Abstract
We introduce two extreme methods to pairwisely compare ordered lists of the same length, viz. the comonotonic and the countermonotonic comparison method, and show that these methods are, respectively, related to the copula T M (the minimum operator) and the Ł ukasiewicz copula T L used to join marginal cumulative distribution functions into bivariate cumulative distribution functions. Given a collection of ordered lists of the same length, we generate by means of T M and T L two probabilistic relations Q M and Q L and identify their type of transitivity. Finally, it is shown that any probabilistic relation with rational elements on a 3-dimensional space of alternatives which possesses one of these types of transitivity, can be generated by three ordered lists and at least one of the two extreme comparison methods.
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References
Brualdi R.A., Hwang G.S. (1992) Generalized transitive tournaments and doubly stochastic matrices. Linear Algebra Applications 172, 151–168
David H.A. (1963), In M.G. Kendall (ed.). The Method of Paired Comparisons. Griffin’s Statistical Monographs & Courses. Griffin & Co., London
De Baets B., De Meyer H. (2005) Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity. Fuzzy Sets and Systems 152, 249–270
De Baets B., De Meyer H., De Schuymer B., Jenei S. (2006) Cyclic evaluation of transitivity of reciprocal relations. Social Choice and Welfare 26, 217–238
De Schuymer B., De Meyer H., De Baets B., Jenei S. (2003) On the cycle-transitivity of the dice model. Theory and Decision 54, 261–285
De Schuymer B., De Meyer H., De Baets B. (2005) Cycle-transitive comparison of independent random variables. Journal of Multivariate Analysis 96, 352–373
Fishburn P.C. (1973), Binary choice probabilities: on the varieties of stochastic transitivity. Journal of Mathematical Psychology 10, 327–352
Fodor J., Roubens M. (1994). Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer, Dordrecht
Klement E., Mesiar R., Pap E. (2000). Triangular Norms. Trends in Logic, Studia Logica Library, Vol 8. Kluwer, Dordrecht
Monjardet B. (1988), A generalisation of probabilistic consistency: linearity conditions for valued preference relations. In: Kacprzyk J., Roubens M.(eds). Non-conventional Preference Relations in Decision Making, Lecture Notes in Economics and Mathematical Systems, Vol. 301. Springer, Berlin, pp. 36–53
Nelsen R. (1998). An Introduction to Copulas, Lecture Notes in Statistics, Vol. 139. Springer, New York
Switalski Z. (1999), Rationality of fuzzy reciprocal preference relations. Fuzzy Sets and Systems 107, 187–190
Switalski Z. (2003a) General transitivity conditions for fuzzy reciprocal preference matrices. Fuzzy Sets and Systems 137, 85–100
Switalski Z. (2003b), Half-integrality of vertices of the generalized transitive tournament polytope (n = 6). Discrete Mathematics 271, 251–260
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De Schuymer, B., De Meyer, H. & De baets, B. Extreme Copulas and the Comparison of Ordered Lists. Theor Decis 62, 195–217 (2007). https://doi.org/10.1007/s11238-006-9012-4
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DOI: https://doi.org/10.1007/s11238-006-9012-4