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

  • Nicolas Houy

    (GATE Lyon Saint-Etienne - Groupe d'analyse et de théorie économique - CNRS : UMR5824 - Université Lumière - Lyon II - École Normale Supérieure - Lyon)

  • William S. Zwicker

    (Union College - Union College)

Registered author(s):

    In cases where legislators represent districts that vary in population, the design of fair legislative voting rules requires an understanding of how the number of votes cast by a legislator is related to a measure of her influence over collective decisions. We provide three new characterizations of weighted voting, each 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 hyperellipsoid 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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    Paper provided by HAL in its series Working Papers with number halshs-00772953.

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    Date of creation: 11 Jan 2013
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    Handle: RePEc:hal:wpaper:halshs-00772953
    Note: View the original document on HAL open archive server: http://halshs.archives-ouvertes.fr/halshs-00772953
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    1. Einy, Ezra & Lehrer, Ehud, 1989. "Regular Simple Games," International Journal of Game Theory, Springer, vol. 18(2), pages 195-207.
    2. 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.
    3. 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.
    4. Gvozdeva, Tatiana & Slinko, Arkadii, 2011. "Weighted and roughly weighted simple games," Mathematical Social Sciences, Elsevier, vol. 61(1), pages 20-30, January.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. Moshé Machover & Dan S. Felsenthal, 2001. "The Treaty of Nice and qualified majority voting," Social Choice and Welfare, Springer, vol. 18(3), pages 431-464.
    10. 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.
    11. Dan Felsenthal & Moshé Machover, 2005. "Voting power measurement: a story of misreinvention," Social Choice and Welfare, Springer, vol. 25(2), pages 485-506, December.
    12. repec:cup:cbooks:9780521182638 is not listed on IDEAS
    13. Taylor, Alan & Zwicker, William, 1997. "Interval measures of power," Mathematical Social Sciences, Elsevier, vol. 33(1), pages 23-74, February.
    14. repec:cup:cbooks:9780521873871 is not listed on IDEAS
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