In a Maker-Breaker game, there is a finite set of elements $X$, and a family $F$ of subsets of $X$ called the "winning sets". Two players, Maker and Breaker, take turns picking untaken elements from $X$. Maker wins by holding a full winning-set while Breaker wins by holding at least one element in each winning-set.

Consider the following generalization. Instead of the family $F$ of winning-sets,
we have a family $F$ of *winning-formulas*.
Each element of $F$ is a logic formula.
The game is played by two players, Satisfier and Falsifier. Each player in turn picks an unset variable and sets it to either True or False.
The goal of Satisfier is to set variables such that at least one formula is satisfied; the goal of Falsifier is to set variables such that all formulas are unsatisfied.

The Maker-Breaker game is a special case in which each formula in $F$ is a conjunction of positive variables. For example, the winning-set $\{x,y,z\}$ corresponds to the formula "$x$ and $y$ and $z$". Note that in this special case, Satisfier always prefers to set his picked variables to True and Falsifier always prefers to set his picked variables to False.

Is anything known about the general case?

familyof formulas, as you can as well take their disjunction. $\endgroup$