Limits of Personalization: Prophet Inequalities for Revenue-Ordered Assortments

Abstract

Consider a clairvoyant firm that knows the products' valuations of each arriving consumer and offers them only the most profitable product they are willing to buy. How much more can such a firm make relative to a firm that offers all consumers the assortment that maximizes expected revenues? We show that for general discrete choice models, the ratio can be exponential in the number of products, but at most equal to the number of products for random utility models. We show that the ratio is at most 2 for the $\alpha$-shaken multinomial logit ($\alpha$-MNL) which includes the MNL and the general attraction model (GAM) as special cases. We also provide sufficient conditions for the ratio of at most 2 to hold for the latent class MNL, and in fact show that in the limit as the coefficient of variation of the utilities goes to infinity the bound is at most 1.5. For all of these cases the revenue-ordered heuristic yields the stated guarantees relevant to the clairvoyant firm.