Deciding if a quantified boolean formula such as
$\forall x_1 \exists x_2 \forall x_3\cdots \exists x_n \varphi(x_1, x_2,\ldots , x_n),$
always evaluates to true is a classical PSPACE-complete problem. This can be viewed as a game between two players, with alternating moves. The first player decides the truth value of the odd-numbered variables and the second player decides the truth value of the even-numbered variables. The first player tries to make $\varphi$ false and the second player tries to make it true. Deciding who has a winning strategy is PSPACE-complete.
I am considering a similar problem with two players, one trying to make a boolean formula $\varphi$ true and the other trying to make it false. The difference is that on a move, a player can choose a variable and a truth value for it (for example, as the very first move, player one might decide to set $x_8$ to true and then in the next move, player two might decide to set $x_3$ to false). This means that the players can decide which of the variables (of those that have not yet been assigned a truth value) they want to assign a truth value, instead of havinng to play the game in the order $x_1 , \ldots , x_n$.
The problem is given a boolean formula $\varphi$ on $n$ variables to decide if player one (trying to make it false) or player two (trying to make it true) has a winning strategy. This problem is clearly still in PSPACE, since the game tree has linear depth.
Does it remain PSPACE complete?