Mathematical Matchmaking for Jobs in Math

"Listen," a friend once told me. "When I'm reading a textbook and come across one of those big ugly equations, I slam the book shut. I want to skoosh it like a bug."

He had my complete sympathy, which is one reason I'm not a scientist. It's also the reason that you'll seldom read about subjects mathematical in this column.

However, now and then I find something I can understand--such as a recent article in Science magazine describing a situation in which mathematicians may have to do their homework if they want to find work at all.

Few new graduates want to join the ranks of academic mathematicians each year; they are competing for even fewer jobs. However, just as in any field of work, it usually happens that person X wants employment with university A, but university A would prefer to recruit person Y, who really would love to join the staff of university B--and so on through the alphabet. This leads to a certain amount of shilly-shallying in hiring, as everybody stalls while they wait for the decision of their most wanted employer or employee.

Donald Lewis, a mathematician at the University of Michigan, has suggested that his colleagues turn their professional skills toward solving the problem of gridlock in the job market. He wants them to build on a well-studied model known as the stable marriage problem.

To solve this problem, a mathematician must assign an equal number of men and women to mates so that everyone is willing to stay married. The matchmaking doesn't work if any two prefer each other to the partners they've been assigned; the assumption is that they'd leave their mates and ruin the symmetrical pairings. To assure nobody's roving eye is reciprocated, the mathematician must find the overall pairing in which all partners are content.

For starters, all the men and all the women must list possible partners in order of preference. Then the mathematician must select the best algorithm--in effect, a descriptor of action in mathematical language--to assign partners. The algorithm also speeds up the trial-and error aspect of matching; that's vitally important, since with just 10 couples, there are more than 3 million possible matchings. (That's what "Science" said, at least; remember, I'm no mathematician.)

One algorithm would have each man (or employer) propose to the woman (or job candidate) at the top of his list. Each woman tentatively accepts the best offer (the one from the man highest on her list) and sends word to the other men not to bother her any more. The woman can break the engagement if a better offer comes along; the man can keep working his way down his list until someone accepts. The result is stable because each man has been rejected by the women higher on his list than the one who accepts him.

This particular algorithm works well enough--it's been used in medical job placement, with hospitals taking the men's roles. But it is, so to speak, sexist. Each man (or employer) ends up with his best possible partner among the stable matches, but that's not true for the women (or job seekers). They'd rather do the proposing.

The challenge facing mathematicians is to come up with a bias-free algorithm, one that favors neither men nor women, employers nor employees. It also must offer the lowest possible amount of dissatisfaction, judged by measuring how far down each person's list the selected partner is.

Then, Lewis believes, the American Mathematical Society should play the role of matchmaker, getting lists from all would-be employees and employers in academic math and applying the best algorithm. Employment assignments would follow, with both job-seeking mathematicians and hiring universities truly putting their money where their math is.