The geometry of voting power: Weighted voting and hyper-ellipsoids
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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- Laruelle,Annick & Valenciano,Federico, 2011.
"Voting and Collective Decision-Making,"
Cambridge University Press, number 9780521182638, November.
- Chang, Pao-Li & Chua, Vincent C.H. & Machover, Moshe, 2006.
"L S Penrose's limit theorem: Tests by simulation,"
Mathematical Social Sciences,
Elsevier, vol. 51(1), pages 90-106, January.
- Hosli, Madeleine O., 1993. "Admission of European Free Trade Association states to the European Community: effects on voting power in the European Community Council of Ministers," International Organization, Cambridge University Press, vol. 47(04), pages 629-643, September.
- Moshé Machover & Dan S. Felsenthal, 2001. "The Treaty of Nice and qualified majority voting," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(3), pages 431-464.
- Laruelle, Annick & Widgren, Mika, 1998.
"Is the Allocation of Voting Power among EU States Fair?,"
Springer, vol. 94(3-4), pages 317-39, March.
- Annick Laruelle & Mika Widgrén, 1998. "Is the allocation of voting power among EU states fair?," Public Choice, Springer, vol. 94(3), pages 317-339, March.
- Laruelle, Annick & Widgren, Mika, 1996. "Is the allocation of voting power among EU states fair?," Discussion Papers (IRES - Institut de Recherches Economiques et Sociales) 1996022, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
- Einy, Ezra & Lehrer, Ehud, 1989. "Regular Simple Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(2), pages 195-207.
- Cervone, Davide P. & Dai, Ronghua & Gnoutcheff, Daniel & Lanterman, Grant & Mackenzie, Andrew & Morse, Ari & Srivastava, Nikhil & Zwicker, William S., 2012. "Voting with rubber bands, weights, and strings," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 11-27.
- Lindner, Ines & Owen, Guillermo, 2007. "Cases where the Penrose limit theorem does not hold," Mathematical Social Sciences, Elsevier, vol. 53(3), pages 232-238, May.
- Taylor, Alan & Zwicker, William, 1997. "Interval measures of power," Mathematical Social Sciences, Elsevier, vol. 33(1), pages 23-74, February.
- Pierre Barthelemy, Jean & Monjardet, Bernard, 1981. "The median procedure in cluster analysis and social choice theory," Mathematical Social Sciences, Elsevier, vol. 1(3), pages 235-267, May.
- Lindner, Ines & Machover, Moshe, 2004. "L.S. Penrose's limit theorem: proof of some special cases," Mathematical Social Sciences, Elsevier, vol. 47(1), pages 37-49, January.
- Gvozdeva, Tatiana & Slinko, Arkadii, 2011. "Weighted and roughly weighted simple games," Mathematical Social Sciences, Elsevier, vol. 61(1), pages 20-30, January.
- Johnston, R. J., 1995. "The Conflict over Qualified Majority Voting in the European Union Council of Ministers: An Analysis of the UK Negotiating Stance Using Power Indices," British Journal of Political Science, Cambridge University Press, vol. 25(02), pages 245-254, April.
- Leech, Dennis, 2002. "Designing the Voting System for the Council of the European Union," Public Choice, Springer, vol. 113(3-4), pages 437-64, December.
- Dan Felsenthal & Moshé Machover, 2005. "Voting power measurement: a story of misreinvention," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 25(2), pages 485-506, December.
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