Formula game

A formula game is an artificial game represented by a fully quantified Boolean formula such as.
One player has the goal of choosing values so as to make the formula true, and selects values for the variables that are existentially quantified with. The opposing player has the goal of making the formula false, and selects values for the variables that are universally quantified with. 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 such that Player E has a winning strategy in the game represented by. 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 true, no matter what choice Player A makes.