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.
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References
David, H.: The Method of Paired Comparisons. Griffin, London (1963)
De Baets, B., De Meyer, H.: Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity. Fuzzy Sets and Systems 152, 249–270 (2005)
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)
De Schuymer, B., De Meyer, H., De Baets, B.: Cycle-transitive comparison of independent random variables. Journal of Multivariate Analysis 96, 352–373 (2005)
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)
Fishburn, P.: Binary choice probabilities: on the varieties of stochastic transitivity. Journal of Mathematical Psychology 10, 327–352 (1973)
Gardner, M.: Mathematical games: The paradox of the nontransitive dice and the elusive principle of indifference. Scientific American 223, 110–114 (1970)
Savage, R.: The paradox of nontransitive dice. The American Mathematical Monthly 101, 429–436 (1994)
Switalski, S.: Rationality of fuzzy reciprocal preference relations. Fuzzy Sets and Systems 107, 187–190 (1999)
Wrather, C., Lu, P.: Probability dominance in random outcomes. Journal of Optimization Theory and Applications 36, 315–334 (1982)
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De Loof, K., De Baets, B., De Meyer, H. (2012). Product Triplets in Winning Probability Relations. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances in Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31724-8_31
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DOI: https://doi.org/10.1007/978-3-642-31724-8_31
Publisher Name: Springer, Berlin, Heidelberg
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