Generalized thurstone models for ranking: Equivalence and reversibility

Previous work has determined the conditions under which generalized versions of Thurstone's theory of comparative judgment are formally equivalent (i.e., empirically indistinguishable) for choice experiments. This note solves the analogous problem for ranking experiments: It is shown that if two “Generalized Thurstone Models” are equivalent for choice experiments with n alternatives they are also equivalent for ranking experiments with n alternatives, despite the fact that ranking generates many more preference probabilities. This result in turn allows one to determine which Generalized Thurstone Models are “reversible,” i.e., satisfy the requirement that regardless of whether the subject ranks from best to worst or from worst to best, rankings that express the same preference order will occur with the same probability.