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Multi-winner scoring election methods: Condorcet consistency and paradoxes

Author

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  • Mostapha Diss

    () (Université de Lyon, Lyon F- 69007, France; CNRS, GATE L-SE, Ecully, F- 69130, France; Université J. Monnet, Saint-Etienne, F- 42000, France)

  • Ahmed Doghmi

    () (University of Rabat, Mohammadia School of Engineering, the QSM Laboratory, Avenue Ibn Sina B.P. 765 Agdal, 10100 Rabat, Morocco)

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, which can be considered as generalisations of well-known single-winner scoring rules. The first comparison is based on the Condorcet committee efficiency which is defined as the conditional probability that a given voting rule picks out the Condorcet committee, given 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 leaves. In doing so, using the well-known Impartial Anonymous Culture condition, we extend the results of Kamwa and Merlin (2015) in two directions. First, our paper is concerned with the probability of the paradoxes no matter the ranking of the leaving candidate. Second, we do not only focus on the occurrence of these paradoxes when one wishes to select a committee of size k = 2 out of m = 4 candidates but we consider more values of k and m.

Suggested Citation

  • Mostapha Diss & Ahmed Doghmi, 2016. "Multi-winner scoring election methods: Condorcet consistency and paradoxes," Working Papers 1613, Groupe d'Analyse et de Théorie Economique Lyon St-Étienne (GATE Lyon St-Étienne), Université de Lyon.
    • Handle: RePEc:gat:wpaper:1613
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    References listed on IDEAS

    as
    1. Kamwa, Eric & Merlin, Vincent, 2015. "Scoring rules over subsets of alternatives: Consistency and paradoxes," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 130-138.
    2. Dominique Lepelley & Ahmed Louichi & Hatem Smaoui, 2008. "On Ehrhart polynomials and probability calculations in voting theory," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 30(3), pages 363-383, April.
    3. Peter C. Fishburn, 1974. "Aspects of One-Stage Voting Rules," Management Science, INFORMS, vol. 21(4), pages 422-427, December.
    4. Mostapha Diss & William Gehrlein, 2012. "Borda’s Paradox with weighted scoring rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(1), pages 121-136, January.
    5. Mitchell, Douglas W & Trumbull, William N, 1992. "Frequency of Paradox in a Common n-Winner Voting Scheme," Public Choice, Springer, vol. 73(1), pages 55-69, January.
    6. repec:spr:sochwe:v:50:y:2018:i:1:d:10.1007_s00355-017-1079-z is not listed on IDEAS
    7. Steven Brams & Peter Fishburn, 2005. "Going from theory to practice: the mixed success of approval voting," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 25(2), pages 457-474, December.
    8. William V. Gehrlein & Dominique Lepelley, 1999. "Condorcet efficiencies under the maximal culture condition," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 16(3), pages 471-490.
    9. Mostapha Diss & Vincent Merlin & Fabrice Valognes, 2010. "On the Condorcet efficiency of approval voting and extended scoring rules for three alternatives," Post-Print halshs-00533124, HAL.
    10. William V. Gehrlein & Dominique Lepelley, 2012. "The Value of Research Based on Simple Assumptions about Voters’ Preferences," Post-Print hal-01245273, HAL.
    11. Debord, Bernard, 1993. "Prudent k-choice functions: properties and algorithms," Mathematical Social Sciences, Elsevier, vol. 26(1), pages 63-77, July.
    12. Davide Cervone & William Gehrlein & William Zwicker, 2005. "Which Scoring Rule Maximizes Condorcet Efficiency Under Iac?," Theory and Decision, Springer, vol. 58(2), pages 145-185, March.
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    15. Thomas C. Ratliff, 2003. "Some startling inconsistencies when electing committees," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 21(3), pages 433-454, December.
    16. Mostapha Diss & William V. Gehrlein, 2015. "The True Impact of Voting Rule Selection on Condorcet Efficiency," Economics Bulletin, AccessEcon, vol. 35(4), pages 2418-2426.
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    19. Gehrlein, William V., 2001. "Condorcet winners on four candidates with anonymous voters," Economics Letters, Elsevier, vol. 71(3), pages 335-340, June.
    20. Samuel Merrill, 1985. "A statistical model for Condorcet efficiency based on simulation under spatial model assumptions," Public Choice, Springer, vol. 47(2), pages 389-403, January.
    21. Eric Kamwa, 2013. "The increasing committee size paradox with small number of candidates," Economics Bulletin, AccessEcon, vol. 33(2), pages 967-972.
    22. Gehrlein, William V. & Lepelley, Dominique, 2001. "The Condorcet efficiency of Borda Rule with anonymous voters," Mathematical Social Sciences, Elsevier, vol. 41(1), pages 39-50, January.
    23. Eric Kamwa & Vincent Merlin, 2018. "Coincidence of Condorcet committees," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 50(1), pages 171-189, January.
    24. Steven Brams & D. Kilgour & M. Sanver, 2007. "A minimax procedure for electing committees," Public Choice, Springer, vol. 132(3), pages 401-420, September.
    25. Gehrlein, William V., 1985. "Condorcet efficiency of constant scoring rules for large electorates," Economics Letters, Elsevier, vol. 19(1), pages 13-15.
    26. Fishburn, Peter C., 1978. "Axioms for approval voting: Direct proof," Journal of Economic Theory, Elsevier, vol. 19(1), pages 180-185, October.
    27. Barış Kaymak & M. Remzi Sanver, 2003. "Sets of alternatives as Condorcet winners," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 20(3), pages 477-494, June.
    28. Peter Fishburn & William Gehrlein, 1976. "Borda's rule, positional voting, and Condorcet's simple majority principle," Public Choice, Springer, vol. 28(1), pages 79-88, December.
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    Citations

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    Cited by:

    1. Mostapha Diss & Muhammad Mahajne, 2019. "Social Acceptability of Condorcet Committees," Working Papers 1906, Groupe d'Analyse et de Théorie Economique Lyon St-Étienne (GATE Lyon St-Étienne), Université de Lyon.
    2. Mostapha Diss & Eric Kamwa & Abdelmonaim Tlidi, 2018. "The Chamberlin-Courant Rule and the k-Scoring Rules: Agreement and Condorcet Committee Consistency," Working Papers hal-01757761, HAL.
    3. Daniela Bubboloni & Mostapha Diss & Michele Gori, 2018. "Extensions of the Simpson voting rule to the committee selection setting," Working Papers 1813, Groupe d'Analyse et de Théorie Economique Lyon St-Étienne (GATE Lyon St-Étienne), Université de Lyon.
    4. repec:spr:sochwe:v:48:y:2017:i:3:d:10.1007_s00355-017-1026-z is not listed on IDEAS
    5. repec:eee:matsoc:v:89:y:2017:i:c:p:70-82 is not listed on IDEAS
    6. repec:spr:grdene:v:27:y:2018:i:4:d:10.1007_s10726-018-9580-z is not listed on IDEAS

    More about this item

    Keywords

    Multi-winner voting rules; committee; Condorcet committee efficiency; paradoxes; probability;

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior

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