What these games have in common is that the definition of a winning strategy for the player who moves first involves the alternation of existential and universal quantifiers in a manner which mimics the definition of the classes \(\Sigma^P_n\) and \(\Pi^P_n\) which comprise \(\textbf{PH}\). Taking this into account, suppose we define \(\sc{TWO}\ \sc{PLAYER}\ \sc{SAT}_n\) to be the variant of \(\sc{TWO}\ \sc{PLAYER}\ \sc{SAT}\) wherein there are at most \(n\) alternations of quantifiers in \(\phi\) (it thus follows that all of the games for formulas in this class will be of at most \(n\) rounds). \(\sc{TWO}\ \sc{PLAYER}\ \sc{SAT}_n\) may be shown to be complete for the class \(\Sigma^P_n\) in the Polynomial Hierarchy. Note, however, as the value of \(n\) increases, we expect that it should become more difficult to decide membership in \(\sc{TWO}\ \sc{PLAYER}\ \sc{SAT}_n\) in much the same way that it appears to become more difficult to determine whether a given player has a winning strategy for increasingly long games of Go or chess.