A data broker sells market segmentations to a producer with private cost who sells a product to a unit mass of consumers. This paper characterizes the revenue-maximizing mechanisms for the data broker. Every optimal mechanism induces quasi-perfect price discrimination. All the consumers with values above a cost-dependent cutoff buy by paying their values while the rest of consumers do not buy. The characterization implies that market outcomes remain unchanged even if the data broker becomes more powerful—either by gaining the ability to sell access to consumers or by becoming a retailer who purchases the product and sells to the consumers exclusively.

A signal is privacy‐preserving with respect to a collection of privacy sets if the posterior probability assigned to every privacy set remains unchanged conditional on any signal realization. We characterize the privacy‐preserving signals for arbitrary state space and arbitrary privacy sets. A signal is privacy‐preserving if and only if it is a garbling of a reordered quantile signal . Furthermore, distributions of posterior means induced by privacy‐preserving signals are exactly mean‐preserving contractions of that induced by the quantile signal . We discuss the economic implications of our characterization for statistical discrimination, the revelation of sensitive information in auctions and price discrimination.

A monotone function interval is the set of monotone functions that lie pointwise between two fixed, -monotone functions. We characterize the set of extreme points of monotone function intervals and apply this to a number of economic settings. First, we leverage the main result to characterize the set of distributions of posterior quantiles that can be induced by a signal, with applications to political economy, Bayesian persuasion, and the psychology of judgment. Second, we combine our characterization with properties of convex optimization problems to unify and generalize seminal results in the literature on security design under adverse selection and moral hazard.