DValue for Boolean games is EXP-complete

We show that the following problem is EXP-complete: given a rational v and a two player, zero-sum Boolean gameG determine whether the value of G is at least v. The proof is via a translation of the proof of the same result for Boolean circuit games in [1]. 1 Preliminaries We will be using the encoding of [2] to replicate the proof of [1]. A familiarity with [2] will make the proof much easier to follow. The specific breed of Boolean games that concerns us here has two players, and since their goals are purely conflicting only one goal formula is necessary. Definition 1.1. A two player, zero-sum Boolean game consists of a set of variables, Φ, partitioned into two sets, Φ1 and Φ2, as well as a propositional logic formula γ1 over Φ. The game is played by letting Player One choose a truth assignment to the variables in Φ1 and Player Two to the variables in Φ2. If the resulting truth assignment satisfies γ1, Player One wins the game. If it does not, Player Two wins the game. The algorithmic question of interest is a decision version of Value. DValue Input: A two player zero-sum Boolean game G and a rational v. Output: YES if the value of G is at least v, NO otherwise.