Skip to main content
SpringerLink
Log in

The Median Procedure as an Example of Penalty-Based Aggregation of Binary Relations

  • Conference paper
  • First Online:
Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations (IPMU 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 854))

  • 1082 Accesses

Abstract

The aggregation of binary relations is a common topic in many fields of application such as social choice and cluster analysis. In this paper, we discuss how the median procedure – probably the most common method for aggregating binary relations – fits in the framework of penalty-based data aggregation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The irreflexive part of a linear order relation is often called a ranking. Since the (ir)reflexivity of a relation does not usually play a role in the aggregation process, both linear order relations and rankings are often used interchangeably.

  2. 2.

    The term tournament relation is sometimes used for referring to the irreflexive part of what we call a tournament relation in this paper.

  3. 3.

    Note that the notions of \(\mathcal {B}\)-median and median are equivalent.

References

  1. Yager, R.R.: Toward a general theory of information aggregation. Inf. Sci. 68, 191–206 (1993)

    Article  MathSciNet  Google Scholar 

  2. Bustince, H., Beliakov, G., Dimuro, G.P., Bedregal, B., Mesiar, R.: On the definition of penalty functions in data aggregation. Fuzzy Sets Syst. 323, 1–18 (2017)

    Article  MathSciNet  Google Scholar 

  3. Calvo, T., Beliakov, G.: Aggregation functions based on penalties. Fuzzy Sets Syst. 161, 1420–1436 (2010)

    Article  MathSciNet  Google Scholar 

  4. Pérez-Fernández, R., De Baets, B.: On the role of monometrics in penalty-based data aggregation (submitted)

    Google Scholar 

  5. Kemeny, J.G.: Mathematics without numbers. Daedalus 88(4), 577–591 (1959)

    Google Scholar 

  6. Black, D.: Partial justification of the Borda count. Public Choice 28, 1–15 (1976)

    Article  Google Scholar 

  7. Monjardet, B.: An axiomatic theory of tournament aggregation. Math. Oper. Res. 3(4), 334–351 (1978)

    Article  MathSciNet  Google Scholar 

  8. Grötschel, M., Wakabayashi, Y.: A cutting plane algorithm for a clustering problem. Math. Program. 45, 59–96 (1989)

    Article  MathSciNet  Google Scholar 

  9. Barthelemy, J.P., Monjardet, B.: The median procedure in cluster analysis and social choice theory. Math. Soc. Sci. 1, 235–267 (1981)

    Article  MathSciNet  Google Scholar 

  10. Pitcher, E., Smiley, M.F.: Transitivities of betweenness. Trans. Am. Math. Soc. 52(1), 95–114 (1942)

    MATH  Google Scholar 

  11. Fishburn, P.C.: Betweenness, orders and interval graphs. J. Pure Appl. Algebra 1(2), 159–178 (1971)

    Article  MathSciNet  Google Scholar 

  12. Huntington, E.V., Kline, J.R.: Sets of independent postulates for betweenness. Trans. Am. Math. Soc. 18(3), 301–325 (1917)

    Article  MathSciNet  Google Scholar 

  13. Pasch, M.: Vorlesungen über neuere Geometrie, vol. 23. Teubner, Leipzig (1882)

    MATH  Google Scholar 

  14. Calvo, T., Mesiar, R., Yager, R.R.: Quantitative weights and aggregation. IEEE Trans. Fuzzy Syst. 12, 62–69 (2004)

    Article  Google Scholar 

  15. Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  16. Wakabayashi, Y.: The complexity of computing medians of relations. Resenhas 3(3), 323–349 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A.D. (eds.): Handb. Comput. Soc. Choice. Cambridge University Press, New York (2016)

    Google Scholar 

  18. Kendall, M.G.: A new measure of rank correlation. Biometrika 30, 81–93 (1938)

    Article  Google Scholar 

  19. Lerer, E., Nitzan, S.: Some general results on the metric rationalization for social decision rules. J. Econ. Theory 37, 191–201 (1985)

    Article  MathSciNet  Google Scholar 

  20. Andjiga, N.G., Mekuko, A.Y., Moyouwou, I.: Metric rationalization of social welfare functions. Math. Soc. Sci. 72, 14–23 (2014)

    Article  MathSciNet  Google Scholar 

  21. Elkind, E., Faliszewski, P., Slinko, A.: Distance rationalization of voting rules. Soc. Choice Welf. 45(2), 345–377 (2015)

    Article  MathSciNet  Google Scholar 

  22. Viappiani, P.: Characterization of scoring rules with distances: application to the clustering of rankings. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence, Buenos Aires (2015)

    Google Scholar 

Download references

Acknowledgments

Raúl Pérez-Fernández is supported as a postdoc by the Research Foundation of Flanders (FWO17/PDO/160).

Author information

Authors and Affiliations

  1. KERMIT, Department of Data Analysis and Mathematical Modelling, Ghent University, Coupure links 653, 9000, Ghent, Belgium

    Raúl Pérez-Fernández & Bernard De Baets

Authors
  1. Raúl Pérez-Fernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernard De Baets
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raúl Pérez-Fernández .

Editor information

Editors and Affiliations

  1. Universidad de Cádiz, Cádiz, Cadiz, Spain

    Jesús Medina

  2. Universidad de Málaga, Málaga, Málaga, Spain

    Manuel Ojeda-Aciego

  3. Universidad de Granada, Granada, Spain

    José Luis Verdegay

  4. Universidad de Granada, Granada, Spain

    David A. Pelta

  5. Universidad de Málaga, Málaga, Málaga, Spain

    Inma P. Cabrera

  6. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  7. Iona College, New Rochelle, New York, USA

    Ronald R. Yager

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Pérez-Fernández, R., De Baets, B. (2018). The Median Procedure as an Example of Penalty-Based Aggregation of Binary Relations. In: Medina, J., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations. IPMU 2018. Communications in Computer and Information Science, vol 854. Springer, Cham. https://doi.org/10.1007/978-3-319-91476-3_30

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-91476-3_30

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-91475-6

  • Online ISBN: 978-3-319-91476-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation