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On minimum integer representations of weighted games

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  • Freixas, Josep
  • Kurz, Sascha
Abstract
We study minimum integer representations of weighted games, i.e. representations where the weights are integers and every other integer representation is at least as large in each component. Those minimum integer representations, if they exist at all, are linked with some solution concepts in game theory. Closing existing gaps in the literature, we prove that each weighted game with two types of voters admits a (unique) minimum integer representation, and give new examples for more than two types of voters without a minimum integer representation. We characterize the possible weights in minimum integer representations and give examples for t≥4 types of voters without a minimum integer representation preserving types, i.e. where we additionally require that the weights are equal within equivalence classes of voters.

Suggested Citation

  • Freixas, Josep & Kurz, Sascha, 2014. "On minimum integer representations of weighted games," Mathematical Social Sciences, Elsevier, vol. 67(C), pages 9-22.
    • Handle: RePEc:eee:matsoc:v:67:y:2014:i:c:p:9-22
    DOI: 10.1016/j.mathsocsci.2013.10.005
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    References listed on IDEAS

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    1. Sascha Kurz, 2012. "On minimum sum representations for weighted voting games," Annals of Operations Research, Springer, vol. 196(1), pages 361-369, July.
    2. H. W. Lenstra, 1983. "Integer Programming with a Fixed Number of Variables," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 538-548, November.
    3. Carreras, Francesc & Freixas, Josep, 1996. "Complete simple games," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 139-155, October.
    4. Josep Freixas & Dorota Marciniak, 2009. "A minimum dimensional class of simple games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(2), pages 407-414, December.
    5. Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
    6. 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.
    7. Sascha Kurz & Nikolas Tautenhahn, 2013. "On Dedekind’s problem for complete simple games," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(2), pages 411-437, May.
    8. Josep Freixas & Xavier Molinero & Salvador Roura, 2012. "Complete voting systems with two classes of voters: weightedness and counting," Annals of Operations Research, Springer, vol. 193(1), pages 273-289, March.
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    Cited by:

    1. Maaser, Nicola & Stratmann, Thomas, 2024. "Costly voting in weighted committees: The case of moral costs," European Economic Review, Elsevier, vol. 162(C).
    2. Serguei Kaniovski & Sascha Kurz, 2018. "Representation-compatible power indices," Annals of Operations Research, Springer, vol. 264(1), pages 235-265, May.
    3. Flavio Pressacco & Laura Ziani, 2018. "Proper strong-Fibonacci games," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 41(2), pages 489-529, November.
    4. Izabella Stach, 2022. "Reformulation of Public Help Index θ Using Null Player Free Winning Coalitions," Group Decision and Negotiation, Springer, vol. 31(2), pages 317-334, April.
    5. Maaser, Nicola & Paetzel, Fabian & Traub, Stefan, 2019. "Power illusion in coalitional bargaining: An experimental analysis," Games and Economic Behavior, Elsevier, vol. 117(C), pages 433-450.
    6. Josep Freixas & Marc Freixas & Sascha Kurz, 2017. "On the characterization of weighted simple games," Theory and Decision, Springer, vol. 83(4), pages 469-498, December.
    7. Kurz, Sascha, 2021. "A note on the growth of the dimension in complete simple games," Mathematical Social Sciences, Elsevier, vol. 110(C), pages 14-18.

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