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Scoring rules over subsets of alternatives: Consistency and paradoxes

Author

Listed:
  • Eric Kamwa

    (LC2S - Laboratoire caribéen de sciences sociales - CNRS - Centre National de la Recherche Scientifique - UA - Université des Antilles)

  • Vincent Merlin

    (CREM - Centre de recherche en économie et management - UNICAEN - Université de Caen Normandie - NU - Normandie Université - UR - Université de Rennes - CNRS - Centre National de la Recherche Scientifique)

Abstract

We know since the works of Gehrlein and Fishburn (1980, 1981), Fishburn (1981) and Saari (1987, 1988, 1990) that, the collective rankings of scoring rules are not stable when some alternatives are dropped from the set of alternatives. However, in the literature, attention has been mainly devoted to the relationship between pairwise majority vote and scoring rules rankings. In this paper, we focus on the relationships between four-candidate and three-candidate rankings. More precisely, given a collective ranking over a set of four candidates, we determine under the impartial culture condition, the probability of each of the six possible rankings to occur when one candidate is dropped. As a consequence, we derive from our computations, the likelihood of two paradoxes of committee elections, the Leaving Member Paradox (Staring, 1986) and the Prior Successor Paradox which occur when an elected candidate steps down from a two-member committee.

Suggested Citation

  • Eric Kamwa & Vincent Merlin, 2015. "Scoring rules over subsets of alternatives: Consistency and paradoxes," Post-Print hal-01702492, HAL.
    • Handle: RePEc:hal:journl:hal-01702492
    DOI: 10.1016/j.jmateco.2015.08.008
    Note: View the original document on HAL open archive server: https://hal.univ-antilles.fr/hal-01702492
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    References listed on IDEAS

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    1. Dan S. Felsenthal, 2012. "Review of Paradoxes Afflicting Procedures for Electing a Single Candidate," Studies in Choice and Welfare, in: Dan S. Felsenthal & Moshé Machover (ed.), Electoral Systems, chapter 0, pages 19-91, Springer.
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    5. 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.
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    14. Vincent R. Merlin & Donald G. Saari, "undated". "The Copeland Method I; Relationships and the Dictionary," Discussion Papers 1111, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    15. Mostapha Diss & Vincent Merlin & Fabrice Valognes, 2010. "On the Condorcet Efficiency of Approval Voting and Extended Scoring Rules for Three Alternatives," Studies in Choice and Welfare, in: Jean-François Laslier & M. Remzi Sanver (ed.), Handbook on Approval Voting, chapter 0, pages 255-283, Springer.
    16. Chevaleyre, Yann & Lang, Jérôme & Maudet, Nicolas & Monnot, Jérôme & Xia, Lirong, 2012. "New candidates welcome! Possible winners with respect to the addition of new candidates," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 74-88.
    17. Tataru, Maria & Merlin, Vincent, 1997. "On the relationship of the Condorcet winner and positional voting rules," Mathematical Social Sciences, Elsevier, vol. 34(1), pages 81-90, August.
    18. Merlin, V. & Tataru, M. & Valognes, F., 2000. "On the probability that all decision rules select the same winner," Journal of Mathematical Economics, Elsevier, vol. 33(2), pages 183-207, March.
    19. Donald G. Saari & Vincent R. Merlin, 1996. "The Copeland method (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 51-76.
    20. 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. Eric Kamwa & Vincent Merlin, 2019. "The Likelihood of the Consistency of Collective Rankings Under Preferences Aggregation with Four Alternatives Using Scoring Rules: A General Formula and the Optimal Decision Rule," Computational Economics, Springer;Society for Computational Economics, vol. 53(4), pages 1377-1395, April.
    2. Mostapha Diss & Ahmed Doghmi, 2016. "Multi-winner scoring election methods: Condorcet consistency and paradoxes," Public Choice, Springer, vol. 169(1), pages 97-116, October.
    3. Erik Friese & William V. Gehrlein & Dominique Lepelley & Achill Schürmann, 2017. "The impact of dependence among voters’ preferences with partial indifference," Quality & Quantity: International Journal of Methodology, Springer, vol. 51(6), pages 2793-2812, November.
    4. Mostapha Diss & Eric Kamwa & Abdelmonaim Tlidi, 2020. "On Some k -scoring Rules for Committee Elections: Agreement and Condorcet Principle," Revue d'économie politique, Dalloz, vol. 130(5), pages 699-725.
    5. 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.
    6. Fatma Aslan & Hayrullah Dindar & Jean Lainé, 2022. "When are committees of Condorcet winners Condorcet winning committees?," Review of Economic Design, Springer;Society for Economic Design, vol. 26(3), pages 417-446, September.
    7. Mostapha Diss & Eric Kamwa & Abdelmonaim Tlidi, 2019. "On some k-scoring rules for committee elections: agreement and Condorcet Principle," Working Papers hal-02147735, HAL.
    8. 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.
    9. Daniela Bubboloni & Mostapha Diss & Michele Gori, 2020. "Extensions of the Simpson voting rule to the committee selection setting," Public Choice, Springer, vol. 183(1), pages 151-185, April.
    10. Mostapha Diss & Clinton Gubong Gassi & Issofa Moyouwou, 2023. "Combining diversity and excellence in multi winner elections," Working Papers 2023-05, CRESE.
    11. Edith Elkind & Piotr Faliszewski & Piotr Skowron & Arkadii Slinko, 2017. "Properties of multiwinner voting rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 48(3), pages 599-632, March.
    12. Egor Ianovski, 2022. "Electing a committee with dominance constraints," Annals of Operations Research, Springer, vol. 318(2), pages 985-1000, November.
    13. Eric Kamwa & Vincent Merlin, 2018. "The Likelihood of the Consistency of Collective Rankings under Preferences Aggregation with Four Alternatives using Scoring Rules: A General Formula and the Optimal Decision Rule," Working Papers hal-01757742, HAL.
    14. Ritu Dutta & Rajnish Kumnar & Surajit Borkotokey, 2023. "How to choose a Compatible Committee?," Papers 2308.03507, arXiv.org.

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    More about this item

    Keywords

    Committee; Paradox; Collective ranking; Consistency; Scoring rule; Impartial Culture;
    All these keywords.

    JEL classification:

    • D70 - Microeconomics - - Analysis of Collective Decision-Making - - - General
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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