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)


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.


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


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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

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