Many-to-many matching and price discrimination

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 are assortative in the weak-order (set inclusion) sense. We then deliver testable predictions relating the optimal matching rules and price schedules to the distribution of the agents' characteristics. The analysis has implications for the design of large matching intermediaries, such as advertising exchanges, business-to-business platforms, and online job-matching agencies.