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Dummy Players and the Quota in Weighted Voting Games

Author

Listed:
  • Fabrice Barthelemy

    (Cemotev - Centre d'études sur la mondialisation, les conflits, les territoires et les vulnérabilités - UVSQ - Université de Versailles Saint-Quentin-en-Yvelines)

  • Dominique Lepelley

    (CEMOI - Centre d'Économie et de Management de l'Océan Indien - UR - Université de La Réunion)

  • Mathieu Martin

    (THEMA - Théorie économique, modélisation et applications - CNRS - Centre National de la Recherche Scientifique - CY - CY Cergy Paris Université)

  • Hatem Smaoui

    (CEMOI - Centre d'Économie et de Management de l'Océan Indien - UR - Université de La Réunion)

Abstract

In a weighted voting game, each voter has a weight and a proposal is accepted if the sum of the weights of the voters in favor of that proposal is at least as large as a certain quota. It is well-known that, in this kind of voting process, it can occur that the vote of a player has no efect on the outcome of the game; such a player is called a "dummy" player. This paper studies the role of the quota on the occurrence of dummy players in weighted voting games. Assuming that every admissible weighted voting game is equally likely to occur, we compute the probability of having a player without voting power as a function of the quota for three, four and fve players. It turns out that this probability is very sensitive to the choice of the quota and can be very high. The quota values that minimize (or maximize) the likelihood of dummy players are derived (Some technical details are voluntarily omitted in this version of our study. These details can be found in the online appendix associated with this paper at https://bit.ly/2MVVuBW).

Suggested Citation

  • Fabrice Barthelemy & Dominique Lepelley & Mathieu Martin & Hatem Smaoui, 2021. "Dummy Players and the Quota in Weighted Voting Games," Post-Print hal-03797495, HAL.
    • Handle: RePEc:hal:journl:hal-03797495
    DOI: 10.1007/s10726-020-09705-y
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    References listed on IDEAS

    as
    1. Fabrice Barthelemy & Mathieu Martin & Bertrand Tchantcho, 2011. "Some conjectures on the two main power indices," THEMA Working Papers 2011-14, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    2. Mostapha Diss & Eric Kamwa & Abdelmonaim Tlidi, 2018. "A Note on the Likelihood of the Absolute Majority Paradoxes," Economics Bulletin, AccessEcon, vol. 38(4), pages 1727-1734.
    3. Dominique Lepelley & Ahmed Louichi & Hatem Smaoui, 2008. "On Ehrhart polynomials and probability calculations in voting theory," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 30(3), pages 363-383, April.
    4. Eric Kamwa, 2019. "On the Likelihood of the Borda Effect: The Overall Probabilities for General Weighted Scoring Rules and Scoring Runoff Rules," Group Decision and Negotiation, Springer, vol. 28(3), pages 519-541, June.
    5. Fabrice Barthélémy & Dominique Lepelley & Mathieu Martin, 2013. "On the likelihood of dummy players in weighted majority games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(2), pages 263-279, July.
    6. Sascha Kurz, 2012. "On minimum sum representations for weighted voting games," Annals of Operations Research, Springer, vol. 196(1), pages 361-369, July.
    7. Dan S. Felsenthal & Moshé Machover, 1998. "The Measurement of Voting Power," Books, Edward Elgar Publishing, number 1489.
    8. Wilson, Mark C. & Pritchard, Geoffrey, 2007. "Probability calculations under the IAC hypothesis," Mathematical Social Sciences, Elsevier, vol. 54(3), pages 244-256, December.
    9. Josep Freixas & Xavier Molinero, 2009. "On the existence of a minimum integer representation for weighted voting systems," Annals of Operations Research, Springer, vol. 166(1), pages 243-260, February.
    10. Alexander I. Barvinok, 1994. "A Polynomial Time Algorithm for Counting Integral Points in Polyhedra When the Dimension is Fixed," Mathematics of Operations Research, INFORMS, vol. 19(4), pages 769-779, November.
    11. Straffin, Philip Jr., 1994. "Power and stability in politics," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 32, pages 1127-1151, Elsevier.
    12. Moyouwou, Issofa & Tchantcho, Hugue, 2017. "Asymptotic vulnerability of positional voting rules to coalitional manipulation," Mathematical Social Sciences, Elsevier, vol. 89(C), pages 70-82.
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    More about this item

    Keywords

    Cooperative game theory; Weighted voting games; Dummy player; Probability of voting paradoxes;
    All these keywords.

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D7 - Microeconomics - - Analysis of Collective Decision-Making

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