When is Assortment Optimization Optimal?

A classical question in economics is whether complex, randomized selling protocols can improve a firm's revenue beyond that of simple, take-it-or-leave-it posted prices. In 1981, Myerson answered this question with an emphatic "No" for a monopolist selling a single product. By contrast, there is no crisp answer for multiple products, and a major focus of Bayesian mechanism design has been understanding the increase in revenue from pricing lotteries over products instead of deterministic products, under different classes of valuation functions and different families of prior customer distributions. In this paper, we ask the same question for assortment optimization, where products have exogenously-fixed prices, and the decision is on a set of substitute products to offer. To formalize such a question, we introduce a Bayesian mechanism design problem where customers have ordinal preferences for products with fixed prices, in which assortments correspond to deterministic mechanisms. Meanwhile, randomized mechanisms correspond to lotteries whose payment is restricted equal the fixed price of the randomly-allocated product, a significant departure from the lotteries currently studied in Bayesian mechanism design. We first show that for general ordinal preference distributions, lotteries under this restriction can still earn higher revenue than any deterministic assortment. We then derive a natural sufficient condition on the distribution which ensures the optimality of assortments. Importantly, this sufficient condition captures common distributions used in the assortment optimization literature, including Multi-Nomial Logit (MNL), Markov Chain choice models, Tversky's Elimination by Aspects model, and mixtures of any one of these with Independent Demand Models. The takeaway is that unless a firm has a very sophisticated model for consumer choice, assortments are optimal.