The pseudo-transitivity of preference relations: Strict and weak (m,n)-Ferrers properties

Traditionally, a preference on a set A of alternatives is modeled by a binary relation R on A satisfying suitable axioms of pseudotransitivity, such as the Ferrers condition (aRb and cRd imply aRd or cRb) or the semitransitivity property (aRb and bRc imply aRd or dRc). In this paper we study (m, n)-Ferrers properties, which naturally generalize these axioms by requiring that a_1 R...R a_m and b_1 R...R b_n imply a_1 R b_n or b_1 R a_m. We identify two versions of (m, n)-Ferrers properties: weak, related to a reflexive relation, and strict, related to its asymmetric part. We determine the relationship between these two versions of (m, n)-Ferrers properties, which coincide whenever m+n=4 (i.e., for the classical Ferrers condition and for semitransitivity), otherwise displaying an almost dual behavior. In fact, as m and n increase, weak (m, n)-Ferrers properties become stronger and stronger, whereas strict (m, n)-Ferrers properties become somehow weaker and weaker (despite failing to display a monotonic behavior). We give a detailed description of the finite poset of weak (m,n)-Ferrers properties, ordered by the relation of implication. This poset depicts a discrete evolution of the transitivity of a preference, starting from its absence and ending with its full satisfaction.