Complexity of mixed equilibria in Boolean games

Boolean games are a succinct representation of strategic games wherein a player seeks to satisfy a formula of propositional logic by selecting a truth assignment to a set of propositional variables under his control. The framework has proven popular within the multiagent community, however, almost invariably, the work to date has been restricted to the case of pure strategies. Such a focus is highly restrictive as the notion of randomised play is fundamental to the theory of strategic games -- even very simple games can fail to have pure-strategy equilibria, but every finite game has at least one equilibrium in mixed strategies. To address this, the present work focuses on the complexity of algorithmic problems dealing with mixed strategies in Boolean games. The main result is that the problem of determining whether a two-player game has an equilibrium satisfying a given payoff constraint is NEXP-complete. Based on this result, we then demonstrate that a number of other decision problems, such as the uniqueness of an equilibrium or the satisfaction of a given formula in equilibrium, are either NEXP or coNEXP-complete. The proof techniques developed in the course of this are then used to show that the problem of deciding whether a given profile is in equilibrium is coNP^#P-hard, and the problem of deciding whether a Boolean game has a rational-valued equilibrium is NEXP-hard, and whether a two-player Boolean game has an irrational-valued equilibrium is NEXP-complete. Finally, we show that determining whether the value of a two-player zero-sum game exceeds a given threshold is EXP-complete.