We study centralized many-to-many matching in markets where agents have private information about (vertical) characteristics that determine match values. Our analysis reveals how matching patterns reflect cross-subsidization between sides. Agents are endogenously partitioned into consumers and inputs. At the optimum, the costs of procuring agents-inputs are compensated by the gains from agents-consumers. We show how such cross-subsidization can be achieved through matching rules that have a simple threshold structure, and deliver testable predictions relating the optimal price schedules to the distribution of the agents' characteristics. The analysis sheds light on the practice of large matching intermediaries, such as media and business-to-business platforms, advertising exchanges, and commercial lobbying firms.
Vertical matching markets many-to-many matching asymmetric information mechanism design cross-subsidization