Pure Strategies in Imperfect Information Stochastic Games

TL;DR

This paper studies pure strategies in imperfect information stochastic games with various objectives, providing decidability results for some cases and showing undecidability for others, advancing understanding of strategy existence in such games.

Contribution

It offers new decidability results for pure strategies in certain objectives and highlights undecidability in others, filling gaps in the theory of imperfect information stochastic games.

Findings

Decidability for positive reachability and almost-sure B"uchi with pure strategies.

Undecidability of positive safety with pure strategies even when the second player is perfectly informed.

Algorithms developed for deciding the existence of pure winning strategies.

Abstract

We consider imperfect information stochastic games where we require the players to use pure (i.e. non randomised) strategies. We consider reachability, safety, B\"uchi and co-B\"uchi objectives, and investigate the existence of almost-sure/positively winning strategies for the first player when the second player is perfectly informed or more informed than the first player. We obtain decidability results for positive reachability and almost-sure B\"uchi with optimal algorithms to decide existence of a pure winning strategy and to compute one if exists. We complete the picture by showing that positive safety is undecidable when restricting to pure strategies even if the second player is perfectly informed.