A Class of Dissimilarity Semimetrics for Preference Relations

We propose a class of semimetrics for acyclic preference relations, any one of which is an alternative to the classical Kemeny-Snell-Bogart metric. These semimetrics are based solely on the implications of preferences for choice behavior and thus appear more suitable in economic contexts and choice experiments. We obtain a fairly simple axiomatic characterization for the class we propose. The apparently most important member of this class, which we dub the “top-difference semimetric,” is characterized separately. We also obtain alternative formulae for it and, relative to this particular metric, compute the diameter of the space of complete and transitive preferences, as well as the best transitive extension of a given acyclic preference relation.