Journal article: We analyze a three-sided matching market where institutions own objects and individuals belong to institutions. Institutions pool their objects to enlarge the choice set of individuals. For any institution, the number of individuals who receive an object must be equal to the number of objects initially owned. Under this distributional constraint, individually rational and fair assignments may fail to exist. However, when the number of individuals is sufficiently large, fair assignments exist and can be found using a new algorithm, called the Nested Deferred Acceptance algorithm with interrupters (NDAI). This procedure nests a one-to-one matching between agents and objects and a one-to-many matching between objects and institutions. We show that it outputs a matching which is Pareto optimal among fair matchings and strategy-proof for individuals. When agents belong to several institutions, the NDAI results in assignments which are fair for agents of the same institution.