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

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## Abstract

The assortment problem in revenue management is the problem of 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 \(\pi \) and then choosing all products with revenue at least \(\pi \). This is known as the *revenue-ordered assortments* strategy. In this paper we study the approximation guarantees provided by revenue-ordered assortments when customers are rational in the following sense: the probability of selecting a specific product from the set being offered cannot increase if the set is enlarged. This rationality assumption, known as *regularity*, is satisfied by almost all discrete choice models considered in the revenue management and choice theory literature, and in particular by random utility models. The bounds we obtain are tight and improve on recent results in that direction, such as for the Mixed Multinomial Logit model by Rusmevichientong et al. (Prod Oper Manag 23(11):2023–2039, 2014). An appealing feature of our analysis is its simplicity, as it relies only on the regularity condition. We also draw a connection between assortment optimisation and two pricing problems called *unit demand envy-free pricing* and *Stackelberg minimum spanning tree*: these problems can be restated as assortment problems under discrete choice models satisfying the regularity condition, and moreover revenue-ordered assortments correspond then to the well-studied *uniform pricing* heuristic. When specialised to that setting, the general bounds we establish for revenue-ordered assortments match and unify the best known results on uniform pricing.

## Keywords

Assortment problem Envy-free pricing Stackelberg games## Notes

### Acknowledgements

We thank Victor Aguiar, Adrian Vetta, and Gustavo Vulcano for their insightful comments that greatly improved the paper. We are also grateful to the anonymous referees for their careful reading of the paper and helpful remarks.

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