Abstract
In a weighted voting game, each voter has a given weight, and a coalition of voters is successful if the sum of their individual weights exceeds a given quota. Such voting systems translate the idea that voters are not all equal by assigning them different weights. In such a situation, two voters are symmetric in a game if interchanging the two voters leaves the outcome of the game unchanged. Two voters with the same weight are naturally symmetric in every weighted voting game, but the converse statement is not necessarily true. We call the latter scenario inconsistent weighting. We investigate herein the conditions that give rise to such a phenomenon within the class of weighted voting games, and we study how the choice of the quota, the total weight, and the number of voters can affect the probability of observing inconsistent weighting.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In the context of empirical politics, when more variables matter than just numbers of votes, the situation gets complicated once we take into account such variables, e.g., spatial location of voters/parties, which typically is relevant for cabinet formation. In France, for instance, such considerations eliminate a coalition between the National Rally Party and the French Communist Party.
All of the infinitely many distinct weighted forms \([q;w_1,\dots ,w_n]\) for a weighted voting game \(G=(N,W)\) yield the same ordered partition \(\mathcal {SYM}\left( G\right)\). If a weighted form \([q;w_1,\dots ,w_n]\) is consistent, then for all \(t\in \{1,\dots ,s\}\) and all \(i,j\in N_t\), \(w_i=w_j\), while the latter equality does not necessarily hold if \([q;w_1,\dots ,w_n]\) is inconsistent, explaining why we can write \(\mathcal {SYM}\left( G\right)\) instead of \(\mathcal {SYM}\left( q;w_1,\dots ,w_n\right)\).
See, for instance, INJEP, 2019, p. 4: https://injep.fr/wp-content/uploads/2019/07/Chiffres-cles-Vie-associative-2019.pdf.
For more information, we refer the reader to http://ute-asso.fr/index-2.html.
For more details on cooperative administrative structures in France, especially EPCI, we refer the reader to https://www.collectivites-locales.gouv.fr/intercommunalite-1.
Code général des collectivités territoriales.
See, for instance, https://sig.ville.gouv.fr/Territoire/242500338. Note that, based on the data and analysis of Blancard et al. (2020), it is also possible to show the same inconsistency in some EPCIs on the French island La Réunion.
See, for instance, https://www.senat.fr/rap/1963-1964/i1963_1964_0063.pdf.
The internal agreement can be found, for instance, in https://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:01964A1229(01)-20040501&from=EN.
The free software to calculate the integer points under the parameterized Barvinok algorithm can be found at http://freecode.com/projects/barvinok. The algorithm allows one to quantify the number of integer solutions for systems of (in)equalities with parameters.
We gratefully acknowledge Sascha Kurz for having provided us with the lists of weighted voting games up to seven voters. The lists are available upon request.
The representations of the 25 cases are available upon request from the authors.
https://github.com/inconsistentweighting/maple_code. A MATLAB version of the code can likewise be obtained from the authors upon a simple request. In order to check whether or not a given weighted game is consistent, the code calculates the Banzhaf power index of every voter in the game. Note that the results will be the same if we use the Banzhaf index instead the Shapley-Shubik power index.
In Germany, for example, according to Art. 44 of that nation’s constitution, only 25% of the members of parliament need to vote favorably in order to start a parliamentary inquiry commission. In the 2017–2021 legislative period, three such commissions were initiated. A second example concerns referendums at the request of an authority—or extraordinary referendums—which exist in quite a number of states. Sometimes, a minority of parliamentarians can refer partial revisions of the constitution to the people, as in Denmark (one third of the members of Parliament) or Spain (10% of the members of either chamber). In some states, a referendum can be requested by a number of constituent entities—for instance, eight of 26 cantons in Switzerland and five of 20 regions in Italy (by decision of the Regional Council).
Recall that it is well known in the literature that in weighted games with three or four voters having unequal weights, at least two voters must be symmetric. That is not true if we allow some voters to have the same weight.
References
Algaba, E., Bilbao, J. M., & Fernandez, J. (2007). The distribution of power in the European Constitution. European Journal of Operational Research, 176(3), 1752–1755.
Alonso-Meijide, J. (2005). Generating functions for coalition power indices: An application to the IMF. Annals of Operations Research, 137, 21–44.
Arcaini, G., & Gambarelli, G. (1986). Algorithm for automatic computation of the power variations in share tradings. Calcolo, 23(1), 13–19.
Banzhaf, J. F. (1965). Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Review, 19, 317–343.
Barthélemy, F., Lepelley, D., Martin, M., & Smaoui, H. (2021). Dummy voters and the quota in weighted voting games. Group Decision and Negotiation, 30, 43–61.
Barthélemy, F., Lepelley, D., & Martin, M. (2013). On the likelihood of dummy voters in weighted majority games. Social Choice and Welfare, 41, 263–279.
Barthélemy, F., & Martin, M. (2021). Dummy voters and the quota in weighted voting games: Some further results. In M. Diss & V. Merlin (Eds.), Evaluating Voting Systems with Probability Models, Essays by and in honor of William Gehrlein and Dominique Lepelley (Chapter 13). Berlin: Springer.
Barthélemy, F., & Martin, M. (2007). Critères pour une meilleure répartition des sièges du Val d’Oise au sein des structures intercommunales: une application au cas du Val d’Oise. Revue Economique, 58, 399–426.
Barvinok, A. (1994). Polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Mathematics of Operations Research, 19, 769–779.
Barvinok, A., & Pommersheim, J. (1999). An algorithmic theory of lattice points in polyhedra, in: New perspectives in algebraic combinatorics, Berkeley, Ca, 1996-1997. Mathematical Sciences Research Institute Publications, 38, 91–147.
Bilbao, J., Fernandez, J., Jimenez, N., & Lopez, J. (2002). Voting power in the European Union Enlargement. European Journal of Operational Research, 143(1), 181–196.
Bison, F., Bonnet, J., & Lepelley, D. (2004). The determination of the number of delegates inside the inter-communities structures: an application of the Banzhaf power index. Revue d’Economie Régionale et Urbaine, 2, 259–282.
Blancard, S., Lepelley, D., & Smaoui, H. (2020). L’évolution du pouvoir de vote des communes au sein des communautés d’agglomération de La Réunion. Working paper CEMOI.
Bonnet, J., & Lepelley, D. (2001). Pouvoir de vote et Intercommunalité : le cas des Etablissements Publics de Coopération Intercommunale de la région de Basse-Normandie. In R. Le Duff, J. P. Rigalet, & G. Schmidt (Eds.), Démocratie et management local (pp. 519–532). France: Dalloz.
Bubboloni, D., Diss, M., & Gori, M. (2020). Extensions of the Simpson voting rule to the committee selection setting. Public Choice, 183, 151–185.
Cervone, D. P., Gehrlein, W. V., & Zwicker, W. S. (2005). Which scoring rule maximizes Condorcet efficiency under IAC? Theory and Decision, 58(2), 145–185.
de Keijzer, B., Klos, T., & Zhang, Y. (2010). Enumeration and exact design of weighted voting games. In: Proceedings of the 9th international conference on autonomous agents and multiagent systems, (Vol. 1, pp. 391-398).
Dia, I., & Kamwa, E. (2020). Le Pouvoir de Vote dans les Etablissements Publics de Coopération Intercommunales de la Martinique et de la Guadeloupe. Economie et Institutions, 28.
Diss, M., & Doghmi, A. (2016). Multi-winner scoring election methods: Condorcet consistency and paradoxes. Public Choice, 169, 97–116.
Diss, M., & Steffen, F. (2018). The Distribution of Power in the Lebanese Parliament Revisited. Working Paper, GATE.
Diss, M., & Zouache, A. (2015). Une étude de la répartition du pouvoir confessionnel au Liban. Revue d’Economie Politique, 125, 527–546.
El Ouafdi, A., Lepelley, D., & Smaoui, H. (2020). Probabilities of electoral outcomes: from three-candidate to four-candidate elections. Theory and Decision, 88, 205–229.
Felsenthal, D. S., & Machover, M. (2003). (2004) Analysis of QM rules in the draft constitution for Europe proposed by the European Convention. Social Choice and Welfare, 23(1), 1–20.
Freixas, J., & Molinero, X. (2010). Weighted games without a unique minimal representation in integers. Optimization Methods& Software, 25(2), 203–215.
Freixas, J., & Molinero, X. (2009). On the existence of a minimum integer representation for weighted voting games. Annals of Operations Research, 166, 243–260.
Freixas, J., & Zwicker, W. S. (2003). Weighted voting, abstention, and multiple levels of approval. Social Choice and Welfare, 21, 399–431.
Gambarelli, G. (1994). Power indices for political and financial decision making: A review. Annals of Operations Research, 51, 1572–9338.
Gehrlein, W. .V., & Lepelley, D. (2011). Voting paradoxes and group coherence. Switzerland: Studies in Choice and Welfare - Springer.
Gehrlein, W. .V. ., & Lepelley, D. (2017). Elections, Voting Rules and Paradoxical Outcomes. Switzerland: Studies in Choice and Welfare - Springer.
Kamwa, E. (2019). On the likelihood of the Borda effect: the overall probabilities for general weighted scoring rules and scoring runoff rules. Group Decision and Negotiation, 28(3), 519–541.
Kamwa, E., & Moyouwou, I. (2020). Susceptibility to manipulation by sincere truncation: the case of scoring rules and scoring runoff systems. In M. Diss & V. Merlin (Eds.), Evaluating Voting Systems with Probability Models, Essays by and in honor of William Gehrlein and Dominique Lepelley. Berlin: Springer.
Koki, C., & Leonardos, S. (2019). Coalitions and Voting Power in the Greek Parliament of 2012: A Case-Study. Homo Oeconomicus, 35, 295–313.
Kurz, S. (2012). On minimum sum representations for weighted voting games. Annals of Operations Research, 196, 361–369.
Kurz, S. (2018). Correction to: On minimum sum representations for weighted voting games. Annals of Operations Research, 271, 1087–1089.
Lane, J.-E., & Maeland, R. (2000). Constitutional analysis: The power index approach. European Journal of Political Research, 37, 31–56.
Laruelle, A., & Widgren, M. (1998). Is the allocation of voting power among EU states fair? Public Choice, 94(3–4), 317–339.
Leech, D. (1992). Empirical analysis of the distribution of a priori voting power: some results of the British Labour Party conference and Electoral College. European Journal of Political Research, 21(3), 245–65.
Leech, D. (2002a). Designing the voting system for the council of the European Union volume. Public Choice, 113(3), 437–464.
Leech, D. (2002b). Voting power in the governance of the international monetary fund. Annals of Operations Research, 109(1), 375–397.
Lepelley, D., Moyouwou, I., & Smaoui, H. (2018). Monotonicity paradoxes in three-candidate elections using scoring elimination rules. Social Choice and Welfare, 50(1), 1–33.
Lepelley, D., Louichi, A., & Smaoui, H. (2008). On Ehrhart polynomials and probability calculations in voting theory. Social Choice and Welfare, 30, 363–383.
Lepelley, D., & Smaoui, H. (2019). Comparing two ways for eliminating candidates in three-alternative elections using sequential scoring rules. Group Decision and Negotiation, 28(4), 787–804.
Mann, I., & Shapley L. S. (1960). Values of large games IV: Evaluating the electoral college by Monte-Carlo techniques. Technical report, The RAND Corporation.
Moyouwou, I., & Tchantcho, H. (2017). Asymptotic vulnerability of positional voting rules to coalitional manipulation. Mathematical Social Sciences, 89, 70–82.
O’Neill, B. (1996). Power and satisfaction in the United Nations Security Council. Journal of Conflict Resolution, 40, 219–237.
Shapley, L. S., & Shubik, M. (1954). A method for evaluating the distribution of power in a committee system. American Political Science Review, 48, 787–792.
Strand, J. R., & Rapkin, D. P. (2010). Weighted Voting in the United Nations Security Council: A Simulation. Simulation and Gaming, 42(6), 1–30.
Taylor, A. D., & Zwicker, W. S. (1992). A characterization of weighted voting. Proceedings of the American Mathematical Society, 115, 1089–1094.
Van Deemen, A., & Rusinowska, A. (2003). Paradoxes of Voting Power in Dutch Politics. Public Choice, 115, 109–137.
Verdoolaege, S., Seghir, R., Beyls, K., Loechner, V., & Bruynooghe, M. (2004). Analytical computation of Ehrhart polynomials: Enabling more compiler analysis and optimizations. Proceedings of International Conference on Compilers, Architecture, and Synthesis for Embedded Systems, (pp. 248-258). Washington D.C.
Wilson, M. C., & Pritchard, G. (2007). Probability calculations under the IAC hypothesis. Mathematical Social Sciences, 54, 244–256.
Acknowledgements
We thank an associate editor and two anonymous reviewers for helpful comments. We also would like to thank Kamel Mazouzi from Mésocentre de Calcul at the university of Franche-Comté and Didier Rebeix from Centre de Calcul de l’université de Bourgogne. Mostapha Diss would like to acknowledge financial support from Région Bourgogne Franche-Comté within the program ANER 2021-2024 (project DSG), from Université de Lyon (project INDEPTH Scientific Breakthrough Program of IDEX Lyon) within the program Investissement d’Avenir (ANR-16-IDEX-0005) and from Université de Franche-Comté within the program Chrysalide-2020. We also thank participants of Mohammed VI Polytechnic University research seminar (Rabat, Morocco), Mohammadia School of Engineering research seminar (Rabat, Morocco), and the 16th Spain-Italy-Netherlands Meeting on Game Theory (Granada, Spain).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
See Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, Fig. 1, Fig. 2 and Fig. 3.
Probability of having inconsistent weighted forms with a number of voters \(n\in \{2,3,4,5,6,7\}\) and some fixed values of w and q. The probabilities are exact for \(n\in \{2,3,4\}\), and they are based on simulations for \(n\ge 5\) with \(10^6\) iterations for \(10\le w \le 50\) and \(10^5\) iterations for \(w>50\)
Probability of having inconsistent weighted forms with a number of voters \(n\in \{8,9,10,15,20\}\) and some fixed values of w and q. The probabilities are simulated with \(10^6\) iterations for \(10\le w \le 50\) and \(10^5\) iterations for \(w>50\)
The limiting probability \(I(n, \infty ,r)\) of having inconsistent weighted forms. The probabilities are exact for \(n=2,5\), and they are based on simulations for \(n\ge 6\) with \(w=5000\) and 100,000 iterations. From Proposition 6, \(I(3, \infty ,r)=I(4, \infty ,r)=1\) whatever the values of r
Rights and permissions
About this article
Cite this article
Béal, S., Deschamps, M., Diss, M. et al. Inconsistent weighting in weighted voting games. Public Choice 191, 75–103 (2022). https://doi.org/10.1007/s11127-021-00951-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11127-021-00951-5