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Power indices expressed in terms of minimal winning coalitions

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  • Fabien Lange

    ()
    (Óbuda University)

  • László Á. Kóczy

    ()
    (Óbuda University)

Abstract

A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the Shapley-Shubik index and the Banzhaf value, show the influence of the individual players. We used to calculate them by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the rules of the legislation. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies calculations. The technique generalises directly to all semivalues. Keywords. Shapley-Shubik index, Banzhaf index, semivalue, minimal winning coalition, Möbius transform.

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File URL: http://uni-obuda.hu/users/vecseya/RePEc/pkk/wpaper/1002.pdf
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Paper provided by Óbuda University, Keleti Faculty of Business and Management in its series Working Paper Series with number 1002.

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Length: 14 pages
Date of creation: 2010
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Handle: RePEc:pkk:wpaper:1002.rdf

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Keywords: Shapley-Shubik index; Banzhaf index; semivalue; minimal winning coalition; Möbius transform.;

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  1. Márkus, Judit & Pintér, Miklós & Radványi, Anna, 2011. "The Shapley value for airport and irrigation games," MPRA Paper 30031, University Library of Munich, Germany.
  2. BILLOT, Antoine & THISSE, Jean-François, 2002. "How to share when context matters: The Möbius value as a generalized solution for cooperative games," CORE Discussion Papers, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) 2002025, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  3. Federico Valenciano & Annick Laruelle, 2000. "- Shapley-Shubik And Banzhaf Indices Revisited," Working Papers. Serie AD, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie) 2000-02, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
  4. Kóczy, László Á., 2012. "Beyond Lisbon: Demographic trends and voting power in the European Union Council of Ministers," Mathematical Social Sciences, Elsevier, Elsevier, vol. 63(2), pages 152-158.
  5. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer, Springer, vol. 17(2), pages 89-99.
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