When is Assortment Optimization Optimal?

Assortment optimization describes a retailer's general problem of deciding which variants in a product category to offer. In a typical formulation, there is a universe of substitute products whose prices have been pre-determined, and a model for how customers choose between these products. The goal is to find a subset to offer that maximizes aggregate revenue. In this paper we ask whether offering an assortment is actually optimal, given the recent emergence of more sophisticated selling practices, such as offering certain products 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. The seller has a Bayesian prior on the buyer's ranking of the items along with an outside option. Under our formulation, revenue maximization over deterministic mechanisms is equivalent to assortment optimization, while randomized mechanisms allow for lotteries that sell fixed-price items. We derive a sufficient condition, based purely on the buyer's ranking distribution, that guarantees assortments to be optimal within this larger class of randomized mechanisms. Our sufficient condition captures many preference distributions commonly studied in the assortment optimization literature -- Multi-Nomial Logit (MNL), Markov Chain, Tversky's Elimination by Aspects model, 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 in comparison to lotteries. Finally, from these results emerge two findings 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.