On the structure of minimal winning coalitions in simple voting games

According to Coleman's index of collective power, a decision rule that generates a larger number of winning coalitions imparts the collectivity a higher a priori power to act. By the virtue of the monotonicity conditions, a decision rule is totally characterized by the set of minimal winning coalitions. In this paper, we investigate the structure of the families of minimal winning coalitions corresponding to maximal and proper simple voting games (SVG). We show that if the proper and maximal SVG is swap robust and all the minimal winning coalitions are of the same size, then the SVG is a specific (up to an isomorphism) system. We also provide examples of proper SVGs to show that the number of winning coalitions is not monotone with respect to the intuitively appealing system parameters like the number of blockers, number of non-dummies or the size of the minimal blocking set.