August 24, 2004
Tweaking the Math to Make Happier Medical Marriages
arlier this month, a federal district court dismissed an antitrust challenge to the Residency Match, the program that assigns medical students to residency positions. The plaintiffs say they will continue their legal efforts, but whatever the outcome, the case has already captured the attention of mathematical economists, who are wrestling with a question at the heart of it: By making it difficult for students to negotiate the terms of employment, does the match program enable hospitals to underpay residents?
The match was developed in the 1950's to bring order to a frenzied market for residents, where competition was manifested in the timing of offers. Hospitals would pressure top students to commit as early as sophomore year, or make offers that expired in hours.
The solution was a matching process, which evolved into what was shown to be a classic mathematical recipe, the so-called stable marriage algorithm. Given equal numbers of boys and girls, each with a list of preferences, the algorithm pairs them so that the relationships are likely to last.
The algorithm, frequently taught in math, computer science and economics courses, has proved powerful in real life. Variants are used for assigning clinical psychologists to internships, and students to slots at New York City public high schools.
The version used for the residency match does a good job of placing students in appropriate programs, economists say. But unlike marriage markets, job markets are influenced by salary negotiations, which the algorithm does not take into account.
Dr. Vincent Crawford, an economics professor at the University of California, San Diego, has come up with a way to build salary negotiation into the algorithm. This, he says, would preserve the best aspects of the program and potentially raise wages.
The marriage algorithm was devised in 1962 by the mathematicians Dr. David Gale, now a professor emeritus at the University of California, Berkeley, and Dr. Lloyd Shapley, an emeritus professor at the University of California at Los Angeles. But it was not until 20 years later that Dr. Alvin E. Roth, a professor of economics and business at Harvard, observed that it was essentially the same as the one used in the match. It works like this:
Each boy ranks all the girls in order of his preference, and each girl does the same. Then, each boy asks his first choice for a date. Each girl with one or more offers dates her favorite and says "no" to the rest.
In the next round, the boys who were rejected move on to their second-choice girl. The girls again date their favorites, possibly throwing over their date from the earlier round for someone better.
Continuing in this way, the mathematicians showed, the dating frenzy eventually subsides into a stable situation where each girl has only one boy, and there is no boy and girl who prefer each other to the people they are dating. That is, every time a boy does not get his first choice, he has no hope of getting anything better. Each of the girls he prefers is paired with someone she prefers to him. The same is true for a girl.
In the match, the students are the boys, and the residency positions are the girls. Each year, after interviews, students and hospitals rank one another in secret preference lists. A computer program then automatically plays out the rounds of dating, producing a match participants must accept.
Because salaries are determined before the results of the match are known, programs cannot easily target the best candidates with higher offers. This may stifle competition, said Dr. Jeremy I. Bulow, an economics professor at Stanford University.
Dr. Crawford's proposal, described in a paper submitted for publication, stems from research he did in the early 1980's that built on the marriage algorithm, but in the context of matching workers to jobs at appropriate salaries. His proposal, a simplified version of this work, would incorporate salary considerations into the rounds of the match.
Each residency program could offer up to three salary levels. Students would rank each level as if it were a separate program. Hospitals would include each student three times in their ranking lists, once at each level.
Dr. Bulow called Dr. Crawford's proposal "potentially a significant step forward" but said further study was needed to see how well it might work.
Mona Signer, director of the National Resident Matching Program, a private, nonprofit corporation that runs the match, said the board would look at the proposal.
"We would consider it like any other suggestion for change," she said.
Dr. Bulow and Dr. Jonathan D. Levin, another Stanford economics professor, suggest, in a paper submitted for publication, that the match may depress residents' salaries. They describe a simplified, hypothetical job market that in some ways resembles the residency match. In this market, they say, the inability to personalize offers results in salaries that are lower than they would be in an ideal free market.
"Say I'm a second-tier hospital," Dr. Bulow said. "I'd like to offer a premium wage to try to attract some of the top people on the market, but if that doesn't work out, I'd like to pay a normal wage."
Because the residency market is complex, it will deviate from a perfect free market for other reasons, Dr. Levin cautioned. Still, he said, the model suggests that an inability to personalize offers can affect wages.
Other scholars disagree. Dr. Roth, who designed the current version of the match, said wages were low because residents cared more about a program's prestige than about salary, and because there were many more applicants than positions.
Also, Dr. Roth added, because residents' salaries are linked to Medicare reimbursements, hospitals do not have much flexibility in their budgets.
Dr. Levin disagreed that the high value of prestige stifled salary competition. "In our model, competition happens, not between the top-tier schools and the middle-ranked schools, but between schools of similar prestige," he said.
Dr. Roth and Dr. Muriel Niederle, a professor of economics at Stanford, published a letter to the editor in The Journal of the American Medical Association in September 2003 showing that salaries for post-residency training appointments did not differ between subspecialties that did and did not use a match. From this, they concluded that salaries were largely determined by other issues.
Other researchers disagree with this conclusion, citing other differences between the markets.
Sherman Marek, a lawyer for the plaintiffs in the lawsuit, said it was unclear what form the continuing legal action would take. He added that the plaintiffs would welcome proposals to introduce salary competition.
"The doctors have been wrestling with this; so have the lawyers and even the politicians," Mr. Marek said. "Perhaps the solution rests with the mathematicians."