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Power indices expressed in terms of minimal winning coalitions

Author

Listed:
  • 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.

Suggested Citation

  • Fabien Lange & László Á. Kóczy, 2010. "Power indices expressed in terms of minimal winning coalitions," Working Paper Series 1002, Óbuda University, Keleti Faculty of Business and Management.
  • Handle: RePEc:pkk:wpaper:1002.rdf
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    File URL: http://uni-obuda.hu/users/vecseya/RePEc/pkk/wpaper/1002.pdf
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    References listed on IDEAS

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    1. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
    2. Annick Laruelle & Federico Valenciano, 2001. "Shapley-Shubik and Banzhaf Indices Revisited," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 89-104, February.
    3. Billot, Antoine & Thisse, Jacques-Francois, 2005. "How to share when context matters: The Mobius value as a generalized solution for cooperative games," Journal of Mathematical Economics, Elsevier, vol. 41(8), pages 1007-1029, December.
    4. Márkus, Judit & Pintér, Miklós & Radványi, Anna, 2011. "The Shapley value for airport and irrigation games," MPRA Paper 30031, University Library of Munich, Germany.
    5. Kóczy, László Á., 2012. "Beyond Lisbon: Demographic trends and voting power in the European Union Council of Ministers," Mathematical Social Sciences, Elsevier, vol. 63(2), pages 152-158.
    6. repec:cup:apsrev:v:48:y:1954:i:03:p:787-792_00 is not listed on IDEAS
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    1. repec:kap:theord:v:82:y:2017:i:4:d:10.1007_s11238-016-9583-7 is not listed on IDEAS

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    Keywords

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

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior

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