In a [Maker-Breaker game](https://en.wikipedia.org/wiki/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?



Satisfier-Falsifier games
This question was bumped to the homepage by Community, it sounded interesting to me so I did a little searching. The only thing I found was the paper



> 
> L. Zhao and M. Müller. Game-SAT: A preliminary report. In Seventh International Conference on Theory and Applications of Satisfiability Testing (SAT 2004), pages 357-362, Vancouver, Canada, 2004.
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which you can find a pdf of in a few places. For example on [one of the author's webpage](https://webdocs.cs.ualberta.ca/~mmueller/ps/gamesat.pdf). I could only find this preliminary report and not anything further. According to google a few paper have cited this one, but it seems only as related work and nothing more in this direction that I could find.


They consider what they call the Game-SAT problem of determining who wins assuming perfect play from each player. Reports on some experiments indicating a "phase transition" between being a win for the satisfier and a win for the falsifier are given. The paper also claims the problem has the same complexity as the quantified Boolean formula problem (QBF). However, no reductions are given and it is stated



> 
> "We have not found any methods to convert
>  between QBF and Game-SAT without introducing an enormous blow-up of the number of variables
>  and clauses."
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> 
> 


