The Domain of RSD Characterization by Efficiency, Symmetry, and Strategy-Proofness

Given a set of $n$ individuals with strict preferences over $m$ indivisible objects, the Random Serial Dictatorship (RSD) mechanism is a method for allocating objects to individuals in a way that is efficient, fair, and incentive-compatible. A random order of individuals is first drawn, and each individual, following this order, selects their most preferred available object. The procedure continues until either all objects have been assigned or all individuals have received an object. RSD is widely recognized for its application in fair allocation problems involving indivisible goods, such as school placements and housing assignments. Despite its extensive use, a comprehensive axiomatic characterization has remained incomplete. For the balanced case $n=m=3$, Bogomolnaia and Moulin have shown that RSD is uniquely characterized by Ex-Post Efficiency, Equal Treatment of Equals, and Strategy-Proofness. The possibility of extending this characterization to larger markets had been a long-standing open question, which Basteck and Ehlers recently answered in the negative for all markets with $n,m\geq5$. This work completes the picture by identifying exactly for which pairs $\left(n,m\right)$ these three axioms uniquely characterize the RSD mechanism and for which pairs they admit multiple mechanisms. In the latter cases, we construct explicit alternatives satisfying the axioms and examine whether augmenting the set of axioms could rule out these alternatives.