Modeling Intransitive Indifference Algebraically and Numerically

Jun Zhang and Yitong Sun

Abstract

Intransitive indifference relation is prevalent in social and behavioral sciences. One of its manifestation is the "threshold" phenomena characteristic of comparative judgments in psychophysics. A formal axiomatic approach to intransitive indifference was pioneered by Luce (1956), who introduced the concept of "semi-order". It was later known that any set endowed with a semi-order relation carries a numerical representation of fixed-intervals (Scott-Suppes representation) justifying the commonly adopted "just noticeable difference" representation of threshold in sensory psychophysics. Moreover, generalizing semi-order is the so-called interval order (Fishburn) with a variable-interval representation for its elements; the latter closely related to biorder (Falmagne, Doignon, etc) that underlies Guttman scale. I will review the literature, and discuss how an interval order induces a “nesting” relation, the absence of which characterizes semi-order. I will then show how pairwise separation of elements of an interval-ordered set (including semi-ordered set as a special case) can be linked to topological separation of a suitably constructed topology on such posets (partially-ordered sets).