Shapley-Scarf Markets with Objective Indifferences

Top trading cycles with fixed tie-breaking (TTC) has been suggested to deal with indifferences in object allocation problems. Unfortunately, under general indifferences, TTC is neither Pareto efficient nor group strategy-proof. Furthermore, it may not select an allocation in the core of the market, even when the core is non-empty. However, when indifferences are agreed upon by all agents (``objective indifferences’’), TTC maintains Pareto efficiency, group strategy-proofness, and core selection. Further, we characterize objective indifferences as the most general setting where TTC maintains these properties.

💡 Research Summary

The paper investigates the classic Shapley‑Scarf housing market—where each agent initially owns a single indivisible object—and asks what happens when agents are indifferent between some objects. In the most general setting, where each agent’s weak preferences may contain arbitrary indifferences, the widely used Top Trading Cycles (TTC) algorithm with a fixed ex‑ante tie‑breaking rule fails to retain three cornerstone properties of a good allocation mechanism: Pareto efficiency, group‑strategy‑proofness, and core‑selection. Prior work (Ehlers 2002, 2014) has shown that under general indifferences no mechanism can simultaneously achieve Pareto efficiency and group‑strategy‑proofness, and that TTC can even produce allocations outside the core.

The authors introduce a structured form of indifference they call “objective indifferences.” The set of houses H is partitioned into K blocks (or types) (H_1,\dots,H_K). All agents are required to be indifferent among houses that belong to the same block, while they must have strict preferences across different blocks. Formally, for any two houses (h,h’) in the same block, (h\sim_i h’) for every agent i; if they belong to different blocks, the agents rank them strictly. This domain is denoted (\mathcal{R}(H)).

The main contributions are twofold. First, the authors prove that on the domain (\mathcal{R}(H)) the standard TTC algorithm—run after breaking ties arbitrarily but consistently—satisfies all three desirable properties:

• Pareto efficiency: Because TTC always selects a cycle of agents pointing to their most‑preferred house, and indifferences never affect the ordering across blocks, the resulting allocation cannot be Pareto‑dominated.
• Group‑strategy‑proofness: Since every agent’s indifference pattern is common knowledge and identical, any coalition that misreports preferences cannot create a new cycle that makes every member at least as well‑off and some strictly better‑off; at least one member would be forced to accept a less‑preferred block.
• Core‑selection: Any blocking coalition would need to reallocate houses within a block to improve members, but such reallocations are impossible because all agents view houses inside a block as identical. Consequently, the TTC outcome lies in the (weak) core whenever the core is non‑empty.

Second, the paper establishes a maximality result: (\mathcal{R}(H)) is the largest preference domain on which any TTC variant (i.e., any fixed tie‑breaking rule) can retain the three properties. If the domain is enlarged—by allowing agents to have different indifference relations within a block, or by permitting indifferences that cross block boundaries—then at least one of Pareto efficiency, group‑strategy‑proofness, or core‑selection is lost. Thus, objective indifferences characterize the most general environment where TTC remains “well‑behaved.”

The authors motivate the model with several real‑world examples. In university dorm assignments, rooms of the same floor plan are essentially interchangeable for students; in public school choice, seats in a language‑immersion program that differ only by physical location are often viewed as identical; and in military occupational assignments, positions of the same rank and function are interchangeable. In each case, agents’ indifferences are naturally common across the population, fitting the objective indifferences framework.

A concrete policy illustration is provided using San Francisco’s school‑choice system. Some families are indifferent between bilingual and regular seats, while others have a strict preference for the bilingual track. When such heterogeneous indifferences coexist, TTC with a fixed tie‑breaker can be manipulated by coalitions and may produce allocations outside the core, which helps explain why the district is moving away from TTC toward mechanisms like Deferred Acceptance.

Methodologically, the paper formalizes the partition‑based preference domain, defines the TTC algorithm with an arbitrary but fixed tie‑breaking rule, and then proves the three properties using adaptations of classic TTC arguments. The Pareto‑efficiency proof follows the usual contradiction: any Pareto‑improving allocation would imply a cycle that TTC omitted, which cannot happen. The group‑strategy‑proofness proof leverages the fact that any deviation that benefits a coalition must create a new cycle involving a strictly better block for at least one member, contradicting the common indifference structure. The core‑selection proof shows that any blocking coalition would need to rearrange houses within a block, which yields no strict improvement because all agents are indifferent there.

Finally, the paper discusses the practical relevance of the maximality result. Since many public allocation problems naturally generate objective indifferences, policymakers can safely employ the simple TTC mechanism without fearing efficiency loss or strategic manipulation. Conversely, when indifferences are heterogeneous, designers should consider alternative mechanisms (e.g., variants of Deferred Acceptance or randomized serial dictatorship) that are robust to such complexity.

In sum, the article makes two key theoretical contributions: (1) it identifies the exact preference domain—objective indifferences—under which TTC with fixed tie‑breaking retains Pareto efficiency, group‑strategy‑proofness, and core‑selection; (2) it proves that this domain is maximal, meaning any broader domain inevitably breaks at least one of these properties. The work bridges a gap between abstract mechanism design theory and concrete allocation settings where identical copies of objects are common, offering clear guidance for mechanism designers in housing, school choice, and similar markets.