Matching and Allocation in Medicine and Health Care

Alvin E. Roth
George Gund Professor of Economics and Business Administration
Harvard University
  (this version June 16, 2003)

Presented at the

National Academy of Engineering

Engineering and the Health Care System Workshop

March 10th, 2003


Published as

Roth, Alvin E. �Matching and Allocation in Medicine and Health Care,� Building a Better Delivery System: A New Engineering/Health Care Partnership,� National Academy of Engineering and Institute of Medicine , National Academies Press, 2005, 237-239


         Many of the previous speakers have considered hospitals as analogous to factories.  But, unlike factories, hospitals are highly decentralized, and many of the important decision makers, including doctors (not to mention patients), aren�t employees of the hospital; they come to the hospital on their own patient-care missions, and they have their own objectives.  

       To efficiently allocate resources to serve these different objectives, it becomes necessary to elicit information from the people who have it.  But eliciting information isn�t always simple, because the information we can elicit depends in part on what we plan to do with it.  When you ask me about something I know, I want to ask you why you are asking.  What you intend to do with my answers will influence the way I answer you.

        That is, what information we can reliably obtain depends in part on how we use it, and what incentives this gives the people from whom we must get the information.  My own most relevant experience of these issues in a medical context comes from redesigning the resident match, so I�ll start my discussion there.  Then I�ll suggest how similar �strategic� issues might arise in organ transplantation, scheduling operating rooms, etc.


       Hospitals only began offering internships about a hundred years ago. Typically, a student graduated from medical school, then looked for a job at a hospital.  By the 1920s, interns had become a significant part of the labor force in hospitals; and internships had become an important part of the career path of doctors.  Hospitals began to try to get good interns by hiring them a little bit earlier than their competitors.     Gradually, hiring began earlier and earlier, and by 1945, hospitals were hiring medical students as early as the end of their sophomore year of medical school, for internships that would begin only after graduation. As a result, residents were being hired so early in their education that it was very hard for residency programs to distinguish the best candidates, or even for candidates to be sure what kind of residency program they would be interested in.  In 1945, medical schools intervened by refusing to release any student information before a certain date�no transcripts, no letters of recommendation, no confirmation that a student was in good standing in medical school. It may have been risky to hire someone just on the basis of sophomore-year grades, but it was even riskier to hire someone just because he said he was a medical student�  So, this intervention was successful at controlling the dates of appointment, and as this became apparent, the date of appointment was successfully moved later, into the senior year of medical school, when more information about students� abilities and preferences was available for finding appropriate matches of students and hospitals.

       But, between 1945 and 1950, a new problem appeared.  In 1945, hospitals were all supposed to wait until a given day to make offers, and give students ten days to accept or reject those offers.  What happened?  Consider a student who got an offer from his third-choice hospital, and had ten days to decide. Suppose that student also heard from his first- or second-choice hospital, saying they like the student, but were not making an offer yet; the student had been placed on a waiting list in case some of the offers they had made were rejected. So, the student waited, which was easy to do because he had ten days to decide about the offer from his third choice hospital.  If all students waited those ten days, the waiting lists didn�t move, and on the tenth day bad things happened. The student might have accepted his third-choice offer and then, later in the day, received a more preferred offer.  The student might have accepted that too.  If, after even only a modest delay to gather up his courage, he informed his third choice hospital of his change of heart, students whom that hospital would have liked to hire may have already committed to other hospitals.  (Obviously the hospital�s problem could be even worse if a long time passed before they realized they had an unfilled position.) On the other hand, even if the student felt honor bound to decline a late, more preferred offer, he might have spent the next year very unhappily at his third choice hospital, explaining to all his colleagues why a talented doc like him shouldn�t have been working in a place like this.  Either way, there was a lot of unhappiness.

   Given that all these troubles had occurred on the tenth day, in 1946 hospitals agreed to allow only eight days for offers to remain open.    As you might imagine, this didn�t solve the problem.  By 1949, residency programs were giving exploding offers�students had to accept or reject immediately, without knowing what other offers might be forthcoming.  So, once again, decisions were being made without all the information that might be available.

      In the early 1950s, a radical innovation was tried--a centralized clearinghouse.  Graduating medical students submitted to the clearinghouse a list, in order of preference, of the residency programs at which they had interviewed.  Residency programs similarly ranked students they had interviewed. These rank order lists�that is, this information elicited from the participants in the market for residents�were than used to match students to residency programs.  And although this system has evolved over the years to take account of changes in the medical marketplace, it has survived to the present day in something close to its original form, as the National Resident Matching Program. (I had the privilege of directing the most recent redesign of the matching algorithm.) 

     The surprising thing that was observed in the 1950�s is that most positions were filled as matched: that is, students and residency programs submitted their rank order lists, and then went on to sign the employment contract suggested by the match.  We now understand that this wasn�t inevitable, but came about because the match algorithm that was chosen in the 1950�s produced matches that were stable, in the sense that there were never �blocking pairs� consisting of a student and a residency program who were not matched to one another but who would both have preferred to be matched to one another rather than matched to their actual partners.

     It is easy to see in principle why i a clearinghouse that produces unstable matches might not succeed.  A student who receives a match with her third choice hospital, for example, only has to make two phone calls to find out if she is part of a blocking pair.  She calls her first and second choice hospitals, and says �before I accept my match outcome, I just wanted to check if you might have a position for me.�  If she is part of a blocking pair, then one of the hospitals will see that they prefer her to someone with whom they are supposed to match.  They might say something like, �by chance we have an extra position��and then call up the candidate they liked less and say they�ve had a budget shortfall and are one position short.  But if the match is stable, when the hospital looks at the list of people with whom it is supposed to match, it sees that it would prefer to go ahead with the match.       To put it another way, if the match is stable, no candidate can find a hospital that she would prefer to go to that is willing to take her.

     One way the importance of stability became evident had to do with the growing number of couples graduating from American medical schools who wanted to find two residency positions in the same city.  The number of couples increased in the 1970�s, as medical schools stopped being overwhelmingly male.  An attempt was made to accommodate couples by allowing them (after being certified by their dean as a �genuine� couple) to indicate that they wished to be matched to residencies in the same city.  They then each submitted a rank order list, as if they were single, except that they were asked to specify one member of the couple as the �leading� member.  The leading member went through the match as if single, and the rank order list of the other member was then edited to remove options that were not in the same city as the residency to which the leading member had matched. 

      Although this procedure did give couples two jobs in the same city, many couples started to find their residencies outside of the match, and it is easy to see why.  Suppose that my wife and I have as our first choice two particular, good positions in Boston .  Our second choice would be to get two particular positions in New York .  If instead we get one good job in Boston and one bad one, we�re not going to be very happy (because of the Iron Law of Marriage, which says you can�t be happier than your spouse).   So an instability may exist: when we call the two residency programs in New York , they may be happy to take us, which now leaves the Boston jobs unfilled, and some people who were matched to the New York jobs scrambling to find new ones.

       So, a failure to elicit the right kind of information (the preferences of couples) contributed to a decline in the effectiveness of the match, by giving couples incentives to circumvent the clearinghouse.  The present match deals with that by allowing couples to submit rank order lists of pairs of positions.  Last year about 550 couples (1100 people) participated in the match as couples.


      Another way in which the importance of stability became clear was through the experience of British doctors.  In the 1960�s, the British began to experience the same kind of troubles the American medical market had experienced before 1945.  But in Britain , different regions of the National Health Service adopted different kinds of centralized clearinghouses.  Some produced stable matches and some did not. The stable systems are still working; but most of the unstable ones failed, sometimes quite dramatically, even though the National Health Service can mandate that jobs must be filled through the centralized clearinghouse.

     But participants learned to circumvent unstable clearinghouses.  In the Birmingham area, for example, after a few years, the majority of the rank order lists submitted by students contained only a single position, hospital programs in turn listed only the students who listed them in this way.  In other words, by the time the lists were submitted, the matching of students to positions had already been determined privately, in advance, by the parties, and they wrote each other�s names down and that was that.  That is, people can often find ways to circumvent even compulsory systems, if they have incentives to do so.   In contrast, stable mechanisms that do not give people incentives to get around them can function efficiently for years.

        Before I move on to topics more directly related to patient care, let me just mention that no design of an allocation or matching system can be successful unless it is first adopted for use.  So part of the design process is the adoption process. The question of how radical changes are adopted is ultimately political. Those who want to see their work implemented need to understand the objections to it, the fears it may arouse, and what constituencies are concerned.  Because complex  systems in which information is decentralized are subject to being gamed, and circumvented, these �political� concerns need to be addressed carefully.

   What are the lessons of these kinds of matching processes for allocation issues more directly concerned with patient care?  People don�t get sick because of incentives, so you might think that incentives, which are such a big deal in labor markets, won�t play a big role in allocation decisions directly concerned with health care.

        But consider organ transplants.  There are about 80,000 candidates on various waiting lists for organs.  Last year, about 22,000 organs were harvested from 11,000 donors. There is scarcity here and real questions about allocation. Over time, the United Network for Organ Sharing has made many modifications in the system allocating these scarce organs.  There are waiting lists, with priority based on criteria such as time on the list and  current health.

        While the details of the allocation rule will certainly effect who gets which organ (or who gets an organ and who does not), it might not be clear how the incentives created by different allocation rules can affect the overall efficiency of allocation.  To get an idea of this, consider the case of pediatric heart transplants. 

Congenital heart defects can now be discovered in utero.  When priority started to be given to patients with greater time on the waiting list, pediatric cardiologists began to put their patients on the waiting list while they were still in the womb.  If a heart became available before the pregnancy was full term, it was often nevertheless in the patient�s interest to perform a  C-section, so that the baby would get the heart.  That meant that donor hearts started going into babies who were not full term and were lower birth weight, which isn�t good for the overall survival rate.  Now the system has been modified so that fetuses can be on a waiting list, but in a different category than already born pediatric patients. But giving more priority to time on the waiting list changed the incentives of pediatric cardiologists, and changed the flow of hearts into babies in an unanticipated and not necessarily positive way.

This brings me back to the game theory observation with which I began--when agents have different objectives (e.g. when each doctor is concerned with managing his own patients), how information is used to make allocations affects the incentives of those who have the information in ways that can alter the allocations in unintended ways. Many aspects of the allocation process for organs involves these issues, from the debate over regional versus national waiting lists, to the priorities that should be given to different kinds of candidates (e.g. chronic versus acute illness). And patients, as well as doctors, can act strategically based on their incentives, as when a given patient may be able to place himself on multiple regional lists, for example.

Similarly, other medical allocation issues involve information that must be elicited from interested participants.  For instance, one of the big issues in scheduling an operating room that is used by many surgeons is how long a given operation will take.  How an operation is described can influence its estimated duration, which in turn influences what resources it is allocated.  To make appropriate allocation and  scheduling decisions, it is first necessary to elicit information, and what information is delivered depends on how that information will be used.

This is of course a common issue in markets.  And because doctors often run their own businesses, the business of the hospital interacts with the business of the market.  So we need to remain aware that anything done inside a hospital interacts with all the other things that go on in the medical marketplace outside of the hospital.

In summary, to do allocation well, information is needed.  When information is decentralized, it still must be found. One of the things that makes systems in which information is decentralized different from those in which it is centralized is the importance of incentives, and the constraints that incentives put on what can be done.  In the medical market for residents, there is a lot of evidence to support the contention that the stability constraint is binding.  As we start to think about how to elicit information to make allocation decisions in other systems, we will have to pay attention to the incentive constraints.

Related papers

Matching docs to jobs:

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.

Roth, A.E., "New Physicians: A Natural Experiment in Market Organization," Science, 250, 1990, 1524-1528.

Roth, A.E., "A Natural Experiment in the Organization of Entry Level Labor Markets: Regional Markets for New Physicians and Surgeons in the U.K.," American Economic Review, vol. 81, June 1991, 415-440

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.

Roth, Alvin E., "The origins, history, and design of the resident match," JAMA. Journal of the American Medical Association, vol. 289, No. 7, February 19, 2003, 909-912.

Organ transplantation:

Michaels, Marian G, Joel Frader, and John Armitage [1993], "Ethical Considerations in Listing Fetuses as Candidates for Neonatal Heart Transplantation," JAMA. Journal of the American Medical Association, January 20, vol. 269, no. 3, 401-403.

Roth, Alvin E., Tayfun S�nmez, and M. Utku �nver, "Kidney Exchange," Quarterly Journal of Economics, 119, 2, May, 2004, 457-488. (Originally published as NBER Paper w10002, September 2003).

(There have been several news stories about the work on implementing kidney exchange in New England that Sonmez and Unver and I are doing with local surgeons and tissue typing experts, especially Frank Delmonico and Susan Saidman. The Boston Globe ran a front page story called "Cross-donor system planned for region's kidney patients,"" June 5, 2004, and The Wall Street Journal ran a story on page B1 titled "Easing the Kidney Shortage," June 17, 2004. The SIAM News ran a slightly more technical story.)

The National Science Foundation has a story highlighting the basic research underlying the organization of kidney exchange, "Kidney Exchange: A Life-Saving Application of Matching Theory."

Roth, Alvin E., Tayfun S�nmez, and M. Utku �nver, "Pairwise Kidney Exchange," Journal of Economic Theory, forthcoming 1. (Originally published as NBER Paper w10698, August 2004.)