Stable Matching with Proportionality Constraints

The problem of finding stable matches that meet distributional concerns is usually formulated by imposing various side constraints. Prior work has focused on 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. We violate the proportions by an amount proportional to the reciprocal of the number of students assigned to the school. For example, if a school is assigned 100 students, then the actual proportion will differ from the desired proportion by at most 2%. Our technique requires an extension of Scarf's lemma, which is of independent interest.