Minimal winning coalitions in weighted-majority voting games

Riker's size principle for n-person zero-sum games predicts that winning coalitions that form will be minimal in that any player's defection will negate the coalition's winning status. Brams and Fishburn (1995) applied Riker's principle to weighted-majority voting games in which players have voting weights w1\geqw2\geq\dots\geqwn, and a coalition is winning if its members' weights sum to more than half the total weight. We showed that players' bargaining power tends to decrease as their weights decrease when minimal winning coalitions obtain, but that the opposite trend occurs when the minimal winning coalitions that form are "weight-minimal", referred to as least winning coalitions. In such coalitions, large size may be more harmful than helpful. The present paper extends and refines our earlier analysis by providing mathematical foundations for minimal and least winning coalitions, developing new data to examine relationships between voting weight and voting power, and applying more sophisticated measures of power to these data. We identify all sets of minimal and least winning coalitions that arise from different voting weights for n\leq6 and characterize all coalitions that are minimal winning and least winning for every n. While our new analysis supports our earlier findings, it also indicates there to be less negative correlation between voting weight and voting power when least winning coalitions form. In this context, players' powers are fairly insensitive to their voting weights, so being large or small is not particularly important for inclusion in a least winning coalition.