A Representation Theorem for Finite Best-Worst Random Utility Models

TL;DR

This paper establishes a fundamental representation theorem for finite best-worst random utility models, linking choice probabilities to utility representations through polynomial non-negativity conditions.

Contribution

It extends existing proof techniques to characterize when best-worst choice probabilities can be represented by a random utility model.

Findings

Non-negativity of best-worst Block-Marschak polynomials is necessary and sufficient.

Provides a representation theorem for finite best-worst choice models.

Extends Falmagne's proof techniques to best-worst choices.

Abstract

This paper investigates best-worst choice probabilities (picking the best and the worst alternative from an offered set). It is shown that non-negativity of best-worst Block-Marschak polynomials is necessary and sufficient for the existence of a random utility representation. The representation theorem is obtained by extending proof techniques employed by Falmagne (1978) for a corresponding result on best choices (picking the best alternative from an offered set).