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.

翻译：给定 $n$ 个个体对 $m$ 个不可分物品具有严格偏好，随机序列独裁（RSD）机制是一种将物品分配给个体的方法，该方法具有效率性、公平性与激励相容性。该机制首先随机抽取个体的顺序，随后每个个体依照此顺序选择其最偏好且仍可获得的物品。此过程持续进行，直至所有物品均被分配或所有个体均获得物品。RSD 因其在涉及不可分物品的公平分配问题（如学校录取与住房分配）中的应用而广受认可。尽管其应用广泛，但完整的公理化刻画一直未能完成。对于平衡情形 $n=m=3$，Bogomolnaia 与 Moulin 已证明 RSD 可由事后效率、平等对待与策略证明性唯一刻画。将这一刻画推广至更大规模市场是否可行，长期以来一直是一个悬而未决的问题，而 Basteck 与 Ehlers 最近对所有 $n,m\geq5$ 的市场给出了否定的答案。本文通过精确确定哪些数对 $\left(n,m\right)$ 下这三个公理能唯一刻画 RSD 机制，以及哪些数对下它们允许多种机制，从而完善了这一图景。对于后一种情况，我们构造了满足这些公理的显式替代机制，并探讨了通过扩充公理集是否能够排除这些替代方案。