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Paired comparisons analysis: an axiomatic approach to ranking methods

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  • Julio González-Díaz
  • Ruud Hendrickx
  • Edwin Lohmann

Abstract

In this paper we present an axiomatic analysis of several ranking methods for general tournaments. We find that the ranking method obtained by applying maximum likelihood to the (Zermelo-)Bradley-Terry model, the most common method in statistics and psychology, is one of the ranking methods that perform best with respect to the set of properties under consideration. A less known ranking method, generalised row sum, performs well too. We also study, among others, the fair bets ranking method, widely studied in social choice, and the least squares method. Copyright Springer-Verlag Berlin Heidelberg 2014

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  • Julio González-Díaz & Ruud Hendrickx & Edwin Lohmann, 2014. "Paired comparisons analysis: an axiomatic approach to ranking methods," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(1), pages 139-169, January.
    • Handle: RePEc:spr:sochwe:v:42:y:2014:i:1:p:139-169
    DOI: 10.1007/s00355-013-0726-2
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    3. László Csató, 2019. "An impossibility theorem for paired comparisons," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(2), pages 497-514, June.
    4. L'aszl'o Csat'o & Csaba T'oth, 2018. "University rankings from the revealed preferences of the applicants," Papers 1810.04087, arXiv.org, revised Feb 2020.
    5. Devlin Stephen & Treloar Thomas, 2018. "A network diffusion ranking family that includes the methods of Markov, Massey, and Colley," Journal of Quantitative Analysis in Sports, De Gruyter, vol. 14(3), pages 91-101, September.
    6. László Csató, 2017. "Measuring centrality by a generalization of degree," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 25(4), pages 771-790, December.
    7. Carmen Herrero & Antonio Villar, 2013. "On the Comparison of Group Performance with Categorical Data," PLOS ONE, Public Library of Science, vol. 8(12), pages 1-7, December.
    8. Osei, Prince P. & Davidov, Ori, 2022. "Bayesian linear models for cardinal paired comparison data," Computational Statistics & Data Analysis, Elsevier, vol. 172(C).
    9. Daniela Bubboloni & Mostapha Diss & Michele Gori, 2020. "Extensions of the Simpson voting rule to the committee selection setting," Public Choice, Springer, vol. 183(1), pages 151-185, April.
    10. Luis C. Dias & Humberto Rocha, 2023. "A stochastic method for exploiting outranking relations in multicriteria choice problems," Annals of Operations Research, Springer, vol. 321(1), pages 165-189, February.
    11. Daniela Bubboloni & Michele Gori, 2018. "The flow network method," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 51(4), pages 621-656, December.
    12. Csató, László & Petróczy, Dóra Gréta, 2021. "On the monotonicity of the eigenvector method," European Journal of Operational Research, Elsevier, vol. 292(1), pages 230-237.
    13. Michele Gori, 2023. "Families of abstract decision problems whose admissible sets intersect in a singleton," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 61(1), pages 131-154, July.
    14. László Csató, 2015. "A graph interpretation of the least squares ranking method," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 44(1), pages 51-69, January.
    15. Csató, László, 2017. "European qualifiers to the 2018 FIFA World Cup can be manipulated," MPRA Paper 82652, University Library of Munich, Germany.
    16. Joseph Gogodze, 2019. "Innovative advantages ranking. A new approach," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 29(1), pages 5-15.
    17. S. S. Dabadghao & B. Vaziri, 2022. "The predictive power of popular sports ranking methods in the NFL, NBA, and NHL," Operational Research, Springer, vol. 22(3), pages 2767-2783, July.
    18. Devlin Stephen & Treloar Thomas & Creagar Molly & Cassels Samuel, 2021. "An iterative Markov rating method," Journal of Quantitative Analysis in Sports, De Gruyter, vol. 17(2), pages 117-127, June.
    19. Carmen Herrero & Antonio Villar, 2018. "The Balanced Worth: A Procedure to Evaluate Performance in Terms of Ordered Attributes," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 140(3), pages 1279-1300, December.
    20. László Csató, 2018. "Characterization of the Row Geometric Mean Ranking with a Group Consensus Axiom," Group Decision and Negotiation, Springer, vol. 27(6), pages 1011-1027, December.
    21. Csató, László & Tóth, Csaba, 2020. "University rankings from the revealed preferences of the applicants," European Journal of Operational Research, Elsevier, vol. 286(1), pages 309-320.
    22. Éva Orbán-Mihálykó & Csaba Mihálykó & László Koltay, 2019. "Incomplete paired comparisons in case of multiple choice and general log-concave probability density functions," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(2), pages 515-532, June.
    23. László Csató, 2017. "On the ranking of a Swiss system chess team tournament," Annals of Operations Research, Springer, vol. 254(1), pages 17-36, July.

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