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Scoring rules over subsets of alternatives: Consistency and paradoxes

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  • Kamwa, Eric
  • Merlin, Vincent

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

  • 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.
    • Handle: RePEc:eee:mateco:v:61:y:2015:i:c:p:130-138
    DOI: 10.1016/j.jmateco.2015.08.008
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    References listed on IDEAS

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    1. 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.
    2. Van Newenhizen, Jill, 1992. "The Borda Method Is Most Likely to Respect the Condorcet Principle," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(1), pages 69-83, January.
    3. Eric Kamwa, 2013. "The increasing committee size paradox with small number of candidates," Economics Bulletin, AccessEcon, vol. 33(2), pages 967-972.
    4. Merlin, Vincent & Valognes, Fabrice, 2004. "The impact of indifferent voters on the likelihood of some voting paradoxes," Mathematical Social Sciences, Elsevier, vol. 48(3), pages 343-361, November.
    5. Saari, Donald G., 1987. "The source of some paradoxes from social choice and probability," Journal of Economic Theory, Elsevier, vol. 41(1), pages 1-22, February.
    6. 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.
    7. Smith, John H, 1973. "Aggregation of Preferences with Variable Electorate," Econometrica, Econometric Society, vol. 41(6), pages 1027-1041, November.
    8. 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.
    9. 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.
    10. Dutta, Bhaskar & Jackson, Matthew O & Le Breton, Michel, 2001. "Strategic Candidacy and Voting Procedures," Econometrica, Econometric Society, vol. 69(4), pages 1013-1037, July.
    11. 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.
    12. Donald G. Saari & Maria M. Tataru, 1999. "The likelihood of dubious election outcomes," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 13(2), pages 345-363.
    13. 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.
    14. William Gehrlein & Peter Fishburn, 1981. "Constant scoring rules for choosing one among many alternatives," Quality & Quantity: International Journal of Methodology, Springer, vol. 15(2), pages 203-210, April.
    15. 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.
    16. 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, 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," Post-Print hal-01757742, HAL.
    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. repec:spr:qualqt:v:51:y:2017:i:6:d:10.1007_s11135-016-0446-7 is not listed on IDEAS
    4. 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.
    5. 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.
    6. 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.
    7. repec:spr:sochwe:v:48:y:2017:i:3:d:10.1007_s00355-017-1026-z is not listed on IDEAS
    8. 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.

    More about this item

    Keywords

    Scoring rule; Consistency; Collective ranking; Committee; Paradox; Impartial culture;

    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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