Top Trading Cycles in Large Markets: The Asymptotic Irrelevance of Priorities

Abstract: Top Trading Cycles (TTC) is Pareto efficient and strategy-proof and explicitly uses agents' priorities. Although TTC favors higher-priority agents in each round, we show that this priority advantage vanishes as the market grows large under a canonical random model of preferences and priorities. In the limit, TTC produces assignments with virtually the same incidence of justified envy as Random Serial Dictatorship (RSD) -- a mechanism entirely blind to priorities. This stark asymptotic equivalence implies that TTC effectively fails to satisfy standard fairness criteria in large markets, casting significant doubt on its practical appeal for balancing efficiency and fairness.

• The paper shows that as market size increases, the priority advantage in TTC converges to zero, yielding fairness outcomes comparable to RSD.
• It employs a novel Markov chain analysis to track cycle dynamics, distinguishing between the roles of short and long cycles in allocations.
• Simulation results confirm that justified envy and blocking pair metrics under TTC align with those of RSD in large, randomized markets.

Asymptotic Irrelevance of Priorities in Top Trading Cycles

Introduction and Context

The paper "Top Trading Cycles in Large Markets: The Asymptotic Irrelevance of Priorities" (2607.10819) addresses the interaction between efficiency and fairness in the assignment of indivisible resources. Specifically, it focuses on the Top Trading Cycles (TTC) mechanism, a canonical algorithm for one-to-one prioritized allocation problems such as public school assignments. TTC is both Pareto efficient and strategy-proof, and uses exogenous object priorities, in contrast to Random Serial Dictatorship (RSD), which is indifferent to priorities.

While TTC’s design directly embeds priorities—ostensibly favoring higher-priority agents and minimizing justified envy—the paper provides a rigorous quantitative analysis showing that, in large random markets, this priority advantage vanishes. In the asymptotic regime, TTC produces allocations whose fairness properties (in terms of justified envy) are virtually indistinguishable from RSD, contradicting the standard intuition about TTC's fairness advantage.

Mechanisms, Fairness, and Priorities

Both TTC and RSD achieve Pareto efficiency and strategy-proofness, but they differ fundamentally in how they treat priorities. TTC, at every intermediate allocation round, ensures that each object points to its highest-priority remaining agent, thus embedding priority into the assignment process. RSD, by construction, ignores priorities entirely, relying on a random serial order of agents. The conventional wisdom is that TTC should substantially outperform RSD in respecting institutional priorities, especially by minimizing justified envy.

The paper challenges and ultimately overturns this intuition. It defines justified envy as the situation where an agent prefers the assignment of another agent and holds a higher priority for the corresponding object. A critical point in the paper is distinguishing between short cycles (cycles of length 2, in which an agent points to an object and the object points back at her) and long cycles (cycles of greater length). Only the assignments via short cycles guarantee strict priority compliance; long cycles allow agents to bypass local priorities as they trade rights along the cycle.

Markov Characterization of TTC and Large-Market Analysis

A central technical contribution is the characterization of TTC’s round-by-round clearing process as a Markov chain. At each round with $n$ agents and $o$ objects, the number of agents assigned $m$ has probability:

$$p_{n,o;m} = \frac{m}{(o n)^{m+1}} \cdot \frac{n!}{(n-m)!} \cdot \frac{o!}{(o-m)!} (o+n-m)$$

This Markov property is nontrivial, since the set of agents and objects remaining after each round is highly nonlinear in past history. Nevertheless, at the aggregate level, this result enables precise tracking of how the pool of remaining agents and objects shrinks over time.

The analysis shows that in a balanced market with $n$ agents and $n$ objects, the expected number of agents assigned in the first round is $O(\sqrt{n})$, and the number of rounds before TTC ceases is $o(n)$. Critically, the expected number of short-cycle assignments per round is always at most 2, regardless of the market size.

Figure 1: Markov dynamics of TTC in a balanced market—expected agents cleared per round and sublinear market collapse rate.

From these facts, it follows that as $n \rightarrow \infty$, the fraction of agents assigned via short cycles converges to zero. The vast majority of assignments are thus finalized through long cycles, which, by the paper's symmetry arguments, randomize realized priority ranks nearly as if agents were assigned uniformly at random.

Asymptotic Equivalence of TTC and RSD

Building on the Markov analysis, the paper formally proves that the empirical distribution of realized priority ranks under TTC converges to the same uniform distribution as under RSD. More precisely, the proportion of agents with justified envy converges to $1/2$ (the same limit as under RSD), and the expected fraction of blocking pairs vanishes at the same rate ($o$0) under both mechanisms:

• For every $o$1, under RSD, the justified envy incidence ratio is exactly $o$2; under TTC this ratio converges to $o$3 as $o$4 grows.
• The expected fraction of agents experiencing any justified envy converges to $o$5 for both mechanisms.
• The joint distribution of preference and realized priority ranks under TTC converges to the product of a uniform random assignment over both agents and objects, matching that of RSD.

Figure 2: Convergence of justified envy and expected priority ranks between TTC and RSD as market size increases.

Numerical and Simulation Results

Simulations corroborate the theoretical findings. For small market sizes, TTC’s use of priorities reduces justified envy relative to RSD. However, as $o$6 increases, the differences become negligible:

Figure 3: Incidence of justified envy among all envy instances decreases rapidly, converging between TTC and RSD as $o$7 increases.

These simulation patterns persist under moderate levels of correlation in agent preferences or priority structures. The irrelevance of priorities under TTC breaks only in degenerate cases (e.g., perfectly correlated preferences, large numbers of identical objects per type, or coarse priority tiers with substantial mass).

Implications and Extensions

The results demonstrate a stark limitation of TTC for fairness in large, random assignment markets. Although TTC is justified-envy minimal among efficient, strategy-proof mechanisms, its procedural respect for priorities does not translate into meaningful empirical differences with RSD at scale, except in markets with strong structural correlations. The practical implication is that, in large assignment markets with i.i.d. preferences and priorities:

• Policymakers cannot expect TTC to offer a significant advantage over RSD in terms of realized priority respect or justified envy metrics.
• The efficiency/fairness trade-off becomes sharply constrained; strict Pareto efficiency and strategy-proofness together force the observed fairness deficit.

Importantly, the Markov characterization of TTC developed in the paper opens avenues for further analytical tractability. This aggregate dynamic is likely to be useful in the study of convergence rates, robustness to correlated preferences, and comparative statics for other allocation mechanisms.

Conclusion

The paper rigorously establishes that, in large random markets, the TTC mechanism’s explicit treatment of priorities is asymptotically irrelevant. As the population grows, the observed distribution of justified envy and priority ranks under TTC converges to those of RSD—a mechanism that ignores priorities completely. This result holds under generic i.i.d. preferences and priorities, and is robust to moderate correlation. Only with significant structure—such as high-priority tiers or small numbers of large objects—does TTC’s theoretical fairness edge persist. Consequently, the pursuit of strict Pareto efficiency and strategy-proofness in large assignment markets effectively neutralizes the practical role of priorities in TTC, making the case for alternative approaches that relax or modify one of these desiderata if institutional fairness is a primary concern.

Figure 4: Empirical difference in justified envy measures vanishes; TTC and RSD become empirically indistinguishable in large markets.