IInterview Hoarding *Vikram Manjunath
University of [email protected]
Thayer Morrill
North Carolina State [email protected]
February 22, 2021
Abstract
Many centralized matching markets are preceded by interviews betweenthe participants. We study the impact on the final match of an increase tothe number of interviews one side of the market can participate in. Ourmotivation is the match between residents and hospitals where, due to theCOVID-19 pandemic, interviews for the 2020-21 season of the NRMP matchhave switched to a virtual format. This has drastically reduced the cost toapplicants of accepting interview offers. However, the reduction in cost is notsymmetric since applicants, not programs, bore most of the costs of in-personinterviews. We show that if doctors are willing to accept more interviews butthe hospitals do not increase the number of interviews they offer, no doctorwill be better off and potentially many doctors will be harmed. This adverseconsequence results from a mechanism we describe as interview hoarding.
We prove this analytically and characterize optimal mitigation strategies forspecial cases. We use simulations to extend the insights from our analyticalresults to more general settings.
Keywords:
NRMP, Deferred acceptance, Interviews, Hoarding
Perhaps the most well known application of matching theory is the entry-level labormarket for physicians. In 2020, 37,256 positions were matched through the Na- * We thank Anna Sorensen and Alkas Baybas for asking us the question that sparked this paper.We also thank Adrienne Quirouet for helpful comments and discussions. a r X i v : . [ ec on . T H ] F e b ional Resident Matching Program (NRMP). The matching process consists of twosteps. First, each physician interviews with a set of residency programs. Second,programs and physicians submit rank-order lists—of only those they interview—to a centralized clearinghouse. This clearinghouse, run by the NRMP, matchesphysicians to residency programs using a version of Gale and Shapley (1962)’sDeferred Acceptance algorithm (Roth and Peranson, 1999).In practice, both programs and applicants are constrained in the number ofinterviews they can take part in. Prior to the COVID-19 pandemic of 2020-21,interviews were done in person. These interviews were particularly costly forphysicians since they not only had to bear travel expenses but had to use scarcevacation days. The cost to programs was mainly in terms of the time it takes.For the 2020-21 matching season, interviews were conducted virtually. While thisdramatically decreased the cost of interviews for physicians, it did not change thecosts much for the programs. We are interested in the implications of this changeon the eventual match.We focus our study particularly on the effects of doctors accepting more in-terview invitations without a corresponding increase to the number of invitationsextended by programs. If some doctors accept more interviews and if the totalnumber of invitations extended does not change, then some of the doctors nec-essarily receive fewer invitations. It seems natural to intuit that at least doctorswith more interviews benefit from the lower costs even if those with fewer inter-views are harmed. However, we show a surprising result: No physician is betteroff when more interviews can be accepted than under the previous arrangement.We prove this for a starting arrangement where the final matching is stable. Weview stability as an equilibrium concept that describes a steady state of a market.The intuition for this result is as follows. Consider a highly sought after physi-cian; one who is offered interviews at the leading programs and who ultimately ismatched with her favorite program. When interviews becomes cheaper, she willaccept more interviews. However, as she would already have matched with herfavorite program, the interviews she accepts are from inferior programs. Theseinterviews do not help her: she ultimately matches with the same program asbefore. The interviews are, in effect, wasted. We refer to this as interview hoard-ing . Interview hoarding has a cascading affect. The physicians who otherwisewould have filled these wasted interview slots now interview with programs theyconsider inferior. These physicians may have more interviews, but they do nothave better interviews in a precise sense: every new interview a doctor has, sherates worse than the program she matched with before. Physicians are ultimatelydivided into three categories: physicians who hoard interview worse than their2ventual match; physicians who receive more but worse interviews; and physi-cians who receive fewer and worse interviews. The first category is indifferentbetween the new costs and the old. The latter two categories are harmed underthe new cost. Thus, when physicians accept more interviews but programs do notreact, the ultimate match is Pareto inferior from the physicians’ perspective. Having shown that increases to doctors’ abilities to accept interviews has ad-verse consequences, we turn to mitigation policies. We consider policies that limitthe numbers of interviews that programs can offer and candidates can accept.Though there are, essentially, no such policies that always (for every preferenceprofile) yield a stable final matching (Proposition 1), we characterize such policiesfor “common preferences” (Proposition 2). These are salient preference profileswhere every doctor ranks the programs the same way and every program ranksthe doctors the same way. The policies we characterize are such that there is acommon cap on the number of interviews any program can offer or candidate canaccept. We also show that if the programs interview capacities are fixed, say at l ,then the number of blocking pairs increases and the match rate decreases as thedoctors’ interview cap gets further away from l (Proposition 3).Our analytical results are suggestive of policies for more general settings wherepreferences are not quite common, but have a common component. We use sim-ulations to show that the lessons from our analytical results hold up under weakerassumptions. Though the optimal cap on doctors’ interview capacities dependson the parameters of the model—and in practice would have to be determinedempirically—our simulations indicate that it is no higher than the number of inter-views that the programs offer. Clearly, the main purpose of interviews is information acquisition. Throughthis channel, having more interviews can benefit a physician. For example, aphysician benefits if by participating in more interviews she learns that a programwhose interview she previously would have rejected is actually her favorite pro-gram. However, our aim is to isolate the effect of congestion in the interview step.We therefore assume that all agents have perfect information regarding their pref-erences.We show that an increase in this congestion can have an adverse impacton performance of the match itself unless either the residency programs or theNRMP reacts. The motivating example in Section 1.2 demonstrates that there need not exist a Pareto rankingfrom the programs’ perspective. This is true whether we define optimality of a policy as maximizing the average proportion ofpositions that are filled or minimizing the average number of blocking pairs. .1 Related Literature While there is a large literature on the post-interview NRMP match, there arerelatively few papers that incorporate the pre-match interview process. One ofthe first to explicitly model interviews in the classic one-to-one matching model isLee and Schwarz (2017). In their model, before participating in a centralized, two-sided match, firms learn their preferences over workers by first engaging in costlyinterviews. They show that even if firms and workers interview with exactly thesame number of agents, the extent of unemployment in the final match dependscritically on the overlap between the sets of workers firms interview. Three otherrecent papers that incorporate pre-match interviews are Kadam (2015), Beyhaghi(2019), and Echenique et al. (2020).Like us, Kadam (2015) considers the implications of loosened interview con-straints for doctors. However, the focus is on strategic allocating scarce interviewslots. For the sake of tractability, the analysis is for a stylized model of large mar-kets. Under the assumption of common preferences, he shows that increasingstudent capacities may increase total surplus, but not in a Pareto-improving way.Moreover, match rate decreases. He also highlights that when preferences are notnecessarily common, the effect is ambiguous since increased interview capacitiesdilutes doctors’ signaling ability.Beyhaghi (2019) also performs a strategic analysis of a stylized large marketmodel. However, she considers a slightly different set up with application capsfor doctors and interview caps for programs. While similar, application caps arenot exactly the same as interview caps: they constrain the number of programs adoctor can express interest in at the outset of the interview matching phase, butnot the number of interviews she can accept at the end. In her model, inequity inthe application caps decreases expected total surplus. Moreover, when interviewcapacity is low, low application caps are socially desirable.In our model, the agents do not choose interviews strategically. Determiningthe optimal set of interviews is closely related to the portfolio choice problem con-sidered by Chade and Smith (2006). They solve for the optimal portfolio whenan agent chooses a portfolio of costly, stochastic options, but ultimately may onlyaccept one of the options. If one were to apply the optimal solution to the inter-view scheduling problem, one would have to pin down precisely the probability ofany given pair matching. This is what makes strategic analysis of the problemintractable without severe simplifying assumptions (such as those in the paperswe have mentioned above). See the multitude of papers following Roth and Peranson (1999). among those theyinterviewed with .Our work is complementary with these works in the sense that they highlightthe importance of understanding the prematch interviews to properly evaluatingthe NRMP match itself.
We present the intuition behind the welfare loss from increased interview capacityfor doctors with a simple example. Consider a market with four doctors { d , ... , d } and four hospitals { h , ... , h } . The agents’ preferences are as follows: d d d d h h h h h h h h d d d d h h h h d d d d h h h h d d d d h h h h d d d d Suppose that the interview capacities of the doctors and hospitals are: d d d d h h h h Interviews are initially offered by hospitals: h invites d , h invites d , h invites d and d , and h invites d and d . As d can accept only one invitation, sherejects h , h , and h . Doctors d and d do not reject their invitations from h and h , respectively. Hospitals h , h , and h then offer interviews to d , d , and d , respectively. Doctor d rejects h ’s invitation. After h invites and is rejectedby d , the final interviews are: 5 d d d h { h , h } h h The final matching is computed by applying the doctor-proposing Deferred Ac-ceptance algorithm to the agent preferences (restricted to agents they interviewwith). The outcome is therefore: d d d d h h h h The interview schedule is well functioning in the sense that the ultimate outcomeis stable with regards to the actual (as opposed to restricted) preferences.Now suppose doctor 1 is able to accept an additional interview (and all other in-terview capacities remain the same). Doctor 1 now accepts h ’s invitation. Doctor2 eventually accepts hospital 4’s invitation, and the interview schedule is: d d d d { h , h } { h , h } h h This leads to the final matching: d d d d h h h Doctor d does not benefit from the additional interview. Since the original match-ing was stable, the interview she adds is with a hospital that she finds worse thanher original match. However, her acceptance of hospital 2’s invitation comes atthe expense of both doctors 2 and 3: both now receive worse assignments. Infact, the final matching is no longer stable. That none of the doctors are better offis not special to this example—we show that this is generally true (Theorem 1).The programs, however, are not unanimously better or worse off: h is better offwhile h is worse off. A market consists of a triple (D, H, P) , where: D is a finite set of doctors ; H isa finite set of hospitals ; and P is a profile of strict preferences for the doctorsand hospitals. We assume that there are at least two doctors and two hospitals: | D | ≥ and | H | ≥ . For each h ∈ H , P h is the set of strict preferences over D ∪ { h } ,6nd for each d ∈ D , P d is the set of strict preferences over H ∪ { d } . The set ofpreference profiles is P ≡ × i ∈ H ∪ D P i .Like Echenique et al. (2020), we combine an interview phase with a matchingphase. The former involves many-to-many matching while the latter is a standardone-to-one matching problem (Roth and Sotomayor, 1990). A matching is a func-tion µ : H ∪ D → H ∪ D such that µ ( h ) ∈ D ∪ { h } , µ ( d ) ∈ H ∪ { d } , and µ ( d ) = h if andonly if µ ( h ) = d . We say that ( d , h ) are a blocking pair to matching µ if h P d µ ( d ) and d P h µ ( h ) . A matching is stable if it does not have a blocking pair.A many-to-many matching is a function ν : H ∪ D → H ∪ D such that ν ( d ) ⊆ H , ν ( h ) ⊆ D , and h ∈ ν ( d ) if and only if d ∈ ν ( h ) .For each h ∈ H , let ι h ∈ N be h ’s interview capacity . Similarly, for each d ∈ D ,let κ d ∈ N be d ’s interview capacity . We call the profile ( ι , κ ) = (( ι h ) h ∈ H ( κ d ) d ∈ D ) the interview arrangement . An interview matching is a many-to-many matching ν such that for every doctor d , | ν ( d ) | ≤ κ d and for every hospital h , | ν ( h ) | ≤ ι h .An interview matching ν is pairwise stable if there is no doctor-hospital pair ( d , h ) such that h (cid:60) ν ( d ) but:• either | ν ( h ) | < ι h or there exists a d (cid:48) ∈ ν ( h ) such that d P h d (cid:48) , and• either | ν ( d ) | < κ d or there exists a h (cid:48) ∈ ν ( d ) such that h P d h (cid:48) . Two-step process:
Given ( ι , κ ) , we call the final matching, which is the culmina-tion of the following two step process, the ( ι , κ ) -matching . In the first step, wherehospitals interview doctors, we ignore the informational effects and focus solelyon congestion. Given P ∈ P :Step 1: Interview matching ν is the hospital-optimal many-to-many stable matchingwhere the capacities of the hospitals and doctors are given by ι and κ re-spectively. This can be computed by applying the hospital-proposing DA:each h ∈ H is matched with up to ι h doctors and each d ∈ D is matchedwith up to κ d hospitals. Since we ignore the informational aspect of theproblem, the input to DA is a choice function for each agent that is respon-sive to her preference relation and constrained by her interview capacity. Hospital-proposing DA is an approximation of the decentralized process bywhich hospitals invite doctors in rounds, extending invitations to further doc-tors when invitations are declined. For the sake of completeness, we define in Appendix A the acceptant and responsive choicefunctions that we appeal to while running DA to compute the interview matching. ( ι , κ ) -matching is chosen by doctor-proposing DA. The input to DA is thetrue preference profile restricted to the interview match, ( P i | ν ( i ) ) i ∈ D ∪ H .Given P ∈ P , we say that ( ι , κ ) is adequate if the ( ι , κ ) -matching at P is stable.If κ is adequate at each P ∈ P , then we say that it is globally adequate . Weinterpret ( ι , κ ) being adequate at a profile P as a sign that the market is functioningwell. Otherwise, a blocking pair could alter their behavior to improve their lot. Inother words, using stability as our notion of equilibrium, ( ι , κ ) being adequate isequivalent to the market being in equilibrium.Finally, we define a welfare comparison between matchings. Given a pair ofmatchings µ and µ (cid:48) , we say that no doctor prefers µ (cid:48) to µ if, for each d ∈ D , µ ( d ) R d µ (cid:48) ( d ) . Our aim is to study how a change in the cost of interviewing impacts a market.The starting point is a market that is at equilibrium. Starting from such an equilib-rium, the goal is to understand the welfare consequences of a shock that permitsdoctors to accept more interviews. That is, starting with P ∈ P and ( ι , κ ) that isadequate at P , we consider an increase to the doctors’ interview capacities to κ (cid:48) and compare the ( ι , κ ) -matching to the ( ι , κ (cid:48) ) -matching.The doctor who accepted more interviews in our example in Section 1.2 didnot benefit from it. Our main result shows that this is true in general. Theorem 1.
Starting at an adequate arrangement, doctors do not benefit fromincreases to their interview capacities. That is, if ( ι , κ ) is adequate at P and κ (cid:48) issuch that, for each d ∈ D , κ (cid:48) d ≥ κ d , then the no doctor prefers the ( ι , κ (cid:48) ) -matchingto the ( ι , κ ) -matching.Proof. Let ν and µ be the interview and final matchings respectively, under ( ι , κ ) .Similarly, let ν (cid:48) and µ (cid:48) be the interview and final matchings under ( ι , κ (cid:48) ) . We framethe temporal language below in reference to a hypothetical change in doctors’interview capacities from κ (“before”) to κ (cid:48) (“after”).We first establish a number of properties of the outcome from the interviewstep. The intuition for these results comes from one of the classical results in two-sided matching theory: When the set of men increases, no man benefits from thisincreased competition while no woman is harmed. In our setting, an increase in See Theorem 2.25 of Roth and Sotomayor (1990).
Lemma 1.
No doctor rejects a hospital it previously interviewed with.Proof.
Suppose not. In the interview matching step (under capacities κ (cid:48) ), let d be the first doctor to reject a hospital h that she interviewed with under capacities κ . As d has at least as much interview capacity, she must have received a newproposal from some hospital h (cid:48) . As h (cid:48) did not propose to d before, it must havebeen rejected by some doctor d (cid:48) ∈ ν ( h ) , a doctor it previously interviewed. Butthis contradicts d being the first doctor to reject a hospital it previously interviewedwith.We cannot say whether a doctor prefers her interviews under κ versus κ (cid:48) aswe only have a doctor’s preferences over individual hospitals and not sets of hos-pitals. However, we show—in a specific sense—that while a doctor may get newinterviews, she does not get better interviews. Lemma 2.
No doctor has a new interview better than her previous matching: if h ∈ ν (cid:48) ( d ) \ ν ( d ) , then µ ( d ) P d h .Proof. Suppose not. Let d be the first doctor when DA is run during the interviewstep under capacities κ (cid:48) to receive a proposal from a hospital h (cid:60) ν ( d ) such that h P d µ ( d ) . As h did not previously propose to d , h must have been rejected by adoctor that it previously interviewed. This contradicts Lemma 1.In the classical result, no man benefits from the increased competition due toadditional men and also no woman is harmed. An analogous result holds in ourframework. A hospital either has the same set of interviews; additional interviews;or she interviews new doctors who she prefers to her previous interviews. In anyscenario, the hospital’s set of interviews (weakly) improves. Lemma 3.
Suppose a hospital h interviews a doctor d under κ (cid:48) . If h previouslyinterviewed d (cid:48) and prefers d (cid:48) to d , then h continues to interview d (cid:48) : if d ∈ ν (cid:48) ( h ) , d (cid:48) ∈ ν ( h ) , and d (cid:48) P h d , then d (cid:48) ∈ ν (cid:48) ( h ) .Proof. As d (cid:48) P h d , h proposes to d (cid:48) before it proposes to d when DA is run in theinterview step under κ (cid:48) . By Lemma 1, h is not rejected by any doctor it previouslyinterviewed. As h proposes to d under κ (cid:48) , it must have already proposed but notbe rejected by d (cid:48) . Therefore, h continues to interview d (cid:48) .9o complete the proof of Theorem 1, we show that if a doctor is rejected by ahospital during the matching step under κ , then she is not matched to that hospitalunder κ (cid:48) . We proceed by induction on the round (of DA in the interview step under κ ) in which the doctor was rejected, and our inductive hypothesis is that if doctor d was rejected by hospital h in round k under κ , then under κ (cid:48) , either she no longerinterviews with h or she is rejected in round k or earlier.For the base step, consider a doctor d that was rejected by hospital h in thefirst round under κ , and let d (cid:48) be the doctor h tentatively accepts. If d no longerinterviews with h ( d (cid:60) ν (cid:48) ( h ) ), then we are done. Therefore, suppose d ∈ ν (cid:48) ( h ) .By Lemma 3, since h prefers d (cid:48) to d and it interviews d , it also interviews d (cid:48) ( d (cid:48) ∈ ν (cid:48) ( h ) ). Doctor d (cid:48) does not have any new interviews with a hospital it prefersto h since h R d (cid:48) µ ( d (cid:48) ) and by Lemma 2 she does not get a new interview with ahospital she prefers to µ ( d (cid:48) ) . Therefore, d (cid:48) continues to propose to h in the firstround even under the new capacities and d continues to be rejected by h in favorof d (cid:48) or possibly a doctor h prefers even more.To complete the inductive argument, suppose that doctor d was rejected byhospital h in favor of doctor d (cid:48) in round k under κ . If h (cid:60) ν (cid:48) ( d ) , then we are done.Otherwise, again by Lemma 3, d (cid:48) ∈ ν (cid:48) ( h ) . Under κ , d (cid:48) proposes to h in round k or earlier. Therefore, d (cid:48) was rejected by all hospitals she interviewed with andprefers to h in an earlier round. By the inductive hypothesis, for any hospital h (cid:48) that rejected d (cid:48) under κ , either d (cid:48) no longer interviews with h (cid:48) or h (cid:48) has alreadyrejected d (cid:48) by round k under κ (cid:48) . Therefore, under κ (cid:48) , either d (cid:48) proposes to h inround k or in a previous round. In either case, by round k , under κ (cid:48) , h has alreadyreceived a proposal it prefers to d . Therefore, doctor d will be rejected by hospital h under κ (cid:48) in round k or earlier.This shows that if d was rejected by hospital h under the old capacities, then d is not matched to h under the new capacities. Note that d has no new interviewswith a hospital she prefers to µ ( d ) . Therefore, if h P d µ ( d ) and h ∈ ν (cid:48) ( d ) , then h ∈ ν ( d ) and h rejected d in some round under the old capacities. Therefore, h also rejects d under the new capacities. In particular, under the new capacities, d is not matched to a hospital she prefers to µ ( d ) .Theorem 1 tells us that doctors increasing the number of interviews they ac-cept will either have no impact on the resulting matching or will make the newmatching Pareto worse from the doctors’ perspective. The example in Section 1.2illustrates that there are instances where increasing the interview capacity doesresult in a Pareto inferior outcome. Note that this example is not pathological.Lemmas 1 and 2 demonstrate the root cause of the inferior match. Doctors at the“top” of the market—those that are highly sought after—add interviews but every10ew interview they accept is with a hospital that is worse than the one they willeventually be assigned to. This is what we refer to as interview hoarding . A poorfinal matching occurs when a hospital’s interviews are all accepted by doctors whowill eventually reject it.Our conclusion from Theorem 1 is that the impact of virtual interviews for the2020-21 season of the NRMP, if the residency programs do not react sufficiently,will be a lower match rate. We expect the most highly sought after physicians tohave more interviews than usual while less demanded physicians will have few orno interviews. It is essential that residency programs either increase the numberof interviews they conduct or rethink their strategies for offering interviews. Theorem 1 assumes that the initial profile of interviews was adequate in the sensethat the outcome of the two-step process is a stable matching. We interpret thisassumption as a characteristic of a well-functioning market in steady state equi-librium. A natural question is how many interviews need to take place and whatdoes the distribution of interviews need to be in order for an interview profile to beadequate. Of course, in general, the answer will depend on characteristics of themarket such as the ratio of doctors to hospitals and how correlated or aligned pref-erences are. However, we are able to provide tight characterizations for certain“end-point” cases which provide intuition for more general markets.
In studying adequate arrangements, we first ask about worst case performance:what arrangements are adequate for every preference profile? It turns out thatonly very extreme arrangements satisfy this property. We characterize these ar-rangements in our next result.
Proposition 1.
Arrangement ( ι , κ ) is globally adequate, if and only if either1. every doctor and every hospital has only unit interview capacity—that is, foreach d ∈ D , κ d = 1 and for each h ∈ H , ι h = 1 —or2. every doctor and every hospital has high interview capacity—that is, for each d ∈ D , κ d ≥ min {| D | , | H |} and for each h ∈ H , ι h ≥ min {| D | , | H |} . roof. We first prove necessity. Suppose that ( ι , κ ) is globally adequate.We start by establishing that if one doctor or hospital has greater than unitinterview capacity, then every doctor and hospital has interview capacity of atleast two. Stated differently, if any doctor or hospital has unit capacity, then alldoctors and hospitals have unit capacity. We denote by ν the interview matchingand by µ the ( ι , κ ) -matching. Claim 1.
1. If there is d ∈ D such that κ d > , then for each d (cid:48) ∈ D , κ d ≥ andfor each h ∈ H , ι h ≥ , and2. If there is h ∈ H such that ι h > , then for each h (cid:48) ∈ H , ι h ≥ and for each d ∈ D , κ d ≥ .Proof. We prove only the first statement as the proof of the second statement isanalogous—it requires only a reversal of the roles of doctors and hospitals.Suppose, for the sake of contradiction, there is a globally adequate ( ι , κ ) wherethere exists a d ∈ D such that κ d > and a h ∈ H such that ι h = 1 . Let h ∈ H \ { h } and d ∈ D \ { d } . Consider P ∈ P where each doctor ranks h firstand h second, and each hospital ranks d first and d second. All hospitals offeran interview to d and as κ d > , d accepts interviews from at least h and h .Since h = 1 , h only interviews d . Let µ be the ( ι , κ ) -matching. Since ( ι , κ ) isadequate, µ is stable, so µ ( d ) = h , as h and d are mutual favorites. Therefore, µ ( h ) = h as h only interviews d . Note that ( d , h ) forms a blocking pair of µ as h P d µ ( d ) , since µ ( d ) (cid:60) { h , h } , and d P h h . This contradicts the stabilityof µ and thus the assumption that ( ι , κ ) is globally adequate. We have thereforeestablished that if there is d ∈ D such that κ d > , then for each h ∈ H , ι h ≥ .We now prove that if there d ∈ D such that κ d > , then for each d ∈ D , κ d ≥ .Suppose for the sake of contradiction, that there is d ∈ D such that κ d = 1 . Let h , h ∈ H . Consider P ∈ P such that each doctor ranks h first and h second, andeach hospital ranks d first and d second. As we have shown above, ι h , ι h ≥ ,so both h and h offer interviews to both d and d . Since h is her favoritehospital, d accepts its offer. Thus, ν ( d ) = { h } . However, µ ( d ) = h since d and h are mutual favorites, so µ ( d ) = d . This means that ( d , h ) form a blockingpair of µ as the only hospital d prefers to h is h . This contradicts the stability of µ and thus the assumption that ( ι , κ ) is globally adequate.We complete the proof of necessity by showing neither a doctor nor a hospitalcan have an intermediate capacity. Claim 2.
There is no d ∈ D such that < κ d < min {| D | , | H |} , and there is no hospital h such that < ι h < min {| D | , | H |} . roof. We prove this statement for the case where | D | ≤ | H | . The proof when | H | < | D | is symmetric.Suppose for contradiction that d ∈ D is such that κ d = k where < k < | D | .Let P ∈ P be such that for i from 1 through k + 1 : P d : h , h , ... , h k +1 , h , ... P h i : h i , h , ... , h i − , h i +1 , ... P h : d , d , ... P h i : d i , d , ... , d i − , d i +1 , ... We have constructed the preference profile P such that:• For each i from 1 through k +1 , d i and h i are matched in every stable match-ing.• Each of the k + 1 hospitals h , ... , h k +1 offers d an interview.• Doctor d accepts interview offers from hospitals h , ... , h k +1 , but not from h .The first and third points are immediate consequences of the preferences. Thesecond is a consequence of the first part of Claim 1: since κ d > , every hospitalhas interview capacity of at least two and d is its second favorite doctor. However,this contradicts the definition of µ as the ( ι , κ ) -matching, since h (cid:60) ν ( d ) yet bystability, h = µ ( d ) .A similar construction shows that there is no h ∈ H such that < ι h < | D | .Suppose for contradiction that h ∈ H is such that ι h = l where < l < | D | . Let P ∈ P be such that for i from 1 through l + 1 : P d : h , h , ... P d i : h i , h , ... , h i − , h i +1 , ... P h : d , d , ... , d l +1 , d P h i : d i , d , ... , d i − , d i +1 , ... By the second part of Claim 1, since ι h > , every doctor has capacity of at leasttwo. Therefore:• For each i from 1 through l + 1 , d i and h i are matched in every stable match-ing.• Each of the l doctors d , ... , d k +1 accepts an interview from h an interview.13 Hospital h does not offer d an interview.Thus, h (cid:60) ν ( d ) , so h (cid:44) µ ( d ) . This contradicts the stability of µ , the ( ι , κ ) -matching, and in turn the assumption that ( ι , κ ) is globally adequate.We now turn to sufficiency. If every agent has an interview capacity of one,then the interview matching is actually a matching. Moreover, it is a stable match-ing. So, suppose that each agent has an interview capacity of at least min {| D | , | H |} .If | D | = | H | , then the interview matching involves an interview between every mu-tually acceptable doctor-hospital pair. This means that the ( ι , κ ) -matching is thedoctor optimal stable matching under the unrestricted preferences, which is sta-ble. We now show, that even if | D | < | H | or | D | > | H | , the ( ι , κ ) -matching, µ , isstable. Suppose the doctor-hospital pair ( d , h ) blocks µ . By defintion of µ as the ( ι , κ ) -matching, if h P d µ ( d ) and d P h µ ( h ) , then h (cid:60) ν ( d ) .Suppose | D | < | H | . Since ι h ≥ | D | , h would have offered an interview to d and have been rejected during the interview matching step, so ν ( d ) contains κ d hospitals that d prefers to h . Since h P d µ ( d ) , and µ ( d ) ∈ ν ( d ) ∪ { d } , this means µ ( d ) = d . Then, d is rejected by every hospital in ν ( d ) during the application ofDA in the second step. However, | ν ( d ) | = κ d ≥ | D | and since d is acceptable toevery hospital in ν ( d ) , she is only rejected when another doctor applies. However,this implies that when DA terminates in the second step, every hospital in ν ( d ) hastentatively accepted some doctor other than d , which is a contradiction—there arenot enough such doctors.Suppose | H | < | D | . Since κ d ≥ | H | , d does not reject any interviews she isoffered. Since h (cid:60) ν ( d ) , h offers interviews to and has them accepted by ι h ≥ | H | doctors whom it prefers to d . Since d P h µ ( h ) , h does not receive a proposal fromany d (cid:48) ∈ ν ( h ) during the application of DA in the second step since it finds all such d (cid:48) better than d . This implies that each d (cid:48) ∈ ν ( h ) is tentatively accepted by somehospital other than h when DA terminates, which is a contradiction—there are notenough such hospitals.Proposition 1 highlights a previously overlooked role that the interview stepplays in determining whether or not the ultimate NRMP match is stable. While in-terviews are necessary for agents to gain information, we learn from Proposition 1that interviews can also act as a bottleneck. Even with complete information, onceany agent is capable of participating in more than one interview, all agents mustinterview with essentially the entire market to be certain that the ultimate match isstable. 14 .2 Homogeneous Arrangements The distribution of interviews is an essential factor in the stability of the NRMPmatch. So, it is natural to consider market interventions when the interview stepis out of balance. Our motivating question is what happens when there is anincrease to the number of interviews doctors can accept. The most straightforwardintervention is to cap the number of interviews an agent can participate it. Herewe consider a homogeneous arrangement : all doctors face the same cap andall hospitals face the same cap, but we allow the doctor and hospital caps topotentially differ. In other words, the arrangement would be described by twonumbers: an interview capacity l ∈ N for hospitals and an interview capacity k ∈ N for doctors. The pair ( l , k ) corresponds to the arrangement ( ι , κ ) where for each h ∈ H , ι h = l and for each d ∈ D , κ d = k .By Proposition 1 a homogenous arrangement ( l , k ) can only be globally ade-quate if l = k = 1 or l , k ≥ min {| D | , | H |} . Nonetheless, ( l , k ) may be adequate for aspecific profile of preferences. One might ask whether, starting at a profile P ∈ P and arrangement ( l , k ) that is adequate at P , if the comparative statics with re-spect to l and k are consistent. The following examples demonstrate that this isnot so. It may be that, depending on P , increasing k by 1 renders a previouslyadequate arrangement inadequate, or the opposite. In other words, the effect ofthe increase to k is specific to P . Similarly for l . Example 1.
Either incrementing or decrementing l or k can render an adequatearrangement inadequate. Suppose | D | = | H | = 3 and consider P ∈ P such that for each i = 1, 2, 3 , P h i d d d h i P d i h h h d i For P , (2, 2) is adequate: the interview matching is ν such that ν ( h ) = ν ( h ) = { d , d } and ν ( h ) = { d } . So, the ( l , k ) -matching is µ such that for each i = 1, 2, 3 , µ ( h i ) = d i , which is the unique stable matching.We now observe that if we increment or decrement either l or k by one, thearrangement is no longer adequate for P . In other words, none of (1, 2), (3, 2), (2, 1), This can be embedded into a larger problem instance. (2, 3) is adequate for P . We summarize the interview matching and the ( l , k ) -matching for each of these below. ( l , k ) interview matching ( l , k ) -matching (1, 2) ν ( h ) = ν ( h ) = { d } , ν ( h ) = { d } µ ( h ) = d , µ ( h ) = d , µ ( h ) = h , µ ( d ) = d (3, 2) ν ( h ) = ν ( h ) = D , ν ( h ) = {} µ ( h ) = d ! , µ ( h ) = d , µ ( h ) = h , µ ( d ) = d (2, 1) ν ( h ) = { d , d } , ν ( h ) = { d } , ν ( h ) = {} µ ( h ) = d , µ ( h ) = d , µ ( h ) = h , µ ( d ) = d (2, 3) ν ( h ) = ν ( h ) = ν ( h ) = { d , d } µ ( h ) = d , µ ( h ) = d , µ ( h ) = h , µ ( d ) = d All four of the final matchings are unstable. ◦ The mechanics of Example 1 are robust and it is not by accident that (2, 2) isadequate to start with. The preferences in the example have a particular salientconfiguration, which we focus on here. A profile P ∈ P has common preferences if all doctors rank the hospitals in the same way, and all hospitals rank the doctorsin the same way. To further restrict the definition, we also require that each doctorfinds each hospital acceptable and each hospital finds each doctor acceptable.That is, for each pair d , d (cid:48) ∈ D and each pair h , h (cid:48) ∈ H , P d | H = P d (cid:48) | H , P h | D = P h (cid:48) | D , d P h h , and h P d d . As we see from Example 1, a result like Proposition 1 does not hold if werestrict ourselves to common preferences. Our next result is a characterization ofhomogeneous arrangements that are adequate for common preferences. Proposition 2.
Under common preferences, a homogeneous arrangement ( l , k ) is adequate if and only if l = k or l , k ≥ min {| D | , | H |} .Proof. Let P ∈ P be such that there are common preferences. Let { d t } | D | t =1 and { h t } | H | t =1 be enumerations of D and H respectively such that every hospital prefers d t to d t +1 and every doctor prefers h t to h t +1 . Let m = min {| D | , | H |} . There is aunique stable matching µ ∗ , such that for each t = 1, ... , m , µ ∗ ( h t ) = d t .Let ν be the interview matching under ( l , k ) and µ be the ( l , k ) -matching.First, we show that ( l , k ) is adequate for P only if l = k or l , k ≥ min {| D | , | H |} .Suppose l (cid:44) k . If l < k and l < min {| D | , | H |} , then for each t = 1, ... , k , ν ( h t ) = { d , ... , d l } . In particular, d k (cid:60) ν ( h k ) so µ ( h k ) (cid:44) d k . On the other hand, if l > k and k < min {| D | , | H |} , then for each t = 1, ... , l , ν ( d t ) = { h , ... , h k } . In particular, Under common preferences there is, obviously, a unique stable matching. The characterization of Proposition 2 does not hold for arrangements that are not homoge-neous. For a counterexample, suppose | D | = 4 , | H | = 3 , there is d ∈ D such that κ d = 3 , for each d (cid:48) ∈ D \ { d } , κ d (cid:48) = 2 , there is h ∈ H such that ι h = 4 , and for each h (cid:48) ∈ H \ { h } , ι h (cid:48) = 2 . For anycommon preferences, ( ι , κ ) is adequate. l (cid:60) ν ( d l ) so µ ( d l ) (cid:44) h l . In either case, the ( l , k ) -matching is not stable so ( l , k ) isnot adequate.Now, we show that if l = k ≤ m , then ( l , k ) is adequate. For each t = 1, ... , m , let t = (cid:98) t − l (cid:99) . Then, for each t = 1, ... , i , ν ( h t ) = { d t +1 , ... , d t + l } and ν ( d t ) = { d t +1 , ... , d t + l } .Thus, for each t = 1, ... , m , µ ( h t ) = d t . So ( l , k ) is adequate at P .Finally, if l , k ≥ m , then for each t = 1, ... , m , ν ( d t ) ⊇ { h , ... , h m } . Since h t ∈ ν ( d t ) , h t = µ ( d t ) . So ( l , k ) is adequate at P .If the hospitals’ interview capacity is fixed at some specific l , an importantpolicy decision is where to set the doctors’ interview cap, k . Proposition 2 saysthat the optimal value for k is exactly at l whether the objective is to minimize thenumber of blocking pairs or to maximize the match rate (the proportion of positionsthat are filled). Our next result sheds light on this objective. Proposition 3.
Fix the hospitals’ interview capacity at l and consider k and k (cid:48) such that either k (cid:48) < k ≤ l or l ≤ k < k (cid:48) . Suppose P ∈ P has common preferences.The ( l , k (cid:48) ) -matching has more blocking pairs and a lower match rate than the ( l , k ) -matching.Proof. Let P ∈ P be such that there are common preferences. Let { d t } | D | t =1 and { h t } | H | t =1 be enumerations of D and H respectively such that every hospital prefers d t to d t +1 and every doctor prefers h t to h t +1 .Let m = min (cid:110)(cid:106) | H | k (cid:107) , (cid:106) | D | l (cid:107)(cid:111) . The interview matching is such that for each d t , if t ≤ ml , ν ( d t ) = { h ( n − k +1 , ... , h nk } where n is such that (1 − n ) l < t ≤ nl if ml < t ≤ ( m + 1) l , ν ( d t ) = (cid:40) { h mk +1 , ... , h n } if | H | ≥ mk + 1 ∅ otherwise where n = min {| H | , ( m + 1) k } and if ( m + 1) n < t , ν ( d t ) = ∅ .We first consider the case where when k > l and show that the number ofmatched hospitals is decreasing in k and that the number of blocking pairs isincreasing in k .Given P and its restriction to ν , the ( l , k ) -matching, µ , at P is such that for each d t , if t ≤ ml , µ ( d t ) = h ( n − k +( t mod l ) , where n is such that ( n − l < t ≤ nl , ml < t ≤ ( m + 1) l , µ ( d t ) = (cid:40) h mk +( t mod l ) if | H | ≥ mk + ( t mod l ) d t otherwise , and if ( m + 1) l < t , µ ( d t ) = d t .Let n = min {| H | − mk , | D | − ml } . Given the ( l , k ) -matching above, the set ofmatched hospitals is { h ik + s : i = 0, ... , m − s = 1 ... , l } ∪ { h t : t = mk + 1, ... , mk + n } . Therefore, the number of matched hospitals is ml + n . Holding l fixed, this isdecreasing in k .The ( l , k ) -matching, is blocked by all pairs consisting of an unmatched hospitaland any doctor with a higher index. That is, ( h t , d t (cid:48) ) such that t ≤ mk , t − k ≥ l and t (cid:48) > t . These are the only pairs the block it. Thus, the number of blockingpairs is m − (cid:88) n =0 k (cid:88) i = l +1 | D | − ( nk + i ). Holding l fixed, this is increasing in k .Now, we consider the case where k < l and show that the number of matchedhospitals is increasing in k and the number of blocking pairs is decreasing in k .Given P and its restriction to ν , the ( l , k ) -matching at P is such that for each h t ,if t ≤ mk , µ ( h t ) = d ( n − l +( t mod k ) , where n is such that ( n − l < t ≤ nl , if mk < t ≤ ( m + 1) k , µ ( h t ) = (cid:40) h ml +( t mod k ) if | D | ≥ ml + ( t mod k ) h t otherwiseand if ( m + 1) k < t , µ ( h t ) = h t .Let n = min {| H | − mk , | D | − ml } . Given ( l , k ) -matching above, the set of matchedhospitals is { h t : t ≤ mk + n . Therefore, the number of matched hospitals is mk + n .Since k < l , this is weakly increasing in k .The ( l , k ) -matching, is blocked by all pairs consisting of an unmatched doctorand any hospital with a higher index. That is, ( d t , h t (cid:48) ) such that t ≤ ml , t − l ≥ and t (cid:48) > t . These are the only pairs the block it. Thus, the number of blockingpairs is m − (cid:88) n =0 l (cid:88) i = k +1 | D | − ( nl + i ). Holding l fixed, this is decreasing in k . Our analytical results are of two sorts. On one hand, Theorem 1 applies withoutrestrictions on preferences. However, it only has something to say about doctors’welfare and only in regards to perturbations to an equilibrium arrangement. Onthe other hand, when we focus on common preferences, Proposition 2 and Propo-sition 3 deliver a clearcut policy prescription. In this section, we use simulations tobridge the gap. This allows us to consider how changes in the doctors’ interviewcapacities affect hospitals’ welfare, match rates, stability, and so on, in a moregeneral setting.While there is evidence that preferences do indeed have a common component(Agarwal, 2015; Rees-Jones, 2018), agents care about “fit” as well. Moreover, anidiosyncratic component is to be expected. We adopt the random utility modelof Ashlagi et al. (2017). Each hospital h ∈ H has a common component to itsquality, x Ch and a “fit” component, x Fh . Similarly, each doctor d ∈ D has a commoncomponent to her quality, x Cd and a fit component, x Fd . The utilities that h and d enjoy from being matched to one another are u h ( d ) = β x Cd − γ (cid:16) x Fh − x Fd (cid:17) + ε hd and u d ( h ) = β x Ch − γ (cid:16) x Fh − x Fd (cid:17) + ε dh respectively, where ε hd and ε dh are drawn independently from the standard logis-tic distribution. Each x Ch , x Fh , x Cd , and x Fd is drawn independently from the uniformdistribution over [0, 1] . The coefficients β and γ weight the common and fit compo-nents respectively. When β and γ are both zero, preferences are drawn uniformlyat random. As β → ∞ , these approach common preferences. As γ increases,preferences become more “aligned”: the fit, which is orthogonal to the commoncomponent, becomes more important. This, in turn, is adapted from Hitsch et al. (2010). a) The average match rate is highest at k = 5 ( pairs). (b) The average number of blocking pairs islowest at k = 6 (
19, 183.43 pairs).
Figure 1 : We vary k from to with l fixed at .Our simulated market has hospitals. We have chosen the number ofdoctors to be . The parameters for the random utility model are β = 40 and γ = 20 . Since our interest is in the effects of changes to doctors’ interviewcapacities, we fix hospital interview capacities at l = 25 .Our first simulation results involve varying k from to . Figure 1a showsthat the match rate increases and then decreases. On the other hand, Figure 1bshows that the number of blocking pairs decreases and then increase. Theseresults are consistent with what we learn from Proposition 3. However, since pref-erences are not common, the match rate does not reach 100% and the number ofblocking pairs remains positive even at the optimal k —that is, the arrangement isnot adequate. Moreover, the optimal k does not equal to l .Our next set of results evaluate a hypothetical policy of restricting doctors to amaximum of interviews. We choose this as a candidate policy as it is optimal in The NRMP match is broken down into smaller matches by specialty. In 2020, among 50specialties for PGY-1 programs, the largest had 8,697 positions, 10th largest had 849 positions,the 25th largest had 38 positions, the 49th largest had one position and the smallest had nopositions. This data is available from the NRMP. Our chosen number of hospitals is comparable tothe 70th percentile among specialties. There were, on average, 0.85 PGY-1 positions per applicant in the 2020. Our chosen numberof students reflects this ratio. We have chosen this upper bound to be high enough that further increases have little effect.Thus, we interpret this as doctors being essentially unconstrained in how many interviews theycan accept. k over the benchmark as well as the distribution of thosewith the opposite preference. We see that the former is considerably far to theright of the latter. Though Theorem 1 applies only to when the starting arrange-ment is adequate, Figure 2a shows that the lesson from that result does extendbeyond. Figure 2b shows the same distributions, except for hospitals. Despitethe fact that Theorem 1 does not address hospitals’ welfare, our simulations showthat more hospitals prefer the optimal k than leaving the doctors unconstrained.The policy also has the benefit of drastically decreasing the number of blockingpairs. Figure 2c shows the distribution of excess blocking pairs when we comparethe matching under k = 5 to the benchmark matching where doctors are uncon-strained. Finally, we compare the distribution of interviews among the doctorsbetween the two arrangements in Figure 2d. The constraint limiting doctors to k = 5 interviews binds for many doctors. An implication of this is that significantlymore doctors receive zero interviews when doctors are unconstrained. This isconsistent with the intuition that if interviews were costless for doctors, then highlysought after doctors would hoard interviews and others would be left with nothing.We next compare the proposed intervention ( k = 5 ) to a hypothetical “ideal”world where there is no interview stage and the final match is based on actualpreferences. Figure 3a displays the distributions of the numbers of doctors whoprefer the match under the intervention as well as the numbers with the oppositepreference. Figure 3b displays the analogous distributions for hospitals. Thoughthe hospitals are typically worse off, the match under interview constraints and theintervention is not Pareto dominated by the “ideal” match: typically, many doctorsare better off. Finally, we consider the possibility that the NRMP could set not only a cap oninterviews that doctors can accept, but can also control the number of interviewsthat hospitals offer. From Proposition 2, we know that if preferences are common,the match rate would be maximized where l = k . Figure 4 shows that, even whenpreferences are not exactly common, there are still optimal combinations of l and k , but they typically involve l > k . 21 a) Distribution of the number of doctorswho prefer their match at k = 5 over beingunconstrained and vice versa. (b) Distribution of the number of hospitalswho prefer their match at k = 5 over the doc-tors being unconstrained and vice versa.(c) Distribution of the number of excessblocking pairs when doctors are uncon-strained over the number of such pairs at k = 5 . (d) Distribution of interviews at k = 5 withouta cap. The uncapped distribution vanisheswith the number of interviews reaching thehundreds. Figure 2:
Comparisons of the intervention of capping doctors’ interview capacitiesat k = 5 to leaving doctors unconstrained.22 a) Distribution of the number of doctorswho prefer their match at k = 5 over theirmatch without an interview stage and viceversa. (b) Distribution of the number of hospitalswho prefer their match at k = 5 over theirmatch without an interview stage and viceversa. Figure 3:
Comparisons of welfare under the intervention k = 5 to the hypotheticalscenario where there is no interview step. The 2020-21 global pandemic has had a significant impact on the way interviewsare conducted. We anticipate that it will also impact the number of interviewsdoctors participate in. Through our simulations and theoretical results, we predictthat unless hospitals also increase the number of interviews they offer, the 2021NRMP match will result in a lower percentage of positions being filled and a lessstable matching.In future years, the NRMP should consider policies to mitigate these effects.Our analysis supports the idea of interview caps and our simulations provide ev-idence that such a policy would reduce the bottleneck created by the interviewstep. Such caps can be implemented with very limited centralization, for instance,using a ticket system.Even if such interventions are not possible in the very short run, our policyprescription is that residency programs should be advised to increase the numberof candidates they interview relative to previous years.Design of a fully centralized clearinghouse, is an area that remains open. Asearlier work on the interview pre-markets have shown, strategic analysis has onlybeen tractable under very stringent assumptions (Kadam, 2015; Lee and Schwarz,23 igure 4 : Match rate as a function of l and k References
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Appendices
A Choice Functions for Interview Step
In the interview step, we compute a many-to-many matching. However, eachdoctor and each hospital only ultimately matches to at most one other partner,and each has strict preferences over partners. This necessitates the definition ofa choice function over sets of partners. For our analytical results, we focus onacceptant choice functions that are responsive to preferences over partners andconstrained by interview capacity. That is, given P ∈ P ,• From the set H (cid:48) ⊆ H , each d ∈ D chooses the κ d best elements of H (cid:48) ac-cording to P d : C d ( H (cid:48) ) = { h ∈ H (cid:48) : h P d d } if |{ h ∈ H (cid:48) : h P d d }| ≤ κ d and B ⊆ { h ∈ H (cid:48) : h P d d } such that | B | = κ d and for each h ∈ B and each h (cid:48) ∈ H (cid:48) \ B , h P d h (cid:48) otherwise.• From the set D (cid:48) ⊆ D , each h ∈ H chooses the κ h best elements of D (cid:48) accord-26ng to P h : C h ( D (cid:48) ) = { d ∈ D (cid:48) : d P h h } if |{ d ∈ D (cid:48) : d P h h }| ≤ ι h and B ⊆ { d ∈ D (cid:48) : d P h h } such that | B | = ι h and for each d ∈ B and each d (cid:48) ∈ D (cid:48) \ B , d P h d (cid:48)(cid:48)