A frequentist view on cycle-transitivity of reciprocal relations

We establish a connection between two transitivity frameworks: the transitivity of fuzzy relations based on a commutative quasi-copula and the cycle-transitivity of reciprocal relations w.r.t. the dual quasi-copula as an upper bound function. Loosely speaking, it turns out that the latter can be characterized by imposing a lower bound on the relative frequency with which the former is fulfilled, when applied to reciprocal relations. We provide two compelling cases: the 4/6 theorem, expressing that the winning probability relation of a set of independent random variables is at least 66.66% product-transitive, and the 5/6 theorem, expressing that the mutual rank probability relation associated with a given poset is at least 83.33% product-transitive. Moreover, these lower bounds turn out be rather conservative, illustrating that, from a frequentist point of view, transitivity is abundant. (C) 2015 Elsevier B.V. All rights reserved.