Priority-Augmented House Allocation

We study the indivisible object allocation problem without monetary transfer, in which each object is endowed with a weak priority ordering over agents. It is well known that stability is generally not compatible with efficiency in this problem. We characterize the priority structures for which a stable and efficient assignment always exists, as well as the priority structures that admit a stable, efficient and (group) strategy-proof rule. While house allocation problems and housing markets are two classic families of allocation problems that admit a stable, efficient and group strategy-proof rule, any priority-augmented allocation problem with more than three objects admits such a rule if and only if it is decomposable into a sequence of subproblems, each of which has the structure of a house allocation problem or a housing market. One corollary of this result is that there exists a stable hierarchical exchange rule (Pápai, 2000) if and only if there exists a stable, efficient and group strategy-proof rule