On the Efficiency and Fairness of Deferred Acceptance With Single Tie-Breaking

Abstract

As a random allocation rule for indivisible object allocation problems under weak priorities, deferred acceptance with single tie-breaking (DA-STB) is not ex-post constrained efficient. We first observe that it also fails to satisfy a natural fairness notion, symmetry at the top, which requires that two agents be assigned their common top choice with equal probability if they have equal claim to it. Then it is shown that DA-STB is ex-post constrained efficient, if and only if it is symmetric at the top, if and only if the priority structure satisfies a certain acyclic condition. We further characterize the priority structures under which DA-STB is ex-post fully efficient. Based on the characterized priority domains, and using a mild fairness notion called local envy-freeness, some new theoretical support is provided for this widely used rule in practice. Among strategy-proof, symmetric and locally envy-free rules, there exists an ex-post constrained efficient (ex-post stable and efficient) one if and only if DA-STB is ex-post constrained efficient (ex-post stable and efficient). Among strategy-proof, ex-post stable and locally envy-free rules, there exists one that is symmetric at the top if and only if DA-STB is symmetric at the top.