A class of dissimilarity semimetrics for preference relations

TL;DR

This paper introduces a new class of semimetrics for preference relations based on choice implications, offering a more suitable alternative to classical metrics in economic and experimental contexts.

Contribution

It provides an axiomatic characterization of the new semimetric class, especially the top-difference semimetric, and explores its properties and limitations.

Findings

Characterized the top-difference semimetric axiomatically

Computed the diameter of preference spaces under this metric

Proved non-embeddability into Euclidean space

Abstract

We propose a class of semimetrics for preference relations any one of which is an alternative to the classical Kemeny-Snell-Bogart metric. (We take a fairly general viewpoint about what constitutes a preference relation, allowing for any acyclic order to act as one.) These semimetrics are based solely on the implications of preferences for choice behavior, and thus appear more suitable in economic contexts and choice experiments. In our main result, we obtain a fairly simple axiomatic characterization for the class we propose. The apparently most important member of this class (at least in the case of finite alternative spaces), which we dub the top-difference semimetric, is characterized separately. We also obtain alternative formulae for it, and relative to this metric, compute the diameter of the space of complete preferences, as well as the best transitive extension of a given acyclic preference relation. Finally, we prove that our preference metric spaces cannot be isometically embedded in a Euclidean space.