The Limits of Personalization in Assortment Optimization

To study the limits of personalization, we introduce the notion of a clairvoyant firm that can read the mind of consumers and sell them the highest revenue product that they are willing to buy. We show how to compute the expected revenue of the clairvoyant firm for a class of rational discrete choice models, and develop prophet-type inequalities that provide performance guarantees for the expected revenue of the traditional assortment optimization firm (a TAOP firm) relative to the clairvoyant firm, and therefore to any effort to personalize assortments. In particular, we show that the expected revenue of the clairvoyant firm cannot exceed twice the expected revenue of the TAOP for the RCS model, the MNL, the GAM and the Nested Logit Model. On the other hand, there are random utility models for which personalized assortments can earn up to $n$ times more than a TAOP firm, where $n$ is the number of products. Our numerical studies indicate that when the mean utilities of the products are heterogeneous among consumer types, and the variance of the utilities is small, firms can gain substantial benefits from personalized assortments. We support these observations, and others, with theoretical findings. While the consumers surplus can potentially be larger under personalized assortments, clairvoyant firms with pricing power can extract all surplus, and earn arbitrarily more than traditional firms that optimize over prices but do not personalize them. For the price-aware MNL, however, a clairvoyant firm can earn at most $e$ times more than a traditional firm.