IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v215y2014i1p215-23010.1007-s10479-013-1374-8.html
   My bibliography  Save this article

Banzhaf index for multiple voting systems. An application to the European Union

Author

Listed:
  • Luisa Monroy
  • Francisco Fernández

Abstract

Multi-criteria simple games constitute an extension of the basic framework of voting systems and collective decision-making. The study of power index plays an important role in the theory of multi-criteria simple games. Thus, in this paper, we propose the extended Banzhaf index for these games, as the natural generalization of this index in conventional simple games. This approach allows us to compare various criteria simultaneously. An axiomatic characterization of this power index is established. The Banzhaf index is computed by taking into account the minimal winning coalitions of each class. Since this index depends on the number of ways in which each player can effect a swing, one of the main difficulties for finding this index is that it involves a large number of computations. We propose a combinatorial procedure, based on generating functions, to obtain the Banzhaf index more efficiently for weighted multi-criteria simple games. As an application, the distribution of voting power in the European Union is calculated. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Luisa Monroy & Francisco Fernández, 2014. "Banzhaf index for multiple voting systems. An application to the European Union," Annals of Operations Research, Springer, vol. 215(1), pages 215-230, April.
    • Handle: RePEc:spr:annopr:v:215:y:2014:i:1:p:215-230:10.1007/s10479-013-1374-8
    DOI: 10.1007/s10479-013-1374-8
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10479-013-1374-8
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10479-013-1374-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Roland Pongou & Bertrand Tchantcho & Lawrence Diffo Lambo, 2011. "Political influence in multi-choice institutions: cyclicity, anonymity, and transitivity," Theory and Decision, Springer, vol. 70(2), pages 157-178, February.
    2. Josep Freixas & William S. Zwicker, 2003. "Weighted voting, abstention, and multiple levels of approval," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 21(3), pages 399-431, December.
    3. J.R. Fernández & E. Algaba & J.M. Bilbao & A. Jiménez & N. Jiménez & J.J. López, 2002. "Generating Functions for Computing the Myerson Value," Annals of Operations Research, Springer, vol. 109(1), pages 143-158, January.
    4. Monroy, Luisa & Fernández, Francisco R., 2011. "The Shapley-Shubik index for multi-criteria simple games," European Journal of Operational Research, Elsevier, vol. 209(2), pages 122-128, March.
    5. Josep Freixas, 2005. "Banzhaf Measures for Games with Several Levels of Approval in the Input and Output," Annals of Operations Research, Springer, vol. 137(1), pages 45-66, July.
    6. J. Alonso-Meijide & C. Bowles, 2005. "Generating Functions for Coalitional Power Indices: An Application to the IMF," Annals of Operations Research, Springer, vol. 137(1), pages 21-44, July.
    7. Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
    8. Algaba, E. & Bilbao, J. M. & Fernandez Garcia, J. R. & Lopez, J. J., 2003. "Computing power indices in weighted multiple majority games," Mathematical Social Sciences, Elsevier, vol. 46(1), pages 63-80, August.
    9. Shapley, L. S. & Shubik, Martin, 1954. "A Method for Evaluating the Distribution of Power in a Committee System," American Political Science Review, Cambridge University Press, vol. 48(3), pages 787-792, September.
    10. R. Amer & F. Carreras & A. Magaña, 1998. "Extension of values to games withmultiple alternatives," Annals of Operations Research, Springer, vol. 84(0), pages 63-78, December.
    11. Bolger, Edward M, 1993. "A Value for Games with n Players and r Alternatives," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(4), pages 319-334.
    12. Hsiao Chih-Ru & Raghavan T. E. S., 1993. "Shapley Value for Multichoice Cooperative Games, I," Games and Economic Behavior, Elsevier, vol. 5(2), pages 240-256, April.
    13. Rubinstein, Ariel, 1980. "Stability of decision systems under majority rule," Journal of Economic Theory, Elsevier, vol. 23(2), pages 150-159, October.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Josep Freixas & Roberto Lucchetti, 2016. "Power in voting rules with abstention: an axiomatization of a two components power index," Annals of Operations Research, Springer, vol. 244(2), pages 455-474, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kurz, Sascha & Mayer, Alexander & Napel, Stefan, 2021. "Influence in weighted committees," European Economic Review, Elsevier, vol. 132(C).
    2. René van den Brink & Agnieszka Rusinowska & Frank Steffen, 2009. "Measuring Power and Satisfaction in Societies with Opinion Leaders: Dictator and Opinion Leader Properties," Tinbergen Institute Discussion Papers 09-052/1, Tinbergen Institute.
    3. Courtin, Sébastien & Nganmeni, Zéphirin & Tchantcho, Bertrand, 2017. "Dichotomous multi-type games with a coalition structure," Mathematical Social Sciences, Elsevier, vol. 86(C), pages 9-17.
    4. Pongou, Roland & Tchantcho, Bertrand & Tedjeugang, Narcisse, 2014. "Power theories for multi-choice organizations and political rules: Rank-order equivalence," Operations Research Perspectives, Elsevier, vol. 1(1), pages 42-49.
    5. René Brink & Agnieszka Rusinowska & Frank Steffen, 2013. "Measuring power and satisfaction in societies with opinion leaders: an axiomatization," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(3), pages 671-683, September.
    6. Michel Grabisch & Agnieszka Rusinowska, 2010. "A model of influence with an ordered set of possible actions," Theory and Decision, Springer, vol. 69(4), pages 635-656, October.
    7. Luisa Monroy & Francisco Fernández, 2012. "Stable sets and cores for multi-criteria simple games and for their extensions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(1), pages 1-22, June.
    8. Sascha Kurz, 2014. "Measuring Voting Power in Convex Policy Spaces," Economies, MDPI, vol. 2(1), pages 1-33, March.
    9. Monroy, Luisa & Fernández, Francisco R., 2011. "The Shapley-Shubik index for multi-criteria simple games," European Journal of Operational Research, Elsevier, vol. 209(2), pages 122-128, March.
    10. Josep Freixas & Montserrat Pons, 2022. "A critical analysis on the notion of power," Annals of Operations Research, Springer, vol. 318(2), pages 911-933, November.
    11. Josep Freixas, 2020. "The Banzhaf Value for Cooperative and Simple Multichoice Games," Group Decision and Negotiation, Springer, vol. 29(1), pages 61-74, February.
    12. Freixas, Josep, 2012. "Probabilistic power indices for voting rules with abstention," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 89-99.
    13. Antônio Francisco Neto, 2019. "Generating Functions of Weighted Voting Games, MacMahon’s Partition Analysis, and Clifford Algebras," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 74-101, February.
    14. Kurz, Sascha & Mayer, Alexander & Napel, Stefan, 2020. "Weighted committee games," European Journal of Operational Research, Elsevier, vol. 282(3), pages 972-979.
    15. Antônio Francisco Neto & Carolina Rodrigues Fonseca, 2019. "An approach via generating functions to compute power indices of multiple weighted voting games with incompatible players," Annals of Operations Research, Springer, vol. 279(1), pages 221-249, August.
    16. Josep Freixas & Montserrat Pons, 2021. "An Appropriate Way to Extend the Banzhaf Index for Multiple Levels of Approval," Group Decision and Negotiation, Springer, vol. 30(2), pages 447-462, April.
    17. Josep Freixas & Roberto Lucchetti, 2016. "Power in voting rules with abstention: an axiomatization of a two components power index," Annals of Operations Research, Springer, vol. 244(2), pages 455-474, September.
    18. Sébastien Courtin & Zéphirin Nganmeni & Bertrand Tchantcho, 2016. "The Shapley–Shubik power index for dichotomous multi-type games," Theory and Decision, Springer, vol. 81(3), pages 413-426, September.
    19. Grabisch, Michel & Rusinowska, Agnieszka, 2011. "A model of influence with a continuum of actions," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 576-587.
    20. M. Musegaas & P. E. M. Borm & M. Quant, 2018. "Three-valued simple games," Theory and Decision, Springer, vol. 85(2), pages 201-224, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:annopr:v:215:y:2014:i:1:p:215-230:10.1007/s10479-013-1374-8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.