A Markov Chain Approximation to Choice Modeling

Assortment optimization is an important problem that arises in many industries such as retailing and airlines where the decision maker needs to decide on an optimal subset of products to offer to maximize the expected revenue. The demand and the revenue of any product depends on the complete set of offered products since customers potentially substitute to an available product if their most preferred product is not available. Such a substitution behavior is captured by a customer choice model that can be thought of as a distribution over preference lists (or permutations of products). A customer with a particular preference list purchases the most preferable product that is available. Therefore, the choice model specifies the probability that a customer selects a particular product for every offer set. One of the key challenges of any assortment planning problem is to find the “right choice model” to describe the substitution behavior when we only observe historical sales data for a small number of assortments. The underlying customer preferences are latent and unobservable. A choice model, in the most general setting, can be thought of as a distribution over permutations that arise from preferences. In the random utility model of preferences, each customer has a utility that depends on the attributes of the product and a random idiosyncratic component, i.i.d according to some distribution. The preference list of the customer is given by the decreasing order of utilities of products. This model was introduced by Thurstone [7] in the early 1900s. An important special case of the above model is obtained assuming idiosyncratic components are i.i.d according to an extreme value distribution such as Gumbel. This model also referred to as the Plackett-Luce model and was proposed independently by Luce [2] and Plackett [4] and later McFadden [3] referred to it as a Multinomial logit model. The MNL model is by far the most popular model as both the estimation as well as the optimization problems are tractable for this model (see [6]). However, the MNL model is not able to capture heterogeneity in substitution behavior and also suffers from the Independence from Irrelevant Alternatives (IIA) property which limit the applicability of the MNL model. More complex choice models such as the Nested logit model and the mixture of MNL models have been studied in the literature. However, both the estimation and the resulting optimization problem become difficult when we use a richer class of parametric models. Rusmevichientong et al. [5] show that the assortment optimization problem is NP-hard for a mixture of MNL model even for the case of mixture of only two MNL models. The work by Farias et al. [1] and van Ryzin and Vulcano [8] are most closely related to our work. Farias et al. [1] consider a non-parametric approach where they use the distribution over permutations with the sparsest support that is consistent with the data. However, the resulting assortment optimization problem can not be solved efficiently for the sparsest support distribution. In van Ryzin and Vulcano [8], the authors consider an iterative expectation maximization algorithm to learn a non-parametric choice model