Couples on the labor market

One of the longstanding puzzles in market design is why we have been as lucky as we have been in the design and operation of labor market clearinghouses that allow couples to state preferences over pairs of jobs. You can't get a stable matching without allowing couples to state their preferences this way, but when they do, the set of stable matching can be empty. But it almost never is, in practice.

Here's a first step towards understanding that:
Kojima, Fuhito, Parag A. Pathak, and Alvin E. Roth, " Matching with Couples: Stability and Incentives in Large Markets," working paper, April 8 2010.

Abstract: Complementarities pose problems in models of two-sided matching markets. This has been a longstanding issue in the design of centralized labor market clearinghouses that need to accommodate complementarities due to couples, as in the US market for medical doctors. These clearinghouses aim for a stable matching but a stable matching does not necessarily exist when
couples are present. This paper provides conditions under which a stable matching exists with high probability in large markets. Moreover, we present a mechanism that finds a stable matching with high probability and in which truth-telling by all participants is an approximate equilibrium. We relate these theoretical results to data from the labor market for clinical psychologists, in which stable matchings exist for all years of our data, despite the presence of couples.

New clearinghouse for new doctors in Scotland

There have been some changes in the SCOTTISH FOUNDATION ALLOCATION SCHEME, the program that matches new medical graduates to their first positions (which would be called residencies in the US) in Scotland.

Two features stand out in comparison to the American system (the NRMP).

First, employers may not submit preferences, but rather are all constrained to rank potential employees by their exam scores.

Second, couples cannot submit preferences over pairs of positions. Instead, each member of the couple submits a rank ordering of individual positions, and the algorithm combines these into a joint preference over pairs that is a function of the submitted rank order list and a table of compatibilities of positions.

"To accommodate linked applicants, a joint preference list is formed for each such pair, using their individual preference lists and the programme compatibility information. If such a pair, a and b, have individual preferences p1, p2, . . . , p10 and q1, q2, . . . , q10 respectively (with a the higher scoring applicant), then the joint preference list of the pair (a,b) is (p1,q1), (p1,q2), (p2,q1), (p2,q2), (p1,q3), (p3,q1), (p2,q3), (p3,q2), . . ., (p9,q10), (p10,q9), (p10,q10) (except that incompatible pairs of programmes are omitted)
In the main body of the algorithm, the members of a linked pair are handled together, so the match of the pair (a,b) to the programmes (p,q) will be accepted only if each of these programmes either has an unfilled place or a lower scoring applicant who can be displaced. A complication arises when one member x of a linked pair has to be withdrawn from a programme p because his/her partner was displaced from their current assigned programme. In this case, some other applicants may have been rejected by p because of the presence of x, and any such applicant a must be withdrawn from their current programme, if any, and have their best achievable preference reset to p. (A corresponding, but more complex reset operation is needed if a is a member of a linked pair). This reset operation thereby allows a further opportunity for applicant a to be matched to programme p.
The algorithm terminates when every single applicant and linked pair is either matched or has been rejected by, or displaced from, every entry in their preference list with no possibility of reconsideration by a programme that has had a withdrawal.
The final matching is stable for single applicants, as before, but also for linked pairs, in the sense that:
there can be no linked pair (a,b) of applicants who would prefer to be matched to compatible programmes (p,q), and at the same time, each of p and q has an unfilled place or an assigned applicant with a lower score than a and b respectively."

HT: Rob Irving, who has designed and implemented the algorithm.

Here are some related papers by members of the Scottish matching group.

Keeping partners together: algorithmic results for the hospitals/residents problem with couples by Eric J. McDermid and David F. Manlove in
Journal of Combinatorial Optimization, (2009)

R.W. Irving, D.F. Manlove and S. Scott, The stable marriage problem with master preference lists, Discrete Applied Mathematics vol. 156 (2008), pp. 2959-2977.

Matching with contracts

When Marilda Sotomayor and I wrote our 1990 book on matching, we divided the book into models without money (like the marriage model) and models with money (like auction models), and showed how similar results obtained for both kinds of models. There's been lots of recent progress in unifying those models and understanding why the results were so similar. The most recent paper in this literature is by John Hatfield and Scott Kominers, Many-to-Many Matching with Contracts, and I asked them to write a blog post about it, which appears below.

"A few years ago, Hatfield and Milgrom (2005) introduced "(many-to-one) matching with contracts," a generalization of classical two-sided matching in which contracts between agents specify both (1) a matching and (2) conditions of the match (such as wages, hours, or specific responsibilities). The results of Hatfield and Milgrom (2005) are surprisingly general--although presented using matching-theoretic language, the Hatfield and Milgrom (2005) model encompasses not only two-sided matching but also several package auction models.Hatfield and Milgrom (2005) assumed a many-to-one matching market, that is, a market in which agents on one side of the market (the "doctors") were never allowed to sign more than one contract. Although reasonable for some applications of matching (such as school choice), some matching applications (such as the United Kingdom Medical Intern match) allow "many-to-many matching," in which every agent can be assigned to multiple agents on the other side of the market.In "Many-to-Many Matching with Contracts," we develop a theory of "many-to-many” version of matching with contracts which extends the model of Hatfield and Milgrom (2005) to allow all agents to accept multiple contracts.This framework extends contract matching to a host of applications not previously covered by generalized matching theory, including the United Kingdom Medical Intern match (discussed above), the United States National Resident Matching Program (which allows agents to pair together and match as "couples" who can receive two jobs), and the matching of consultants to firms. Additionally, many-to-many matching with contracts generalizes the theory of bilateral "buyer-seller markets".
We prove that substitutability of preferences (for agents on both sides of the market) is both sufficient and necessary to guarantee the existence of stable many-to-many contract allocations; in many-to-one applications, in contrast, weaker conditions than substitutability guarantee existence. This result shows that, in general, a stable match is not guaranteed to exist in the matching with couples problem, since couples' preferences are generally assumed to be non-substitutable by nature. These results also provide insight into why the necessity result does not hold in the many-to-one matching case, and allows us to identify a new class of non-substitutable many-to-one preferences which are in some sense projections of substitutable many-to-many preferences, and for which the existence of a stable match is guaranteed."

Two-career job searches

When a couple needs two career-track jobs, they face a hard problem of coordination with each other and with their prospective employers. If they are in different industries, they need to find a four-way match, between the two of them and two different employers. If they are academics, they can at least try to find two jobs at the same university, but if they are in different disciplines the negotiations will involve different departments (and maybe different schools, i.e. different deans), and so the search and negotiation process can be complex, and can still involve potentially very different timing of searches and hiring.

The Chronicle of Higher Ed has a first person account of one such struggle, that ended successfully with two tenure-track assistant professorships at the same university: Lessons of a Dual Hire.

The (pseudonymous) author writes:"After three years of job searching for me in the geological sciences, and four years for my husband in engineering, we successfully maneuvered this year to find two tenure-track positions at the same university. Here's how it happened."

The article goes on to explain some of the difficulties that were overcome in the most recent, successful job search.

Here are two earlier related posts, both of which touch on my work on making the clearinghouse for new doctors, the National Resident Matching Program, more friendly to couples.

Job market for couples (which concerns law schools hiring of couples); and

Match Day for new doctors, which is specifically about couples who are both seeking jobs as new doctors.

Even the medical clearinghouse doesn't do much to help doctors whose spouses have non-medical careers (or even doctors whose spouses have medical careers with different years of graduation from medical school). Some years ago, I was asked to respond to an essay from a doctor's spouse which suggested that maybe the market would work better without a match, i.e. without any centralized clearinghouse. That essay, and my reply, were published in an online student edition of JAMA that no longer exists, on web pages that are no longer maintained. However I am linking to them below, on the remarkable internet archive also known as the Wayback Machine.

Mismatch, by Betsy Brody, University of Notre Dame

Response to Betsy Brody's "Mismatch" by Alvin E. Roth (both originally in MSJAMA, April 7, 1999.

Rereading my response, I would have written it a bit differently today, but the basic point still seems right. But two-career searches are tough, no doubt about it.

Fertility and religion

The Bible encourages reproduction, but puts bounds on sexuality, and so religious couples seeking treatment for infertility need to navigate carefully. For orthodox Jews, the Puah Institute (named after one of the two Hebrew midwives in the story of the birth of Moses) helps with this navigation. Here's a story about rabbinical supervision of medical fertility treatments involving in vitro fertilization: How to make a kosher baby.

Some of the particular issues that arise with in vitro fertilization involve how to be sure, in a legal sense, who the parents are. (This is presumably also one of the reasons behind the restriction of sex to marriage in so many cultures, when sex was the only reproductive option.)

The problem of how to integrate new technological and commercial possibilities with ancient customs, rules, and practices is very clear when those are religious in nature. But similar problems also present themselves in secular society, in dealing e.g. with issues like surrogacy, or compensation for sperm and egg donors in ways that navigate around repugnance.

In only slightly related news (but still on the subject of being fruitful, and making sure your children have the right parents), the cost of arranged marriages is going up among Israel's most orthodox Jews: Haredi matchmaking rates skyrocketing

Same sex marriage in New Hampshire

Once again, an ancient repugnance is annulled by legislators, not just judges: New Hampshire Legalizes Same-Sex Marriage .
"The New Hampshire legislature approved revisions to a same-sex marriage bill on Wednesday, and Gov. John Lynch promptly signed the legislation, making the state the sixth to let gay couples wed. "

Match Day for new doctors

Today is the third Thursday in March, Match Day for young doctors seeking their first job through the National Resident Matching Program (NRMP).

Here is the NRMP press release. Almost 30,000 applicants (11,000 from foreign medical schools) sought the 22,427 first year positions available in this year's match. About 95% of the available positions were filled through the match, the rest are filled in a post-match "scramble." (The organization of the scramble is under discussion, and here's the WSJ's account of it.)

The NY Times covers match day with a story and a picture: A Medical Student’s Rite of Passage . As in many discussions of labor markets, the author finds that many medical students wish they had more control over where they are going. Some of them attribute this loss of control to the match process, while others know something about how the medical market worked before the match. (Among the online comments on the story is this one:
"I wish they did something like this for law students. The job experiences and training available to new lawyers are extremely uneven. Plus it is on the law student to secure that first job on the open market, with no real guarantees of getting hired."

There's a new book by Brian Eule, Match Day: One Day and One Dramatic Year in the Lives of Three New Doctors, that follows three women through the match, one of them now his wife. I haven't seen the book yet, but I talked to him a number of times while he was writing it, and he once gamely sat through a lecture in my Market Design class.

788 couples went through the match this year as couples (some others may go through without identifying themselves that way, and the NY Times story remarks that some were disappointed not to have gotten jobs together). For the technically inclined, here's an account of how the new couples algorithm works to allow couples to express their preferences over pairs of jobs (and of the design process that led to the new match algorithm that's been in use since 1998):
Roth, A. E. and Elliott Peranson, "The Redesign of the Matching Market for American Physicians: Some Engineering Aspects of Economic Design," American Economic Review, 89, 4, September, 1999, 748-780.

Market for matchmakers: sorting by price

Dating and matchmaking services vary widely on a number of dimensions, one of which is price. Below I'll talk about services whose prices vary from zero to an initial fee of $20,000. One question is how much if anything does that already tell you about who might use which services?

Penny-Pinchers Might Unite at Free Dating Site
"Match.com, which is owned by Internet company IAC/InterActiveCorp and also runs dating site Chemistry.com, was set to announce Thursday the launch of DownToEarth.com. ...DownToEarth.com joins other free dating sites like Plentyoffish.com and OkCupid.com, and expects to bring in revenue from ads. It is geared toward Web dating newcomers and lets users put up post-rendezvous ratings regarding the truthfulness of others' pictures and profiles."

Online Dating Putting You Off? Try a Matchmaker
"Matchmakers prescreen potential matches, focusing on long-term compatibility rather than “short-term chemistry,” Ms. Clampitt said.
While online sites allow unlimited fantasizing, matchmakers encourage clients to take their heads out of the clouds. “Sometimes we will get a guy who is a good-looking man, but no Brad Pitt, and he wants a thin model,” said Shoshanna Rikon, the owner of Shoshanna’s Matches, a Yenta-style matchmaking service in Manhattan that includes an in-person interview and a Web presence, and charges about $1,500 for eight dates. “We try to be more realistic with who we set him up with.""

The New Arranged Marriage
"Janis Spindel Serious Matchmaking Incorporated's fees begin -- begin! -- at $20,000 for an initiation fee, plus $1,000 for a one-year membership that includes 12 dates. That also includes a background check and a home visit, during which Janis spends time with the client, to get a sense of him and verify that he is who he says he is (i.e., rich or very rich). Her image consultant also comes to inspect his wardrobe and, if necessary, make plans to revamp his look. Janis has many clients outside the New York area (in Tampa, Miami, Los Angeles, Toronto, Las Vegas). An out-of-town client must fly Janis and an assistant first class and put them up in a hotel for the home visit. Additionally, a marriage bonus is expected -- sometimes it's a car or extravagant jewelry; other times it's cash. She has received gifts in the $75,000-to-$250,000 range. "

This latter service primarily charges fees to men, and actively recruits attractive women to match them to. This reminds me of a 1993 paper by Mark Bagnoli and Ted Bergstrom called "Courtship as a Waiting Game" which considers why husbands are often older than wives. In their model, people live for two periods. In period 1, men and women are each endowed with a "quality" between 0 and 1, and a woman's quality is common knowledge at period 1, but, although men know their own quality at period 1, it only becomes common knowledge at period 2. So, in their model, the highest quality men wait until period 2, and marry the highest quality women. I guess that, in this model, the $20,000 above would be a signal of male quality... :)

Of course, the value of a match could be a subject of dispute; e.g. here's an 1885 report from the NY Times about a matchmaker suing to receive his full fee after a marriage was arranged but called off. Needless to say, matrimony need not be the only object of matchmaking; Daniel Hamermesh has a Freakonomics post describing an internet site "Ashley Madison, which matches up married women and men who wish to have a quick fling. " (I couldn't figure out their fee structure from the easy to access parts of their web page, but they do offer a $249 refund under their "Affair Guarantee Program" if you fail to have one...)

Marriage market in Iran

The marriage market in Iran is not proceeding as planned, the Guardian reports: Premarital sex on rise as Iranians delay marriage, survey finds

"The survey also revealed that the average marrying age had risen to 40 for men and 35 for women, a blow to the government's goal of promoting marriage to shore up society's Islamic foundations."

The rise in age of marriage might be a result of religious barriers being raised to courtship between unmarried men and women. But there are other hypotheses to consider:

"Many blame economic circumstances for their failure to marry, citing high inflation, unemployment and a housing shortage along with cultural traditions that expect brides' families to provide dowries and husbands to commit themselves to mehrieh, an agreed cash gift."

"However, Hojatoleslam Ghasem Ebrahimipour, a sociologist, told Shabestan news agency that the trend was due to the availability of premarital sex, and feminism among educated women. "When a woman is educated and has an income, she does not want to accept masculine domination through marriage," he said."

Job market for couples

Two-career households are an important part of the labor force, particularly in academia. Hiring them can be part of a department or school's strategy. See Law Schools Hiring Faculty Couples, from Brian Leiter's Law School Reports.

(Note to students: we'll be covering couples in the medical labor market in either lecture 2 or 3....see
Roth, A.E., "The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory," Journal of Political Economy, 92, 1984, 991-1016.
and
Roth, A. E. and Elliott Peranson, "The Redesign of the Matching Market for American Physicians: Some Engineering Aspects of Economic Design," American Economic Review, 89, 4, September, 1999, 748-780.