Consider the following infinite perfect information game with two players (the name I gave in the title of the post it totally made up): at each round $i \in \omega$, player $\mathrm{I}$ picks a natural number $x_i$ and a $\boldsymbol{\Delta}_2^0$ subset of the Baire space $\omega^\omega$ denoted by $A_i$ such that $A_{i-1}\subseteq A_i$ for all $i$; player $\mathrm{II}$ then picks a boolean value $y_i \in \{0,1\}$.

So at the end of the game player $\mathrm{I}$ will have built an infinite sequence $x = (x_n)_{n \in \omega}$ and an increasing chain (wrt set inclusion) of $\boldsymbol{\Delta}_2^0(\omega^\omega)$ sets $(A_n)_{n \in \omega}$ whilst player $\mathrm{II}$ will have built a boolean sequence $y = (y_n)_{n \in \omega}$ (an element of the Cantor space). Player $\mathrm{II}$ wins the play if *at least one* of these conditions is satisfied:

- $\bigcup_{n \in \omega} A_n \not\in \boldsymbol{\Delta}_2^0(\omega^\omega)$
- The sequence $y$ is eventually constant, i.e. $\exists k \ \forall n \ge k \ y_n = y_k$. Moreover The sequence $y$ is eventually equal to $1$ if and only if $x \in \bigcup_{n \in \omega} A_n$. Intuitively we require player $\mathrm{II}$ to "guess" whether the real played by $\mathrm{I}$ will or won't belong to the set $\mathrm{I}$ is building.

Now I'm wondering whether player $\mathrm{II}$ has a winning strategy in this game, or if the game is determined, and, eventually, under which hypotheses.

In a simpler game, in which player $\mathrm{I}$ does not keep changing the sets $A_i$, player $\mathrm{II}$ has a winning strategy
(see, for this result and a wider discussion, this paper by Raphael Carroy **Playing in the first Baire class**, specifically Proposition 3.10).

Any idea? Thanks