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 ψ size‐wise dominates another mechanism ϕ if the latter never assigns more agents than the former, and at some problem, ψ assigns more agents than ϕ. 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.