Assortment Optimization under the Sequential Multinomial Logit Model

We study the assortment optimization problem under the Sequential Multinomial Logit (SML), a discrete choice model that generalizes the multinomial logit (MNL). Under the SML model, products are partitioned into two levels, to capture differences in attractiveness, brand awareness and, or visibility of the products in the market. When a consumer is presented with an assortment of products, she first considers products in the first level and, if none of them is purchased, products in the second level are considered. This model is a special case of the Perception-Adjusted Luce Model (PALM) recently proposed by Echenique et al (2018). It can explain many behavioural phenomena such as the attraction, compromise, similarity effects and choice overload which cannot be explained by the MNL model or any discrete choice model based on random utility. In particular, the SML model allows violations to regularity which states that the probability of choosing a product cannot increase if the offer set is enlarged. This paper shows that the seminal concept of revenue-ordered assortment sets, which contain an optimal assortment under the MNL model, can be generalized to the SML model. More precisely, the paper proves that all optimal assortments under the SML are revenue-ordered by level, a natural generalization of revenue-ordered assortments that contains, at most, a quadratic number of assortments. As a corollary, assortment optimization under the SML is polynomial-time solvable. This result is particularly interesting given that the SML model does not satisfy the regularity condition and, therefore, it can explain choice behaviours that cannot be explained by any choice model based on random utility.