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 good. By contrast, there is no crisp answer for multiple goods, and a major focus of Bayesian mechanism design since then has been to understand the power of randomized lotteries over deterministic pricing in different settings. In this paper, we ask the same question for assortment optimization, where goods have exogenously-fixed prices, and the decision is a set of substitute goods to offer. To formalize this question, we introduce a Bayesian mechanism design problem with fixed prices and ordinal preferences, in which assortments correspond to deterministic mechanisms. Meanwhile, randomized mechanisms correspond to lotteries whose payment is restricted to equal the fixed price of the randomly-allocated good. This models the ``fixed-price lotteries'' trending in designer fashion, a significant departure from traditional lotteries based on price discounts. We first show that for general ordinal distributions, lotteries under this restriction can still earn greater 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 commonly-used distributions in the assortment optimization literature, including Multi-Nomial Logit, Markov Chain, Tversky's Elimination by Aspects, and mixtures with Independent Demand Models. The takeaway is that unless a firm has a sophisticated model for consumer choice, fixed-price lotteries are no better than assortments.