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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

Author

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  • Eric Kamwa

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

  • Vincent Merlin

    () (CREM - Centre de recherche en économie et management - UNICAEN - Université de Caen Normandie - NU - Normandie Université - UR1 - Université de Rennes 1 - UNIV-RENNES - Université de Rennes - CNRS - Centre National de la Recherche Scientifique)

Abstract

In most of the social choice literature dealing with the computation of the exact probability of voting events under the impartial culture assumption, authors deal with no more than four constraints to describe voting events. With more than four constraints, most of the authors rely on Monte-Carlo simulations. It is usually more tricky to estimate the probability of events described by five constraints. Gehrlein and Fishburn (1980) have tried, but their conclusions are based on conjectures. In this paper, we circumvent this conjecture by having recourse to the technique suggested by Saari and Tataru (1999) in order to compute the limit probability of the consistency of collective rankings when there are four competing alternatives given that the decision rule is a scoring rule. We provide a general formula for the limit probability of the consistency and we determine the optimal decision rules among the scoring rules that provide the best guarantee of consistency. Given the collective ranking on a set A, we have consistency if the collective ranking on B a proper subset of A is not altered after some alternatives are removed from A.

Suggested Citation

  • 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," Post-Print hal-01757742, HAL.
  • Handle: RePEc:hal:journl:hal-01757742
    DOI: 10.1007/s10614-018-9816-7
    Note: View the original document on HAL open archive server: https://hal.univ-antilles.fr/hal-01757742
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    References listed on IDEAS

    as
    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. Antonio D’Ambrosio & Willem J. Heiser, 2016. "A Recursive Partitioning Method for the Prediction of Preference Rankings Based Upon Kemeny Distances," Psychometrika, Springer;The Psychometric Society, vol. 81(3), pages 774-794, September.
    3. 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.
    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. Fabrice Valognes & Vincent Merlin & Monica Tataru, 2002. "On the likelihood of Condorcet's profiles," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(1), pages 193-206.
    6. Eric Kamwa & Vincent Merlin, 2015. "Scoring rules over subsets of alternatives: Consistency and paradoxes," Post-Print halshs-01238563, HAL.
    7. 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.
    8. 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.
    9. Gehrlein, William V. & Fishburn, Peter C., 1976. "The probability of the paradox of voting: A computable solution," Journal of Economic Theory, Elsevier, vol. 13(1), pages 14-25, August.
    10. William V. Gehrlein & Dominique Lepelley, 2017. "Elections, Voting Rules and Paradoxical Outcomes," Studies in Choice and Welfare, Springer, number 978-3-319-64659-6, February.
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