The geometry of voting power: Weighted voting and hyper-ellipsoids

• 成果类型: Article
• 署名作者: Houy, Nicolas; Zwicker, William S.
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Union College
• 刊物名称: GAMES AND ECONOMIC BEHAVIOR
• ISSN/ISSBN: 0899-8256
• DOI: 10.1016/j.geb.2013.12.001
• 发表日期: 2014
• 页码: 7-16
• 关键词: Weighted voting; Voting power; simple games; Ellipsoidal separability

摘要

Suppose legislators represent districts of varying population, and their assembly's voting rule is intended to implement the principle of one person, one vote. How should legislators' voting weights appropriately reflect these population differences? An analysis requires an understanding of the relationship between voting weight and some measure of the influence that each legislator has over collective decisions. We provide three new characterizations of weighted voting that embody this relationship. Each is based on the intuition that winning coalitions should be close to one another. The locally minimal and tightly packed characterizations use a weighted Hamming metric. Ellipsoidal separability employs the Euclidean metric: a separating hyper-ellipsoid contains all winning coalitions, and omits losing ones. The ellipsoid's proportions, and the Hamming weights, reflect the ratio of voting weight to influence, measured as Penrose Banzhaf voting power. In particular, the spherically separable rules are those for which voting powers can serve as voting weights. (C) 2013 Elsevier Inc. All rights reserved.