Banach-Mazur Games on Graphs

Abstract

We survey determinacy, definability, and 
complexity issues of Banach-Mazur games on finite and 
infinite graphs.

Infinite games where two players take turns to move a token 
through a directed graph, thus tracing out an infinite path, 
have numerous applications in different branches of mathematics
and computer science. In the usual format,
the possible moves of the players are given by
the edges of the graph; in each move
a player takes the token from its current position 
along an edge to a next position. In Banach-Mazur games 
the players instead select in each move a \emph{path} 
of arbitrary finite length rather than just an edge. 
In both cases the outcome of a play is an infinite 
path. A winning condition is thus given by a set of 
infinite paths which is often specified by a logical formula,
for instance from S1S, LTL, or first-order logic. 

Banach-Mazur games have a long tradition in descriptive 
set theory and topology, and they have recently been shown to 
have interesting applications also in computer science, 
for instance for planning in nondeterministic domains,
for the study of fairness in concurrent systems, and
for the semantics of timed automata.  

It turns out that Banach-Mazur games behave quite differently than 
the usual graph games. Often they admit simpler winning strategies
and more efficient algorithmic solutions. For instance, Banach-Mazur 
games with $\omega$-regular winning conditions always have 
positional winning strategies, and winning positions
for finite Banach-Mazur games with Muller winning condition
are computable in polynomial time.