Banach-Mazur Games with Simple Winning Strategies

Abstract

We discuss several notions of "simple" winning strategies for
Banach-Mazur games on graphs, such as positional strategies,
move-counting or length-counting strategies, and strategies with a
memory based on finite appearance records (FAR). We investigate
classes of Banach-Mazur games that are determined via these kinds of
winning strategies. 

Banach-Mazur games admit stronger determinacy results than classical
graph games. For instance, all Banach-Mazur games with omega-regular
winning conditions are positionally determined. Beyond the
omega-regular winning conditions, we focus here on Muller conditions
with infinitely many colours. We investigate the infinitary Muller
conditions that guarantee positional determinacy for Banach-Mazur
games. Further, we determine classes of such conditions that require
infinite memory but guarantee determinacy via move-counting
strategies, length-counting strategies, and FAR-strategies. We also
discuss the relationships between these different notions of determinacy.