When is Assortment Optimization Optimal?

Given many different items in a product category, each with its own fixed price point, which subset should a retailer offer to its customers? Assortment optimization describes the process of finding a subset that maximizes average revenue, based on a model for how customers choose between items in that product category. In this paper we ask whether offering an assortment is actually optimal, given the emergence of more sophisticated selling practices, such as offering certain items only through lotteries. To formalize this question, we introduce a mechanism design problem where the items have fixed prices and the seller optimizes over (randomized) allocations, given a distribution for how a buyer ranks the items. Under our formulation, deterministic mechanisms correspond to assortments, while randomized mechanisms correspond to lotteries for selling items with fixed prices. We derive a sufficient condition, based purely on the buyer's preference distribution, that guarantees assortments to be optimal within the larger class of randomized mechanisms. Our sufficient condition captures many preference distributions commonly studied in assortment optimization, including Multi-Nomial Logit (MNL), Markov Chain, a mixture of MNL with an Independent Demand model, and simple cases of Nested Logit. When our condition does not hold, we also bound the suboptimality of assortments compared to lotteries. Finally, from our paper emerges two results of independent interest: an example showing that Nested Logit is not captured by Markov Chain choice models, and a tighter Linear Programming relaxation for assortment optimization.