A winning strategy for Verifier in the verification game for \(\phi\) is a sequence of moves and countermoves for all possible replies and counter-replies of Falsifier which is guaranteed to result in a win for Verifier. Let us call the problem of deciding whether Verifier has such a winning strategy for \(\phi\) \(\sc{TWO}\ \sc{PLAYER}\ \sc{SAT}\). As the moves of the two players mirror the interpretations of \(\exists\) and \(\forall\) for QBF formulas, it is not difficult to see that \(\phi \in \sc{TWO}\ \sc{PLAYER}\ \sc{SAT}\) just in case \(\phi \in \sc{TQBF}\). And for any QBF-formula \(\phi\) with \(n\) initial quantifiers we may efficiently construct an equivalent formula with at most \(2n\) quantifiers in the required alternating form by interspersing dummy quantifiers as needed. It thus follows that like \(\sc{TQBF}\), \(\sc{TWO}\ \sc{PLAYER}\ \sc{SAT}\) is also \(\textbf{PSPACE}\)-complete.