Relations between best, worst, and best–worst choices for random utility models

Abstract

For random utility models and under very mild assumptions, using the inclusion–exclusion principle, we derive an identity which expresses the probability that an alternative is the worst choice within a finite set of alternatives as an alternating sum of best choice probabilities. Under slightly stronger assumptions on the distribution of the vector of random utilities, we also derive an identity for the joint best–worst choice probability that an alternative is the best choice while another alternative is the worst one. These identities are applied to specific models with independent extreme value and even generalized extreme value distributed utilities.