Revealed Preference Analysis of Shapley-Scarf Housing Markets
This paper establishes graph-theoretic conditions to determine whether observed reallocations of indivisible objects in Shapley-Scarf housing markets are rationalizable as Pareto efficient, individually rational, or weak core-stable, demonstrating that while checking weak core stability is NP-complete, these relaxed criteria effectively eliminate a broader set of market possibilities and are successfully applied to social housing data in England.
Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer
Imagine a city where every family is given a home by the government, but the rules of the neighborhood allow them to swap houses with one another if they both agree. This happens in many places, from social housing in the United Kingdom to job postings within government agencies. The central authority that assigns the initial homes does not know what the families actually want; it only sees who ends up where after the swaps are finished. The big question for the people in charge is simple but difficult to answer: did these swaps make things better for everyone, or did they just shuffle people around without fixing the underlying problems? Without being able to ask the families directly what they prefer, officials cannot simply look at the final list of who lives where and declare the system a success or a failure. They need a way to look at the outcome and ask, "Is there any possible set of desires that could explain this result as a fair and efficient one?"
This is the puzzle tackled by a team of researchers who studied how to judge these real-world exchanges using only the data of who started with what and who ended up with what. They focused on a specific type of market where items cannot be divided, like a house or a specific job, and where no money changes hands to balance the trade. In such a world, a swap is only considered good if everyone involved is happier than they were before, and if no group of people could have traded their original items among themselves to make everyone in that group strictly better off. The researchers developed a method to test whether the observed swaps fit these standards, even though they never saw the preferences that drove the decisions.
The team applied their method to real data from social housing in England, looking at thousands of families across eighteen different local areas. They treated each local authority as its own separate market and reconstructed the swaps by comparing two snapshots of the housing system taken six months apart. Because they could not track individual families over time, they grouped people into categories based on shared characteristics, such as age and household size, assuming that people in the same category would have similar preferences. They then ran three different tests on the data. The first test checked if the swaps were efficient, meaning no one could have been made better off without making someone else worse off. The second test checked for stability, ensuring that no group of families could have broken away and traded their original homes among themselves to improve their situation. The third test added a layer of realism by assuming that preferences followed a logical order, such as higher-income families generally preferring larger homes.
The results showed that the tests were powerful enough to distinguish between markets that worked well and those that did not. In some areas, the data suggested that the swaps were consistent with a fair and efficient system. In others, the patterns of movement were so contradictory that no reasonable set of preferences could explain them as a success. For instance, in one market, the data showed a cycle where a family with a small house moved to a large one, while another family with a large house moved to a small one, and a third family moved in a way that created a loop of dissatisfaction that could never be resolved. The researchers found that the two tests often gave different answers for the same market; a market might pass the efficiency test but fail the stability test, or vice versa. This means that looking at only one aspect of the exchange would give an incomplete picture.
A significant part of their work involved figuring out how hard it is to perform these tests. They proved that while checking for simple efficiency is straightforward and can be done quickly, checking for the more complex stability condition is computationally difficult, belonging to a class of problems that are notoriously hard to solve as the number of people and houses grows. However, they discovered that in practice, this difficulty rarely matters. In almost every case where their tests rejected a market as unstable, the reason was obvious from the start: the data itself contained a contradiction that required no complex calculation to spot. The complex search was only needed for a tiny fraction of the cases, and even then, the computer solved them in less than a second.
When they applied the most restrictive test, which assumed that preferences followed a single, logical order based on the quality of the homes, only one of the eighteen markets passed all the checks. This suggests that while some local housing markets may be functioning well, many others are producing outcomes that cannot be explained by a simple, rational system of preferences. The researchers did not conclude that the English social housing system is broken, but they did show that their method can identify which markets are consistent with a well-functioning exchange and which are not. Their work provides a new tool for policymakers to look at the results of housing swaps and ask whether the system is truly serving the people it is meant to help, or if a different approach is needed.
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