Characterizing priorities for deferred acceptance with or without outside options

In a model of priority-based allocation of indivisible objects where there may not be outside options, we characterize the priority structures under which the deferred acceptance algorithm (DA) satisfies various desiderata. We first identify an acyclicity condition that is necessary and sufficient for DA to be group strategy-proof, robustly stable, weakly group robustly stable, or to implement the stable allocation correspondence in Nash equilibria. When there is no outside option and there are more agents than total resources, the condition becomes considerably weaker, and no longer requires the priorities between any pair of objects to be similar. We further find a condition on priorities that is necessary and sufficient for the efficiency or consistency of DA, which is in general stronger than the above incentive properties for this mechanism.