June 4–6, 2012: Matching Problems: Economics meets Mathematics

First joint conference between the Becker Friedman Institute and the Stevanovich Center for Financial Mathematics.




This three-day workshop brought together experts in mathematical and economic theory, econometrics, and empirical study of matching to explore the latest theory and a wide array of applications.

In opening remarks, Institute Chair Gary S. Becker noted that most economic problems can be looked at as matching problems. “It’s a very important topic,” he said. “Matching theory started in labor markets with simple models. As economists probed more deeply into the theory, they needed more sophisticated mathematics. This is the first conference that I know of that brings together economists and mathematicians to address this matching problem.”

The conference sessions began with tutorials on matching theory and the underlying mathematics, to provide a common knowledge base for scholars from different fields.

John Hatfield of Stanford University began with Gale and Shapely’s work (1962) that explored how marriage matches could be made so that no individual wished for a divorce. Such models are usefully employed in programs that match medical residents with hospitals, school choice systems, labor markets, and auctions, Hatfield said.

In these applications, the “woman” side of the match may take many partners and the relationships take many forms. The quality or success of the relationship or match depends on the specific characteristics offered and desired. “If I’m bidding on baseball card, I want to know more than if I’m getting a card. I want to know if I’m getting a 1972 Hank Aaron card.”

Becker Friedman Institute Research Scholar Scott Kominers extended the tutorial to explore recent models that go beyond two-sided matches to cases that included intermediaries, multiple partners (or buyers), and transfers of utility. “The main results are that in arbitrary trading networks with multilateral contracts, transferable utility, and concave preferences, these [matching models] work.” He explored implications for cyclic contracts where agents are both buyers and sellers in a manufacturing cycle.

In the next tutorial, conference co-organizer Robert McCann of the University of Toronto defined the mathematics of matching and geometry of matching spaces. He showed how optimal transport, which mathematically calculates the most cost-efficient way to distribute mass or materials from one point to another, can be applied to determine the optimal match between buyers and sellers.

Jean-Marc Robin of Sciences Po and Robert Shimer of the University of Chicago reviewed literature and theory of search models that help understand labor markets and equilibrium wages. They reviewed partial search models, equilibrium search models, wage posting models. Pierre-André Chiappori and Bernard Salanie of Columbia University explored the econometrics of matching in the final tutorial.

Following the tutorials, presentations blended theory with data and real-world applications, highlighting new work that explored topics such as marriage trends, labor markets, and school choice assignments.