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

• We introduce two versions of (m,n)(m,n)-Ferrers properties, weak and strict.
• The two versions of (m,n)(m,n)-Ferrers properties only coincide for interval orders and semiorders.
• As mm and nn increase, strict (m,n)(m,n)-Ferrers properties become weaker and weaker (under transitivity).
• As mm and nn increase, weak (m,n)(m,n)-Ferrers properties become stronger and stronger.
• Weak (m,n)(m,n)-Ferrers properties form a finite poset under implication.

Traditionally, a preference on a set AA of alternatives is modeled by a binary relation RR on AA satisfying suitable axioms of pseudo-transitivity, such as the Ferrers condition (aRbaRb and cRdcRd imply aRdaRd or cRbcRb) or the semitransitivity property (aRbaRb and bRcbRc imply aRdaRd or dRcdRc). In this paper we study (m,n)(m,n)-Ferrers properties, which naturally generalize these axioms by requiring that a1R…Rama1R…Ram and b1R…Rbnb1R…Rbn imply a1Rbna1Rbn or b1Ramb1Ram. We identify two versions of (m,n)(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)(m,n)-Ferrers properties, which coincide whenever m+n=4m+n=4 (i.e., for the classical Ferrers condition and for semitransitivity), otherwise displaying an almost dual behavior. In fact, as mm and nn increase, weak (m,n)(m,n)-Ferrers properties become stronger and stronger, whereas strict (m,n)(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)(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.