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

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  • Houy, Nicolas
  • Zwicker, William S.

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.

Suggested Citation

  • Houy, Nicolas & Zwicker, William S., 2014. "The geometry of voting power: Weighted voting and hyper-ellipsoids," Games and Economic Behavior, Elsevier, vol. 84(C), pages 7-16.
    • Handle: RePEc:eee:gamebe:v:84:y:2014:i:c:p:7-16
    DOI: 10.1016/j.geb.2013.12.001
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    Cited by:

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    2. Bertrand Mbama Engoulou & Lawrence Diffo Lambo, 2019. "Amplitude of weighted representation of voting games with several levels of approval," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(4), pages 1111-1137, December.
    3. Kurz, Sascha & Mayer, Alexander & Napel, Stefan, 2020. "Weighted committee games," European Journal of Operational Research, Elsevier, vol. 282(3), pages 972-979.
    4. Sam Jones, 2022. "Extending multidimensional poverty identification: from additive weights to minimal bundles," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 20(2), pages 421-438, June.
    5. Josep Freixas & Marc Freixas & Sascha Kurz, 2017. "On the characterization of weighted simple games," Theory and Decision, Springer, vol. 83(4), pages 469-498, December.
    6. Serguei Kaniovski & Sascha Kurz, 2018. "Representation-compatible power indices," Annals of Operations Research, Springer, vol. 264(1), pages 235-265, May.
    7. Sascha Kurz & Nicola Maaser & Stefan Napel & Matthias Weber, 2014. "Mostly Sunny: A Forecast of Tomorrow's Power Index Research," Tinbergen Institute Discussion Papers 14-058/I, Tinbergen Institute.

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    More about this item

    Keywords

    Weighted voting; Voting power; Simple games; Ellipsoidal separability;
    All these keywords.

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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