Consistent strategy-proof assignment by hierarchical exchange

We characterize the family of efficient, consistent, and strategy-proof rules in house allocation problems. These rules follow an endowment inheritance and trade procedure as in Pápai’s hierarchical exchange rules (Pápai in Econometrica 68, 1403–1433, 2000) and closely resemble Ergin’s priority rules (Ergin in Econometrica 70, 2489–2497, 2002). We prove that if there are at least four objects, these are the only rules that are efficient in two-agent problems, $2$ -consistent, and strategy-proof. A corollary is that these three basic properties together imply the full requirements of efficiency, consistency, group strategy-proofness, and reallocation-proofness.