Boolean combinations of weighted voting games

Weighted voting games are a natural and practically important class of simple coalitional games, in which each agent is assigned a numeric weight, and a coalition is deemed to be winning if the sum of weights of agents in that coalition meets some stated threshold. We study a natural generalisation of weighted voting games called Boolean Weighted Voting Games (BWVGs). BWVGs are intended to model decision-making processes in which components of an overall decision are delegated to committees, with each committee being an individual weighted voting game. We consider the perspective of an individual who has some overall goal that they desire to achieve, represented as a propositional logic formula over the decisions controlled by the various committees. We begin by formulating the framework of BWVGs, and show that BWVGs can provide a succinct representation scheme for simple coalitional games, compared to other representations based on weighted voting games. We then consider the computational complexity of problems such as determining the power of a particular player with respect to some goal, and investigate how the power of a player with respect to the overall goal is related to the power of that player in individual games. We show trade-offs between the complexity of these problems, the nature of underlying Boolean formulas used, and representations of weights (binary versus unary) in our games.