Top Trading Cycles, Consistency, and Acyclic Priorities for House Allocation with Existing Tenants

We study the house allocation with existing tenants model (introduced by Abdulkadiroglu and Sonmez, 1999) and consider rules that allocate houses based on priorities. We introduce a new acyclicity requirement for the underlying priority structure which is based on the acyclicity conditions by Ergin (2002) and Kesten (2006) for house allocation with quotas and without existing tenants. We show that for house allocation with existing tenants a top trading cycles rules is consistent if and only if its underlying priority structure satisfies our acyclicity condition. Moreover, even if no priority structure is a priori given, we show that a rule is a top trading cycles rule based on ownership adapted acyclic priorities if and only if it satisfies Pareto-optimality, individual-rationality, strategy-proofness, reallocation-proofness, and consistency.