Simple strategies for Banach-Mazur games and fairly correct systems

TL;DR

This paper explores simple strategies in Banach-Mazur games on graphs, introducing a probabilistic version to characterize sets of probability 1 and establishing determinacy results for certain winning sets.

Contribution

It generalizes classical Banach-Mazur games to probabilistic settings and analyzes the relation between probability 1 sets and simple strategies, providing new determinacy results.

Findings

Probabilistic Banach-Mazur game characterizes sets of probability 1.

Determinacy holds for countable intersections of open sets.

Connections between strategies and measure-theoretic properties are established.

Abstract

In 2006, Varacca and V\"olzer proved that on finite graphs, omega-regular large sets coincide with omega-regular sets of probability 1, by using the existence of positional strategies in the related Banach-Mazur games. Motivated by this result, we try to understand relations between sets of probability 1 and various notions of simple strategies (including those introduced in a recent paper of Gr\"adel and Lessenich). Then, we introduce a generalisation of the classical Banach-Mazur game and in particular, a probabilistic version whose goal is to characterise sets of probability 1 (as classical Banach-Mazur games characterise large sets). We obtain a determinacy result for these games, when the winning set is a countable intersection of open sets.