Assortment Optimization and Pricing Under a Nonparametric Tree Choice Model

We consider assortment and pricing problems when customers purchase products according to a nonparametric choice model. Each customer arrives with a preference list and will purchase the highest-ranking offered product in her preference list. We assume the set of customer classes is derived from paths in a tree, in which the order of nodes visited along each path gives the corresponding preference list. First, we study assortment problems, in which the goal is to find which products to offer to maximize expected revenue. We give a dynamic programming solution, which can be extended to versions of the assortment problem in which there are fixed costs for offering a product, shelf constraints, or substitution costs. Second, we study the joint assortment and pricing problem, in which the goal is to simultaneously select the set of offered products as well as their prices. We solve the pricing problem optimally when customers have some universal ranking of the products, and hence the tree takes the form of a single path. We also solve the problem optimally on the general tree when the prices are restricted to be quality consistent; higher-quality products must be priced above lower-quality products. Finally, we present computational experiments on both synthetic data and real hotel purchase data. Our estimation procedure shows both how to build the tree of products and how to estimate the underlying arrival probabilities of each customer type from historical sales data. These experiments show that the tree choice model captures customer purchasing behavior more accurately than the multinomial logit choice model in the majority of test cases.