Formula game

A formula game is an artificial game represented by a fully quantified Boolean formula such as $$\exists x_1 \forall x_2 \exists x_3 \ldots \psi$$.

One player (E) has the goal of choosing values so as to make the formula $$\psi$$ true, and selects values for the variables that are existentially quantified with $$\exists$$. The opposing player (A) has the goal of making the formula $$\psi$$ false, and selects values for the variables that are universally quantified with $$\forall$$. The players take turns according to the order of the quantifiers, each assigning a value to the next bound variable in the original formula. Once all variables have been assigned values, Player E wins if the resulting expression is true.

In computational complexity theory, the language FORMULA-GAME is defined as all formulas $$\Phi$$ such that Player E has a winning strategy in the game represented by $$\Phi$$. FORMULA-GAME is PSPACE-complete because it is exactly the same decision problem as True quantified Boolean formula. Player E has a winning strategy exactly when every choice they must make in a game has a truth assignment that makes $$\psi$$ true, no matter what choice Player A makes.