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On the structure of minimal winning coalitions in simple voting games

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  • Maria Axenovich
  • Sonali Roy

Abstract

According to Colemanï¾’s index of collective power, a decision rule that generates a larger number of winning coalitions imparts the collectivity a higher a priori power to act. By the virtue of the monotonicity conditions, a decision rule is totally characterized by the set of minimal winning coalitions. In this paper, we investigate the structure of the families of minimal winning coalitions corresponding to maximal and proper simple voting games (SVG).We show that if the proper and maximal SVG is swap robust and all the minimal winning coalitions are of the same size, then the SVG is a specific (up to an isomorphism) system.We also provide examples of proper SVGs to show that the number of winning coalitions is not monotone with respect to the intuitively appealing system parameters like the number of blockers, number of non-dummies or the size of the minimal blocking set.
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  • Maria Axenovich & Sonali Roy, 2010. "On the structure of minimal winning coalitions in simple voting games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 34(3), pages 429-440, March.
    • Handle: RePEc:spr:sochwe:v:34:y:2010:i:3:p:429-440
    DOI: 10.1007/s00355-009-0408-2
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    1. J. Freixas & M.A. Puente, 1998. "Complete games with minimum," Annals of Operations Research, Springer, vol. 84(0), pages 97-109, December.
    2. Dan S. Felsenthal & Moshé Machover, 1998. "The Measurement of Voting Power," Books, Edward Elgar Publishing, number 1489.
    3. 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.
    4. 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.
    5. Carreras, Francesc & Freixas, Josep, 1996. "Complete simple games," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 139-155, October.
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    1. Frits Hof & Walter Kern & Sascha Kurz & Kanstantsin Pashkovich & Daniël Paulusma, 2020. "Simple games versus weighted voting games: bounding the critical threshold value," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 54(4), pages 609-621, April.

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