Imperfect discrimination, similarity, and stochastic transitivity

We study the relationship between imperfect discrimination, similarity, and stochastic transitivity in generalized versions of Perturbed Utility (Fudenberg et al. in Econometrica 83(6):2371–2409, 2015) and Fechnerian models (Debreu in Econometrica 26(3):440–444, 1958). We show that these models are equivalent and that, within them, the properties of a similarity function can characterize all notions of stochastic transitivity (as Weak, Moderate, and Strong). Specifically, He and Natenzon (Am Econ Rev Insights 6(2):176–195, 2024) have recently shown that choice probabilities are moderately transitive if and only if the similarity function is a metric. We provide a counterpoint and show that unless choice probabilities are strongly transitive, the similarity function can violate the triangle inequality.