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Figure Skating and the Theory of Social Choice


  • Truchon, Michel



The rule used by the United States Figure Skating Association and the International Skating Union, hereafter the ISU Rule, to aggregate individual rankings of the skaters by the judges into a final ranking, is an interesting example of a social welfare function. This rule is examined thoroughly in this paper from the perspective of the modern theory of social choice. The ISU Rule is based on four different criteria, the first being median ranks of the skaters. Although the median rank criterion is a majority principle, it is completely at odd with another majority principle introduced in this paper and called the Extended Condorcet Criterion. It may be translated as follows: If a competitor is ranked consistently ahead of another competitor by an absolute majority of judges, he should be ahead in the final ranking. Consistency here refers to the absence of a cycle in the majority relation involving these two skaters. There are actually many cycles in the data of four Olympic Games that were examined. The Kemeny rule may be used to break these cycles. This is not only consistent with the Extended Condorcet Criterion but the latter also proves useful in finding Kemeny orders over large sets of alternatives, by allowing decomposition of these orders. The ISU, the Kemeny, the Borda rankings and the ranking according to the raw marks are then compared on 24 olympic competitions. The four rankings disagree in many instances. Finally it is shown that the ISU Rule may be very sensitive to small errors on the part of the judges and that it does not escape the numerous theorems on manipulation. Some considerations are also offered as to whether the ISU Rule is more or less prone to manipulation than others. La règle utilisée par la United States Figure Skating Association et l'International Skating Union, ci-après la règle de l'ISU, pour agréger les classements des patineurs par chacun des juges en un classement final, est un exemple intéressant de fonction de bien-être social. Cette règle est examinée en détail dans cet article du point de vue de la théorie moderne des choix sociaux. Cette règle repose sur quatre critères, le premier étant le rang médian des patineurs. Bien que ce critère soit en fait un principe majoritaire, il va à l'encontre d'un autre principe majoritaire introduit ici et appelé le Critère de Condorcet généralisé. Il peut être traduit ainsi: Si un compétiteur est classé avant un autre de manière cohérente par une majorité de juges, il devrait l'être dans le classement final. La cohérence réfère à l'absence de cycle dans la relation majoritaire impliquant ces deux compétiteurs. De fait, plusieurs cycles ont été rencontrés dans les données de quatre Jeux olympiques qui ont été examinées. La règle de Kemeny peut être utilisée pour briser ces cycles. Non seulement cette règle est-elle cohérente avec le Critère de Condorcet généralisé mais ce dernier s'avère utile dans la recherche d'ordres de Kemeny sur un grand nombre d'alternatives, en permettant la décomposition de ces ordres. Les classements des patineurs selon les règles de l'ISU, de Kemeny, de Borda et selon les notes brutes sont ensuite comparés pour 24 compétitions olympiques. Les quatre classements sont souvent différents. Finalement, il est démontré que la règle de l'ISU peut être très sensible à de petites erreurs de la part des juges et qu'elle n'échappe pas aux nombreux théorèmes d'impossibilité sur la manipulation. Quelques remarques sont aussi offertes sur la plus ou moins grande susceptibilité de cette règle à la manipulation par rapport à d'autres règles.

Suggested Citation

  • Truchon, Michel, 1998. "Figure Skating and the Theory of Social Choice," Cahiers de recherche 9814, Université Laval - Département d'économique.
  • Handle: RePEc:lvl:laeccr:9814

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    References listed on IDEAS

    1. Le Breton, Michel & Truchon, Michel, 1997. "A Borda measure for social choice functions," Mathematical Social Sciences, Elsevier, vol. 34(3), pages 249-272, October.
    2. Truchon, Michel, 1998. "An Extension of the Concordet Criterion and Kemeny Orders," Cahiers de recherche 9813, Université Laval - Département d'économique.
    3. Barthelemy, J. P. & Guenoche, A. & Hudry, O., 1989. "Median linear orders: Heuristics and a branch and bound algorithm," European Journal of Operational Research, Elsevier, vol. 42(3), pages 313-325, October.
    4. Young, H. P., 1974. "An axiomatization of Borda's rule," Journal of Economic Theory, Elsevier, vol. 9(1), pages 43-52, September.
    5. Muller, Eitan & Satterthwaite, Mark A., 1977. "The equivalence of strong positive association and strategy-proofness," Journal of Economic Theory, Elsevier, vol. 14(2), pages 412-418, April.
    6. Jonathan Levin & Barry Nalebuff, 1995. "An Introduction to Vote-Counting Schemes," Journal of Economic Perspectives, American Economic Association, vol. 9(1), pages 3-26, Winter.
    7. Peyton Young, 1995. "Optimal Voting Rules," Journal of Economic Perspectives, American Economic Association, vol. 9(1), pages 51-64, Winter.
    8. Saari, Donald G, 1990. "Susceptibility to Manipulation," Public Choice, Springer, vol. 64(1), pages 21-41, January.
    9. I. Good, 1971. "A note on condorcet sets," Public Choice, Springer, vol. 10(1), pages 97-101, March.
    10. Gibbard, Allan, 1973. "Manipulation of Voting Schemes: A General Result," Econometrica, Econometric Society, vol. 41(4), pages 587-601, July.
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    Cited by:

    1. Truchon, Michel & Gordon, Stephen, 2009. "Statistical comparison of aggregation rules for votes," Mathematical Social Sciences, Elsevier, vol. 57(2), pages 199-212, March.
    2. Giuseppe Munda, 2012. "Intensity of preference and related uncertainty in non-compensatory aggregation rules," Theory and Decision, Springer, vol. 73(4), pages 649-669, October.
    3. Boudreau, James & Ehrlich, Justin & Sanders, Shane & Winn, Adam, 2014. "Social choice violations in rank sum scoring: A formalization of conditions and corrective probability computations," Mathematical Social Sciences, Elsevier, vol. 71(C), pages 20-29.
    4. James Boudreau & Justin Ehrlich & Mian Farrukh Raza & Shane Sanders, 2018. "The likelihood of social choice violations in rank sum scoring: algorithms and evidence from NCAA cross country running," Public Choice, Springer, vol. 174(3), pages 219-238, March.
    5. Azzini, Ivano & Munda, Giuseppe, 2020. "A new approach for identifying the Kemeny median ranking," European Journal of Operational Research, Elsevier, vol. 281(2), pages 388-401.
    6. William Gehrlein, 2002. "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences ," Theory and Decision, Springer, vol. 52(2), pages 171-199, March.
    7. Matthew Harrison-Trainor, 2020. "An Analysis of Random Elections with Large Numbers of Voters," Papers 2009.02979,
    8. Mohamed Drissi-Bakhkhat & Michel Truchon, 2004. "Maximum likelihood approach to vote aggregation with variable probabilities," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 23(2), pages 161-185, October.
    9. Giuseppe Munda & Michela Nardo, 2009. "Noncompensatory/nonlinear composite indicators for ranking countries: a defensible setting," Applied Economics, Taylor & Francis Journals, vol. 41(12), pages 1513-1523.
    10. Adrian Deemen, 2014. "On the empirical relevance of Condorcet’s paradox," Public Choice, Springer, vol. 158(3), pages 311-330, March.
    11. Truchon, Michel, 1998. "An Extension of the Concordet Criterion and Kemeny Orders," Cahiers de recherche 9813, Université Laval - Département d'économique.

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    JEL classification:

    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • D70 - Microeconomics - - Analysis of Collective Decision-Making - - - General

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