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Transitivity frameworks for reciprocal relations: cycle-transitivity versus \(FG\)-transitivity. (English) Zbl 1114.91031
Summary: For a reciprocal relation \(Q\) on a set of alternatives \(A\), two transitivity frameworks which generalize both \(T\)-transitivity and stochastic transitivity are compared: the framework of cycle-transitivity, introduced by the present authors et al. [Soc. Choice Welfare 26, No. 2, 217–238 (2006; Zbl 1158.91338)] and which is based upon the ordering of the numbers \(Q(a,b), Q(b,c)\) and \(Q(c,a)\) for all \((a,b,c)\in A^3\), and the framework of \(FG\)-transitivity, introduced by Z. Switalski [Fuzzy Sets Syst. 137, 85–100 (2003; Zbl 1052.91033)] as an immediate generalization of stochastic transitivity. The rules that enable to express FG-transitivity in the form of cycle-transitivity and cycle-transitivity in the form of \(FG\)-transitivity, illustrate that for reciprocal relations the concept of cycle-transitivity provides a framework that can cover more types of transitivity than does the concept of \(FG\)-transitivity.

MSC:
91B08 Individual preferences
91B12 Voting theory
91E45 Measurement and performance in psychology
03E72 Theory of fuzzy sets, etc.
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