Matching with floor constraints

Floor constraints are a prominent feature of many matching markets, such as medical residency, teacher assignment, and military cadet matching. We develop a theory of matching markets under floor constraints. We introduce a stability notion, which we call floor respecting stability, for markets in which (hard) floor constraints must be respected. A matching is floor respecting stable if there is no coalition of doctors and hospitals that can propose an alternative matching that is feasible and an improvement for its members. Our stability notion imposes the additional condition that a coalition cannot reassign a doctor outside the coalition to another hospital (although she can be fired). This condition is necessary to guarantee the existence of stable matchings. We provide a mechanism that is strategy-proof for doctors and implements a floor respecting stable matching.