Multiple Discrete Choice with Order Statistic Marginal Utilities

Abstract

The paper presents a random utility maximization model for individuals selecting discrete quantities from a set of n alternatives. Multiple alternatives with positive quantities may be selected. Diminishing marginal utility to quantity of each alternative is modeled via order statistics of independent Gumbel random variables. The model is parsimonious — requiring n + 1 parameters — and tractable — admitting closed-form expressions for choice probabilities. As such, the model is amenable to maximum likelihood estimation of structural parameters from observed choices.

Probability functions recover binary logit probabilities under binary choice and a maximum quantity of one unit, and probability is monotonic in the quantity of each alternative. The monotonic property likely restricts the application of the model to a narrow range of settings. The property is a manifestation of a simple recursive relationship among Gumbel order statistic probabilities. This relationship, to my knowledge, has not previously been identified in the literature and may lead to new models for capturing important complexities in a tractable manner.