- 2017 Elsevier Inc. In the assignment problem where agents can stay unassigned, the size of the assignment is an important consideration for designers. Bogomolnaia and Moulin (2015) show that there is a tension between size and fairness: the guaranteed size ratio of any envy-free mechanism is at most rm which converges decreasingly to 1 [Formula presented] 63.2% as the maximum size increases. They then ask whether rm is also the guaranteed size ratio for any ordinally efficient and envy-free mechanism. We study this issue and show that the lower bound of the guaranteed size ratio of ordinally efficient and envy-free mechanisms converges to [Formula presented] as the maximum size increases, which means that almost half of the maximum size is wasted at the lower bound. Moreover, the exact lower bound is [Formula presented] when the maximum size m is odd.