式,这三类是不同的。