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

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 π and then choosing all products with revenue at least π. 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. [42]. 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.(

Assortment optimization under the Sequential Multinomial Logit Model

“…As a result, adding extreme options sometimes leads to a positive effect for compromise products, whose probabilities of being selected increase in relative terms compared to other products. This phenomenon violates again the IIA axiom of Luce's model and the regularity axiom satisfied by all random utility models (Berbeglia and Joret, 2016). Again, the compromise effect can be captured with the PALM.…”

“…Recently, Berbeglia and Joret (2016) studied how well revenue-ordered assortments approximate the optimal revenue for a large class of choice models, namely all choice models that satisfy regularity. They provide three types of revenue guarantees that are exactly tight even for the special case of the RUM family.…”

Assortment Optimization Under a Single Transition Model

“…Revenue-ordered assortment has been shown to be optimal in many assortment optimization problems or has a worst-case performance guarantee (see, e.g., Talluri and van Ryzin 2004, Rusmevichientong and Topaloglu 2012, Wang and Wang 2016. Indeed, as shown in Berbeglia and Joret (2015), the optimal revenueordered assortment achieves a worst-case ratio of the optimal revenue under a class of "regular discrete choice models". In this paper, we show that the revenue-ordered assortment is optimal under a special case of the MCST model, and has a worst-case performance guarantee under the general case.…”

The Generalized Stochastic Preference Choice Model

“…Unfortunately, the worst-case performance guarantees of the revenue ordered assortments strategy deteriorates considerably under the GSP class. 3 The bound no longer holds even if the factor is multiplied by (1+ǫ) for any ǫ > 0. Restricted to the RUM class, Aouad et al [1] obtained, independently, essentially the same revenue guarantees (up to a constant factor).…”

“…Unlike the previous examples, this choice model satisfies the regularity condition. However it is not difficult to prove that it is not a RUM (see McFadden and Richter [20], Berbeglia and Joret [3]).…”