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On the existence of a minimum integer representation for weighted voting systems

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  • Josep Freixas
  • Xavier Molinero

Abstract

A basic problem in the theory of simple games and other fields is to study whether a simple game (Boolean function) is weighted (linearly separable). A second related problem consists in studying whether a weighted game has a minimum integer realization. In this paper we simultaneously analyze both problems by using linear programming. For less than 9 voters, we find that there are 154 weighted games without minimum integer realization, but all of them have minimum normalized realization. Isbell in 1958 was the first to find a weighted game without a minimum normalized realization, he needed to consider 12 voters to construct a game with such a property. The main result of this work proves the existence of weighted games with this property with less than 12 voters. Copyright Springer Science+Business Media, LLC 2009

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  • 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.
    • Handle: RePEc:spr:annopr:v:166:y:2009:i:1:p:243-260:10.1007/s10479-008-0422-2
    DOI: 10.1007/s10479-008-0422-2
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    1. Le Breton, Michel & Montero, Maria & Zaporozhets, Vera, 2012. "Voting power in the EU council of ministers and fair decision making in distributive politics," Mathematical Social Sciences, Elsevier, vol. 63(2), pages 159-173.
    2. Michel Le Breton & Karine Van Der Straeten, 2017. "Alliances Électorales et Gouvernementales : La Contribution de la Théorie des Jeux Coopératifs à la Science Politique," Revue d'économie politique, Dalloz, vol. 127(4), pages 637-736.
    3. Maria Montero & Alex Possajennikov, 2021. "An Adaptive Model of Demand Adjustment in Weighted Majority Games," Games, MDPI, vol. 13(1), pages 1-17, December.
    4. Monisankar Bishnu & Sonali Roy, 2012. "Hierarchy of players in swap robust voting games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(1), pages 11-22, January.
    5. Fabrice Barthelemy & Dominique Lepelley & Mathieu Martin & Hatem Smaoui, 2021. "Dummy Players and the Quota in Weighted Voting Games," Group Decision and Negotiation, Springer, vol. 30(1), pages 43-61, February.
    6. Sascha Kurz & Stefan Napel, 2014. "Heuristic and exact solutions to the inverse power index problem for small voting bodies," Annals of Operations Research, Springer, vol. 215(1), pages 137-163, April.
    7. Serguei Kaniovski & Sascha Kurz, 2018. "Representation-compatible power indices," Annals of Operations Research, Springer, vol. 264(1), pages 235-265, May.
    8. Sylvain Béal & Marc Deschamps & Mostapha Diss & Issofa Moyouwou, 2022. "Inconsistent weighting in weighted voting games," Public Choice, Springer, vol. 191(1), pages 75-103, April.
    9. Molinero, Xavier & Riquelme, Fabián & Roura, Salvador & Serna, Maria, 2023. "On the generalized dimension and codimension of simple games," European Journal of Operational Research, Elsevier, vol. 306(2), pages 927-940.
    10. Xavier Molinero & Maria Serna & Marc Taberner-Ortiz, 2021. "On Weights and Quotas for Weighted Majority Voting Games," Games, MDPI, vol. 12(4), pages 1-25, December.
    11. 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.
    12. Freixas, Josep & Kaniovski, Serguei, 2014. "The minimum sum representation as an index of voting power," European Journal of Operational Research, Elsevier, vol. 233(3), pages 739-748.
    13. Montero, Maria, 2017. "Proportional Payoffs in Legislative Bargaining with Weighted Voting: A Characterization," Quarterly Journal of Political Science, now publishers, vol. 12(3), pages 325-346, October.
    14. Akihiro Kawana & Tomomi Matsui, 2022. "Trading transforms of non-weighted simple games and integer weights of weighted simple games," Theory and Decision, Springer, vol. 93(1), pages 131-150, July.
    15. 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.
    16. Tanaka, Masato & Matsui, Tomomi, 2022. "Pseudo polynomial size LP formulation for calculating the least core value of weighted voting games," Mathematical Social Sciences, Elsevier, vol. 115(C), pages 47-51.
    17. Le Breton, Michel & Montero, Maria & Zaporozhets, Vera, 2012. "Voting power in the EU council of ministers and fair decision making in distributive politics," Mathematical Social Sciences, Elsevier, vol. 63(2), pages 159-173.
    18. Sascha Kurz, 2012. "On minimum sum representations for weighted voting games," Annals of Operations Research, Springer, vol. 196(1), pages 361-369, July.
    19. Freixas, Josep & Kurz, Sascha, 2014. "On minimum integer representations of weighted games," Mathematical Social Sciences, Elsevier, vol. 67(C), pages 9-22.
    20. Molinero, Xavier & Riquelme, Fabián & Serna, Maria, 2015. "Forms of representation for simple games: Sizes, conversions and equivalences," Mathematical Social Sciences, Elsevier, vol. 76(C), pages 87-102.
    21. Molinero, Xavier & Riquelme, Fabián & Serna, Maria, 2015. "Cooperation through social influence," European Journal of Operational Research, Elsevier, vol. 242(3), pages 960-974.

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