Assortment Optimisation Under a General Discrete Choice Model : A Tight Analysis of Revenue-Ordered Assortments

A central problem in revenue management, known as the assortment problem, consists in deciding which subset of products to offer to consumers in order to maximise revenue. A simple and natural strategy is to select the best assortment out of all those that are constructed by fixing a threshold revenue r and then choosing all products with revenue at least r. This is known as the revenue-ordered assortments strategy. The main contribution of this paper is a precise analysis of how well revenue-ordered assortments approximate the optimum revenue when customers are rational in the following sense: the probability of selecting a specific product (including the no-choice option) from the set being offered cannot increase if the offer set is enlarged. The corresponding discrete choice models form a broad class of models which includes all discrete choice models based on random utility. Our analysis of revenue-ordered assortments match and unify known results for certain models, and improves the best known results for others, such as for the Mixed Multinomial Logit model recently studied by Rusmevichientong et al (2014). An appealing feature of our analysis is that it is simple and relies only on the above-mentioned rationality property, and yet it is best possible even for very specific models within the class. We also show that a large class of problems known as envy-free pricing problems can be seen as assortment problems for a specifically constructed discrete choice model that satisfies the rationality property. In this context, revenue-ordered assortments turn out to be equivalent to the well-studied uniform pricing strategy