I have sat in lectures on set theory and I have seen the use of games cropping up in many places. I don't really understand what was going on and how useful games are in set theory, but here I have a question.
Take a typical example in descriptive set theory where we associate initial segments of a branch of a tree to other sets. For instance, let $M$ be a perfect Polish space and we are assigning (a function) each initial segment in the Cantor space a nonempty open neighborhood of $M$ such that
- if $u < v$, then $N_u\subseteq N_v$,
- if $u$ and $v$ are incompatible, then $N_u\cap N_v=\emptyset$, and
- if $lh(u)=n$, then the radius of $N_u$ is no larger than $2^{-n}$.
I find it convenient to picture the construction as two players playing a game: Player I gives a finite binary sequence and Player II collaborate by producing a neighborhood satisfying 1, 2, and 3.
This game doesn't require any player to win, but it seems to give a useful way to describe the process that's taking place.
In set theory, do we ever consider games without having either player to win, perhaps for constructing sets or functions as given above? Or perhaps my description above didn't make much sense, or there are aspects of winning condition in it that was implicit or for which I failed to see?
