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.

💡 Research Summary

**
The paper investigates the exact market sizes for which the three classic axioms—Ex‑Post Efficiency, Equal Treatment of Equals (ETE), and Strategy‑Proofness—uniquely characterize the Random Serial Dictatorship (RSD) mechanism in the assignment of indivisible objects to agents with strict preferences. While Bogomolnaia and Moulin (2001) showed that these axioms single out RSD for the balanced case of three agents and three objects, and Basteck and Ehlers (2025) proved non‑uniqueness for all markets with at least five agents and five objects, the full picture for intermediate sizes remained open.

The authors first reformulate the axioms in weaker but equivalent forms: “support efficiency” (an agent can receive an object with positive probability only if that object appears in some Pareto‑efficient deterministic assignment) and the “full‑assignment property” (if an agent always receives a particular object in every serial‑dictatorship outcome, the mechanism must assign that object with probability one). Strategy‑proofness is decomposed into “upper variance” and “lower variance,” two conditions that together with swap monotonicity are equivalent to the original SP requirement.

Using these formulations, the paper proves uniqueness in two regimes:

1. Any number of objects with at most three agents (n ≤ 3, arbitrary m).
   By analyzing adjacent swaps in agents’ rankings, the authors show that the three axioms propagate constraints across the entire preference space, forcing a single assignment matrix for every profile. The argument relies only on the weakened versions of the axioms.

2. The balanced case with four agents and four objects (n = m = 4).
   A computer‑assisted exhaustive case analysis (detailed in Appendix B) confirms that no alternative mechanism can satisfy the three axioms in this specific market.

For the case of two objects (m = 2), the authors give a complete parameterization: every mechanism satisfying the axioms can be described by a single function that assigns, for each subset of agents, the probability that members of the subset receive their top object when they rank the two objects identically and the rest rank them oppositely. This representation makes it easy to identify when RSD is the unique mechanism (which occurs only for certain parameter values) and when a continuum of alternatives exists.

In all remaining market sizes—i.e., n ≥ 4 with m ≠ n, and any market with n, m ≥ 5—the three axioms are insufficient. The authors construct explicit alternative mechanisms as follows: they start from a deterministic rule whose symmetrization yields RSD, then modify the underlying deterministic rule on a carefully chosen family of preference profiles while preserving support efficiency and strategy‑proofness. Symmetrizing the modified rule produces a new randomized mechanism that still satisfies Ex‑Post Efficiency, ETE, and SP, yet differs from RSD. This construction generalizes the method of Basteck and Ehlers and works for every market outside the uniqueness domain.

The paper also examines whether strengthening the axiom system can restore uniqueness. Adding the bounded variance axiom (which limits the magnitude of probability changes caused by adjacent swaps) eliminates alternatives only in very small markets; for n ≥ 5 and m ≥ 5, non‑unique mechanisms still exist. Further augmentations with non‑bossiness and cross‑monotonicity (introduced in Appendix A) also fail to recover uniqueness in sufficiently large markets. Thus, even fairly strong fairness and incentive‑compatibility requirements do not suffice to single out RSD beyond the identified regimes.

Overall, the contribution is a complete classification of (n, m) pairs into “uniqueness” and “non‑uniqueness” zones for the three‑axiom system, together with constructive proofs for both sides. The results clarify why the balanced case n = m = 4 was historically difficult: uniqueness holds only at that precise point for n = 4, whereas any deviation (more objects or fewer objects) immediately admits alternative mechanisms. Practically, the findings inform designers of school‑choice, housing, or other allocation platforms when RSD can be justified solely by the three classic axioms and when additional criteria must be explicitly imposed.