Cycle-transitivity is all around

In this paper, the transitivity properties of reciprocal relations, also called probabilistic relations, are investigated within the framework of cycle-transitivity, which generalizes the concepts of T-transitivity, stemming from fuzzy set theory, and of stochastic transitivity, common to many mathematical models in psychology, social choice and welfare, financial mathematics, etc. It is emphasized that this unifying framework is tailor-made for characterizing the transitivity of reciprocal relations that originate from the comparison of random variables. Interesting types of transitivity are highlighted and shown to be realizable in applications. For example, given a collection of random variables (X_k)_kisin1, pairwisely coupled by means of a same copula C isin {T_M,Tp,T_L}, the transitivity of the reciprocal relation Q defined by Q(X_i,X_j) = Prob{X_i > X_j} + 1/2 Prob{X_i = X_j} can be characterized within the cycle-transitivity framework. Similarly, given a poset (P, <) with P = {x₁,..., x_n}, the transitivity of the mutual rank probability relation Qp, where Qp(X_i,X_j) denotes the probability that X_i precedes x_j in a random linear extension of P, is characterized as a type of cycle-transitivity for which no realization had been found so far.