On the dimensionality of voting games

In a yes/no voting game, a set of voters must determine whether to accept or reject a given alternative. Weighted voting games are a well-studied subclass of yes/no voting games, in which each voter has a weight, and an alternative is accepted if the total weight of its supporters exceeds a certain threshold. Weighted voting games are naturally extended to A:-vector weighted voting games, which are intersections of A: different weighted voting games: a coalition wins if it wins in every component game. The dimensionality, k, of a k-vector weighted voting game can be understood as a measure of the complexity of the game. In this paper, we analyse the dimensionality of such games from the point of view of complexity theory. We consider the problems of equivalence, (checking whether two given voting games have the same set of winning coalitions), and minimality, (checking whether a given k-vector voting game can be simplified by deleting one of the component games, or, more generally, is equivalent to a k'-weighted voting game with k' < k). We show that these problems are computationally hard, even if k = 1 or all weights are 0 or 1. However, we provide efficient algorithms for cases where both k is small and the weights are polynomially bounded. We also study the notion of monotonicity in voting games, and show that monotone yes/no voting games are essentially as hard to represent and work with as general games.