An analyst observes the frequency with which a decision maker (henceforth, DM) takes actions, but not the frequency with which she takes actions conditional on a payoff relevant state. In this setting, we characterize the information structures that rationalize the marginal distribution over actions as the result of the DM first observing the information structure's realization and then choosing actions optimally. Our main result identifies a core condition that must be satisfied between the marginal action distribution and the distribution over the DM's posterior beliefs for the information structure to rationalize the DM's choices.

The trade-off between product personalization and price discrimination is at the center of the debate about the use of consumer data. Whereas firms use consumer data for product personalization, allowing the firm to better meet consumers' needs, consumer data in the form of purchase histories is also used for personalized pricing. Policy makers note that rich product lines may compensate for the costs of personalized pricing. Absent from this debate is that consumer data in the form of purchase histories is endogenously determined. On the one hand, the data is selected as it reflects the consumer's trade-off between a better product match and the costs of price discrimination. On the other hand, the firm designs which products the consumer can choose from, thereby controlling how informative purchase histories may be about the consumer's preferences.