Strategy-proof size improvement: is it possible?*

The number of assigned agents (i.e., size) is an important parameter in object allocations. While size maximality clashes with individual rationality and strategy-proofness, it can still be possible to increase the size over a mechanism while keeping these properties. To pursue this research, we devise a size comparison criterion to investigate the possibility of size increase. A mechanism $\psi$ size-wise dominates another mechanism $\varphi$ if the latter never assigns more agents than the former, and at some problem, $\psi$ assigns more agents than $\varphi$. We obtain arguably mild conditions for a mechanism to not be dominated size-wise by an individually rational and strategy-proof mechanism. Moreover, whenever there are at least as many objects as the total number of agents, we find conditions, different from those previously mentioned, for a mechanism to not be dominated size-wise by an individually rational and group strategy-proof mechanism. These results have implications for deferred-acceptance, top trading cycles, efficiency-adjusted deferred-acceptance, serial dictatorship, and Boston mechanisms.