Skip to main content
SpringerLink
Log in

Multi-winner scoring election methods: Condorcet consistency and paradoxes

  • Published:
Public Choice Aims and scope Submit manuscript

Abstract

The goal of this paper is to propose a comparison of four multi-winner voting rules, k-Plurality, k-Negative Plurality, k-Borda, and Bloc. These four election methods are extensions of usual scoring rules designed for electing a single winner and are compared on the basis of two criteria. The first comparison is based on the Condorcet committee efficiency which is defined as the conditional probability for a peculiar voting rule to select the Condorcet committee, provided that such a committee exists. The second comparison is based on the likelihood of two paradoxes of committee elections: the Prior Successor Paradox and the Leaving Member Paradox, which occur when a member of an elected committee exits. Aside from these two extensions, this paper is one of the very rare contributions giving exact results under the Impartial Anonymous Culture (IAC) condition for the case of four candidates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The Condorcet efficiency of a single-winner voting rule is the conditional probability that this given voting rule picks out the Condorcet winner, given that such a candidate exists.

  2. In addition to the LMP, the other paradox described by Staring (1986) in his original paper is the Increasing Committee Size Paradox (ICSP) which occurs if, by increasing the target size k, we can find that the candidates that are elected in the committees change for each k. Under some conditions, both committees may be totally disjoint. Notice that, for the constant scoring rule as voters cast votes for their k most preferred candidates, Kamwa (2013) provided the probability of the ICSP when the size of a committee increases from 1 to 2 in three-candidate elections under the IAC condition as a function of the number of voters. Mitchell and Trumbull (1992) considered slightly more general voting methods: to select the k members of the committee, each voter cast votes for their \(k^*\) most preferred candidates. Two particular paradoxes are then considered using Monte–Carlo simulations under different assumptions, including the IC hypothesis. The first paradox is a version of the ICSP in which a candidate is elected under a vote for \(k^*\) candidates procedure, and then that candidate is not elected for larger values of \(k^*\). The second paradox considered in Mitchell and Trumbull (1992) occurs when the Condorcet winner is not selected among the k members of the committee.

  3. Also called Single Nontransferable Vote (SNTV).

  4. Also called Limited Voting (Kamwa and Merlin 2015) or Constant Scoring Rule (Gehrlein 1985a). We use this new designation since it corresponds to the one in the recent literature (e.g., Elkind et al. (2015)).

  5. Approval voting (AV) is a voting procedure in which voters may vote for as many candidates as they wish. It asks each voter to distinguish the candidates she approves of from the ones she considers as unacceptable. The alternative with the highest degree of approbation is then selected. The proponents of this voting procedure (e.g., Brams (1980); Brams and Fishburn (1978, 2005); Fishburn and Brams (1981)) discussed several advantages that it has over other electoral systems.

  6. Bloc is equivalent to k-Plurality and k-Negative Plurality when \(k=1\) and \(k=m-1\), respectively.

  7. IC is a second well-known model which considers the set of all preference profiles as a sample space where a voter preference profile identifies the specific preference ranking that each voter has on the candidates. In other words, when strict preferences over the set of m candidates are assumed, the IC assumption (Guilbaud 1952) assumes that each voter is equally likely to pick any of the m! preferences. Notice that individual voter’s preferences are not anonymous under IC condition while they are under IAC assumption. In addition, in the IC model, the votes are totally independent whereas under IAC, the votes are correlated in a specific way. So, moving from IC to IAC allows to evaluate the impact of introducing some degree of homogeneity in voters’ preferences on the probability of paradoxes.

  8. We can also generate simulations with more than six candidates. However, this option is ignored since these simulations cannot affect the conclusions that we obtain in our paper with 3, 4, 5, and 6 candidates. In other words, the behaviour of voting rules is the same for more than 6 candidates.

  9. We would like to thank an anonymous referee to an earlier version of this paper for pointing out this useful formula.

  10. \(x_3\) is a Condorcet loser: a candidate that loses all head-to-head comparisons with other candidates.

  11. Our volumes are found with the use of the algorithm implemented in Gawrilow and Joswig (2000).

  12. We can also generate simulations with 1,000,000 voters and 1,000,000 elections or more. We have ignored this option since, even with 100,000 voters and 100,000 elections, our results guarantee a very low margin of error. We give another argument of the choice of 100,000 voters and 100,000 elections in footnote 13.

  13. The simulations in Lepelley et al. (2000a) are provided with parameters greater than 100,000 voters and 100,000 elections. The small difference between our probabilities and those of Lepelley et al. (2000a) show that simulations with only 100,000 voters and 100,000 elections are sufficient.

  14. In this framework, one supposes that voters keep their preferences unchanged over the rest of candidates no matter the candidate who leaves the elected committee.

  15. This is important since for each paradox the event describing the first probability and the second one are not disjoint.

  16. Evidently, for \(m=5\) and \(m=6\), we are not interested in the ranking neither between candidates in the committee nor between the candidates not in the committee, with the exception of the prior successor.

  17. In other words, the reduced profile defining the preferences of voters over the rest of candidates will be unchanged in comparison with the original one, with \(x_1\) removed.

References

  • Aziz, H., Brill, M., Conitzer, V., Elkind, E., Freeman, R. & Walsh, T. (2015). Justified representation in approval-based committee voting. In Proceedings of the 29th AAAI conference on artificial intelligence

  • Berg, S., & Lepelley, D. (1994). On probability models in voting theory. Statistica Neerlandica, 48(2), 133–146.

    Article  Google Scholar 

  • Brams, S. J. (1980). Approval voting in multicandidate elections. Policy Studies Journal, 9, 102–108.

    Article  Google Scholar 

  • Brams, S. J., & Fishburn, P. C. (1978). Approval voting. American Political Science Review, 72, 831–847.

    Article  Google Scholar 

  • Brams, S. J., & Fishburn, P. C. (2002). Voting procedures, pages 173–236. In K. Arrow, A. Sen, & K. Suzumura (Eds.). Handbook of social choice and welfare. Elsevier.

  • Brams, S. J., & Fishburn, P. C. (2005). Going from theory to practice: The mixed success of approval voting. Social Choice and Welfare, 25, 457–474.

    Article  Google Scholar 

  • Brams, S. J., Kilgour, D., & Sanver, M. (2006). How to elect a representative committee using approval balloting. In B. Simeone & F. Pukelsheim (Eds.), Mathematics and democracy. Studies in choice and welfare, pp. 83–95. Berlin: Springer.

  • Brams, S. J., Kilgour, M., & Sanver, R. (2007). A minimax procedure for electing committees. Public Choice, 132, 401–420.

    Article  Google Scholar 

  • Brandt, F., Geist, C., & Strobel, M. (2016). Analyzing the practical relevance of voting paradoxes via Ehrhart theory, computer simulations and empirical data. In Proceedings of the 15th international conference on AAMAS.

  • Bueler, B., Enge, A., & Fukuda, K. (2000). Exact volume computation for polytopes: A practical study. Polytopes—Combinatorics and Computation, DMV Seminar, 29, 131–154.

    Google Scholar 

  • Cervone, D., Gehrlein, W. V., & Zwicker, W. (2005). Which scoring rule maximizes Condorcet efficiency under IAC? Theory and Decision, 58, 145–185.

    Article  Google Scholar 

  • Cohen, J., & Hickey, T. (1979). Two algorithms for determining volumes of convex polyhedra. Journal of the ACM, 26, 401–414.

    Article  Google Scholar 

  • Marquis de Condorcet (1785) Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. Paris.

  • Debord, B. (1992). An axiomatic characterization of Borda’s k-choice function. Social Choice and Welfare, 9, 337–343.

    Article  Google Scholar 

  • Debord, B. (1993). Prudent \(k-\)choice functions: Properties and algorithms. Mathematical Social Sciences, 26, 63–77.

    Article  Google Scholar 

  • Diss, M., & Gehrlein, W. V. (2012). Borda’s paradox with weighted scoring rules. Social Choice and Welfare, 38, 121–136.

    Article  Google Scholar 

  • Diss, M., & Gehrlein, W. V. (2015). The true impact of voting rule selection on Condorcet efficiency. Economics Bulletin, 35, 2418–2426.

    Google Scholar 

  • Diss, M., Merlin, V., & Valognes, F. (2010). On the Condorcet efficiency of approval voting and extended scoring rules for three alternatives. In J. F. Laslier, & M. R. Sanver (Eds.) Handbook of approval voting, pp. 255–283. Heidelberg: Springer.

  • El Ouafdi, A. (2016). Probabilities of electoral outcomes in 4-candidate elections. Unpublished working-paper. CEMOI, University of La Réunion.

  • Elkind, E., Faliszewski, P., Skowron, P. & Slinko, A. (2015). Properties of multiwinner voting rules. Mimeo.

  • Elkind, E., Lang, J., & Saffidine, A. (2011). Choosing collectively optimal sets of alternatives based on the Condorcet criterion. In IJCAI’11, pp. 186–191.

  • Felsenthal, D. S., & Maoz, Z. (1992). Normative properties of four single-stage multi-winner electoral procedures. Behavioral Science, 37, 109–127.

    Article  Google Scholar 

  • Fishburn, P. C. (1974a). Aspects of one-stage voting rules. Management Science, 21, 422–427.

    Article  Google Scholar 

  • Fishburn, P. C. (1974b). Simple voting systems and majority rule. Behavioral Science, 19, 166–179.

    Article  Google Scholar 

  • Fishburn, P. C. (1981). An analysis of simple voting systems for electing committees. SIAM Journal on Applied Mathematics, 41, 499–502.

    Article  Google Scholar 

  • Fishburn, P. C., & Brams, S. J. (1981). Approval voting, Condorcet’s principle, and runoff elections. Public Choice, 36, 89–114.

    Article  Google Scholar 

  • Fishburn, P. C., & Gehrlein, W. V. (1976a). An analysis of simple two stage voting systems. Behavioral Science, 21, 1–12.

    Article  Google Scholar 

  • Fishburn, P. C., & Gehrlein, W. V. (1976b). Borda’s rule, positional voting, and Condorcet’s simple majority principle. Public Choice, 28, 79–88.

    Article  Google Scholar 

  • Fishburn, P. C., & Gehrlein, W. V. (1977). An analysis of voting procedures with non-ranked voting. Public Choice, 22, 178–185.

    Google Scholar 

  • Gawrilow, E., & Joswig, M. (2000). Polymake: A framework for analyzing convex polytopes. In G. Kalai & G. M. Ziegler (Eds.), Polytopes–Combinatorics and computation, pp. 43–74. Birkhäuser.

  • Gehrlein, W. V. (1985a). The Condorcet criterion and committee selection. Mathematical Social Sciences, 10, 199–209.

    Article  Google Scholar 

  • Gehrlein, W. V. (1985b). Condorcet efficiency of constant scoring rules for large electorates. Economics Letters, 19, 13–15.

    Article  Google Scholar 

  • Gehrlein, W. V. (1995). Condorcet efficiency and social homogeneity. In W. Barnett, H. Moulin, M. Salles & N. Schofield (Eds.), Social choice, welfare and ethics, pp. 127–143. Cambridges: Cambridge University Press.

  • Gehrlein, W. V. (1999). The Condorcet efficiency of Borda rule under the dual culture condition. Social Science Research, 28, 36–44.

    Article  Google Scholar 

  • Gehrlein, W. V. (2001). Condorcet winners on four candidates with anonymous voters. Economics Letters, 71, 335–340.

    Article  Google Scholar 

  • Gehrlein, W. V., & Fishburn, P. C. (1976). Condorcet’s paradox and anonymous preference profiles. Public Choice, 26, 1–18.

    Article  Google Scholar 

  • Gehrlein, W. V., & Lepelley, D. (1999). Condorcet efficiencies under the maximal culture condition. Social Choice and Welfare, 16, 471–490.

    Article  Google Scholar 

  • Gehrlein, W. V., & Lepelley, D. (2001). The Condorcet efficiency of Borda rule with anonymous voters. Mathematical Social Sciences, 41, 39–50.

    Article  Google Scholar 

  • Gehrlein, W. V., & Lepelley, D. (2010). Voting paradoxes and group coherence: The Condorcet efficiency of voting rules. Springer.

  • Gehrlein, W. V., & Lepelley, D. (2012). The value of research based on simple assumptions about voters’ preferences. In D. S. Felsenthal & M. Machover (Eds.), Electoral systems: Paradoxes, assumptions and procedures (pp. 173–200). Publisher: Springer.

    Chapter  Google Scholar 

  • Gehrlein, W. V., & Lepelley, D. (2015). The Condorcet efficiency advantage that voter indifference gives to Approval Voting over some other voting rules. Group Decision and Negotiation, 24, 243–269.

    Article  Google Scholar 

  • Guilbaud, G. T. (1952). Les théories de l’intérêt général et le problème logique de l’agrégation. Economie Appliquée, 5, 501–584.

    Google Scholar 

  • Hill, I. D. (1988). Some aspects of elections: To fill one seat or many. Journal of the Royal Statistical Society, 151, 243–275.

    Article  Google Scholar 

  • Kamwa, E. (2013). The increasing committee size paradox with a small number of candidates. Economics Bulletin, 33, 967–972.

    Google Scholar 

  • Kamwa, E., & Merlin, V. (2013). Coincidence of Condorcet committees. Mimeo: Normandie Univ.

    Google Scholar 

  • Kamwa, E., & Merlin, V. (2015). Scoring rules over subsets of alternatives: Consistency and paradoxes. Journal of Mathematical Economics, 61, 130–138.

    Article  Google Scholar 

  • Kaymak, B., & Sanver, R. (2003). Sets of alternatives as Condorcet winners. Social Choice and Welfare, 20, 477–494.

    Article  Google Scholar 

  • Kilgour, M. (2010). Approval balloting for multi-winner elections. In J. F. Laslier & M. R. Sanver (Eds.), Handbook of approval voting, pp. 105–124. Heidelberg: Springer.

  • Kilgour, M., & Marshall, E. (2012). Approval balloting for fixed-size committees, . In D. S. Felsenthal & M. Machover (Eds.), Electoral systems, studies in choice and welfare, pp. 305–326. Springer.

  • Lawrence, J. (1991). Polytope volume computation. Mathematics of Computation, 57, 259–271.

    Article  Google Scholar 

  • Lepelley, D., Louichi, A., & Smaoui, H. (2008). On Ehrhart polynomials and probability calculations in voting theory. Social Choice and Welfare, 30, 363–383.

    Article  Google Scholar 

  • Lepelley, D., Louichi, A., & Valognes, F. (2000a). Computer simulations of voting systems. Advances in Complex Systems, 03, 181–194.

    Article  Google Scholar 

  • Lepelley, D., Pierron, P., & Valognes, F. (2000b). Scoring rules, Condorcet efficiency and social homogeneity. Theory and Decision, 49, 175–196.

    Article  Google Scholar 

  • Merrill, S. (1984). A comparison of efficiencies of multicandidate electoral systems. American Journal of Political Science, 28, 23–84.

    Article  Google Scholar 

  • Merrill, S. (1985). A statistical model for Condorcet efficiency based on simulation under spatial model assumptions. Public Choice, 47, 389–403.

    Article  Google Scholar 

  • Mitchell, D. W., & Trumbull, W. N. (1992). Frequency of paradox in a common \(n-\)winner voting scheme. Public Choice, 73, 55–69.

    Article  Google Scholar 

  • Pérez, J., Jimeno, J.-L., & García, E. (2012). No show paradox in Condorcet \(k-\)voting procedures. Group Decision and Negotiation, 21, 291–303.

    Article  Google Scholar 

  • Ratliff, T. (2003). Some startling inconsistencies when electing committees. Social Choice and Welfare, 21, 433–454.

    Article  Google Scholar 

  • Ratliff, T. (2006). Selecting committees. Public Choice, 126, 343–355.

    Article  Google Scholar 

  • Schürmann, A. (2013). Exploiting polyhedral symmetries in social choice. Social Choice and Welfare, 40, 1097–1110.

    Article  Google Scholar 

  • Staring, M. (1986). Two paradoxes of committee elections. Mathematics Magazine, 59, 158–159.

    Article  Google Scholar 

  • Young, H. (1975). Social choice scoring functions. SIAM Journal on Applied Mathematics, 28, 824–838.

    Article  Google Scholar 

Download references

Acknowledgments

The authors wish to express gratitude to Professor Peter Kurrild-Klitgaard, Associate Editor, as well as two anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. The first author gratefully acknowledges financial support by the National Agency for Research (ANR)—research program “Dynamic Matching and Interactions: Theory and Experiments” (DynaMITE) ANR-BLANC.

Author information

Authors and Affiliations

  1. Univ Lyon, UJM Saint-Etienne, CNRS, GATE L-SE UMR 5824, 42023, Saint-Étienne, France

    Mostapha Diss

  2. University of Rabat, Mohammadia School of Engineering, the QSM Laboratory, Avenue Ibnina B.P. 765 Agdal, 10100, Rabat, Morocco

    Ahmed Doghmi

Authors
  1. Mostapha Diss
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmed Doghmi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mostapha Diss.

Appendix

Appendix

The values for 3 and 4 candidates are exact; the others are estimates (Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14).

Table 1 \(P^{CC}_{IAC_\infty } (k, m)\)
Full size table
Table 2 \(CCE_{IAC_\infty }^{k-P} (k, m)\)
Full size table
Table 3 \(CCE_{IAC_\infty }^{k-NP} (k, m)\)
Full size table
Table 4 \(CCE_{IAC_\infty }^{k-B} (k, m)\)
Full size table
Table 5 \(CCE_{IAC_\infty }^{B} (k, m)\)
Full size table
Table 6 \(P_{PSP}^{NB} (k, m)\) for k-Plurality and \(IAC_\infty\)
Full size table
Table 7 \(P_{PSP}^{PB} (k, m)\) for k-Plurality and \(IAC_\infty\)
Full size table
Table 8 \(P_{PSP}^{NB} (k, m)\) for k-Negative Plurality and \(IAC_\infty\)
Full size table
Table 9 \(P_{PSP}^{PB} (k, m)\) for k-Negative Plurality and \(IAC_\infty\)
Full size table
Table 10 \(P_{PSP}^{NB} (k, m)\) for k-Borda and \(IAC_\infty\)
Full size table
Table 11 \(P_{PSP}^{PB} (k, m)\) for k-Borda and \(IAC_\infty\)
Full size table
Table 12 \(P_{PSP}^{NB} (k, m)\) for Bloc and \(IAC_\infty\)
Full size table
Table 13 \(P_{PSP}^{PB} (k, m)\) for Bloc and \(IAC_\infty\)
Full size table
Table 14 \(P_{LMP}^{NB} (k, m)\) under \(IAC_\infty\)
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Diss, M., Doghmi, A. Multi-winner scoring election methods: Condorcet consistency and paradoxes. Public Choice 169, 97–116 (2016). https://doi.org/10.1007/s11127-016-0376-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11127-016-0376-x

Keywords

JEL Classification

Search

Navigation