GGame Theoretic Consequences of Resident Matching
Yue Wu ∗ University of Washington
Abstract
The resident matching algorithm, Gale-Shapley, currently used by SF Matchand the National Residency Match Program, has been in use for over 50years without fundamental alteration. The algorithm is a stable-marriagemethod that favors applicant outcomes. However, in these 50 years, therehas been a big shift in the supply and demand of applicants and programs.These changes along with the way the Match is implemented have induced acostly race among applicants to apply and interview at as many programs aspossible. Meanwhile programs also incur high costs as they maximize theirprobability of matching by interviewing as many candidates as possible.
Keywords: resident matching, game theory, Nash equilibrium, BayesianNash equilibrium
C11, C15, C32
1. Introduction
The tech report is organized as follows: Section 2 Background, Section 4Resident matching with no costs, Section 5 Resident matching with costs. ∗ Corresponding author
Email address: [email protected] (Yue Wu)
Preprint submitted to Journal of Econometrics March 17, 2020 a r X i v : . [ ec on . T H ] M a r . Background Residency matching connects graduating medical students with residencyprograms as they pursue further medical training and specialization. Eachyear, over 43,000 US and international graduates apply for 31,000 residencypositions. Unique to medicine, a probabilistic algorithm determines the fateof graduating students.The residency matching algorithm has evolved over time. The first resi-dency programs, then called internships, were introduced in the 1920s as op-tional postgraduate training. These programs initially attracted only a fewmedical graduates, who wished to gain more exposure to clinical medicine.The inadequate supply of interns led to fierce competition among the pro-grams, which manifested as a race between programs to secure binding com-mitments from potential graduates as early as possible [8]. This resulted inmedical students receiving internship offers up to two years before graduation[11]. This process not only disrupted the last two years of medical school, butwas suboptimal on both sides. Students often had to accept or reject offerswithout having heard back from all programs, to which they applied, includ-ing potentially more preferred ones. Programs had incomplete informationand little medical school performance to judge applicant qualifications whenextending offers this early.To counter this, Turner proposed and the medical schools agreed in 1945to not release transcripts nor give letters of recommendation until a later date.However, this did not fundamentally change the process, but simply pushedback the process until the last year of medical school. The programs andapplicants still played the same adversarial game, but on a shorter time scale.2he programs made time-limited offers, while the students delayed acceptinguntil hearing back from more preferred programs. This led programs tosteadily decrease the offer time limits to reduce the risk of their preferredcandidates successfully waiting and accepting alternate offers. So from aninitially agreed offer time limit of 10 days in 1945, the programs decreased itto less than 12 hours in 1950 [10].In order to avoid this race to the bottom between programs, the NationalInterassociation Committee on Internships (NICI) was formed in 1950 to ex-amine existing matching plans and perform a trial run for a centralized matchsystem. Then in October 1951, 79 medical schools formed the National Stu-dent Internship Committee (NSIC) and decided to adopt the Boston PoolPlan [6] nationally upon the recommendations of the National Interassocia-tion Committee on Internships (NICI). This method was modified to be moreequitable to applicants, and then first used nationally in April 1952 [7]. TheNational Internship Matching Program (NIMP), now the National ResidentMatching Program (NRMP), was incorporated on Jan 7, 1953, to manageand administer the matching process.The NRMP algorithm saw only minor changes from 1952 until 1997 [4].It was a stable-marriage algorithm with the programs being the proposingside. Then in 1997, after a year of preliminary study [1], the NRMP decidedto adopt an applicant-proposing version of the old algorithm [9]. In ophthal-mology, the match process is administered by SF Match, which has used anapplicant proposing stable-marriage algorithm since 1977.In the following section, we discuss the matching algorithms in more detailto better understand their strengths and weaknesses.3 . Matching Algorithms
The initial NIMP algorithm matched the applicants to programs accord-ing to the confidential ranked lists submitted by all participants [6]. Theapplicants rank all the programs they would be willing to attend in ascend-ing ordinal order, with 1 being most preferred. The programs group theapplicants by tiers and then rank within tiers. For example, if a programhad n available positions, its first choice applicants would be tier 1 and rankedwithin tier from one to n, and its next tier students would be ranked n+1to 2n, and so on. The algorithm starts by matching students and programsthat each other 1 or most preferred. Then, the programs, which had notfilled their positions, would be matched with students in their second tier,but who had ranked the programs 1. If still unfilled, the programs would benext matched with applicants in their second tier and who had ranked theprograms 2. The process continues iteratively and the can be representedas paired ranks in the following order [(1,1), (2,1), (1,2), (2,2), (3,1), (3, 2),(1, 3), (2, 3), (3, 3), ], where the ordered rank pair is the program tier andstudent rank respectively.Immediately after this initial NIMP algorithm was proposed, objectionsarose as students showed that it might penalize highly qualified students. Forexample, suppose an applicant did not match his first choice, and his secondchoice program had ranked him in the first tier. However, given the orderof the matching, his second choice program might already have filled all itspositions with candidates in its second tier. This leads to a situation whereboth the applicant and the program prefer each other, but are not matched,4nd is an example of an unstable match.
The final adopted NIMP algorithm was modified to avoid these types ofsituations by using deferred matching [7]. It is a stable-marriage algorithm,and is mathematically equivalent to the Gale-Shapley algorithm, which wonthe Nobel Prize in Economics in 2012.The canonical stable-marriage algorithm finds stable matches between agroup of n men and a group of n women, that have ranked each other interms of preference. A match is stable when for a match tuple of man andwoman, denoted (A, B):1. there is no woman D that man A prefers, and who also prefers A toher current match2. there is no man C that woman B prefers, and who also prefers B to hiscurrent matchAlternatively a match (A, B) is not stable if:1. there is a woman D that man A prefers, and who also prefers A to hercurrent match2. Or there is man C that woman B prefers, and who also prefers B to hiscurrent matchThe stable-marriage problem applies not only to matching men and women,but to many other two-sided stable matching problems. The resident match-ing problem can be reworded as a stable marriage problem with the applicants5s one side and the residency programs the other. The only difference is thatresidency programs can match as many applicants as they have residencypositions.For an equal number of participants on each side, who have ranked everypotential partner, it was proved by Gale and Shapley that a stable matchwomen exist [3], and their eponymous algorithm finds a stable solution. Galeand Shapley had originally applied their algorithm to matching colleges withstudents, but this is mathematically equivalent to matching programs withmedical students.The Gale-Shapley algorithm takes as input the rank lists of all the par-ticipants along with a proposing side. Without loss of generality, assumefor the following discussion that programs are the proposing side. Then thealgorithm first selects a program at random from the pool of programs withunfilled positions. The program will first propose to its most preferred can-didate according to its rank list. If that candidate is unmatched, then atentative match is formed with the program, and the algorithm picks an-other program to start proposing. If the candidate is matched, the algo-rithm checks if they would prefer the proposing program over their currentlymatched program. If they prefer the proposing program, then their previousmatch is annulled and they are matched tentatively with the proposer, andtheir previous partner is added back to the proposing pool. The algorithmcontinues until all program positions have been filled.6 . Resident matching with no costs
Consider the general resident matching setup with N applicants, P pro-grams with each program having s , ..., s P available spots, such that thereis a total of S = (cid:80) pi = i s i available spots. In the current process, the appli-cants apply to the programs, and are then either invited for an interviewor rejected. The applicants then rank the programs in strict ordinal orderin terms of their preference. On the other side, the programs also rankthe applicants, specifically their invited interviewees in ordinal preferenceas well. Finally, applicants’ and programs’ rank lists are matched using astable-marriage algorithm favouring the applicant.First, we show that in the current resident matching process it is optimalfor applicants to rank as many as programs as possible when there is no costto playing the ranking game. We prove this by induction starting with asub-game with one applicant, A , and one program α , at the ranking stageof the resident matching process. Then the actions of the applicant for thissub-game are either rank or not rank the program and vice versa for theprogram. The payout scenario is shown below:Applicant A Program α rank not rankrank f A ( r Aα , M ), g α ( ρ αA , M ) f A ( r Aα , M ), 0not rank 0, g α ( ρ αA , M ) 0, 0This two-by-two payout table has applicant actions as rows and programactions columns. The payout is a 2-tuple of applicant payout and programpayout in that order. 7learly, if neither ranked the other then the payoff for both is zero. Notranking in this sub-game is equivalent to non-participation. Let r Aα denotethe rank applicant A assigned program α . Similarly, let ρ αA be the rankprogram α gave applicant A . Next let f A and g α denote the payout functionsof applicant A and program α respectively. For clarity, the payout f A of thecandidate is a function of the candidate’s rank of the program, denoted r Aα ,and the match status M = (0 , g α of the program isfunction of the program’s rank of the applicant, denoted ρ αA , and the matchstatus M = (0 , r ∈ Z + (1) M ∈ (0 ,
1) (2) f :( Z + , (0 , → R (3) g :( Z + , (0 , → R (4)If r > r , (5)then f ( r , M ) ≥ f ( r , M ) , (6)and g ( r , M ) ≥ g ( r , M ) (7)Next note the match status M = (0 , f A ( r Aα , M ) = f A ( r Aα , M = 0) + f A ( r Aα , M = 1) (8) g α ( ρ αA , M ) = g α ( ρ αA , M = 0) + g α ( ρ αA , M = 1) (9)Clearly, if the participant matched, M = 1, then the payout is positive, as8hey found a partner. If they did not match, M = 0, their payout is atworst zero, as their status has not changed and they have not lost anythingby participating in the current game, where there are no participation costs.However, it is possible that the payouts are non-negative even if a match didnot occur. For example, in the one student and one program sub-game, theapplicant or program might find it rewarding to know if the counter-partyranked them or not. In practice, the payout for this information discoveryshould be intuitively much lower than for matching. In this sub-game, ifboth the applicant and program ranked each other, they will be matchedunder the stable-marriage algorithm; M = 1 for both. However, if only sideranked the other there will be no match under the stable-marriage algorithm; M = 0. Therefore the payout table simplifies to:applicant A program α rank not rankrank f A ( r Aα , M = 1), g α ( ρ αA , M = 1) f A ( r Aα , M = 0), 0not rank 0, g α ( ρ αA , M = 0) 0, 0Thus, for the one applicant and one program sub-game, the highest payoutis achieved when both rank each other, as f A ( r Aα , M = 1) ≥ f A ( r Aα , M =0) ≥ g α ( ρ αA , M = 1) ≥ g α ( ρ αA , M = 0) ≥ α and β , but still one participant A . Then there will be two payout tables. First,there is the payout table for applicant A and program α :9pplicant A Program α rank not rankrank f A ( r Aα , M ), g α ( ρ αA , M ) f A ( r Aα , M ), 0not rank 0, g α ( ρ αA , M ) 0, 0Then there is the payout for A versus program β Applicant A Program β rank not rankrank f A ( r Aβ , M ), g β ( ρ βA , M ) f A ( r Aβ , M ), 0not rank 0, g β ( ρ βA , M ) 0, 0The difference in payouts between this game and the previous game 4 is thatthere is no guarantee the applicant and a program match even if they rankeach other. The total applicant payout is the sum of applicant payouts foreach program: f A = max [ f A ( r Aα , M ) ,
0] + max [ f A ( r Aβ , M ) , ≥ f A = f A ( r Aα , M = 0) + f A ( r Aα , M = 1) + f A ( r Aβ , M = 0) + f A ( r Aβ , M = 1)(11)Then if neither program ranked the applicant, this simplifies to: f A = f A ( r Aα , M = 0) + f A ( r Aβ , M = 0) ≥ α or β but not both ranked applicant A , then A willmatch with the program that ranked A . Without loss of generality supposeonly A ’s less preferred program β ranked A , then the payout is: f A = f A ( r Aα , M = 0) + f A ( r Aβ , M = 1) ≥ (12) ≥ f A ( r Aβ , M = 1) > f A ( r Aβ , M = 0)Finally, if both programs ranked A , A would match the more preferred pro-gram α and the payout is: f A = f A ( r Aα , M = 1) + f A ( r Aβ , M = 0) ≥ (13) ≥ f A ( r Aα , M = 1) ≥ f A ( r Aβ , M = 1)and, f A ( r Aα , M = 0) ≥ f A ( r Aβ , M = 0)Clearly, the applicant achieves the highest payout by ranking both programswhile holding the actions of the programs constant.For the programs the payouts are max [ g α ( ρ αA , M ) ,
0] and max [ g β ( ρ βA , M ) , g α ( ρ αA , M = 0) ≥ g β ( ρ βA , M = 0) ≥ A and B and one program α . This is similar to the gamewith one applicant and two programs. The difference is that the payouts ofthe applicants are: f A = max [ f A ( r Aα , M ) , ≥ f B = max [ f B ( r Bα , M ) , ≥ α ’s payout is: f A = max [ g α ( ρ αA , M ) ,
0] + max [ g α ( ρ αB , M ) , ≥ N applicants and P programs, the payout ofapplicant i ’s is: f i = P (cid:88) p =1 max [ f i ( r ip , M ) , ≥ i to always rank a program. Therefore for applicant i , there isa pure strategy where he or she ranks all the programs that maximizes hisor her payout. Similarly, since the payouts for the other applicants j (cid:54) = i ,are similar to Equation ( ?? ), the other applicants will also maximize theirpayout by ranking all programs.In practice, most applicants and programs will not match against eachother so that the pairwise payoff for applicant i and program p is likely f i ( r ip , M = 0) , g i ( ρ pi , M = 0). Next note that under the stable-marriageGale-Shapley algorithm [3] , applicants will match with their highest ranked12rogram that also preferred them to other candidates, until the number ofspots at the programs are filled. So for applicants that matched there willbe a payoff tuple with M = 1, e.g. f i ( r ip , M = 1) , g i ( ρ pi , M = 1).Now consider the program payouts. The payout for a program ι is: g ι = N (cid:88) i =1 max [ g ι ( ρ ιi , M ) , ≥ ι to rank all applicants.By the same logic, it is optimal for all programs to rank all applicants.Therefore, in a framework where applicants and programs can take theactions a) rank or b) not rank without incurring any costs, it is optimal forthe applicants and programs to rank every potential partner. They shouldrank a potential partner even if there is no guarantee that that partner rankedthem. Of course, this framework with no costs is unrealistic in the followingways:1. There is full knowledge all potential partners. This may be true formedical students knowing all potential residency programs, but theconverse is not true. Clearly, residency programs only know the stu-dents that applied to their program.2. Not all applicants apply to all programs because of i) costs of applica-tions or ii) payoff of matching with certain programs is 0.3. There is an interview process where applicants and programs identifymore likely partners. This interview process serves as mutual signalling,but has high costs.In the next section 5, we consider the optimality of ranking and not rankingwith costs. 13 . Resident matching with costs As we showed in the previous section 4, it is always better to rank. Thereare no costs to ranking for an applicant. They can rank all potential resi-dencies online. However, the applicant ranks only have reasonable potentialpayoff if the counterparty likely ranked them. For example, if an applicantranked a program that they did not even apply to, there is almost no chancethat the program ranked the applicant as they would not be aware of theapplicant’s existence. Similarly, an applicant can rank a program that theyapplied to but were rejected for interview. Again it is unlikely, the programranked a rejected applicant or at least not likely that they ranked a rejectedapplicant highly enough that they potentially matched. Therefore, to in-crease the odds of a better potential match and thereby potential payoff, anapplicant should increase the odds of being ranked by the programs. Theycan start by applying to as many programs as possible, resources permitting.The application cost is a step-wise function of the number of programs ap-plied, Ophthalmology application costs. These costs range from $60 for thefirst ten programs to $35 per program after 40 programs. After an applicantpasses the first round filter and is invited for an in-person interview, thereare the costs of attending interviews. This cost will vary depending on theapplicant home city and the program city. Finally, every applicant will havedifferent monetary and time budgets for the whole application and interviewprocess.Incorporating these costs and budgets into a generic applicant i ’s payout14n Equation (17): f i = q i (cid:88) p =1 (E p ( f i ( r ip , M )] − A ( p )) (19)E p [ f i ( r ip , M )] =( p ( I p = 1) × E[ f i ( r ip , M ) | I p = 1] − E[ C p ])+ p ( I p = 0) × E[ f i ( r ip , M ) | I p = 0] (20)subject to q i (cid:88) p =1 ( p ( I p = 1)E[ C p ] + A ( p )) ≤ B i and q i (cid:88) p =1 t p ≤ T i Equation (19) states that the payout of the whole process for applicant i is the sum of the expected payoff for applying to q i programs. The payoutfor applying to the p th program is the expected payout E p [ f i ( r ip , M )] mi-nus the cost of application A ( p ). The term A ( p ) is the cost for applying tothe p th program and is a deterministic step function in resident matching.The expectated payoff of applying to the p th program (20) is the payoff forbeing accepted for an interview, p ( I p = 1), multiplied the expected payofffor attending the interview, E[ f i ( r ip , M ) | I p = 1], minus the expected costof attending the interview E[ C p ] plus the payout if not accepted for inter-view. The number of applications and interviews applicant i can pay for isconstrained by the monetary budget B i and the time budget T i .First, we analyse the effect of applying to programs. Each term in Equa-tion (20) consists of the probability of being accepted for an interview, andthe expected payoff being ranked by program if interviewed or not inter-viewed. We use Bayesian payoffs to estimate the payoff given empirical data[5]. The probability of matching is incremental with each extra interview15 igure 1: Cumulative probability of matching for number of times applicant was ranked increasing the probability of matching to a program. The empirical cumu-lative distribution of matching for an applicant in Ophthalmology given thenumber of interviews is given in Figure 1. The probability of being acceptedfor an interview is approximately 1 / ≈
14% in SF Match data. Now the costfor applying to all 116 Ophthalmology programs in 2019 is $3170. Applyingto all 116 programs is expected to yield 16.6 interviews. The cost per inter-view was estimated to be $404. The total cost of applying to all and thenattending the 16.6 expected interviews is $9,865. This pales in comparison tothe estimated annual average income of $366,000 for an ophthalmologist in2019 [2], and an expected career income of over $10,000,000, even discountedfor present value. The only thing that should stop applicants applying to allprograms is current budget constraints, but $9,865 should not be the limitingfactor for most applicants given the overall cost of medical education. Thisis borne out by the trend in applications in Figure 2.16 igure 2: Average number of applications and interviews over time
References [1] , a. Design review of the national resident matching program. https://stanford.edu/~alroth/nrmp.html . Accessed: 2020-2-4.[2] , b. Medscape log in.