IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-00780511.html
   My bibliography  Save this paper

Power indices expressed in terms of minimal winning coalitions

Author

Listed:
  • Fabien Lange

    (GREThA - Groupe de Recherche en Economie Théorique et Appliquée - UB - Université de Bordeaux - CNRS - Centre National de la Recherche Scientifique)

  • Laszlo A Koczy

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.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Fabien Lange & Laszlo A Koczy, 2012. "Power indices expressed in terms of minimal winning coalitions," Post-Print hal-00780511, HAL.
    • Handle: RePEc:hal:journl:hal-00780511
    DOI: 10.1007/s00355.012.0685.z
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    Other versions of this item:

    References listed on IDEAS

    as
    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. Michel Grabisch & Jean-Luc Marichal & Marc Roubens, 2000. "Equivalent Representations of Set Functions," Mathematics of Operations Research, INFORMS, vol. 25(2), pages 157-178, May.
    4. 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.
    5. Felsenthal, Dan S & Machover, Moshe, 1996. "Alternative Forms of the Shapley Value and the Shapley-Shubik Index," Public Choice, Springer, vol. 87(3-4), pages 315-318, June.
    6. 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.
    7. Pradeep Dubey & Abraham Neyman & Robert James Weber, 1981. "Value Theory Without Efficiency," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 122-128, February.
    8. 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.
    9. R J Johnston, 1978. "On the Measurement of Power: Some Reactions to Laver," Environment and Planning A, , vol. 10(8), pages 907-914, August.
    10. Josep M. Colomer & Florencio Martínez, 1995. "The Paradox of Coalition Trading," Journal of Theoretical Politics, , vol. 7(1), pages 41-63, January.
    11. Grabisch, M. & Marichal, J.-L. & Roubens, M., 1998. "Equivalent Representations of a Set Function with Applications to Game Theory and Multicriteria Decision Making," Liege - Groupe d'Etude des Mathematiques du Management et de l'Economie 9801, UNIVERSITE DE LIEGE, Faculte d'economie, de gestion et de sciences sociales, Groupe d'Etude des Mathematiques du Management et de l'Economie.
    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. Aleksei Y. Kondratev & Vladimir V. Mazalov, 2020. "Tournament solutions based on cooperative game theory," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(1), pages 119-145, March.
    2. 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.
    3. Debabrata Pal, 2021. "Does everyone have equal voting power?," Indian Economic Review, Springer, vol. 56(2), pages 515-525, December.
    4. Saadia Obadi & Silvia Miquel, 2017. "Clan information market games," Theory and Decision, Springer, vol. 82(4), pages 501-517, April.

    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. Ori Haimanko, 2019. "Composition independence in compound games: a characterization of the Banzhaf power index and the Banzhaf value," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(3), pages 755-768, September.
    2. Ori Haimanko, 2020. "Generalized Coleman-Shapley indices and total-power monotonicity," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(1), pages 299-320, March.
    3. Giulia Bernardi, 2018. "A New Axiomatization of the Banzhaf Index for Games with Abstention," Group Decision and Negotiation, Springer, vol. 27(1), pages 165-177, February.
    4. Margarita Domènech & José Miguel Giménez & María Albina Puente, 2022. "Weak null, necessary defender and necessary detractor players: characterizations of the Banzhaf and the Shapley bisemivalues," Annals of Operations Research, Springer, vol. 318(2), pages 889-910, November.
    5. Francesc Carreras & María Albina Puente, 2012. "Symmetric Coalitional Binomial Semivalues," Group Decision and Negotiation, Springer, vol. 21(5), pages 637-662, September.
    6. Berghammer, Rudolf & Bolus, Stefan & Rusinowska, Agnieszka & de Swart, Harrie, 2011. "A relation-algebraic approach to simple games," European Journal of Operational Research, Elsevier, vol. 210(1), pages 68-80, April.
    7. Carreras, Francesc & Freixas, Josep & Puente, Maria Albina, 2003. "Semivalues as power indices," European Journal of Operational Research, Elsevier, vol. 149(3), pages 676-687, September.
    8. André Casajus & Frank Huettner, 2019. "The Coleman–Shapley index: being decisive within the coalition of the interested," Public Choice, Springer, vol. 181(3), pages 275-289, December.
    9. Julien Reynaud & Fabien Lange & Łukasz Gątarek & Christian Thimann, 2011. "Proximity in Coalition Building," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 3(3), pages 111-132, September.
    10. Barua, Rana & Chakravarty, Satya R. & Roy, Sonali, 2006. "On the Coleman indices of voting power," European Journal of Operational Research, Elsevier, vol. 171(1), pages 273-289, May.
    11. Annick Laruelle & Federico Valenciano, 2005. "A critical reappraisal of some voting power paradoxes," Public Choice, Springer, vol. 125(1), pages 17-41, July.
    12. Annick Laruelle & Federico Valenciano, 2001. "Shapley-Shubik and Banzhaf Indices Revisited," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 89-104, February.
    13. László Á. Kóczy, 2018. "Partition Function Form Games," Theory and Decision Library C, Springer, number 978-3-319-69841-0, December.
    14. Barua, Rana & Chakravarty, Satya R. & Sarkar, Palash, 2009. "Minimal-axiom characterizations of the Coleman and Banzhaf indices of voting power," Mathematical Social Sciences, Elsevier, vol. 58(3), pages 367-375, November.
    15. László Á. Kóczy, 2016. "Power Indices When Players can Commit to Reject Coalitions," Homo Oeconomicus: Journal of Behavioral and Institutional Economics, Springer, vol. 33(1), pages 77-91, August.
    16. Friedman, Jane & Parker, Cameron, 2018. "The conditional Shapley–Shubik measure for ternary voting games," Games and Economic Behavior, Elsevier, vol. 108(C), pages 379-390.
    17. Carreras, Francesc, 2005. "A decisiveness index for simple games," European Journal of Operational Research, Elsevier, vol. 163(2), pages 370-387, June.
    18. Barua, Rana & Chakravarty, Satya R. & Roy, Sonali & Sarkar, Palash, 2004. "A characterization and some properties of the Banzhaf-Coleman-Dubey-Shapley sensitivity index," Games and Economic Behavior, Elsevier, vol. 49(1), pages 31-48, October.
    19. Somdeb Lahiri, 2021. "Pattanaik's axioms and the existence of winners preferred with probability at least half," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 31(2), pages 109-122.
    20. Haimanko, Ori, 2018. "The axiom of equivalence to individual power and the Banzhaf index," Games and Economic Behavior, Elsevier, vol. 108(C), pages 391-400.

    More about this item

    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

    Statistics

    Access and download statistics

    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:hal:journl:hal-00780511. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.