#medstories #intraining

If you had to pick a month that made the most sense with this article’s title, I’d argue that with Valentine’s Day and the romantic festivities that come with it, February would be the clear winner. And, in all honesty, though I intended to publish this post in February, I felt it fitting to delay by a day because, for me and approximately 37,000 other medical students who likely applied through the National Residency Matching Program (NRMP) for residency this year, March is Match.

March is the month that most fourth year medical students build their calendar around. It is the month when we find out where we’ll be spending the next 3-6 years of our lives, and which program will introduce us to new lifelong friendships and a new professional family. During the month of March, Zillow likely has record numbers of medical students searching for places to live in the cities they hope to call home, program’s websites likely have repeat views from students hoping a program will like them too, and significant others start to consider what their lives and careers may look like once all is said and done.

So, let’s talk about this process that holds at its behest so many futures.

The Match is a process that gives students and programs a chance to meet one another in a speed-dating sort of dance and allows for them to choose their favorite dance partners at the end of the event. The host of the event, the NRMP, then employs an algorithm to figure out how to get everyone a match as high up on their list as possible. The Nobel Prize-winning algorithm was first introduced by Lloyd Shapley, who used its theory to express a solution to the Stable Marriage Problem. He explained that it was indeed possible through this method to create stable marriages between heterosexual individuals in a society despite people’s complex preferences for a mate.

Alvin Roth, who won the Nobel Prize in 2012 for his work along with Shapley, built on this by actualizing the theory in economic markets, most notably the Residency Match. Before the Match, the process of getting into residency worked a lot like getting a first job — students applied and interviewed and hospitals would try to offer slots to applicants they liked before other programs did. This was often accompanied by a time crunch, leading medical students to choose a site of training before having an opportunity to consider the other options that may be available to them.

Using the match theory proposed by Shapley, we’ll try to explain the Match in terms of the Stable Marriage Problem:

In this scenario, every unengaged woman in a cohort can propose to her first choice man. The man can then provisionally accept engagement until a proposal that he deemed better came along, at which point the woman he was provisionally engaged to could propose to her second choice man as her new “first choice.”

At no point would any person know what number choice they were on the other person’s list of preferences.

Here are our marriage rank lists (W=Woman, M=Man):

WA: 1) MA, 2) MB, 3) MC

WB: 1) MB, 2) MA, 3) MC

WC: 1) MA, 2) MC, 3) MB

MA: 1) WB, 2) WC, 3) WA

MB: 1) WC, 2) WA, 3) WB

MC: 1) WB, 2) WA, 3) WC

In this case, WA proposes to MA, who provisionally accepts since she is on his list. The two are provisionally matched.

WB proposes to MB, who provisionally accepts since she is on his list. The two are provisionally matched.

WC then proposes to MA, who is also first on her list. Since WC is higher on MA’s list than WA, he would jump ship and break his provisional match to WA. Instead, MA would be matched with WC.

WA, now left without a partner, proposes to her second choice (MB) as her new “first choice.” Since WA is higher on MB’s list than WB, MB would break his provisional match with WB and would instead be matched with WA.

Because of this shuffle, WB is left without a partner and proposes to her second choice (MA) as her new “first choice.” On MA’s list, WB is above his current match. MA, therefore breaks his provisional match with WC, and is provisionally matched with WB.

This switch leaves WC without a match, so she proposes to her second choice (MC) as her new “first choice.” MC has not been matched yet and has ranked WC. Therefore, he is matched with WC.

After all is said and done and the shuffling is over, the pairings are absolute — WA is matched with MB; WB is matched with MA; WC is matched with MC.

All of this shuffling and confusion keeps the process from penalizing people for ranking one person over another. In the example above, we see that a woman who ranks a man second is paired with him over a woman who ranks him first if two things happen: 1) the woman who ranked him second did not match with her first choice and 2) the man prefers the woman who ranked him second over the woman who ranked him first.

It also slightly favors women over men. If the algorithm were done in reverse, with men proposing to women, two men would have matched with their #1 choice, and one with his #2, while two women would have matched with their #3 and one with her #2. As it stands now, all the women matched with their #2, while one man matched with his #1, one with his #2, and one with his #3.

To apply the above scenario to the residency match is easy — think of the women as applicants and the men as programs. WA becomes Applicant A, MA becomes Program A, and so forth through the list. In this translation, every applicant matched at their #2 program. Program A got its #1 applicant, Program B got its #2 applicant, and Program C got its #3 applicant.

I want to acknowledge that this is simplistic for three main reasons:

  1. Every rank list is a different length — like if the number of dates offered created variance in the number of prospectives a man or woman could propose to in the Stable Marriage Problem
  2. Programs have multiple slots to fill before they start declining proposals — polygamy if applied to the Stable Marriage Problem
  3. There are more applicants than slots available, with 37,103 applicants competing for 33,167 slots in 2018like if there wasn’t a 50–50 gender ratio in the Stable Marriage Problem

But the example described illustrates what is at the heart of the black box we all call the Match and the most common reason why applicants don’t match: they do not rank enough programs. If WA had not ranked MC, they would not have matched and MC would go unfilled. They same problem occurs when programs do not rank enough candidates. If MC relied solely on WB, it would have been unfilled and WA would not have matched.

These programs and candidates find each other in a much more disorganized “scramble,” called the SOAP, during which candidates are offered positions or passed on after phone calls with programs. It’s a nightmare shared by both programs and medical students, but it can also match hopefuls to programs they never considered before and introduce programs to candidates they may have been missing. No one wants to SOAP, but the best piece of advice I have been given in my career so far is “plan for the worst, hope for the best.” Every medical student should at least know that there is the possibility of “scrambling,” however improbable.

We’ve worked very hard to get to this point in our lives and Match Day will most definitely be an unforgettable experience. We’ll reap the rewards of a long journey and start a new adventure even more arduous than the last. This process not only signifies an end, but also a new beginning. I wish my colleagues and future co-residents the best of luck this month.

After dating our programs of choice and setting our rank lists, finally all that’s left to do…is match :)

Pavitra Krishnamani is a medical student with a background in global health interested in innovating how we deliver healthcare to our patients at home and abroad. To learn more about her and her work, check out her website.

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