The famous game-theoretic couple, Alice & Bob, live in the set-theoretic universe, $V$, a model of $ZFC$. Just like many other couples they sometimes argue over a statement, $\sigma$, expressible in the language of set theory. (One may think of $\sigma$ as a family condition/decision in the real life, say having kids or living in a certain city, etc.)

Alice wants $\sigma$ to be true in the world that they live but Bob doesn't. In such cases, each of them tries to manipulate the sequence of the events in such a way that makes their desired condition true in the ultimate situation. Consequently, a game of forcing iteration emerges between them as follows:

Alice starts by forcing over $V$, leading the family to the possible world $V[G]$. Then Bob forces over $V[G]$ leading both to another possible world in which Alice responds by forcing over it and so on. Formally, during their turn, Alice and Bob are choosing the even and odd-indexed names for forcing notions, $\dot{\mathbb{Q}}_{0}$, $\dot{\mathbb{Q}}_{1}$, $\dot{\mathbb{Q}}_{2}$, $\cdots$, in a forcing iteration of length $\omega$, $\mathbb{P}=\langle\langle\mathbb{P}_{\alpha}: \alpha\leq\omega\rangle, \langle\dot{\mathbb{Q}}_{\alpha}: \alpha<\omega\rangle\rangle$, where the ultimate $\mathbb{P}$ is made of the direct/inverse limit of its predecessors (depending on the version of the game). Alice wins if $\sigma$ holds in $V^{\mathbb{P}}$, the *ultimate future*. Otherwise, Bob is the winner.

Question 1.Is there any characterization of the statements $\sigma$ for which Alice has a winning strategy in (the direct/inverse limit version of) the described game? How much does it depend on the starting model $V$?

Clearly, Alice has a winning strategy if $\sigma$ is a consequence of $ZFC$, a rule of nature which Bob can't change no matter how tirelessly he tries and what the initial world, $V$, is! However, if we think in terms of *buttons* and *switches* in Hamkins' forcing multiverse, the category of the statements for which Alice has a winning strategy seems much larger than merely the consequences of $ZFC$.

I am also curious to know how big the difference between the direct and inverse limit versions of the described game is:

Question 2.What are examples of the statements like $\sigma$, for which Alice and Bob have winning strategies in the direct and inverse limit versions of the described game respectively?

**Update.** Following Mohammad's comment, it seems a variant of this game in which Alice and Bob are restricted to choose certain types of forcing notions (e.g. c.c.c. or proper) might be of interest as well. So the following version of the question 1 arises:

Question 3.Is there any characterization of the statements $\sigma$ for which Alice has a winning strategy in the described game restricted to forcing notions from the class $\Gamma$ where $\Gamma$ is the class of all c.c.c./proper/... forcings?