📈 economics

Revealed Preferences of One-Sided Matching

This paper develops a nonparametric revealed preference framework to derive testable conditions for the core in one-sided matching economies with unobserved preferences, demonstrating that the core is falsifiable when agents' preferences are determined solely by observable characteristics.

Original authors: Andrew Tai

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrew Tai

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 world where people trade things they cannot split apart: a specific apartment, a particular school seat, or a unique organ for transplant. In these situations, money often cannot simply solve the problem of who gets what, because the items are indivisible and sometimes cannot be bought or sold at all. Economists have long used a concept called the "core" to describe a stable outcome in such markets. Think of the core as a state of perfect group peace: no small group of people could break away, trade their original items among themselves, and make everyone in that group strictly better off. If such a group exists, the current arrangement is unstable. If no such group exists, the market has reached the core. The challenge for researchers is that while we can see who owns what and who ends up with what, we cannot see what people actually prefer. We see the map of the trade, but not the compass of desire that guided it. Without knowing what people want, it is impossible to tell if a market is truly stable or just lucky.

Andrew Tai, an economist, has developed a way to test whether a market has reached this state of stability, even when the preferences of the people involved remain hidden. His work focuses on a specific type of exchange where everyone wants exactly one item, and the items are distinct, like houses in a neighborhood or slots in a school system. The key to his discovery is a simple but powerful assumption: people who look the same on paper—sharing the same age, income, or background—likely want the same things. By grouping individuals into these "types," Tai creates a structure that allows him to look at the final distribution of goods and ask a definitive question: is it possible for this outcome to be stable?

The paper provides a clear set of rules to answer this question. Tai translates the complex web of trades into a simple visual structure, similar to a map of one-way streets connecting different locations. In this map, every person is a point, and a line is drawn from a person to the person who originally owned the item they now hold. The crucial insight is how these lines form loops. Tai found that for a market to be stable, every group of people who are connected in a closed loop must receive the exact same type of item if they belong to the same "type" of person. If two people with identical backgrounds end up in the same trading loop but receive different items, the market is not stable. It implies that one of them could have traded with the other to get a better deal, meaning the current arrangement is not in the core. This condition is not just a suggestion; it is a strict requirement. If the data violates this rule, the market cannot be in a stable equilibrium, regardless of what the hidden preferences might be.

The research also explores what happens when money is allowed to change hands. In these scenarios, people can pay or receive cash to balance the value of the items they trade. Here, the test for stability becomes slightly more nuanced but remains equally rigorous. Tai shows that a market with money is stable if and only if there is no way for a group of people to trade among themselves, keep their original items, and end up with more money than they started with. In the language of the map, this means there cannot be any loop where the total amount of money flowing out is positive. If such a loop exists, the group could simply keep their original items and pocket the extra cash, proving the current deal is unstable. This finding connects the stability of these markets to the existence of a consistent set of prices. If the market is stable, one can always find a set of prices that makes the observed trades look like the result of rational, price-driven decisions.

This work matters because it turns the abstract idea of market stability into something that can be checked against real-world data. In many situations, such as public housing allocation or school assignments, the exact mechanism used to decide who gets what is often a mystery, or it may be a complex, hidden process. Regulators and analysts can now look at the final results—who got which house or school—and apply Tai's rules to see if the outcome is even plausible as a stable equilibrium. If the rules are broken, it proves the market is not in a stable state, suggesting that some groups are being left worse off than they could be. The paper does not claim to find the perfect mechanism for every market, nor does it predict exactly how people will trade in the future. Instead, it offers a powerful tool for auditing the past. It allows us to look at a completed transaction and say with certainty whether it could have been a fair, stable outcome, or if it was fundamentally flawed. By making the core falsifiable, the paper gives economists a way to test the very foundations of how we understand exchange in a world where things cannot be divided.

Technical Summary: Revealed Preferences of One-Sided Matching

Problem Statement
This paper investigates the testable implications of the core in exchange economies characterized by indivisible goods and unit demand, a setting analogous to the house-swapping model of Shapley and Scarf (1974). The central challenge addressed is the "rationalizability" of observed market outcomes: given observable data on agents, their endowments, and final allocations, can one determine whether these outcomes are consistent with the core equilibrium concept?

In classical revealed preference theory, preferences are unobserved, and the goal is to infer if choices maximize utility subject to a budget. In this exchange economy setting, the problem is distinct because the set of feasible alternatives is not exogenously fixed by a budget constraint but is endogenously determined by the preferences of all other agents. Furthermore, the core does not inherently rely on prices, complicating the application of standard revealed preference tests. The paper seeks to derive nonparametric conditions that allow an analyst to falsify the hypothesis that a market is in the core, specifically under the assumption that agents' preferences are solely determined by observable characteristics (agent types).

Methodology
The author develops a model of aggregate matchings where agents and objects are grouped into "types." Agents of the same type share identical (unobserved) preferences. This assumption of common preferences within types serves as the identifying assumption, introducing the necessary discipline to make the core falsifiable; without it, any observed allocation could be rationalized by assigning unique preferences to every individual.

The core methodological innovation is the representation of exchange economies using directed graphs (digraphs). The paper constructs specific graph structures to analyze the economy:

Without Transfers ( $G_{NT}$ ): A digraph where vertices represent individual agents. An arc exists from agent $ik$ to agent $i'k'$ if $ik$ receives an object that $i'k'$ was endowed with ( $x_{ik} = e_{i'k'}$ ). The analysis focuses on the Strongly Connected Components (SCCs) of this graph.
With Transfers ( $G_T$ ): A weighted digraph constructed similarly, but arcs are assigned lengths based on the net monetary flow: $\ell(ik, i'k') = \omega_{ik} - m_{ik}$ , where $\omega$ is the endowment of money and $m$ is the allocated money.

The paper leverages graph-theoretic results, particularly Tarjan's algorithm for finding SCCs, to derive tractable conditions for rationalizability. The logic relies on the interpretation of SCCs as "market segments" where agents can trade among themselves.

Key Results

1. Economies Without Monetary Transfers (NT-Economies)
The paper establishes necessary and sufficient conditions for an NT-economy to be NT-rationalizable (i.e., consistent with the strict core).

Theorem 1: An economy is rationalizable if and only if, for any two agents of the same type who reside in the same SCC of $G_{NT}$ , they receive the same object type.
Interpretation: Within a market segment (an SCC), agents with identical preferences must be treated equally. If two agents of the same type are in the same SCC but receive different objects, a blocking coalition can always be constructed, rendering the allocation unstable.
Corollaries: The paper derives that identical agents (same type, same endowment) must receive the same object. It also provides a method to calculate the minimum number of agent types required to rationalize a given allocation by analyzing the distinct object types within each SCC.

2. Economies With Monetary Transfers (T-Economies)
The paper extends the analysis to economies with quasilinear utility ( $V_i(h, m) = v_i(h) + m$ ).

Theorem 2: Three conditions are equivalent for T-rationalizability:
1. The economy is T-rationalizable.
2. There exists a price vector $p$ such that the net payment for each agent equals the price difference between their allocated and endowed objects: $(x_{ik} - e_{ik}) \cdot p = \omega_{ik} - m_{ik}$ .
3. The graph $G_T$ contains no cycles with a total length greater than zero.
Interpretation: Condition (3) implies that no group of agents can form a cycle of trades that results in a net transfer of money outward while maintaining the same object allocation. If such a "positive length" cycle exists, the group could block the allocation by keeping their money. Condition (2) establishes that rationalizability is equivalent to the existence of a competitive equilibrium supporting the allocation.
Strict Rationalizability: The paper provides conditions for strict T-rationalizability (where preferences are strict over objects), requiring that agents of the same type in the same SCC either receive the same object or pay different amounts (implying different prices).

Significance and Contributions
The paper claims to contribute to the literature on the testable implications of equilibria and revealed preference theory in matching markets in several ways:

Falsifiability of the Core: It develops reasonable conditions under which the core is falsifiable, akin to the falsifiability of utility maximization in classical revealed preferences (Afriat, 1967). This addresses the "scientific content" of the core in settings with indivisible goods.
Tractability: By utilizing graph representations and aggregate matching types, the paper derives conditions that are "meaningful, intuitive, and tractable." The conditions can be checked efficiently (linear time using Tarjan's algorithm), offering a practical nonparametric test for equilibrium.
Auditability: The results offer a framework to evaluate market stability when the allocation mechanism is unknown or hidden. An analyst can determine if an observed outcome is stable without knowing the specific trading process or mechanism used.
Connection to Competitive Equilibrium: The paper establishes equivalence results showing that in this setting, core allocations are supported by competitive equilibrium prices, bridging the gap between cooperative game theory (core) and non-cooperative market theory (competitive equilibrium) in the presence of indivisibilities.

The paper concludes that while the identifying assumption of common preferences within types is strong, it is necessary to prevent the "anything goes" result where any allocation is rationalizable. The derived conditions provide ex ante predictions for equilibrium outcomes and serve as a tool for checking the stability of observed markets in applications such as housing allocation, school assignment, and organ exchange.

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