The law of aggregate demand and welfare in the two-sided matching market

doi:10.1016/j.econlet.2007.10.012

Economics Letters

Volume 99, Issue 3, June 2008, Pages 581-584

https://doi.org/10.1016/j.econlet.2007.10.012 Get rights and content

Abstract

In the college admission problem, we show that the student-optimal stable matching is weakly Pareto optimal for students if colleges' preferences satisfy substitutability and the law of aggregate demand. We also show that both of these properties are important for the result.

Introduction

The theory of two-sided matching considers matching between two types of agents, for example colleges and students. A classical result states that the student-optimal stable matching is weakly Pareto optimal for students if preferences of colleges are responsive.

We show that the above welfare conclusion holds more generally. More specifically, if preferences of colleges satisfy substitutability and the law of aggregate demand (Hatfield and Milgrom, 2005), then the student-optimal stable matching is weakly Pareto optimal for students.

Then we investigate how important substitutability and the law of aggregated demand are for the result to hold. We find that even if a college's preferences violate the law of aggregate demand, there is an instance such that weak Pareto optimality of the student-optimal stable matching can be guaranteed for any preferences of students and other colleges that satisfy substitutability and the law of aggregate demand. However violation of the law of aggregate demand does let us show a weaker converse result. If a college's preference relation violates the law of aggregate demand, then there is a preference profile of students and other colleges with singleton preferences under which there is an individually rational matching that all students weakly prefer and all but certain students strictly prefer to the student optimal stable matching. Taken together, we conclude that the law of aggregate demand is important for the welfare properties of a stable matching.

The weak Pareto optimality of the student-optimal stable matching has been obtained under more restrictive assumptions in the literature. Roth (1982) shows the result when colleges have responsive preferences. Martinez et al. (2004) show the result when colleges have substitutable and q-separable preferences. The result seems to have been unknown under affirmative action constraints, such as those studied by Abdulkadiroğlu (2005). The class of substitutable preferences with the law of aggregate demand subsumes all these domains. Under substitutability and the law of aggregate demand, Hatfield and Kojima (2007a) show that the student optimal stable mechanism is group strategy proof for students, and apply this result to obtain an alternative proof of the weak Pareto optimality. On the other hand, results in the other directions that investigate if these conditions are minimal sufficient conditions for the welfare property are new to the best of our knowledge.

The above conclusions are especially interesting in the context of school choice (Abdulkadiroğlu and Sönmez, 2003). Focusing on welfare of students is relevant in the school choice setting since colleges are regarded as objects to be assigned rather than agents. Moreover extending the domain of preferences is important, since school preferences often violate assumptions such as responsiveness and q-separability under which the conclusion has been known in the literature. For instance, some of the public schools in the New York City are required to admit a certain proportion of students from each of high, middle and low test score populations. Such requirements violate the above simple conditions, but still satisfy substitutability and the law of aggregate demand. Our analysis gives some justification for the use of the student-optimal stable mechanism originally advocated by Balinski and Sönmez (1999) and Abdulkadiroğlu and Sönmez (2003).¹

Finally, the current paper contributes to the growing literature on matching with contracts. Hatfield and Milgrom (2005) present a new framework that generalizes models of matching and auction. They show that the law of aggregate demand they introduce and substitutability are crucial for some of the key results in the matching literature. The current paper gives another instance in which the law of aggregate demand plays a key role in matching theory.

Section snippets

Model

A market is tuple Γ = (S, C, (⪰_i)_i ∈_S ∪ C). S and C are finite and disjoint sets of students and colleges. For each student s ∈ S, ⪰_s is a strict preference relation over C and being unmatched (being unmatched is denoted by ∅). For each college, ⪰_c is a strict preference relation over the set of subsets of students. If j ≻_i ∅, then j is said to be acceptable to i. Given college c and a set of students S′ ⊆ S, define Ch_c(S′) to be a set such that Ch_c(S′) ⊆ S′ and Ch_c(S′)⪰_cS″ for any S″ ⊆ S′. In words, Ch_c(S

Result

Theorem 1

Suppose that preferences of every college satisfy substitutability and the law of aggregate demand. Then μ_S is weakly Pareto optimal for students. That is, there exists no individually rational matching μ such that μ≻_s μ_S for every s ∈ S.

To prove Theorem 1, we assume on the contrary that there exists an individually rational matching μ such that μ≻_s μ_S for every s ∈ S.

Claim 1

μ(s) ∈ C for every s ∈ S. That is, every student is matched at μ.

Proof

Since μ_S is individually rational, μ_S (s) ⪰_s ∅ for every s ∈ S. By

Acknowledgements

I am grateful to John Hatﬁeld, Taisuke Matsubae, Alvin Roth, Tayfun Sönmez and Satoru Takahashi for discussion in early stages, and especially to an anonymous referee for helpful comments.

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There are more references available in the full text version of this article.

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The law of aggregate demand and welfare in the two-sided matching market

Abstract

Introduction

Section snippets

Model

Result

Acknowledgements

Journal of Economic Theory

College admission with affirmative action

International Journal of Game Theory

The New York City High School match

American Economic Review Papers and Proceedings

The Boston Public School match

American Economic Review Papers and Proceedings

School choice: a mechanism design approach

American Economic Review

College admissions and the stability of marriage

American Mathematical Monthly

Matching with contracts

American Economic Review