Advertisement

Product Triplets in Winning Probability Relations

  • Karel De Loof
  • Bernard De Baets
  • Hans De Meyer
Part of the Communications in Computer and Information Science book series (CCIS, volume 300)

Abstract

It is known that the winning probability relation of a dice model, which amounts to the pairwise comparison of a set of independent random variables that are uniformly distributed on finite integer multisets, is dice transitive. The condition of dice transitivity, also called the 3-cycle condition, is, however, not sufficient for an arbitrary rational-valued reciprocal relation to be the winning probability relation of a dice model. An additional necessary condition, called the 4-cycle condition, is introduced in this contribution. Moreover, we reveal a remarkable relationship between the 3-cycle condition and the number of so-called product triplets of a reciprocal relation. Finally, we experimentally count product triplets for several families of winning probability relations.

Keywords

dice representability dice transitivity independent random variables product triplets reciprocal relation winning probability relation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    David, H.: The Method of Paired Comparisons. Griffin, London (1963)Google Scholar
  2. 2.
    De Baets, B., De Meyer, H.: Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity. Fuzzy Sets and Systems 152, 249–270 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    De Baets, B., De Meyer, H., De Schuymer, B., Jenei, S.: Cyclic evaluation of transitivity of reciprocal relations. Social Choice and Welfare 26, 217–238 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    De Schuymer, B., De Meyer, H., De Baets, B.: Cycle-transitive comparison of independent random variables. Journal of Multivariate Analysis 96, 352–373 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    De Schuymer, B., De Meyer, H., De Baets, B., Jenei, S.: On the cycle-transitivity of the dice model. Theory and Decision 54, 261–285 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Fishburn, P.: Binary choice probabilities: on the varieties of stochastic transitivity. Journal of Mathematical Psychology 10, 327–352 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gardner, M.: Mathematical games: The paradox of the nontransitive dice and the elusive principle of indifference. Scientific American 223, 110–114 (1970)Google Scholar
  8. 8.
    Savage, R.: The paradox of nontransitive dice. The American Mathematical Monthly 101, 429–436 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Switalski, S.: Rationality of fuzzy reciprocal preference relations. Fuzzy Sets and Systems 107, 187–190 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Wrather, C., Lu, P.: Probability dominance in random outcomes. Journal of Optimization Theory and Applications 36, 315–334 (1982)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Karel De Loof
    • 1
  • Bernard De Baets
    • 1
  • Hans De Meyer
    • 2
  1. 1.Department of Mathematical Modelling, Statistics and BioinformaticsGhent UniversityGentBelgium
  2. 2.Department of Applied Mathematics and Computer ScienceGhent UniversityGentBelgium

Personalised recommendations