A set $X\subseteq\omega^\omega$ is **unravelable** iff there is a possibly larger set $A$ and a clopen set $Y\subseteq A^\omega$ (with respect to the product topology coming from the discrete topology on $A$) such that winning strategies for the game on $A$ with payoff set $Y$ can be converted to winning strategies for the game on $\omega$ with payoff set $X$ in a particularly simple way (see Martin's paper [*A purely inductive proof of Borel determinacy*](https://www.math.ucla.edu/%7Edam/booketc/purely_inductive.pdf) for the precise definition). By Gale–Stewart, unravelability implies determinacy; however, it is not the only route to proving the determinacy of a pointclass.


My question is whether there is a known upper bound on the complexity of an unravelable pointclass. To make this somewhat precise:



> 
> Is it consistent with $\mathsf{ZFC}$ that every set in $\mathcal{P}(\mathbb{R})^{L(\mathbb{R})}$ is unravelable?
> 
> 
> 


The strongest results I can find are *vastly* weaker than this, namely that under large cardinal assumptions $\Pi^1\_1$ sets are unravelable *(Neeman [1](https://www.sciencedirect.com/science/article/pii/S0168007200000221),[2](https://link.springer.com/article/10.1007/BF02771982); I remember seeing the same result for Borel-on-$\Pi^1\_1$ sets, but I can't find a reference for it at the moment)*. On the other hand, I don't see an easy proof that this is anywhere close to the most unravelability we can expect, nor can I find this stated in the literature.



What sets can be unraveled?
I emailed Itay Neeman, and he told me the following:



> 
> As far as I know it's open. I don't think anything is known about
> unraveling beyond what you can get from my methods. These give the
> Suslin operation on $\Pi^1\_1$ sets, and various iterations of that. In
> terms of the large cardinal hierarchy it's still all just using
> measures. Anything above that is open I think.
> 
> 
> In particular I don't think it's known if $\Sigma^1\_2$ sets can be
> unraveled.
> 
> 
> 


For now I think that settles this question. (Itay later said that it's unclear whether we "should" expect unravelability of $\Sigma^1\_2$ sets from large cardinals or not in the first place. So I think my main takeaway at this point is that unravelability is **not** best thought of as yet another tameness property, where the slogan "large cardinals make definable things tame" is expected to prevail.)


