随机策略中的不同方式：在有限记忆假设下重访库恩定理
Different Strokes in Randomised Strategies: Revisiting Kuhn's Theorem under Finite-Memory Assumptions
AI总结：
本文研究双人并发随机博弈中有限记忆策略的表达能力，通过考察初始化、输出和转移三个组成部分的随机化与否，建立了适用于完美/不完美信息、完美/不完美回忆及多玩家情形的完整策略分类体系，从而在有限记忆假设下重新审视库恩定理。
摘要：
在（可能随机的）图上的双人（对抗性）博弈是理论计算机科学中一种普遍的模型，尤其作为反应式综合的框架。当处理本质上概率性的目标、平衡多个目标或在部分信息的情境中时，最优策略可能需要随机化。定义随机策略的方式并不唯一。例如，可以使用所谓的混合策略或行为策略。在最一般的设定下，这两类策略的表达能力并不相同。博弈论中的一个开创性结果——库恩定理——断言它们在完美回忆博弈中是等价的。这一结果关键依赖于策略可以使用无限记忆的可能性，即对所有过去观测的无限制知识。然而，计算机系统在实践中是有限的。因此，将注意力限制在有限记忆策略上是恰当的，这些策略被定义为带输出的自动机。随机化可以在这些自动机中以不同方式实现：初始化、输出或转移可以分别是随机的或确定性的。根据哪些方面被随机化，相应的有限记忆策略类的表达能力会有所不同。在这项工作中，我们研究双人并发随机博弈，并通过对上述三个组成部分中哪些被随机化进行变化，给出了有限记忆策略类的完整分类。我们的分类适用于具有完美回忆的完美信息和不完美信息博弈，以及超过两个参与者的博弈。我们还为不完美回忆博弈提供了一个适配的分类。
英文摘要：
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 setting, 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 concurrent 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 in games of perfect and imperfect information with perfect recall, and in games with more than two players. We also provide an adapted taxonomy for games with imperfect recall.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
