Throughout, we work in ZF+DC+AD.

The subsets of $\mathbb{R}$ which are comeager on some interval are exactly those which contain a comeager $\Pi^0_2$ set, and the sets which contain a perfect set are exactly those which contain an uncountable $\Pi^0_1$ set. This means that we have a simple description of the class of sets whose corresponding Banach-Mazur game, or corresponding perfect set game, is a win for player I. And similar statements apply to many other kinds of topological games.

By contrast, when we consider general games the situation is much more complicated: there is no snappy description of those sets $A\subseteq\omega^\omega$ such that the classical game on $\omega$ with payoff set $A$ is a win for player I, and indeed it is known that the game quantifier $\Game$ is quite powerful. So we can try to distinguish types of games which build reals in terms of how hard it is to describe those sets which are a win for player I.

I'm specifically interested in the Banach game. Fix a payoff set $A\subseteq\mathbb{R}_{>0}$. In the game $B(A)$, players I and II alternate playing positive reals $a_i$, with each real smaller than the preceding one; player I wins if $\sum a_i$ exists and is in $A$. This is an interesting class of games - for example, although they are a priori games on $\mathbb{R}$, Becker showed that Banach determinacy is implied by AD, which is vastly weaker (both as a principle and in consistency strength) than AD$_\mathbb{R}$; and this followed Freiling's proof that Banach determinacy implies AD, which was also surprising given how specific the Banach game is and how difficult it is to "code" information into other topological or analytical games (Banach-Mazur, perfect set, etc.).

Now, in the paper linked above, Becker makes an interesting guess (I'm hesitant to call it a "conjecture" since it's not clear how strongly he believes it):

The (rather vague) problem which Banach posed was to characterize those sets $A$ for which I (II) has a winning strategy in $B(A)$ . . . This paper will not provide the reader with any answer to Banach's question. I know of no nontrivial way to characterize when player I (or II) wins, and I suspect there is none [emphasis mine].

I'm curious about ways to formalize, and prove or refute, this guess. There are several reasonable candidates for formalization; the simplest, to me, seem to be the following two.

First, following the above discussion we can ask for a simple characterization of the corresponding "largeness" notion:

  • Is there a family $\mathcal{F}_+$ of Borel sets such that $B(A)$ is a win for player I iff $X\subseteq A$ for some $X\in\mathcal{F}_+$?

This would amount to a "from below" characterization. We could also look for a "from above" characterization (e.g. $A$'s Banach-Mazur game is a win for II iff $A$ is meager, and $A$'s perfect set game is a win for II iff $A$ is countable):

  • Is there a family $\mathcal{F}_-$ of Borel sets such that $B(A)$ is a win for player II iff $A\subseteq X$ for some $X\in\mathcal{F}_-$?

(Though similar, these questions aren't equivalent. E.g. consider the class of sets containing uncountable Borel sets. This has a classification from below by Borel (indeed, $\Pi^0_1$) sets, but none from above.)

There are a vast number of further questions. For example, from a negative answer we can continue descriptive-set-theoretically: how complicated must a from-below or from-above description be? Or we can take a different approach entirely, e.g. measure the computational strength (in terms of Kleene recursion or otherwise) of the corresponding game quantifier. But I think the question above is already getting at something interesting, so that's what I'll ask here.

  • $\begingroup$ Note that unless I'm missing something, a priori the family of sets which are won by I (II) in the Banach game is $\Sigma^2_1$, since a strategy is a map from reals. This goes down to $\Sigma^1_2$, of course, if we can argue that any winning strategy must be continuous in the obvious sense, but I don't immediately see why that should be the case. $\endgroup$ – Noah Schweber Sep 15 '17 at 16:37
  • $\begingroup$ Since you put no conditions on $\mathcal F_+$ beyond just consisting of Borel sets, your first question can be formulated in a simpler-looking (to me) way: If $B(X)$ is a win for player I, must $X$ have a Borel subset $Y$ such that $B(Y)$ is also a win for I? Note also that the answer becomes an easy yes if we replace "Borel" with "analytic", since the set of all outcomes of plays in which I uses a specific strategy is analytic. $\endgroup$ – Andreas Blass Sep 30 '17 at 14:37
  • $\begingroup$ @AndreasBlass I agree with the first part. I'm not sure, though, that it becomes trivial if we replace "Borel" with "analytic" - while strategies for Banach games can be coded by reals (Thm. 1.3 of Freiling's paper), this coding is quite complicated; it's not clear to me that "the set of allowed plays by $\Sigma$" is in fact analytic, since while it is $\Sigma^1_1$ in $\Sigma$ I don't see why $\Sigma$ is boldface $\Sigma^1_1$ as a map from $\mathbb{R}^{<\omega}$ to $\mathbb{R}$. $\endgroup$ – Noah Schweber Sep 30 '17 at 15:00
  • $\begingroup$ I think you're right; winning strategies can be more complicated than I assumed. $\endgroup$ – Andreas Blass Sep 30 '17 at 18:31

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