Matching Markets: When Price Is Not Enough

How economists design systems to match students to schools, doctors to hospitals and kidney donors to patients, where prices cannot do the job.

Most markets use prices to decide who gets what. But in some markets, prices cannot or should not be used. You cannot simply buy a place at a public school, a medical residency or a kidney. These are matching markets, where both sides must choose each other.

The Nobel prize

In 2012, Alvin Roth and Lloyd Shapley received the Nobel prize in economics for the theory of stable allocations and the practice of market design.

Stable matching

In 1962, David Gale and Lloyd Shapley developed the deferred acceptance algorithm. It produces a stable matching: no two participants would both prefer each other to the partners they were assigned. Stability matters because unstable matches tend to unravel, as people try to make side deals.

How deferred acceptance works

In a school choice version:

Each student applies to their most preferred school.
Each school tentatively holds its best applicants up to capacity and rejects the rest.
Rejected students apply to their next choice.
Schools reconsider, keeping the best applicants among new and held ones.
The process repeats until no more rejections occur.

A key feature is that students do best by listing their true preferences, rather than gaming the system.

Real applications

Medical residency matching in the United States, redesigned with Roth’s help in the 1990s.
School choice systems in cities such as New York and Boston.
Kidney exchange: a patient with a willing but incompatible donor can swap donors with another pair. Roth helped design kidney exchange programmes that have enabled many more transplants.

A kidney swap

Patient A needs a kidney, and her husband is willing to donate but is incompatible. Patient B's sister is also incompatible with B. But A's husband matches B, and B's sister matches A. By swapping donors, both patients receive transplants. Larger chains of swaps, organised by computer algorithms, can help many more patients.

Thinking markets always need prices

Some important allocations cannot use prices, because buying and selling would be illegal or unfair. Carefully designed matching systems can allocate these fairly and efficiently.

Key takeaways

Matching markets pair participants where prices cannot be used.
Gale and Shapley's deferred acceptance algorithm produces stable matches.
Roth and Shapley won the 2012 Nobel prize for work on matching.
Applications include medical residencies, school choice and kidney exchange.