Making Serial Dictatorships Fair

In priority-based matching, serial dictatorship (SD) is simple, strategyproof, and Pareto efficient, but not free of justified envy (i.e. fair). This paper studies how to fairly order agents in SD as a function of their priorities. I show that if preferences are identical across agents and uniformly distributed, and objects have unit capacities, the serial order that minimizes the expected number of justified envy cases is the Kemeny ranking of agents’ priorities. If any of these assumptions – identical preferences, uniformly distributed preferences, or unit capacities – is relaxed, the optimal SD follows a weighted Kemeny ranking. Broadly, these results demonstrate how insights from social choice theory can inform the design of practical matching mechanisms.

💡 Research Summary

The paper tackles the fairness shortfall of Serial Dictatorship (SD), a widely used mechanism in priority‑based matching markets such as school choice, public housing, and organ allocation. While SD is strategy‑proof and Pareto‑efficient, it often generates “justified envy”: a higher‑priority agent prefers another agent’s assignment and also has a higher priority for that object. The author asks how to order agents in SD, using only the ex‑ante known priority rankings (which are legally fixed), so that the expected number of justified‑envy cases is minimized.

The core insight is to view the problem as a rank‑aggregation task. Each object supplies a strict priority ordering over agents; the planner must combine these multiple orderings into a single serial order. Under three restrictive assumptions—identical preferences across agents, uniformly random preference draws, and unit capacities for objects—the expected number of justified‑envy incidents is minimized by the Kemeny ranking of the agents’ priority profiles. The Kemeny rule selects the permutation that minimizes the sum of pairwise disagreements across all object‑specific rankings, which aligns exactly with minimizing expected envy when each object is equally likely to appear at any position in agents’ preference lists.

The paper then relaxes each assumption in turn and shows that the optimal serial order becomes a weighted Kemeny ranking. If objects are not equally likely to be top‑ranked in preferences, the disagreement contribution of an object’s ranking is weighted by the probability that the object appears at a given position in the preference list (Proposition 1). When agents have heterogeneous preferences, the probability of envy grows with the distance between agents’ positions in the serial order; consequently, disagreements involving agents later in the order receive higher weight (Proposition 2). Finally, when objects have capacities larger than one, the chance that a later dictator envies an earlier one depends on whether the object’s capacity has already been filled. The optimal order then uses object‑ and position‑specific weights that capture both the likelihood of being matched to a particular object and the probability that the object’s capacity is exhausted at each step (Proposition 3).

Methodologically, the author expresses the expected number of justified‑envy cases as a linear combination of pairwise priority disagreements, with coefficients derived from the preference distribution and capacity structure. This formulation maps directly onto the Kemeny objective, establishing a formal equivalence. Because computing an exact Kemeny ranking is NP‑hard, the paper notes that existing approximation algorithms and heuristics from social‑choice literature can be employed in practice, making the approach tractable for real‑world applications.

The contribution is twofold. Theoretically, it connects a classic social‑choice aggregation rule to a central fairness metric in matching markets, extending the relevance of Kemeny’s rule beyond voting. Empirically, it offers a concrete, implementable prescription for policymakers: replace arbitrary or random serial orders with (weighted) Kemeny orders derived solely from legally mandated priority lists, thereby reducing justified envy without sacrificing strategy‑proofness or efficiency. This bridges a gap between the literature on efficient matching mechanisms and the growing body of work seeking fairness enhancements, and it provides a clear, quantitative tool for improving equity in a variety of allocation settings.