Parameter identification in Markov chain choice models

In assortment planning, the seller's goal is to select a subset of products (called an assortment) to offer to a customer so as to maximize the expected revenue. This task can be formulated as an optimization problem given the revenue generated from selling each product, along with a probabilistic model of the customer's preferences for the products. Such a discrete choice model must capture the customer's substitution behavior when, for instance, the offered assortment does not contain the customer's most preferred product. Our focus in this paper is the Markov chain choice model (MCCM) proposed by Blanchet et al. (2016). In this model, the product selected by the customer is determined by a Markov chain over products where the products in the offered assortment are absorbing states. The current state represents the desired product; if that product is not offered, the customer transitions to another product according to the Markov chain probabilities, and the process continues until the desired product is offered or the customer leaves. MCCM generalizes widely-used discrete choice models such as the multinomial logit model (Luce, 1959; Plackett, 1975), as well as other generalized attraction models (Gallego et al., 2014); it also well-approximates other random utility models found in the literature such as mixed multinomial logit models (McFadden and Train, 2000). At the same time, the MCCM permits computationally efficient unconstrained assortment optimization as well as efficient approximation algorithms in the constrained case (Blanchet et al., 2016; Désir et al., 2015); this stands in contrast to some richer models such as mixed multinomial logit models (Rusmevichientong et al., 2010) and the nested logit model (Davis et al., 2014) for which assortment optimization is generally intractable. This combination of expressiveness and computational tractability makes MCCM very attractive for use in assortment planning.