The National Resident Matching Program is a centralized clearinghouse through which new medical graduates in the United States obtain their first positions. The history of this market, from the market failures that the centralized system was designed to address, to the present, is discussed, and a hypothesis about the behavior of such markets is presented. New evidence is then presented from a set of similar centralized markets in the United Kingdom. Because some of these latter markets have failed, while others have succeeded, they provide a natural experiment that permits the hypothesis to be tested. The new evidence also suggests directions in which modifications of existing procedures might be considered.