$2^3$ Quantified Boolean Formula Games and Their Complexities

Consider QBF, the Quantified Boolean Formula problem, as a combinatorial game ruleset. The problem is rephrased as determining the winner of the game where two opposing players take turns assigning values to boolean variables. In this paper, three common variations of games are applied to create seven new games: whether each player is restricted to where they may play, which values they may set variables to, or the condition they are shooting for at the end of the game. The complexity for determining which player can win is analyzed for all games. Of the seven, two are trivially in P and the other five are PSPACE-complete. These varying properties are common for combinatorial games; reductions from these five hard games can simplify the process for showing the PSPACE-hardness of other games.