How should New York City assign students to schools? How should young doctors be matched with residencies? How should lifesaving kidneys be allocated to desperate patients?
For the answers, we have the work of the 2012 winners of the Nobel Memorial Prize in Economic Sciences, Alvin Roth, a Harvard colleague who is soon leaving for Stanford, and Lloyd Shapley, a professor emeritus at the University of California, Los Angeles.
The world is full of allocation problems, and in most cases, economists favor standard markets. How should we allocate petroleum or Chevrolets or old-master paintings?
Economics 101 suggests that these goods should go to the people who are willing to pay the most for them and therefore value them most highly. Sometimes, as with gas stations, the seller just posts a price and sells to anyone willing to pay. Other times, as at a General Motors dealership, there is a bit of haggling. The 1996 and 2007 Nobel prizes acknowledged the pioneering work of William Vickrey and Roger Myerson in designing auctions, but in all these settings the basic objective remains the same: get the good to the person who is willing to pay the most.
In the problems analyzed by Roth and Shapley, the right answer isn’t to let the highest bidder prevail. In some settings, there are two-sided preferences; residents care about which hospital employs them, and hospitals care about which residents they get.
Harvard could sell its undergraduate spots to the highest bidder, but this would reduce the quality of a Harvard education. We don’t sell kidneys, because we are understandably squeamish about auctioning off the right to live.
Shapley began this literature in 1962 with a paper co-written with David Gale, who might have been included in this prize if he hadn’t died in 2008. They focused on a particularly ubiquitous matching problem, the marriage market, where the standard economic answer, a cash auction, isn’t terribly sensible.
A few medieval monarchs might have given their daughters to the highest bidder, but it’s a good bet that if Catherine of Aragon’s preferences had counted for anything, she would have married someone other than Henry VIII. Stable marriages require a matching mechanism that respects the preferences of both spouses.
Gale and Shapley pondered both the marriage question, which involved matching one man and one woman (this was 1962, after all), and the college assignment problem, where multiple applicants were sorted to a single choice. They produced a matching algorithm that is surprisingly plausible, given the authors’ interest in high abstraction.
The Gale-Shapley mechanism has men first propose to their favorite woman. Women who receive more than one proposal string along their favorite candidate and reject the others. The rejected men make new proposals to any woman who has not rejected them already, and the process continues. Whenever a woman receives an offer she likes better than that from the man she is stringing along, she drops her current beau and chooses the new one.
Since the authors assume that the number of men and women is equal and that everyone prefers marriage to solitude, eventually everyone finds a mate. The allocation created by this process is stable: No women or men would like to switch their current spouse for someone else who would also prefer them to their current spouse. The more desirable man would have proposed to the woman first if he thought she was a more desirable mate.
The allocation mechanism also works on Sadie Hawkins Day, if the girls are doing the picking, and it also works for college admissions. Each applicant applies to his or her favorite college, and the college can decide to put the applicant on a waiting list. The rejected applicants then apply to other colleges, until the process sorts itself out. Not only will the eventual outcome be stable, it will also be as good for the applicants as any other stable allocation mechanism.
In 1982, however, Roth pointed out a problem with actually carrying out the Gale-Shapley mechanism: People have an incentive to lie. A woman may reject a man she actually likes more, because that man, if rejected, will end up breaking up the current understanding between some other woman and a man she likes even better. For matching mechanism to work well, they need to elicit honest behavior, not strategic play.
This truth-telling problem remains when we move from a decentralized process of courtship to a centralized clearing-house mechanism, as the medical internship market did in the early 1950s. As Roth describes in a 1984 article, the decentralized market for interns was painfully chaotic in the 1930s and 1940s. Interns would string hospitals along until the last possible date, hoping that a better offer would show up, and hospitals tried to eliminate this uncertainly by requiring students to sign binding agreements early in their medical school careers.
To regularize the market, a central clearing bureau was established, which would elicit a list of preferences from both the students and the hospitals, and then use a matching algorithm to assign interns. The first proposed matching algorithm had hospitals ranking students into five groups and students ranking hospitals, and then matched the top choice of one with the top choice of the other, and so on.
But in this design, students have an incentive to lie, and overstate their desire to work in hospitals that are less likely to fill up immediately. The system was dropped in favor of one with “tentative matches” that can be undone if a better match appears as the process unfolds, and that is closer to the Gale-Shapley mechanism. While Roth proved that this system is unlikely to elicit perfect truth-telling, it is far better than the first proposal.
Roth’s expertise made him a natural fit for New York City to consult in 2003 when it was designing its school matching system. Before 2003, New York’s schools allocated spots through a complicated process involving repeated student applications, rejections and much agony.
The system wasn’t only cumbersome. It also led to strategic behavior, as when people applied first to less popular schools to make sure they got in, leaving students with none of their top choices. Roth pushed toward a clearing-house system, based loosely on the models foreseen by Gale and Shapley and carried out by hospitals. The result wasn’t exactly his ideal. It had two rounds instead of one to address New York’s specialized exam schools, but the new system seems to be a major step forward.
In recent years, Roth has been deeply involved in solving the kidney-exchange problem. Distinguished economists have long advocated a free market, where kidneys go to the highest bidder and organ donors are paid handsomely. As Roth notes, however, “repugnance” can be a powerful “constraint on markets.” We don’t much like the idea of selling life itself, yet without financial markets, allocating kidneys can be as tricky as allocating slots in New York public schools.
The classic problem occurs when an ailing individual has a prospective donor, but that donor’s kidney is immunologically incompatible with the patient’s body. In a 2004 article, which I edited in the Quarterly Journal of Economics, Roth and two co-authors explore sensible kidney exchange systems. Roth proposes a system, which has its roots in earlier work by Shapley and in an idea from Gale.
The basic system is that each sick person, who has a donated kidney that doesn’t work for them, points to their preferred kidney in the exchange. This system must produce a cycle, in which person A prefers kidney B and person with kidney B prefers kidney C and person with kidney C prefers kidney A. The algorithm works by satisfying the bargains implied by the cycle, which must make everyone who trades better off, and starting again. Roth’s algorithm is more complicated, but it still hews to the basic goal of allocating kidneys fairly and in a way that maximizes the number of lives saved.
Roth and Shapley have done much more than just design exchanges. Shapley is the giant of cooperative game theory; Roth is a pioneering experimental economist. Their work is very far from the contentious debates of macroeconomics, and it illustrates economics in action at its best. Not only scholars, but children in New York City and people with kidney diseases should be cheering for this year’s choices for the Nobel.
(Edward Glaeser, an economics professor at Harvard University, is a Bloomberg View columnist. He is the author of “Triumph of the City.” The opinions expressed are his own.)
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