The Match Algorithm

CaRMS uses a globally recognized and award-winning algorithm to match students into postgraduate medical training programs throughout Canada. The Match Algorithm is licensed from the Canadian-based National Matching Services Inc. (NMS) and has been used to conduct medical residency matches in North America for over 50 years. The Match Algorithm, known as the Roth-Peranson algorithm, was designed by Alvin Roth and NMS President Elliott Peranson and was key to Roth winning the Nobel Prize for economics in 2012.

What information does the Match Algorithm use?

The Match Algorithm requires two kinds of data – (1) the number of available positions, or quota, for each program in the match and (2) applicant and program rank order lists (ROLs).

Postgraduate Medical Education (PGME) offices determine which programs participate in the match and assign a number of available positions (quota) to each participating program. PGME offices also have the opportunity to provide CaRMS with instructions for moving quota from one program to another any time prior to the running of the match. These reversion instructions can affect the final allocation of program quota in the match.

ROLs are submitted by applicants and programs through CaRMS Online.

How does the Match Algorithm work?

The Match Algorithm compares applicant and program ROLs and matches applicants to programs based on both parties’ stated preferences. ROLs submitted by applicants indicate a list of programs where they wish to train ranked in their order of preference. ROLs submitted by programs indicate a list of applicants they wish to train ranked in their order of preference. The final preferences of applicants and programs, as indicated in their ROLs, determine the match outcome.

The algorithm is applicant-proposing, meaning it starts with an attempt to place an applicant into his or her most preferred program. In this way, the algorithm provides applicants with their best possible outcome based on the ROL submitted. At the end of the matching process, each applicant has either been matched to the most preferred choice possible from their ROL or all choices submitted by the applicant have been exhausted and they have not been matched.

The algorithm at work: Detailed match scenario

Applicant rank order lists

Eight applicants are applying to four programs. After considering the relative desirability of each program, the applicants submit the following rank order lists to CaRMS.

Anderson Beaudry Chen Davis Eastman Feldman Garcia Hassan
 1. City  1. City  1. City  1. Mercy  1. City  1. City  1. City  1. State
 2. Mercy  2. Mercy  2. City  2. Mercy  2. General  2. Mercy  2. City
 3. General  3. State  3. Mercy  3. State  3. Mercy
 4. State  4. General  4. State  4. General  4. General

Applicant Anderson makes only a single choice, City, because he is under the impression from remarks made by the program director that he would be ranked very highly, and he had in turn assured the director that he would rank City number one. It is acceptable for programs to express a high level of interest in applicants to recruit them into their program and for applicants to say that they prefer one program over others. Such expressions, however, should not be considered as commitments.

Applicant Beaudry ranks only the two programs that were desired by every applicant – Mercy and City. As the recipient of a special award in his third year, he feels that he is a most desirable applicant. However, he has not been assured of a match with either of these programs. Applicants should consider ranking all programs that they are willing to attend to reduce the likelihood of not matching.

Applicant Chen ranks City, which she prefers, and Mercy. Standing first in her class in her third year, she knows that she is a desirable applicant and she has been assured by the program director at Mercy that she will be ranked first. Even though she has been assured of being ranked high at Mercy, she prefers City and knows she does not risk her offer at Mercy even if she ranks Mercy further down her list.

Applicant Feldman would be very pleased to be at State, where she had a very good clerkship, and feels that they will rank her high on their list. Although she does not think she has much of a chance, she prefers City, General, or Mercy so she ranks them higher and ranks State fourth. She is using the match to maximum advantage.

Applicant Hassan is equally sure he will be able to obtain a position at State, but he also prefers the other programs. He ranks State first because he is afraid that State might fill its positions with others if he does not place it first on his list. Applicants should rank programs in order of preference. Their choices should not be influenced by speculations about whether a program will rank them high, low, or not at all. The position of a program on an applicant’s rank order list will not affect that applicant’s position on the program’s rank order list, and therefore will not affect the program’s preference for matching with that applicant as compared with any other of the program’s applicants. During the matching process, an applicant is placed into the most preferred program that ranks the applicant and does not fill all its positions with more preferred applicants. Therefore, rank #1 should be the applicant’s most preferred choice.

Applicants Davis, Eastman, and Garcia have interviewed at the same programs. Like the other applicants, they desire a position at City or Mercy and rank these programs either first or second, depending on preference. However, since they are not assured of a match to either of these desirable programs, these applicants also list State and General lower on their rank order lists. They are using the match well.

Summary of guidelines for the preparation of applicant rank order lists

  1. Applicants should rank, in sequence, those programs that represent their true preferences.
  2. Factors to consider in determining the number of programs to rank include the competitiveness of the specialty, the competition for the specific programs being ranked, and the qualifications that the applicant offers.
  3. Applicants should rank all acceptable programs, i.e., programs in which he or she is willing to undertake residency training. Conversely, if an applicant finds certain programs unacceptable and is not interested in accepting offers from these programs, these program(s) should not be ranked.

Program rank order lists

Two positions are available at each program. The four programs, having determined their preferences for the eight applicants, also submit rank order lists to CaRMS.

Mercy City General State
1. Chen 1. Garcia 1. Beaudry 1. Beaudry
2. Garcia 2. Hassan 2. Eastman 2. Eastman
3. Eastman 3. Hassan 3. Anderson
4. Anderson 4. Anderson 4. Chen
5. Beaudry 5. Chen 5. Hassan
6. Chen 6. Davis 6. Feldman
7. Davis 7. Garcia 7. Davis
8. Feldman 8. Garcia

The program director at Mercy Hospital ranks only two applicants, Chen and Garcia, for his two positions, although several more are acceptable. By ranking only two applicants, he runs the risk of having vacant positions. Programs should rank all applicants who they are willing to train.

The program director at State feels that his program is not the most desirable to most of the applicants, but that he has a good chance of matching Feldman and Hassan. Instead of ranking these two applicants at the top of his list, however, he ranks more desired applicants higher. He also ranks all of the acceptable applicants to his program. He is using the match well.

The program directors at City and General have participated in the matching process before. They include all acceptable applicants on their rank order lists with the most preferred ranked high. These program directors are not concerned about filling their available positions within the first two ranks. They prefer to try to match with the strongest, most desirable candidates. They are using the match to maximum advantage.

Each applicant’s rank order list is traversed “downwards,” from most preferred program to least preferred, until the first program to which the applicant can be tentatively matched is reached, or until the applicant’s list of choices is exhausted. Each program accepts applicants “upwards” on its rank order list, continually removing less preferred matches in favour of more preferred applicants, until the program is matched to the most preferred applicants who wish to be matched to the program.

Anderson Beaudry Chen Davis Eastman Feldman Garcia Hassan
Unmatched Unmatched Matched to Mercy Matched to General Matched to City Matched to State Matched to City Matched to State

At the end of the matching process, each applicant has either been matched to the most preferred choice possible, or all choices submitted by the applicant have been exhausted.

Applicant match results show that Anderson and Beaudry went unmatched because they listed too few choices. Applicant Hassan could have matched at City had Hassan ranked choices in order of preference.

Program match results illustrate that Mercy, which ranked only two applicants, and General, which ranked seven out of eight, had unfilled positions. General could have matched with Feldman, who ranked it #2, had Feldman been ranked.