  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.
