Arena-Independent Finite-Memory Determinacy in Stochastic Games

TL;DR

This paper investigates the complexity of strategies in stochastic zero-sum games, focusing on arena-independent finite-memory strategies, and extends deterministic game results to stochastic settings.

Contribution

It characterizes when arena-independent finite-memory strategies suffice in stochastic games and reduces their analysis to simpler one-player cases.

Findings

Pure AIFM strategies suffice for optimal play and subgame perfect strategies.

Reduction of two-player stochastic game analysis to one-player cases.

Characterization of objectives requiring AIFM strategies based on intuitive properties.

Abstract

We study stochastic zero-sum games on graphs, which are prevalent tools to model decision-making in presence of an antagonistic opponent in a random environment. In this setting, an important question is the one of strategy complexity: what kinds of strategies are sufficient or required to play optimally (e.g., randomization or memory requirements)? Our contributions further the understanding of arena-independent finite-memory (AIFM) determinacy, i.e., the study of objectives for which memory is needed, but in a way that only depends on limited parameters of the game graphs. First, we show that objectives for which pure AIFM strategies suffice to play optimally also admit pure AIFM subgame perfect strategies. Second, we show that we can reduce the study of objectives for which pure AIFM strategies suffice in two-player stochastic games to the easier study of one-player stochastic games (i.e., Markov decision processes). Third, we characterize the sufficiency of AIFM strategies through two intuitive properties of objectives. This work extends a line of research started on deterministic games to stochastic ones.