I study the challenges of integrating many matching markets into a centralized clearinghouse. Examples include merging the admissions processes of private and public schools or merging local kidney exchange programs into one large multi-hospital kidney exchange platform.
I find that stable, or even Pareto optimal matching mechanisms must always hurt some agents after integration occurs. I call this property integration monotonicity. I show that integration monotonicity is at odds with standard efficiency and fairness criteria in economics and distributive justice.
However, I show that the integration of matching markets under stable matching mechanisms never hurts more than one-half of the society, and thus can be implemented via majority voting. Furthermore, using a large market approach, I show that in markets with many agents, stable matching mechanisms are asymptotically integration monotonic. The large market approach allows me to compute the gains of social integration, which I show are almost always positive.