Towards tractable Boolean games

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Boolean games are a compact and expressive class of games, based on propositional logic. However, Boolean games are computationally complex: checking for the existence of pure Nash equilibria in Boolean games is Σp/2-complete, and it is co-NP-complete to check whether a given outcome for a Boolean game is a pure Nash equilibrium. In this paper, we consider two possible avenues to tractability in Boolean games. First, we consider the development of alternative solution concepts for Boolean games. We introduce the notion of k-bounded Nash equilibrium, meaning that no agent can benefit from deviation by altering fewer than k variables. After motivating and discussing this notion of equilibrium, we give a logical characterisation of a class of Boolean games for which k-bounded equilibria correspond to Nash equilibria. That is, we precisely characterise a class of Boolean games for which all Nash equilibria are in fact k-bounded Nash equilibria. Second, we consider classes of Boolean games for which computational problems related to Nash equilibria are easier than in the general setting. We first identify some restrictions on games that make checking for beneficial deviations by individual players computationally tractable, and then show that certain types of socially desirable equilibria can be hard to compute even when the standard decision problems for Boolean games are easy. We conclude with a discussion of related work and possible future work.