Different Strokes in Randomised Strategies: Revisiting Kuhn's Theorem under Finite-Memory Assumptions

TL;DR

This paper investigates the expressiveness of various finite-memory randomized strategies in two-player stochastic games, extending classical results like Kuhn's theorem to more practical, finite-memory settings.

Contribution

It provides a comprehensive taxonomy of finite-memory strategies based on which components are randomized, applicable to a wide range of game types and information structures.

Findings

Complete taxonomy of finite-memory strategies with different randomization components

Applicability to games with perfect/imperfect information and multiple players

Extension of Kuhn's theorem to finite-memory, practical scenarios

Abstract

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.