  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.
