Incomplete paired comparisons in case of multiple choice and general log-concave probability density functions

  • Éva Orbán-MihálykóEmail author
  • Csaba Mihálykó
  • László Koltay
Original Paper


A scoring method based on paired comparison allowing multiple choice is investigated. We allow general log-concave probability density functions for the random variables describing the difference of the objects. This case involves Bradley–Terry models and Thurstone models as well. A sufficient condition is proved for the existence and uniqueness of the maximum likelihood estimation of the parameters in case of incomplete comparisons. The axiomatic properties of the method are also investigated.


Paired comparison Multiple options Maximum likelihood estimation Strictly log-concave probability density function Axiomatic properties 


  1. Agresti A (1992) Analysis of ordinal paired comparison data. Appl Stat 41(2):287–297CrossRefGoogle Scholar
  2. Bradley RA, Terry ME (1952) Rank analysis of incomplete block designs: I. The method of paired comparisons. Biometrika 39(3/4):324–345CrossRefGoogle Scholar
  3. Bozóki S, Fülöp J, Rónyai L (2010) On optimal completion of incomplete pairwise comparison matrices. Math Comput Modell 52(1):318–333CrossRefGoogle Scholar
  4. Cattelan M (2012) Models for paired comparison data: a review with emphasis on dependent data. Stat Sci 27(3):412–433CrossRefGoogle Scholar
  5. Chebotarev PY, Shamis E (1998) Characterizations of scoring methodsfor preference aggregation. Ann Oper Res 80:299–332CrossRefGoogle Scholar
  6. Davidson RR (1970) On extending the Bradley–Terry model to accommodate ties in paired comparison experiments. J Am Stat Assoc 65(329):317–328CrossRefGoogle Scholar
  7. Ford LR (1957) Solution of a ranking problem from binary comparisons. Am Math Mon 64(8):28–33CrossRefGoogle Scholar
  8. González-Díaz J, Hendrickx R (2014) Paired comparisons analysis: an axiomatic approach to ranking methods. Soc Choice Welf 42(1):139–169CrossRefGoogle Scholar
  9. Mosteller F (1951) Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations. Psychometrika 16(1):3–9CrossRefGoogle Scholar
  10. Orbán-Mihálykó É, Mihálykó Cs, Koltay L (2017) A generalization of the Thurstone method for multiple choice and incomplete paired comparisons. Cent Eur J Oper Res.
  11. Prékopa A (1973) Logarithmic concave measures and functions. Acta Sci Math 34(1):334–343Google Scholar
  12. Rubinstein A (1980) Ranking the participants in a tournament. SIAM J Appl Math 38(1):108–111CrossRefGoogle Scholar
  13. Rao PV, Kupper LL (1967) Ties in paired-comparison experiments: a generalization of the Bradley–Terry model. J Am Stat Assoc 62(317):194–204CrossRefGoogle Scholar
  14. Saaty TL (1990) How to make a decision: the analytic hierarchy process. Eur J Oper Res 48(1):9–26CrossRefGoogle Scholar
  15. Stern H (1990) A continuum of paired comparisons models. Biometrika 77(2):265–273CrossRefGoogle Scholar
  16. Stern H (1992) Are all linear paired comparison models empirically equivalent? Math Soc Sci 23(1):103–117CrossRefGoogle Scholar
  17. Thurstone LL (1927) A law of comparative judgement. Psychol Rev 34(4):273–286CrossRefGoogle Scholar
  18. Tutz G (1986) Bradley–Terry–Luce models with an ordered response. J Math Psychol 30(3):306–316CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PannoniaVeszprémHungary

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