Penalty-Based Aggregation Beyond the Current Confinement to Real Numbers: The Method of Kemeny Revisited

  • Raúl Pérez-FernándezEmail author
  • Bernard De Baets
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 643)


The field of aggregation theory addresses the mathematical formalization of aggregation processes. Historically, the developed mathematical framework has been largely confined to the aggregation of real numbers, while the aggregation of other types of structures, such as rankings, has been independently considered in different fields of application. However, one could lately perceive an increasing interest in the study and formalization of aggregation processes on new types of data. Mostly, this aggregation outside the framework of real numbers is based on the use of a penalty function measuring the disagreement with a consensus element. Unfortunately, there does not exist a comprehensive theoretical framework yet. In this paper, we propose a natural extension of the definition of a penalty function to a more general setting based on the compatibility with a given betweenness relation. In particular, we revisit one of the most common methods for the aggregation of rankings – the method of Kemeny – which will be positioned in the penalty-based aggregation framework.


Penalty function Aggregation of rankings Kemeny Monometric 


  1. 1.
    Beliakov, G., Bustince, H., Fernandez, J.: The median and its extensions. Fuzzy Sets Syst. 175, 36–47 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bustince, H., Beliakov, G., Dimuro, G.P., Bedregal, B., Mesiar, R.: On the definition of penalty functions in data aggregation. In: Fuzzy Sets and Systems (2016).
  3. 3.
    Calvo, T., Beliakov, G.: Aggregation functions based on penalties. Fuzzy Sets Syst. 161, 1420–1436 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Calvo, T., Mesiar, R., Yager, R.R.: Quantitative weights and aggregation. IEEE Trans. Fuzzy Syst. 12, 62–69 (2004)CrossRefGoogle Scholar
  5. 5.
    Condorcet, M.: Essai sur l’Application de l’Analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix, De l’Imprimerie Royale, Paris (1785)Google Scholar
  6. 6.
    De Miguel, L., Campión, M.J., Candeal, J.C., Induráin, E., Paternain, D.: Pointwise aggregation of maps: its structural functional equation and some applications to social choice theory. In: Fuzzy Sets and Systems (in press).
  7. 7.
    Gagolewski, M.: Penalty-based aggregation of multidimensional data. In: Fuzzy Sets and Systems (in press).
  8. 8.
    Kemeny, J.G.: Mathematics without numbers. Daedalus 88(4), 577–591 (1959)Google Scholar
  9. 9.
    Kendall, M.G.: A new measure of rank correlation. Biometrika 30, 81–93 (1938)CrossRefzbMATHGoogle Scholar
  10. 10.
    Pasch, M.: Vorlesungen über neuere Geometrie, vol. 23. Teubner, Leipzig, Berlin (1882)zbMATHGoogle Scholar
  11. 11.
    Pérez-Fernández, R., Alonso, P., Díaz, I., Montes, S., De Baets, B.: Monotonicity as a tool for differentiating between truth and optimality in the aggregation of rankings. J. Math. Psychol. 77, 1–9 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pérez-Fernández, R., Rademaker, M., Alonso, P., Díaz, I., Montes, S., De Baets, B.: Monotonicity-based ranking on the basis of multiple partially specified reciprocal relations. In: Fuzzy Sets and Systems (in press).
  13. 13.
    Pérez-Fernández, R., Rademaker, M., De Baets, B.: Monometrics and their role in the rationalisation of ranking rules. Inf. Fusion 34, 16–27 (2017)CrossRefGoogle Scholar
  14. 14.
    Pitcher, E., Smiley, M.F.: Transitivities of betweenness. Trans. Am. Math. Soc. 52(1), 95–114 (1942)zbMATHGoogle Scholar
  15. 15.
    Wilkin, T., Beliakov, G.: Weakly monotonic averaging functions. Int. J. Intell. Syst. 30(2), 144–169 (2015)CrossRefGoogle Scholar
  16. 16.
    Yager, R.R.: Toward a general theory of information aggregation. Inf. Sci. 68, 191–206 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yager, R.R., Rybalov, A.: Understanding the median as a fusion operator. Int. J. Gen. Syst. 26, 239–263 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.KERMIT, Department of Mathematical Modelling, Statistics and BioinformaticsGhent UniversityGhentBelgium

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