The geometry of voting power: Weighted voting and hyper-ellipsoids |
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Affiliation: | 1. Groupe d''Analyse et de Théorie Economique, CNRS – University of Lyon, 69130 Ecully, France;2. Department of Mathematics, Union College, Schenectady, NY 12308, United States;1. Microsoft Research, Herzliya, Israel;2. Department of Economics, The Hebrew University, Jerusalem 91905, Israel;3. Computer and Information Sciences, University of Pennsylvania, Philadelphia, PA 19104, USA;1. University of Houston, 4800 Calhoun Road, Houston, TX 77020, United States;2. University of Colorado at Boulder, Box 216 UCB, Boulder, CO 80309-0216, United States;3. Jet Propulsion Laboratory, M/S 264-640, 800 Oak Grove Drive, Pasadena, CA 91109, United States;1. Department of Physics, Central Washington University, 400 E. University Way, Ellensburg, WA, 98926-7442, USA;2. Lunar and Planetary Laboratory, University of Arizona, 1629 E. University Blvd., Tucson, AZ 85721-0092, USA;1. Kellogg School of Management, Northwestern University, 2001 Sheridan Rd., Evanston, IL 60208, United States;2. Department of Economics, University of Illinois, 1407 W. Gregory Dr., Urbana, IL 61801-3606, United States;3. Department of Finance, London School of Economics, Houghton St., London WC2A 2AE, United Kingdom |
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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. |
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Keywords: | Weighted voting Voting power Simple games Ellipsoidal separability |
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