Stable Matching with Proportionality Constraints

The problem of finding stable matches that meet distributional concerns is usually formulated by imposing side constraints whose "right-hand sides" are absolute numbers specified before the preferences or number of agents on the "proposing" side are known. In many cases, it is more natural to express the relevant constraints as proportions. We treat such constraints as soft but provide ex post guarantees on how well the constraints are satisfied while preserving stability. Our technique requires an extension of Scarf's lemma, which is of independent interest.