Size versus fairness in the assignment problem

When not all objects are acceptable to all agents, maximizing the number of objects actually assigned is an important design concern. We compute the guaranteed size ratio of the Probabilistic Serial mechanism, i.e., the worst ratio of the actual expected size to the maximal feasible size. It converges decreasingly to 1−1e≃63.2% as the maximal size increases. It is the best ratio of any Envy-Free assignment mechanism.