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

Nicolas Houy and William S. Zwicker
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William S. Zwicker: Union College

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Abstract: 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.

Keywords: weighted voting; voting power; simple games; ellipsoidal separability (search for similar items in EconPapers)
Date: 2014
Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00926969v1
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Citations: View citations in EconPapers (6)

Published in Games and Economic Behavior, 2014, 84, pp.7-16. ⟨10.1016/j.geb.2013.12.001⟩

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Related works:
Journal Article: The geometry of voting power: Weighted voting and hyper-ellipsoids (2014)
Working Paper: The geometry of voting power: weighted voting and hyper-­ellipsoids (2013)
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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:halshs-00926969

DOI: 10.1016/j.geb.2013.12.001

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