Enumeration of weighted games with minimum and an analysis of voting power for bipartite complete games with minimum

Abstract

This paper is a twofold contribution. First, it contributes to the problem of enumerating some classes of simple games and in particular provides the number of weighted games with minimum and the number of weighted games for the dual class as well. Second, we focus on the special case of bipartite complete games with minimum, and we compare and rank these games according to the behavior of some efficient power indices of players of type 1 (or of type 2). The main result of this second part establishes all allowable rankings of these games when the Shapley-Shubik power index is used on players of type 1.

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Fig. 1
Fig. 2

Notes

  1. 1.

    Complete games are also known in the literature of simple games as linear games or directed games.

  2. 2.

    In Carreras and Freixas (1996) they are called δ-minimal winning vectors.

  3. 3.

    More precisely, the notation wg(n,∗,r) stands for \(\sum_{t = 1}^{n}{wg}(n,t,r)\).

  4. 4.

    Unfortunately we cannot apply Vandermonde’s identity \(\sum _{j=0}^{k}\binom{m}{j} \binom{n}{ k-j}=\binom{m+n}{ k}\) directly.

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Acknowledgements

The authors are grateful to the two referees of this paper for their interesting comments and also for their exhaustive reports that contributed to improve the original submitted version.

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Correspondence to Josep Freixas.

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Research of the first author partially funded by Grants SGR 2009–1029 of Generalitat de Catalunya and MTM 2012–34426 from the Spanish Economy and Competitiveness Ministry, from the Spanish Science and Innovation Ministry.

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Freixas, J., Kurz, S. Enumeration of weighted games with minimum and an analysis of voting power for bipartite complete games with minimum. Ann Oper Res 222, 317–339 (2014). https://doi.org/10.1007/s10479-013-1348-x

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Keywords

  • Simple game
  • Weighted and complete games
  • Enumerations
  • Shapley-Shubik power index
  • Banzhaf power indices