A technical note on finite best-worst random utility model representations

TL;DR

This paper explores the mathematical foundations of best-worst choice models, providing new insights into their utility representations and extending previous characterizations for sets of four or more options.

Contribution

It constructs a probability measure on rankings that implies a random utility representation, advancing the theoretical understanding of best-worst choice models.

Findings

Complete characterization of the best-worst-choice polytope for four alternatives.

Block-Marschak inequalities are insufficient for sets of four or more alternatives.

Construction of a probability measure on rankings for random utility representation.

Abstract

This paper investigates the random utility representation of best-worst choice probabilities (picking the best and the worst alternative from an offered set). Doignon (2023) presented a complete characterization of the best-worst-choice polytope on four alternatives. Moreover, using polytope methods he showed that the Block-Marschak inequalities for best-worst choices are not sufficient for a random utility representation of best-worst choices for sets of four or more alternatives. Following the approach of Falmagne (1978), we construct a probability measure on the set of rankings for a set of four alternatives implying a random utility representation.