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Weightedness and structural characterization of hierarchical simple games

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  • Gvozdeva, Tatiana
  • Hameed, Ali
  • Slinko, Arkadii

Abstract

In this paper we give structural characterizations of disjunctive and conjunctive hierarchical simple games by characterizing them as complete games with a unique shift-maximal losing coalition, and a unique shift-minimal winning coalition, respectively. We prove canonical representation theorems for both types of hierarchical games and establish duality between them. We characterize the disjunctive and conjunctive hierarchical games that are weighted majority games. This paper was inspired by Beimel et al. (2008) and Farràs and Padró (2010) characterizations of ideal weighted threshold access structures of secret sharing schemes.

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  • Gvozdeva, Tatiana & Hameed, Ali & Slinko, Arkadii, 2013. "Weightedness and structural characterization of hierarchical simple games," Mathematical Social Sciences, Elsevier, vol. 65(3), pages 181-189.
    • Handle: RePEc:eee:matsoc:v:65:y:2013:i:3:p:181-189
    DOI: 10.1016/j.mathsocsci.2012.11.007
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    References listed on IDEAS

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    1. Freixas, Josep & Puente, Maria Albina, 2008. "Dimension of complete simple games with minimum," European Journal of Operational Research, Elsevier, vol. 188(2), pages 555-568, July.
    2. Carreras, Francesc & Freixas, Josep, 1996. "Complete simple games," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 139-155, October.
    3. J. Freixas & M.A. Puente, 1998. "Complete games with minimum," Annals of Operations Research, Springer, vol. 84(0), pages 97-109, December.
    4. Gvozdeva, Tatiana & Slinko, Arkadii, 2011. "Weighted and roughly weighted simple games," Mathematical Social Sciences, Elsevier, vol. 61(1), pages 20-30, January.
    5. J. Freixas, 1997. "Different ways to represent weighted majority games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 5(2), pages 201-211, December.
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    Cited by:

    1. O’Dwyer, Liam & Slinko, Arkadii, 2017. "Growth of dimension in complete simple games," Mathematical Social Sciences, Elsevier, vol. 90(C), pages 2-8.
    2. He, Shawei & Marc Kilgour, D. & Hipel, Keith W., 2017. "A general hierarchical graph model for conflict resolution with application to greenhouse gas emission disputes between USA and China," European Journal of Operational Research, Elsevier, vol. 257(3), pages 919-932.
    3. Josep Freixas & Sascha Kurz, 2014. "Enumeration of weighted games with minimum and an analysis of voting power for bipartite complete games with minimum," Annals of Operations Research, Springer, vol. 222(1), pages 317-339, November.

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