This question is a follow-up to [this one](https://mathoverflow.net/questions/351663/undetermined-banach-mazur-games-in-zf); see that question for the definition of Banach-Mazur games. There James Hanson showed that ZF+DC proves that there is an undetermined Banach-Mazur game; however, the situation with ZF+$\neg$DC is still open *(there was an error in a claimed answer, and this question has been modified from its original form to accommodate that)*. 

I'd like to ask what the ZF-situation is. However, in ZF alone things are rather weird so there are multiple plausible "right" versions of this question (which are all equivalent in ZF+DC). Unfortunately this does make the question rather subjective (presumably based on the answers we somehow pin down which version was "right," but how?), but I think it's within $\epsilon$ of precise for a sufficiently small $\epsilon$.

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The most obvious question to ask is whether the result, phrased exactly as previously, continues to hold:

>**Version 1**: Does ZF prove "There is an undetermined Banach-Mazur game"?

On second thought, however, it's not obvious to me that this is the right question. Without DC determinacy is a bit weird, and it's arguably better to talk about **quasistrategies**. A quasistrategy lays out a nonempty set of "permitted moves" at each stage, with a quasistrategy **s** being winning for a player if there is no play consistent with **s** in which that player loses and a game being quasidetermined if it has a winning quasistrategy.

A game is *quasidetermined* if one player or the other has a winning quasistrategy. In ZF+DC determinacy is the same as quasideterminacy, but in ZF they are distinct. So we can separately ask:

> **Version 2**: Does ZF prove "There is an un-quasidetermined Banach-Mazur game?"

Yet on third thought, *even that* isn't obviously the right question: it's consistent with ZF that *both* players have winning quasistrategies in a given game! *(Consider "Alternate choosing new elements of an amorphous set.")* So  we might want to land somewhere in the middle - say that a game is *weakly determined* if exactly one player has a winning quasistrategy.

> **Version 3**: Is the statement "Every Banach-Mazur game is *weakly determined*" consistent with ZF?