Stability and Strategy-proofness for Matching with Constraints: A Necessary and Sufficient Condition
Distributional constraints are common features in many real matching markets, such as medical residency matching, school admissions, and teacher assignment. We develop a general theory of matching mechanisms under distributional constraints. We identify the necessary and sufficient condition on the constraint structure for the existence of a mechanism that is stable and strategy-proof for the individuals. Our proof exploits a novel connection between a matching problem under distributional constraints and a matching problem with contracts.