Graduate admission with financial support

Journal of Mathematical Economics 87 (2020) 114–127

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Graduate admission with financial support✩ Mustafa Oˇguz Afacan Faculty of Arts and Social Sciences, Sabancı University, 34956, İstanbul, Turkey

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Article history: Received 18 November 2018 Received in revised form 6 January 2020 Accepted 20 January 2020 Available online 31 January 2020 Keywords: Matching Graduate admission Financial support Stability Strategy-proofness Respecting improvement

a b s t r a c t We formulate a graduate admission problem, which features different financial support options, in a matching with contracts setting. We introduce an algorithm, called ‘‘Minimum-Need Adjusted Cumulative Offer Process’’ (MCOP). Under certain mild assumptions on students’ preferences, MCOP is stable, fair, strategy-proof, and respects improvements. Moreover, it limitedly respects departments’ minimum number of teaching and research assistant (TA/RA) needs in the sense that no other stable mechanism honors those needs more than MCOP. It is also efficient within the class of stable mechanisms that limitedly respect the TA/RA needs. Lastly, we offer an axiomatic characterization: A mechanism is stable and strategy-proof, and limitedly respects the TA/RA needs if and only if it is MCOP. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Every year, students who wish to pursue graduate studies apply to graduate programs, and departments conduct admission processes to decide whom to accept and what kind of firstyear financial support to offer to admitted students.1 , 2 There are four admission types, each associated with a different financial support option: admission with fellowship, admission with teaching assistantship (TAship), admission with research assistantship (RAship), and admission without financial support. The current U.S. graduate admissions markets run in a decentralized fashion, which causes problematic outcomes, as detailed later in Section 3.1. In this study, we propose a centralized algorithm that not only avoids all these current problems but also admits some other highly desirable properties. Each admission type entails different conditions. While the admission types of with fellowship and without financial support do not require a workload, students with TAship and RAship admissions are required to supply a certain workload (in general, at most 20-h TA/RAship work per week). The advantages of TAship and RAship are that they lead students to improve their

teaching and research abilities, respectively. Besides, they may differ in their respective stipend amounts.3 Hence, we can say that each admission type has its advantages and disadvantages, and therefore, students may well have heterogeneous preferences over department-admission type tuples.4 Departments rank students based on their subjective evaluations. They are interested in their admitted student body as well as the financial support they offer to the admitted students. First, departments are willing to offer financial support to each of their admitted graduate (doctoral) students as long as they can do so.5 Second, admission without financial support is the least prestigious admission type, and departments would rather offer it to the worst-ranked admitted student body whenever the capacities of all other admission types are exhausted. Another constraint making the admission procedure more challenging is that departments may need a certain number of teaching and research assistants (TA/RA, respectively). Therefore, while deciding who gets what kind of admission, departments need to take care of their TA/RA needs.

✩ I am grateful to the associate editor and the anonymous referees for their comments and suggestions. I would like to thank Fuhito Kojima, Peter Troyan, Bertan Turhan, and Alexander Westkamp for their comments and suggestions. E-mail address: [email protected]. 1 In general, subsequent years’ financial supports are determined after

3 For instance, at the University of Wisconsin, fellowship admission includes a higher stipend than both TAship and RAship (see https://grad.wisc.edu/funding/ fellowships/ and https://grad.wisc.edu/funding/graduate-assistantships/). The converse is true for Princeton (see https://gradschool.princeton.edu/node/1012). On the other hand, financial support options pay the same stipend amount at Cornell’s Arts & Humanities and Social Sciences (see https://gradschool.cornell. edu/financial-support/stipend-rates/). 4 Indeed, Bersola et al. (2014) empirically report that students may have

students start their graduate studies. 2 In referring to graduate programs, we mainly have Ph.D. programs in our mind. However, our model can accommodate master-level graduate admissions by setting the capacities of master program admissions with financial support to zero.

different preferences over those financial support options. 5 This may be because departments’ unused monetary resources go to their universities; hence, their left budget does not bring any good to them. Hence, they may prefer to offer any possible financial support to students to attract them.

https://doi.org/10.1016/j.jmateco.2020.01.006 0304-4068/© 2020 Elsevier B.V. All rights reserved.

M.O. Afacan / Journal of Mathematical Economics 87 (2020) 114–127

We first formulate a graduate admissions problem, including the above aspects, in a matching with contracts setting. While there are related studies on similar problems, to the best of our knowledge, ours is the first to incorporate the financial support term and the TA/RA needs into an admission problem. We introduce a choice function for departments, called ‘‘admission with financial support’’ (Aw FS) choices. We justify the Aw FS choices by observing that they internalize both the departments’ TA/RA needs and the appealing normative condition that the worstranked admitted student body should receive no financial support whenever the capacities of other admission types are exhausted. We then propose a twofold algorithm. Its first stage consists of the well-known ‘‘cumulative offer process (COP)’’ Hatfield and Milgrom (2005) under the Aw FS choices. For each department with a TA or RA shortage (or both) with respect to its TA/RA need at the first-stage outcome, the second stage attempts to mitigate these shortages as much as possible by carefully changing the admissions of the admitted students within each department. One feature of the second stage is that TA shortages are prioritized in that it attempts to mitigate TA shortages first. Once TA shortages are eliminated, only then does it deal with RA shortages. We choose this approach because TA shortages are arguably more critical to departments in carrying out necessary academic activities.6 However, this second stage can be modified to reverse the order of the adjustments at no cost. We refer to our twofold mechanism as the ‘‘Minimum-Need Adjusted Cumulative Offer Process’’ (MCOP). For our analysis, we impose certain assumptions on student preferences. First, we assume that students have a ranking list over departments, and each of them prefers any contract (a contract consists of a department and an admission-type) with the financial support of more favorable departments.7 Second, if a student finds a contract with some financial support better than the outside option, then he continues to prefer contracts with any financial support of the same department to the outside option. Third , if a student would rather have a contract without financial support of a department instead of having a contract with some financial support of some other department , then he continues to prefer the former contract to any contract of the latter department. Lastly, every student prefers contracts that include some financial support to those without financial support of the same departments. While our results would disappear in the absence of these assumptions , as we will formally elaborate more in the Model section, they are empirically supported by several studies (e.g., Bersola et al., 2014; Poock and Love, 2001; Malaney, 1984); hence, we believe they are rather mild in the current context. We study various properties of MCOP. We first show that MCOP produces a stable Gale and Shapley (1962) and Hatfield and Milgrom (2005) allocation. Then, we consider its responsiveness to the departments’ TA/RA needs. To this end, we say that a mechanism respects the TA/RA needs more than another mechanism if, (i) for each problem and each department, the former’s outcome never induces a higher TA shortage than the latter’s, and moreover, if both of them do not yield a TA shortage for the department, then it is the case for RA needs as well, and (ii) for some problem and some department, the latter’s outcome induces a higher TA or RA shortage (or both) than the former’s. We say that a mechanism limitedly respects the TA/RA needs if no other stable mechanism respects the TA/RA needs more than itself. MCOP limitedly respects the TA/RA needs. 6 Research projects can be suspended or done without RAs. However, departments have to offer certain courses. 7 This assumption allows students to prefer less favorable departments’ contracts with financial support to more favorable departments’ contracts without financial support.

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Next, we study the efficiency, fairness, and incentive properties of MCOP. We show that no stable mechanism that limitedly respects the TA/RA needs efficiency-wise dominates MCOP. In other words, MCOP is efficient within the class of stable mechanisms that limitedly respect the TA/RA needs. For our fairness analysis, by following Balinski and Sönmez (1999), we say that an allocation is fair if no student envies someone else’s contract while, at the same time, he has a better ranking at the latter’s department. MCOP is fair as well. In terms of incentives, MCOP is both strategy-proof and respects improvements. While the former guarantees that no student ever benefits from misreporting his preferences, the latter makes sure that no student receives a worse contract after improving his rankings in departments. In our last analysis, we offer an axiomatic characterization of MCOP. We show that a mechanism is stable and strategy-proof, and limitedly respects the TA/RA needs if and only if it is MCOP. The independence of these axioms is also obtained. This study is the first that explicitly incorporates (soft) minimum needs – in other words, (soft) minimum capacities – into a matching with contracts setting. In addition to the current graduate admission problem, minimum capacities appear in many other real-life problems; hence, our algorithm may be useful to future studies.8 One final remark relates to our proof techniques. Some of our proofs are partly based on the results of Kominers and Sönmez (2016). Specifically, we show that Aw FS choices fall within their domain of choice functions. This enables us to utilize some of their results in our findings on the first stage of MCOP, which is COP under the Aw FS choices. We then use these auxiliary findings to prove our ultimate results about MCOP. 2. Related literature Hatfield and Milgrom (2005) were the first to formulate a matching with contracts problem. By generalizing the Gale and Shapley (1962)’s deferred acceptance mechanism (DA), they introduce COP and show that it produces a stable allocation under a substitutes condition. Hatfield and Kojima (2010) introduce two conditions, weaker than substitutability (from weaker to stronger): ‘‘bilateral substitutes’’ (BS) and ‘‘unilateral substitutes’’ (US). They obtain the stability of COP under BS and show that COP coincides with DA and produces the student-optimal stable matching under the stronger US. With an additional ‘‘law of aggregate demand’’ (LAD) condition, they also obtain the strategyproofness of COP.9 Aygün and Sönmez (2012) and Aygün and Sönmez (2013) obtain all of those results with an additional irrelevance of rejected contracts condition (henceforth, IRC ). In a recent study, Hatfield and Kominers (2014) introduce a ‘‘substitutable completability’’ condition and show that COP is both stable and strategy-proof for every choice function profile that satisfies their condition as well as IRC and LAD.10 Hatfield et al. (2015) provide necessary and sufficient conditions for choice functions for the existence of a stable and strategy-proof mechanism and show that if such a mechanism exists, then it is nothing but COP. Hirata and Kasuya (2017) obtain that at most one stable and strategy-proof mechanism exists in the class of choice functions that consists of those satisfying IRC . 8 Some real-life matching markets where explicit or hidden minimum capacities are present include Hungarian college admissions (Biró et al., 2010), cadet-branch assignments in the U.S. Army Sönmez (2013) and Sönmez and Switzer (2013), the Japanese medical residency market (Kamada and Kojima, 2015), and various school-choice programs (Ehlers et al., 2014). 9 Alkan and Gale (2003) introduce a similar condition, called ‘‘size monotonicity’’ in a schedule matching setting. 10 They show that there are choice functions that are BS but that do not satisfy substitutable completability, and vice versa.

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College admission with scholarship has recently received attention. Two important related works from that literature are Abizada (2016) and Abizada and Dur (2017). The former studies college admissions where colleges offer seats and stipends to students, who have preferences over college-money tuples. Colleges are only interested in the student body they admit, that is, they do not value money. Each college has a budget limit that restricts the maximum total stipend it can offer. Even though (Abizada, 2016) and the current study are conceptually related, their formulations are different. We specify four different matching (financial support) terms exogenously, whereas, in Abizada (2016), there is no a priori fixed matching term. Moreover, the respective choice functions are different. Colleges do not have preferences over matching terms-money-in Abizada (2016), whereas, in our formulation, colleges care about their students’ financial support terms. In other words, matching terms enter into the model as a pure constraint on colleges in Abizada (2016), on the other hand, in our study, they are primitives that shape colleges’ choices.11 Besides, there is no minimum capacity sort of constraint in Abizada (2016). Because the formulations are different, the respective notions and mechanisms differ too. Similar to Abizada (2016), Abizada and Dur (2017) consider a college admission setting where colleges decide not only the student body they admit but also the scholarship they offer to the admitted students. In their setting, college choices exhibit complementarities, and they find a condition under which the existence of a stable allocation is guaranteed. This study is also related to the strand of matching literature where institutions have minimum quotas. In a school-choice setting where schools have hard minimum quotas, Fragiadakis et al. (2015) introduce strategy-proof mechanisms. Fragiadakis and Troyan (2017) then generalize (Fragiadakis et al., 2015) to a controlled school-choice setting by letting students have different types and schools have type-specific (hard) minimum quotas. They offer strategy-proof algorithms that also admit desirable fairness properties. In the same setting as Fragiadakis and Troyan (2017), Ehlers et al. (2014) show that satisfying hard type-specific minimum quotas is impossible even with weak efficiency and fairness requirements. Instead, they interpret those bounds as soft constraints and introduce a mechanism that satisfies distributional goals as long as they do not hurt students. Hafalir et al. (2013) consider a simpler controlled school-choice setting where there are only two types of students and study the welfare effects of two different affirmative action policies. In another related study, Westkamp (2013) considers the German university admission system and introduces a matching with complex constraints problem where schools can reserve certain portions of their capacity for special groups of students. There are two main differences between those studies and ours. First, all those prior studies are in the standard matching without contracts setting. Moreover, whenever there is only one type of student, as in our case, all but Fragiadakis et al. (2015) collapse to the base school-choice model of Abdulkadiroğlu and Sönmez (2003). 3. The model and results There are finite sets of students I, departments D, financial support terms S, and contracts X . For ease of exposition, we refer to each university–department pair as a department.12 There are 11 In another conceptually related paper Afacan (2013), students have a budget constraint and colleges have application fees. Students have to pay those fees to apply to colleges. Similar to Abizada (2016), students do not have preferences over money, and application fees only impose constraints on students’ feasible set of colleges that they can apply to with their limited budget. 12 For instance, one department is MIT Economics and another is Harvard Sociology.

four different admission types, each of which is associated with a distinct financial support term: admission with fellowship (F ), admission with TAship (T ), admission with RAship (R), and admission without financial support (N). That is, S = {F , T , R, N }.13 Each contract x ∈ X consists of one student xI ∈ I, one department xD ∈ D, and one financial support term xS ∈ S. Each student can sign at most one contract. The null-contract, denoted by ∅, represents having no contract. In the rest of the paper, we refer to any contract x with financial support term xS ∈ S as a xS -type contract. Each student i has a strict preference relation Pi over {x ∈ X : xI = i} ∪ {∅}. Let P be the set of all preference relations. We write x Ri x′ only if x Pi x′ or x = x′ . By slightly abusing the notation, for any d, d′ ∈ D and s, s′ ∈ S, we write (d, s) Pi (d′ , s′ ) whenever (i, d, s) Pi (i, d′ , s′ ). A contract x is acceptable to student i if x Pi ∅; and otherwise, it is unacceptable. We assume that the students have a rank ordering over the departments. Let ▷i be the student i’s rank ordering over D where for any pair of departments d, d′ , d ▷i d′ means that department d is more favorable than department d′ for student i. In the rest of the paper, we have the following suppositions concerning the students’ preferences: For any student i and pair of departments d, d′ ∈ D, (i) for any s ∈ S, (d, s) Ri (d, N). (ii) If d ▷i d′ , then for any pair s, s′ ∈ S \ {N }, (d, s) Pi (d′ , s′ ) and (d, N) Pi (d′ , N). (iii) If (d, N) Pi (d′ , s) for some s ∈ {F , T , R}, then (d, N) Pi (d′ , s′ ) for any s′ ∈ {F , T , R}. (iv ) If (d, s) Pi ∅ for some s ∈ S \ {N }, then (d, s′ ) Pi ∅ for any s′ ∈ S \ {N }. The first supposition asserts that the students always prefer contracts with financial support to the same department’s without financial support contract. The second assumption says that the students would rather have a contract including financial support from their more favorable departments. Note that the students may still prefer contracts with financial support of their less preferred departments to those without financial support of their more preferred departments. For instance , if a student finds MIT Economics better than Harvard Economics, then he would prefer any contract with financial support of the former to any contract of the latter. And , of course, among both departments’ without financial support contracts, he prefers MIT’s. However, he may very well prefer Harvard Economics with financial support to MIT Economics without financial support. The third assumption says that if a student finds a department’s without financial support contract better than another department’s contract with some financial support, then this preference continues to apply for any type of financial support of the latter department.14 The last condition, on the other hand, imposes that if a contract with some form of financial support is found to be better than having no contract, then so is any contract with financial support of the same department.15 While the first and last conditions are natural , the second and third assumptions imply that in certain cases, students’ preferences depend purely on their ranking over departments. Indeed, this kind of choice behavior has been documented to be observed. For instance, Bersola et al. (2014), Poock and Love (2001), and Malaney (1984) all empirically demonstrate that 13 In Section 4, we discuss how our model can be generalized to include more than four admission types. 14 By invoking assumptions (i) and (ii), if (d, N) P (d′ , s) for some s ∈ {F , T , R}, i

then it has to be that d ▷i d′ . 15 Note that a student can find a contract without financial support unacceptable while he may find the same department’s contracts with financial support acceptable.

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students’ graduate program choices overwhelmingly depend on non-financial matters, such as departments’ reputations, faculty and research quality, and faculty access, which are all main factors determining students’ rankings over departments.16 Moreover, in personal communications with two economics professors at different universities in the United States, based on their graduate admissions experiences, both pointed out that they could not convince students to choose their respective departments even after they had improved their initial offers in monetary terms. The chosen contract of student i from X ′ ⊆ X is defined as follows: Ci (X ′ ) = max[{x ∈ X ′ : xI = i } ∪ {∅}]. Pi

Let CI (X ′ ) = i∈N Ci (X ′ ) be the set of contracts that are chosen from X ′ by some student. We write P = (Pi )i∈I and PI ′ = (Pi )i∈I ′ for the preference profile of all the students and that of a group of students I ′ ⊂ I, respectively. For X ′ ⊆ X , let XI′ = {i ∈ I : ∃ x ∈ X ′ with xI = i}, that is, the set of students having a contract in X ′ . Each department d has a rank order ≻d over I ∪ {∅} where ∅ denotes keeping seat empty. Student i is admissible to department d if i ≻d ∅. Otherwise, he is inadmissible. We write ≻= (≻d )d∈D for the rank order profile. Departments have a capacity constraint for each of their admission types that limits the maximum number of students that can receive those admissions. For department d, let qFd , qTd , qRd , and qNd be the capacities for the admission types of F -type, T -type, R-type, and N-type, respectively.17 Apart from the maximum capacities, each department needs a certain number of TA/RA. Let qT and qR be these TA/RA d d needs of department d, respectively. By construction, 0 ≤ qT ≤ qTd d and 0 ≤ qR ≤ qRd . d For a given set of contracts X ⊆ X , let d(X ) = {X ′ ⊆ X : for each x ∈ X ′ , xD = d, xI ≻d ∅, and for any x′ ∈ X ′ with x ̸ = x′ , xI ̸ = x′I }. Moreover, for any s ∈ S, let Xds = {x ∈ X : xS = s and xD = d}. A choice function for department d is Cd : 2X → 2X such that for each X ⊆ X ,

⋃

Cd (X ) ∈ {X˜ ∈ d(X ) : |X˜ dF | ≤ qFd , |X˜ dT | ≤ qTd , |X˜ dR | ≤ qRd , and

|X˜ dN | ≤ qNd }. ⋃ Let CD (X ) = d∈D Cd (X ) be the set of contracts that are chosen from X by some department. We write Rd (X ) for the set of contracts that are rejected from X by department d, that is, Rd (X ) = X \ Cd (X ). Let C = (Cd )d∈D be the choice function profile of the departments. Note that, by construction, no inadmissible student is ever chosen by any department. Let us point out that our whole analysis will be based on a particular choice function, namely the Aw FS choice function, which will formally be introduced in Section 3.2. An allocation is a subset of contracts X ⊂ X such that each student has at most one contract, and for each department d, (i) |XdF | ≤ qFd , (ii) |XdT | ≤ qTd , (iii) |XdR | ≤ qRd , and (iv ) |XdN | ≤ qNd . We extend the student preferences to the set of allocations in a natural way. By slightly abusing the notation, we write P for that preference profile, and it is defined as follows: For each student i and pair of allocations X , X ′ , X Pi X ′ if and only if {x ∈ X : xI = 16 For instance, using their survey data, Bersola et al. (2014) report that three out of four students who rejected an offer from Western University would have still rejected it if their current institution and Western University had offered equivalent financial support. 17 These capacities are hard constraints in the sense that they are not transferable between the admission types. This restriction is arguably mild in practice because departments have limited office rooms (TAs/RAs, in general, are assigned an office; hence a non-TA/RAship offer cannot easily be converted to a TA/RAship offer), and some fellowships are outside resources (hence, a non-fellowship offer may not be converted to a fellowship offer).

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i} Pi {x′ ∈ X ′ : x′I = i}. For an allocation X and contract x ∈ X where xI = i, let Xi = (xD , xS ) where (Xi )D = xD , and (Xi )S = xS . In words, Xi is student i’s contract at allocation X . We also define Xd = {x ∈ X : xD = d}. Remark 1. Note that as opposed to the capacities, we do not impose the departments’ TA/RA needs as feasibility constraints over allocations. There are two main reasons for that. First, in reality, departments may fall short of their TA/RA needs, but they can still accommodate those needs and continue to operate (they may close some courses or suspend some research projects). Moreover, if we were to impose them as feasibility requirements, then we would have needed some further assumptions on the number of students, their preferences, and departments’ rank orders. Our current approach avoids such extra impositions. Even though TA/RA needs do not enter our model as hard constraints, we will see that our choice function proposal respects those needs more than any other choice function does; and moreover, our algorithm fulfills the TA/RA needs to the extent that stability permits. Allocation X dominates another allocation X ′ if for each student i ∈ I, X Ri X ′ , with strictly holding for some student. Allocation X is efficient if it is not dominated by another allocation. Allocation X is fair if there exists no pair of students i, j such that Xj Pi Xi and i ≻d j where d = (Xj )D . In the rest of the paper, unless otherwise stated, we take all the problem elements, except the students’ preferences P, as fixed primitives and write P to denote the problem. Definition 1. An allocation X is stable if (1) CI (X ) = CD (X ) = X , and (2) there exist no department d and set of contracts X ′ ̸ = Cd (X ) such that X ′ = Cd (X ∪ X ′ ) ⊆ CI (X ∪ X ′ ). A mechanism ψ is a systematic procedure that produces an allocation for every problem P ∈ P |I | . We write ψ (P) to denote its outcome in problem P. Mechanism ψ dominates another mechanism φ if, for each problem P, either ψ (P) dominates φ (P) or they are the same, with the former holding for some problem. Mechanism ψ is < fair, stable, efficient> if ψ (P) is < fair, stable, efficient> for each problem P. Having introduced the formal problem and notions, we next discuss the current graduate admission mechanism’s handicaps below. 3.1. The current graduate admission mechanism The current graduate admission system in the United States is decentralized. The lack of a centralized system itself entails coordination problems, which may result in inefficient, unfair, and unstable allocations. This is because students who receive offers are asked to notify departments about their decisions by April 15. However, by that date, they may not hear from all of their applications owing to various reasons, including being on a wait list. In such a case, students may accept one of their offers before hearing from all of their applications. If a student accepts one of his offers and receives a more preferred offer after that, then this may result in unfair and unstable allocations whenever the more preferred offer is directed to a worse ranked student. Moreover, such late acceptances may yield inefficient matchings as well.18 Indeed, the problem of not hearing from all 18 Consider a situation where Bob accepts Harvard Economics before he hears from MIT Economics. After the decision deadline, MIT Economics sends an offer to him and he declines it as he has already accepted Harvard’s offer. Then, MIT’s offer is directed to Dan and he accepts it. If Bob prefers MIT to Harvard (given the offered financial supports) and Dan’s preferences is the other way around, then this placement would be inefficient.

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applications by the deadline is a common phenomenon that has been discussed on various platforms. For instance, the following excerpt is taken from one of the graduate admission forums:19 ‘‘With the passing of midnight a couple hours ago, the April 15 deadline to accept/decline offers from graduate schools (as decided by the Council of Graduate Schools) has passed. While I have made my choice, there is still at least one (pretty well-respected) school from which I never received a decision. Thus, in order to have waited out a decision from this program, I would have had to risk losing other offers (one of the conundrums that the April 15 resolution aims to avoid).’’ Yet another excerpt from the same page: ‘‘Graduate schools generally do not promise to make a decision by April 15.’’ The decentralized feature of the current system is not the only source of the problems. Another issue is the lack of information as to students’ preferences. In the current practice, every year, students send their applications to departments, and departments rank applicants based on their merit-based criteria. They then offer different admission types depending on their ranking over students and various constraints, such as capacity and TA/RA needs. The problem here is that they make these offers without knowing students’ preferences over different kinds of financial support options. As each admission type has its advantages against the others, students may have heterogeneous preferences over them.20 This lack of information itself (even under a centralized system) may result in severe inefficient, unfair, and unstable allocations. In the rest of the paper, we offer a new centralized algorithm for graduate admission markets. As opposed to the above lack of information in the current system, it solicits students’ full preferences over department and admission types tuples. We ultimately show that not only does it avoid the current mechanism’s severe handicaps above, but it also admits some other desirable properties. To this end, we first introduce a particular choice function profile for the departments. 3.2. The admission with financial support (AwFS) choices For department d, let CdA denote the department d’s Aw FS choice function, which is defined as follows. For any given set of contracts X ⊆ X , Phase 0: Remove all the contracts from X that belong either to the departments other than department d or to the inadmissible students of department d, and add them to the rejected contracts set Rd (X ). Let Xd′ = X \ Rd (X ). Phase 1: Consider only the T -type contracts in Xd′ and starting from the best ranked student’s T -type contract, choose one at a time following the student rank order until min{qT , |Xd′T |} many d of them have been chosen. Remove the chosen students’ other ′ types of contracts from Xd (if any) and add them to Rd (X ). Phase 2: Consider only the R-type contracts in Xd′ and starting from the best ranked student’s R-type contract, choose one at a time following the student rank order until min{qR , |Xd′R |} many d of them have been chosen. Remove the chosen students’ other types of contracts from Xd′ (if any) and add them to Rd (X ). Phase 3: Consider only the F -type contracts in Xd′ and starting from the best ranked student’s F -type contract, choose one at a time following the student rank order until min{qFd , |Xd′F |} many 19 See http://academia.stackexchange.com/questions/43675/phd-programsthat-miss-admission-decision-deadlines. 20 As argued earlier, apart from possible stipend differences, each option has its advantage. TAship and RAship lead students to improve themselves at teaching and research, respectively. Fellowship admission, on the other hand, pays students without requiring any workload.

of them have been chosen. Remove the chosen students’ other types of contracts from Xd′ (if any) and add them to Rd (X ). Phase 4: Consider only the T -type contracts in Xd′ and starting from the best ranked student’s T -type contract, choose one at a time following the student rank order until min{qTd − qT , |Xd′T |} d many of them have been chosen. Remove the chosen students’ other types of contracts from Xd′ (if any) and add them to Rd (X ). Phase 5: Consider only the R-type contracts in Xd′ and starting from the best ranked student’s R-type contract, choose one at a time following the student rank order until min{qRd − qR , |Xd′R |} d many of them have been chosen. Remove the chosen students’ other types of contracts from Xd′ (if any) and add them to Rd (X ). Phase 6: Consider only the N-type contracts in Xd′ and starting from the best ranked student’s N-type contract, choose one at a time following the student rank order until min{qNd , |Xd′N |} many of them have been chosen. Remove the chosen students’ other types of contracts from Xd′ (if any) and add them to Rd (X ). The chosen set of contracts in the course of the above phases constitutes CdA (X ). In words, CdA takes care of department d’s TA/RA needs by attempting to satisfy these needs sequentially, starting with the TA needs. Once these needs are satisfied as much as possible, CdA only then considers F -type contracts and chooses as many of them as its capacity. Then, it turns back to T -type and R-type contracts and selects them in the same manner for the remaining capacities. Lastly, it chooses as many N-type contracts as up to qNd . Let C A = (CdA )d∈D . Now, let us formally justify the Aw FS choices. As the N-type admission is the worst one, a well-sound choice function should be such that whenever no contract or the N-type contract of an admissible student is chosen from a choice set, then the capacities of his non-chosen contract types in the choice set should be exhausted for the better-ranked students. Let us formalize this property. For any X ⊆ X and s ∈ S, let Cd (X , s) = {x ∈ Cd (X ) : xS = s}. In words, it is the set of s-type contracts that Cd chooses from X . We say that⋃ a choice function Cd respects rankings if for any X ⊆ X , i ∈ XI \ [ s∈{F ,T ,R} Cd (X , s)]I such that i ≻d ∅, and any x ∈ Rd (X ) with xI = i, xD = d, and xS = s, we have |Cd (X , s)| = qsd , and moreover, for any j ∈ [Cd (X , s)]I , j ≻d i. Proposition 1. CdA respects rankings. The proof directly comes from that fact that under C A , N-type contracts are chosen in the last phase after no capacity is left for the other admission types, and the contracts are chosen one at a time following the student rankings. Apart from respecting ranking, we will find that C A satisfies the TA/RA needs as much as possible in a certain sense, which will be another desirable property of C A . This will formally be provided in Remark 3. In the rest of the paper, we endow the departments with the Aw FS choices, and our whole analysis will be based on these choices. We first introduce a twofold algorithm. Its first stage consists of the‘‘cumulative offer process’’ (COP) under C A . Hence, let us first outline the COP below. 3.3. The cumulative offer process Below describes how the COP works under C A . Step 1: One arbitrarily chosen student i offers his favorite acceptable contract x1 . The offer-receiving department d holds the contract if x1 ∈ CdA ({x1 }) and rejects it otherwise. Let Yd1 = {x1 } and Yd1′ = ∅ for all d′ ̸ = d. In general, Step t: One arbitrarily chosen student currently having no held contract offers his favorite acceptable contract xt among the ones which have not been rejected in the previous steps. The offerreceiving department d holds xt if xt ∈ CdA (Ydt −1 ∪ {xt }) and rejects it otherwise. Let Ydt = Ydt −1 ∪ {xt } and Ydt′ = Ydt′−1 for all d′ ̸ = d.

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The algorithm terminates whenever each student either has a contract that is held by a department or had all of his acceptable contracts rejected. As there are finitely many contracts, the algorithm terminates in some finite step T . The final outcome is ⋃ A T A A d∈D Cd (Yd ). Let ψ denote the COP under C . 3.4. The minimum-need adjusted cumulative offer process Let X˜ be the allocation where X˜ i = ∅ for each student i ∈ I. Here is our two-stage mechanism formal definition: Given a problem P, Stage 1. Obtain ψ A (P) = X . Stage 2. We run the following steps in this stage. Step 2.1. For each department d with |XdT | ≥ qT and |XdR | ≥ qR , d

d

we let X˜ d = Xd . If it is the case for every department, then the algorithm ends with the final outcome of X˜ . Otherwise, we apply the following steps for each department d with either (or both) |XdT | < qTd or |XdR | < qRd . Let us first order the students having a contract in Xd with respect to their rankings at department d. From best to worst, let (Xd )I = {i1 , . . . , im } be that rank ordering. We cancel all of their contracts with department d under X . Step 2.2. If m > qT , then we skip this step and move to the d next step. Otherwise, we let each student in (Xd )I receive a T -type contract of department d and add those contracts to X˜ d . Step 2.3. If m > qT + qR , then we skip this step and move to d d the next step. Otherwise, that is, qT < m ≤ qT + qR , we consider d d d a contract set Ad that consists of qT and m − qT many T -type and d d R-type contracts of department d, respectively. Starting from the best student and following the student ordering, one by one, we let each student in (Xd )I pick his favorite remaining contract in Ad and then remove it from Ad .21 We add the selected contracts to X˜ d . Step 2.4. If m > qT + qR , then we run the following procedure. d d Let us consider the set of all available contracts of department d in the problem and write Bd for that set.22 Substep 2.4.1. We start with student i1 and let him choose his favorite available contract in Bd and remove it from Bd . Let us add that contract to X˜ d . Let λT1 = max{qT − |X˜ dT |, 0} and d

λR1 = max{qRd − |X˜ dR |, 0}. We then move to the next substep. For k ≤ m, Substep 2.4.k. We consider student ik . If m − (k − 1) > λTk−1 + λRk−1 , then we let him choose his favorite available contract in Bd . Otherwise, that is, m − (k − 1) ≤ λTk−1 + λRk−1 , then we consider the contract set Ed that consists of λTk−1 and λRk−1 many T -type

and R-type contracts of department d, respectively.23 Starting from student ik , one by one and following the student ordering, we let each remaining student in (Xd )I pick his favorite available contract in Ed and remove the selected contract from Ed . We then add their selected contracts to X˜ d . As there are finitely many steps, the algorithm ends in a finite ˜ time. The final set of contracts ⋃ of department d is Xd , and the outcome of the algorithm is d∈D X˜ d . We call the mechanism ‘‘The Minimum-Need Adjusted Cumulative Offer Process’’ and write MCOP for short. In words, after the first stage COP outcome, for each department d with a TA/RA shortage, MCOP attempts to change the contract-types of the students who are assigned to department d to eliminate the shortages as much as possible. To this end, 21 That is, we run a serial dictatorship. 22 More explicitly, for each s ∈ {F , T , R, N }, B consists of qs many s-type d d contracts of department d. 23 Note that as we proceed one by one, it will never be the case that m − (k − 1) < λTk−1 + λRk−1 . That is, the following procedure will be invoked

whenever m − (k − 1) = λTk−1 + λRk−1 .

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it lets those students choose their favorite contract-type (one by one following their rankings at department d) until the shortages can just be ruled out with the students who yet to pick their contracts. After that, the remaining students are forced to choose between T -type and R-type contracts. In this phase, MCOP adjusts the quotas of T -type and R-type contracts in a way that TA shortage is eliminated as much as possible, and only then, RA shortage is mitigated (or completely eliminated) with the remaining students. Remark 2. It is important to observe that no student’s assigned department at the first stage ψ A outcome changes later in MCOP. This implies that the MCOP outcome is independent of the department ordering in which the steps are applied one by one for each department. It also implies that the MCOP outcome remains the same even if its steps are simultaneously applied for all the departments. We further illustrate how MCOP works on a simple problem below. Example 1. Let us consider a problem with three students, say i, j, k, and a department d such that qFd = 2 and qT = qR = d d qTd = qRd = qNd = 1. Let the preferences be such that Pi = Pj = Pk : (d, F ), (d, R), (d, T ), ∅. The rank ordering of the students at department d is such that i ≻d j ≻d k ≻d ∅. The first stage COP outcome of MCOP, say X , is such that Xi = (i, d, F ), Xj = (j, d, F ), and Xk = (k, d, R). Hence, |XdT | < qT . d As |Xd | > qT + qR , we start with considering all the available d d contracts of department d in the problem. We then let student i pick his favorite contract among them, which is the F -type contract of department d. We next consider student j. In this step, λT1 = λR1 = 1. That is, we have just enough students left to fulfill the TA/RA needs (this case corresponds to m − (k − 1) ≤ λT1 + λR1 ). Hence, we consider a set of contracts that consists of one T -type and one R-type contracts of department d. Student j picks his favorite available contract from that set, which is the R-type contract of department d. Then, student k picks the only remaining T -type contract. Hence, the final MCOP outcome for department d is X˜ d where X˜ d = {(i, d, F ), (j, d, R), (k, d, T )}. 3.5. The results In the rest of the paper, we study various properties of MCOP. We start with stability and show that MCOP is stable. For the proof of this result, we first observe that the Aw FS choice functions fall into the choice function class of Kominers and Sönmez (2016), which we call ‘‘slot-specific priority choices’’. Then, by invoking (Kominers and Sönmez, 2016)’s Theorem 1, we conclude that ψ A is stable. Finally, we show that MCOP is stable whenever ψ A is stable. We relegate all the formalities to Appendices A and B and give the ultimate result below. Theorem 1. MCOP is stable with respect to the Aw FS choices. We next investigate the MCOP’s performance in responding to the TA/RA needs of the departments. To this end, we first introduce a notion to compare allocations in terms of that aspect. Definition 2. An allocation X respects the TA/RA needs more than another allocation X ′ if the followings hold. (i) For any department d, |XdT | ≥ min{qT , |Xd′T |}. d (ii) For any department d with either (or both) |Xd′T | ≥ qT or d |XdT | = |Xd′T |, |XdR | ≥ min{qRd , |Xd′R |}. (iii) For some department d and s ∈ {T , R}, |Xds | > |Xd′s | and |Xd′s | < qs . d

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Respecting the TA/RA needs notion above compares allocations based on their performance in satisfying these needs. As we have already discussed before, we consider TA needs more critical, hence they are prioritized (however, this priority could easily be changed). The essence of the comparison notion is to respect TA needs as much as possible, and only then to satisfy the RA needs to the extent possible. In words, if an allocation X respects the TA/RA needs more than another allocation X ′ , then it means that for each department, the former does not admit a greater TA shortage than the latter. If, for some department, X yields a higher RA shortage than X ′ , then the latter admits a greater TA shortage than the former for the same department. Moreover, for some department, X ′ yields a higher TA or RA shortage than X . A mechanism ψ respects the TA/RA needs more than another mechanism φ if there is no problem at which the latter’s outcome respects the TA/RA needs more than the former’s while the converse is true for some problem. It is immediate to see that stability and respecting the TA/RA needs are incompatible in the sense that there exists a problem where for each stable allocation, there exists a non-stable allocation that respects the TA/RA needs more than itself.24 However, because stability is our core solution notion, we look for stable mechanisms that are unbeatable in terms of respecting the TA/RA needs by a stable mechanism. We say that a mechanism is limitedly respecting the TA/RA needs if there is no stable mechanism that respects the TA/RA needs more than the former. Our result below shows that MCOP is limitedly respecting the TA/RA needs.25 Theorem 2. MCOP is limitedly respecting the TA/RA needs. Proof. See Appendix D. □ For the proof of Theorem 2, based on Kominers and Sönmez (2016), we construct an artificial standard many-to-one matching problem (without contracts ) with responsive priorities and define a natural association between allocations in the original and artificial problems. By utilizing this construction, we obtain the key observation that at any stable allocation, each department admits the same number of students, which is the property known as the ‘‘rural hospitals theorem’’. We relegate the description of the artificial problem to Appendix C. Remark 3. We can straightforwardly extend the respecting TA/RA needs comparison notion to choice functions. A choice function profile C = (Cd )d∈D respects the TA/RA needs more than another choice function profile C ′ = (Cd′ )d∈D if, for any set of contracts X ⊆ X and any department d, (i) |Cd (X , T )| ≥ min{qT , |Cd′ (X , T )|}, d

24 For instance, consider one student i and two departments d, d′ with qF = 1, d qTd′ = qT′ = 1, and all other capacities are zero. Student i’s preferences are such d that he prefers the F -type contract of department d to the T -type contract of department d′ , and he would rather sign any of the two contracts than have no contract at all. He is admissible to both departments. The only stable allocation here is such that student i receives the F -type contract of department d. On the other hand, the allocation at which he signs a T -type contract with department d′ respects the TA/RA needs more than the stable one. Note that the latter is not stable. 25 Not every stable mechanism admits this property. For instance, let us consider a pair of stable mechanisms ψ and φ , and a problem where there is only one student, say i, and one department d with qFd = qTd = qT = 1, and d all other capacities are zero. Assume that i prefers the F -type contract to the T -type, and both are acceptable. Moreover, he is admissible to department d. At one stable allocation, he signs a F -type contract, whereas, at the other one, he signs a T -type contract. Let us assume that ψ gives the former (allocation) and φ gives the latter at this problem, and moreover, at any other problem, assume that they produce the same stable allocation. While both of them are stable, φ respects the TA/RA needs more than ψ ; hence ψ is not limitedly respecting the TA/RA needs.

(ii) if |Cd′ (X , T )| ≥ qT or |Cd′ (X , T )| = |Cd (X , T )|, then |Cd (X , R)| ≥ d min{qR , |Cd′ (X , R)|}, and (iii) for some X ′ ⊆ X , department d, and d s ∈ {T , R}, |Cd (X ′ , s)| > |Cd′ (X ′ , s)| and |Cd′ (X ′ , s)| < qs . We can d easily obtain that no choice function profile respects the TA/RA A needs more than C . This is because for any given X ⊆ X , CdA first tries to satisfy the TA needs by using all the department d’s contracts in X , and then it attempts to satisfy the RA need by using all the remaining contracts. Only then does it choose other types of contracts. We next look at the fairness aspect of MCOP. By invoking Kominers and Sönmez (2016) and Roth and Sotomayor (1989), we first obtain that the ψ A outcome associates with the DA outcome in the aforementioned artificial problem, which gives us the fairness of ψ A . Then, by using this auxiliary finding, we show that MCOP is fair as well.26 We present the DA definition in Appendix E and give the proof in Appendix F. Theorem 3. MCOP is fair. Proof. See Appendix F. □ Below, we study the efficiency properties of MCOP. Theorem 4. (i) MCOP is dominated by another stable mechanism. (ii) MCOP is not dominated by a stable mechanism that also limitedly respects the TA/RA needs. Proof. See Appendix G. □ In the rest of the paper, we study the incentive properties of MCOP. A mechanism ψ is strategy-proof if there exist no problem P and student i with false preferences Pi′ such that ψ (Pi′ , P−i ) Pi ψ (P).27 Theorem 5. MCOP is strategy-proof. Proof. See the Appendix H. □ Another important incentive property is that students should never be penalized because of their ranking improvements. This property is known as ‘‘respecting improvement’’ in the literature, which was first introduced by Balinski and Sönmez (1999). For its formalization, let us add the student ranking profile to the problem notation and write (P , ≻) for the given problem. A ranking profile ≻′ is an improvement over ≻ for student i if, for each department d and each pair of students j, k ∈ I \ {i}, (i) i ≻d j ⇒ i ≻′d j and (ii) j ≻d k ⇔ j ≻′d k. A mechanism ψ respects improvements if there exists no pair of problems (P , ≻) and (P , ≻′ ) such that ≻′ is an improvement over ≻ for student i and ψ (P , ≻) Pi ψ (P , ≻′ ). Theorem 6. MCOP respects improvements. Proof. See Appendix H. □ Until now, we show several desirable properties of MCOP. However, one may wonder whether there are other mechanisms 26 Note that stability with respect to the AwFS choices does not imply fairness. To see this, let us consider two students, say i, j, and one department d. Let qFd = qTd = qT = 1, and all the other capacities are zero. Let both students be d admissible while student i has a better ranking than student j. Both contracts are acceptable to the students, while each of them prefers the F -type contract to the T -type contract. Let X be an allocation such that Xi = (i, d, T ) and Xj = (j, d, F ). While X is stable (with respect to the Aw FS choices), it is not fair. 27 P denotes the preference profile of all students other than student i. −i

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admitting those properties. For this purpose, the result below shows that MCOP is the unique stable and strategy-proof mechanism that also limitedly respects the TA/RA needs.28 The independence of the axioms is also provided in Appendix I. Theorem 7. A mechanism is stable and strategy-proof, and limitedly respects the TA/RA needs if and only if it is MCOP. Proof. See Appendix I. □ 4. Conclusion & discussion We study graduate admissions with financial support in a matching with contracts setting. We propose an algorithm and show that it admits desirable fairness, stability, efficiency, and incentive properties. Some recent works have already found important applications of matching with contracts framework and successfully proposed new algorithms. This study identifies another such real-life problem for which a successful algorithm is proposed. Searching for other practical matching markets and studying them in a matching context might be a fruitful future research direction. One final discussion is about a possible extension of the model. In our formulation, we assume four different kinds of contracttypes. This is for ease of exposition, that is, we can easily generalize the problem to have any finite number of contract-types. Basically, we can consider any finite number contract-types with financial support (we can refer to them as ‘‘good’’ types) as well as any finite number of contract-types without financial support (we can refer to them as ‘‘bad’’ types). We can straightforwardly extend our assumptions to that case (specifically, any good-type contract is better than any bad-type contract while students may have heterogeneous preferences over the good-types and badtypes separately). Each type would have a capacity, and at the same time, it may have a minimum need. We can order the types that have minimum needs as we do by putting T -type before R-type. Then, we can easily generalize Aw FS choices and MCOP. The first stage of MCOP would remain the same. Its second stage would run exactly in the same manner, but in that case, we iteratively apply the same steps to the extra types as well.29 Appendix A. Slot-specific priority choices Kominers and Sönmez (2016) introduce a new class of choice functions under which a department (or branch in their terminology) fills each of its slots sequentially by following a certain precedence order over its slots. Each slot has a ranking order (or priority list) over the contracts, and whenever its turn to choose a contract, it picks the contract that is maximal with respect to its ranking order. In what follows, we first observe that the Aw FS choice functions fall into this class of choice functions that we call ‘‘slot-specific priority choices’’. That is, we can define a certain precedence order and a ranking list for the departments’ slots and have the Aw FS choices coinciding with the choices that are induced by the slots’ choices following the precedence order as in slot-specific priority choices. Let us consider a department d and refer to its every seat (capacity) as a slot. Department d has four types of slots, each of them corresponding to a different admission type. Namely, it has 28 This result was suggested by an anonymous referee. I am thankful to the referee, especially for this suggestion. 29 For instance, the shortage types which are to be eliminated first and second (TA/RA shortages in the current modeling, respectively) would replace the TA/RA shortages in the current mechanism, respectively. Shortages in the other later types would be handled similarly.

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F -type, T -type, R-type, and N-type slots. Let us label and order each of these slots. For each s ∈ S, let Bs = {bs1 , . . . , bsqs } be the d

ordering of the s-type slots (recall that qsd is the capacity for stype admissions). Let us endow each different slot with a ranking order over the set of contracts. For a s-type slot, say bsk , let Π (bsk ) be its ranking order which is defined as follows: (i) For any x ∈ X with either (or both) xD ̸ = d or xS ̸ = s, ∅ Π (bsk ) x. (ii) For any x, x′ ∈ X with xD = x′D = d and xS = x′S = s, x Π (bsk ) x′ iff xI ≻d x′I . (iii) For any x ∈ X with xD = d and xS = s, x Π (bsk ) ∅ iff xI ≻d ∅. A contract x is acceptable to slot bsk if x Π (bsk ) ∅. Finally, consider the following precedence order ▷d over the set of all department d’s slots: bT1 , . . . , ▷d bTT ▷d bR1 , . . . , ▷d bRR ▷d bF1 , . . . , bF F ▷d bT T

qd

qd +1

qd

, . . . , ▷d bTqT ▷d bRqR +1 , . . . , ▷d bRqR ▷d bN1 , . . . , bNqN . d

d

d

qd

d

In words, the precedence order is such that the first qT slots d are of T -type following the ordering within themselves (as specified in BT ), then the next qR slots are of R-type following the d ordering within themselves (again as specified in BR ). In the same way, the next qFd slots are of F -type, then the remaining T -type and R-type slots respectively come. The last qNd slots are of N-type. Let Cd′ be the slot-specific priority choice function of department d, which is defined as follows: For any given subset of contracts X ′ ⊆ X ,

• Consider the first slot with respect to the precedence ordering ▷d , which is bT1 . Fill this slot by choosing the Π (bT1 )-

maximal contract in X ′ , and then remove the chosen student’s all other contracts from X ′ . In general, ′ • Consider the next slot with respect to ▷d , say bsk . Fill this ′ slot by choosing the Π (bsk )-maximal contract in X ′ , and then remove the chosen student’s all other contracts from X ′ . • This process continues until either no unchosen acceptable contract left in X ′ or all the slots are filled. From the Aw FS choice definition, it is immediate to see that CdA coincides with Cd′ . That is, for any X ′ ⊆ X , CdA (X ′ ) = Cd′ (X ′ ). Hence, we have the following result. Proposition 2. The Aw FS choices belong to the class of slot-specific priority choice functions of Kominers and Sönmez (2016). Kominers and Sönmez (2016) show that COP is stable under any slot-specific priority choice function profile. Hence, the stability of ψ A immediately follows from that result and Proposition 2 above. Corollary 1. ψ A is stable. Appendix B Proof of Theorem 1. Given a problem P, let us write X = ψ A (P) and X ′ = MCOP(P). In the second stage of the MCOP, students’ contracts at X can only be changed within the same departments, and they may only sign a contract with a financial support. This, along with our assumptions on the student preferences and the definition of the Aw FS choices, implies that CI (X ′ ) = CDA (X ′ ) = X ′ . We now claim that there exist no set of contract X ′′ and department d such that X ′′ ̸ = Xd′ and X ′′ = CdA (X ′ ∪ X ′′ ) ⊆ CI (X ′ ∪ X ′′ ). Assume for a contradiction that there exist such a set of contracts X ′′ and department d. We then have two cases to consider.

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Case 1. First consider the case where XI′′ ⊆ (Xd′ )I . Let i1 be the best ranked student in XI′′ with respect to ≻d . From X ′′ = CdA (X ′ ∪ X ′′ ) and the definition of CdA , student i1 is the best ranked student in (Xd′ )I as well. Let us assume that Xi′′ ̸ = Xi′ . That is, 1 1 (Xi′′ )S = s and (Xi′ )S = s′ where s ̸ = s′ . By our supposition, 1 1 ′′ ′ we have Xi Pi1 Xi . Because of the definition of the Aw FS choices 1 1 along with the facts that i1 is the best ranked student and the first stage of the MCOP consists of COP, we have (Xi1 )S ̸ = s′ . That is, his contract is changed in the course of the second stage of MCOP. Moreover, from the definition of the second stage of MCOP and the fact that i1 is the best ranked student, it implies that every student in (Xd′ )I signs either a T -type or a R-type contract at X ′ , and moreover, |Xd′T | ≤ qT and |Xd′R | ≤ qR . Hence, if s = F , then d d Xi′′ ∈ / CdA (X ′ ∪ X ′′ ). 1 Now assume that s = T and s′ = R. That is, student i1 prefers T -type to R-type, yet he signs a R-type contract at X ′ . In Theorem 3, we prove that MCOP is fair. Let us invoke this result here and conclude that there is no student in (Xd′ )I who signs a T type contract of department d at X ′ (this also implies that qT = 0). d This, along with our above observation that every student signs either a T -type contract or a R-type contract, implies that every student signs a R-type contract, and yet still |Xd′R | ≤ qR . The latter d is because, otherwise, student i1 would not be forced to sign a R-type contract and would keep his contract at the first stage X outcome, which is different than that at X ′′ as shown above. Then, / CdA (X ′ ∪ X ′′ ). by the definition of the Aw FS choices, we have Xi′′ ∈ 1 ′ Let us now assume that s = R and s = T . That is, student i1 prefers R-type to T -type, and yet he signs a T -type contract of department d at X ′ . Due to the fairness of MCOP (see Theorem 3), it implies that no student in (Xd′ )I signs a R-type contract with department d at X ′ . Hence, by the definition of the Aw FS choice, we have |Xd′T | ≤ qT or qRd = 0 (or both). In both cases, we have d / CdA (X ′ ∪ X ′′ ). Xi′′ ∈ 1 Note that s ̸ = N. This is because we know that every student in (Xd′ )I signs either a T -type or a R-type contract with department d at X ′ . Hence, this, along with our assumptions on the student preferences, shows that s ∈ {F , T , R}. The above analysis reveals that Xi′′ = Xi′ . Let i2 be the second 1 1 best ranked student (with respect to ≻d ) in XI′′ . As Xi′′ = Xi′ , by 1 1 the same reasons as above, i2 is the second best ranked student ′′ ′ ′ ′′ in (Xd )I as well. Assume that Xi ̸ = Xi . That is, (Xi )S = s and 2 2 2 (Xi′ )S = s′ where s ̸ = s′ , and by our supposition, Xi′′ Pi2 Xi′ . As 2 2 2 Xi′′ = Xi′ and (Xi′′ )S = s, we have either (Xi′ )S = s and qsd ≥ 2 1 2 1 1 s ′ or (Xi )S ̸ = s and qd > 0. This, along with the facts that i2 is the 1 second best ranked student and the first stage of MCOP consists of COP under the Aw FS choices, implies that (Xi2 )S ̸ = s′ . That is, his contract is changed in the course of the second stage of MCOP. As i2 is forced to change his contract in the second stage of MCOP, we have that every student in (Xd′ )I \ {i1 } signs either a T -type or a R-type contract at X ′ , and the TA/RA needs of the department d cannot be ruled out with the students having a worse ranking than i2 . Hence, this, along with Xi′′ = Xi′ , implies that if s = F , 1 1 then Xi′′ ∈ / CdA (X ′ ∪ X ′′ ). 2 Now assume that s = T and s′ = R. That is, student i2 prefers T -type to R-type, yet he signs a R-type contract at X ′ . Due to the fairness of MCOP, there is no student in XI′ \ {i1 } who signs a T type contract at X ′ . This implies that there is no TA shortage at X ′ and either (or both) |Xd′R | ≤ qR or there is no left T -type capacity d after possibly student i1 receives a T -type contract at X ′ . Then, from Xi′′ = Xi′ and the Aw FS choices definition, it implies that 1 1 Xi′′ ∈ / CdA (X ′ ∪ X ′′ ). 2 Let us now assume that s = R and s′ = T . That is, student i2 prefers R-type to T -type, yet he signs a T -type contract at X ′ . Due to the fairness of MCOP, there is no student in XI′ \ {i1 } who signs a R-type contract at X ′ . This implies that either |Xd′T | ≤ qT d

or |Xd′T | > qT and there is no R-type seat left after possibly d student i1 signs a R-type contract with department d. In both ′′ cases, from Xi = Xi′ and the definition of the Aw FS choices, we 1 1 have Xi′′ ∈ / CdA (X ′ ∪ X ′′ ). 2 The proof of the case of s = N directly follows from the same arguments as in the same case for student i1 . Hence, we conclude that Xi′′ = Xi′ . Once we repeat the same analysis to 2 2 the other students in XI′′ one at a time following their rankings, we eventually conclude that for any i ∈ XI′′ , Xi′′ = Xi′ . If XI′′ = (Xd′ )I , then it implies that X ′′ = Xd′ , contradicting our starting supposition. In the other case, that is, XI′′ ⊂ (Xd′ )I , we have X ′′ ⊂ Xd′ . This, along with the supposition that X ′′ = CdA (X ′′ ∪ X ′ ), contradicts the fact that CDA (X ′ ) = X ′ . Case 2. Let us now consider the case where XI′′ ⊈ (Xd′ )I . Let i ∈ XI′′ \ (Xd′ )I . That is, student i is such that (Xi′′ )D = d, whereas at X ′ , he has no contract or (Xi′ )D = d′ where d ̸ = d′ , and Xi′′ Pi Xi′ . Without loss of generality, let us assume that (Xi′ )D = d′ . The same arguments below show the other case as well. Xi′′ Pi Xi′ , along with our assumptions on the student preferences, shows that if department d had to have a TA or RA shortage at X , then student i would have matched with department d at X ; hence at X ′ as well. This shows that department d does not have a TA or RA shortage at X . This in turn implies that Xd = Xd′ . That is, department d has the same set of contracts at both X and X ′ . This, along with the stability of ψ A and our assumptions on students’ preferences, implies that either (or both) X ′′ ̸ = CdA (X ′ ∪ X ′′ ) or X ′′ ⊈ CI (X ′ ∪ X ′′ ), which finishes the proof. □ Appendix C Here, based on Kominers and Sönmez (2016), we associate each problem to an artificial many-to-one standard matching (without contracts) problem with responsive priorities. In the artificial problem, the set of students I remains the same, and let us refer to the other side as ‘‘slots’’. Let C denote the set of slots. For each department d ∈ D, we define four different types of slots in C ; namely, F -type, T -type, R-type, and N-type slots and denote them by dF , dT , dR , and dN , respectively. For any s ∈ S and department d, slot ds has the capacity given by qds = qsd . Let Pi′ be the student i’s preferences over C and being unassigned, denoted by ∅. It is defined as follows: ′ ′ (i) For any pair of slots ds and dˆ s , ds Pi′ dˆ s if and only if ′ ˆ (d, s) Pi (d, s ), and (ii) for any ds , ds Pi′ ∅ if and only if (d, s) Pi ∅

A slot ds is said to be acceptable to student i if ds Pi′ ∅. Otherwise, it is unacceptable. Each slot ds ∈ C has a ranking list ≻′ds over I ∪ {∅} where ∅ denotes keeping seat empty. They directly come from the student rankings at the associated department. That is, ≻′ds =≻d . Student i is admissible to slot ds if i ≻′ds ∅. Otherwise, he is inadmissible. Let us denote the artificial problem, which is associated to original problem P, by P ′ . In the artificial ′ ′ problem, for a student i with Pi′ , we write ds R′i dˆ s only if ds Pi′ dˆ s ′ or ds = dˆ s . A matching µ in the artificial problem is an assignment of students to slots such that each student is assigned at most one slot, and no slot has more assigned students than its capacity. Let µk be the matching of student or slot k ∈ I ∪ C under µ. We are now ready to associate every allocation in the original problem to a matching in the artificial problem. Given an allocation X , we define the associated matching µX in the artificial problem as follows: For any student i ∈ I,

µXi = ds iff (i, d, s) ∈ X and µXi = ∅ iff i ∈ / XI . Note that as X is an allocation, and for any ds ∈ C , qds = qsd , µX defines a matching in the artificial problem. By our construction,

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for any pair of allocations X , X ′ in the original problem, and for ′ any student i ∈ I, Xi Ri Xi′ if and only if µXi R′i µXi . Appendix D Before moving to the proof, for ease of exposition, we state the stability definition in the artificial problem. Matching µ is stable at artificial problem P ′ if (i) for any i ∈ I, µi R′i ∅, (ii) for any ds and i ∈ µds , i ≻′ds ∅, and (iii) there exists no blocking student–slot pair (i, ds ) in the sense that ds Pi′ µi , i ≻′ds ∅, and either (or both) |µds | < qds or for some j ∈ µds , i ≻′ds j. Proof of Theorem 2. We first show that at any problem and any department d, the number of students signing a contract with department d is the same at every stable allocation.30 First of all, due to the rural hospitals theorem in the standard matching setting with responsive priorities (Roth, 1984), in the class of stable allocations whose associated matchings in the artificial problem are also stable, every department signs a contract with the same number of students at every stable allocation in this class. However, there are stable allocations whose associated matchings fail to be stable in the artificial problem.31 In what follows, we show that it is the case for those stable allocations as well.32 Let X be a stable allocation in the original problem P such that its associated matching µX fails to be stable in the artificial problem P ′ . By the definitions of the artificial problem and µX , µXi R′i ∅ for any i ∈ I, and for any ds ∈ C and i ∈ µXds , i ≻′ds ∅. Hence, matching µX admits a blocking student–slot pair. Let (i, ds ) be a such blocking pair. That is, student i constitutes a blocking pair with a slot of department d. First, it cannot be the case that (Xi )D ̸ = d. In other words, student i is assigned to department d at X . This is because, otherwise, by the definition of the Aw FS choices, matching X would not have been stable in the original problem. Hence, let us consider the set of students who block µX for different slots of their already assigned departments. Let us first consider department d’s slots along with its assigned students who block matching µX . Among such students, consider the best ranked student, say i1 , along with his most favorable slot with which he can block the matching. Let us satisfy that blocking pair by assigning him to that slot. Moreover, if the slot exceeds its capacity after the reassignment, then consider the worst ranked assigned student in this slot and let him be assigned to the student i1 ’s initial slot. Otherwise, keep him at his original slot. By construction, as student i1 is the best ranked one among the blocking students within department d, at the obtained matching after those reassignments, no student who is

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assigned to department d and has a better ranking than student i1 belongs to a blocking-pair. If the newly obtained matching above is stable, then we are done because at that newly obtained matching, every department admits the same number of students as X (the number of students that a department admits is the total number of students who are assigned to one of its slots). Assume that the newly obtained matching is not stable, and there is a student who is assigned to department d and blocks the matching with a slot of department d. Let i2 be the best ranked student among the ones who belong to a blocking-pair with a slot of department d. Note that i2 has a worse ranking than student i1 . Similar to above, let us place him at his most favorable slot with which he forms a blockingpair. If that slot exceeds its capacity after this placement, then consider the worst ranked assigned student in this slot and let him be assigned to the i2 ’s former slot. Otherwise, keep him at his slot. As the same as above, as student i2 is the best ranked one among blocking students within department d, at the obtained matching after those reassignments, no student who is assigned to department d and has a better ranking than student i2 belongs to a blocking-pair. As the same as above, if the obtained matching right above is stable, then we are done. Let us assume that the newly obtained matching is not stable and there is a student who is assigned to department d and blocks the matching with a slot of department d. Let student i3 be the best ranked such student. By construction, that student is such that he has a worse ranking than both i2 and i1 . If we keep obtaining different matchings as the same as above, then as the set of students is finite and the fact that in every step, we consider a different student who has a worse ranking than the previously considered blocking students, after some iteration, the students who block the obtained matching with a slot of department d will exhaust. Once we repeat the above procedure for every department, the final matching that we will obtain will be stable at the artificial problem. By construction, at that obtained stable matching, the total number of students who are assigned to the slots of a department remains the same as at µX . This, along with the rural hospitals theorem in the artificial problem, shows that at any stable allocation, every department admits the same number of students (their contract-types may be different though). Starting with the TA shortage, the second stage of MCOP tries to rule out any TA/RA shortages with the departments’ assigned students to the extent that possible. This, along with our above finding that every department admits the same number of students at any stable allocation, implies that MCOP is limitedly respecting the TA/RA needs. □ Appendix E. The deferred acceptance mechanism (DA)

30 This property is known as ‘‘rural hospitals theorem’’. There are different variants of this theorem. In the current matching with contract setting, the version that we consider is obtained by Hatfield and Kojima (2010) under US and the LAD. However, our Aw FS choice functions do not satisfy either of them. First, let us define them. Cd satisfies US if there are no Y ⊆ X and x, z ∈ X \ Y such that z ∈ / Cd (Y ∪ {z }), zI ∈ / YI , and z ∈ Cd (Y ∪ {x, z }). Cd satisfies the LAD if for any Y ⊂ Y ′ ⊆ X , |Cd (Y )| ≤ |Cd (Y ′ )|. Now, let us consider two students i, j and one department d, with qFd = qTd = qT = 1, and all the other capacities are d zero. Let i ≻d j ≻d ∅, Pi : x, x′ , ∅, Pj : y, y′ , ∅ where xS = yS = F and x′S = y′S = T . A A Then, y ∈ / Cd ({x, y}) but y ∈ Cd ({x} ∪ {y, x′ }), violating US. For the lack of the LAD, observe that CdA ({x, y′ }) = {x, y′ } yet CdA ({x, y′ } ∪ {x′ }) = {x′ }. 31 For instance, consider one student i and one department d. Let qF = qT = d

d

qT = 1, with all the other capacities are zero. Let student i prefer the F -type d contract to the T -type contract and both of them are acceptable. Moreover, let i ≻d ∅. Consider the allocation X at which student i receives a T -type contract. While X is stable at the original problem, the associated matching µX fails to be stable at the artificial problem. 32 Kominers and Sönmez (2016) do not obtain the rural hospitals theorem in their work.

Below outlines how DA (Gale and Shapley, 1962) works in the artificially constructed many-to-one matching problem. Step 1. Each student applies to his first choice acceptable slot. Among its applicants, one by one following its ranking list, each slot tentatively accepts as many admissible students as up to its capacity and rejects the rest. In general, Step t. Each rejected student in the previous step applies to his next best acceptable slot. Among its current step applicants and the tentatively accepted ones in the previous step, one by one following its ranking list, each slot tentatively accepts as many admissible students as up to its capacity and rejects the rest. The algorithm terminates whenever every student is either tentatively assigned to some slot or rejected from each of his acceptable slots. The tentative assignments at the terminal step realize as the final DA assignments.

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Appendix F Kominers and Sönmez (2016) show that the DA outcome in their artificial problem construction associates to the COP outcome in the original problem (they define an association between matchings (allocations) across problems as the same as we do in the Appendix C). While their artificial construction is a oneto-one matching problem and ours is a many-to-one matching with responsive priorities instance, from Roth and Sotomayor (1989), we know that the sets of stable allocations in one-to-one and many-to-one with responsive priorities matching problems coincide with each other in a very natural sense. This, along with Appendix A, shows that Kominers and Sönmez (2016)’s result directly carries over to our setting as well. That is, we have the following corollary. Corollary 2. Given a problem P, the ψ A outcome associates with the A DA outcome at the associated artificial problem P ′ . That is, µψ (P) = ′ DA(P ). Proof of Theorem 3. We first claim that ψ A is fair. Assume for a contradiction that ψ A is not fair at some problem P, and let ψ A (P) = X 0 . This means that there exists a pair of students, say i, j, such that Xj0 Pi Xi0 and i ≻d j where d = (Xj0 )D . By Appendix C, 0

0

it implies that µXj Pi′ µXi and student i has a better ranking than student j at student j’s assigned slot. From Corollary 2, we know 0 that µX is the DA outcome at P ′ . Hence, it implies that the DA outcome is not fair. This, however, contradicts the fairness of DA in the artificial problem (Balinski and Sönmez, 1999). Hence, ψ A is fair. Let MCOP(P) = X 1 . To show the fairness of X 1 , let us consider two students i, j such that Xj1 Pi Xi1 . First, let us consider the case where (Xj1 )D = d and either i ∈ / XI1 (that is, he does not have a contract at X 1 ) or (Xi1 )D = d′ where d ̸ = d′ .33 Let us assume that (Xi1 )D = d′ . The same arguments below work for the other case of i ∈ / XI1 as well. As (Xj1 )D = d and (Xi1 )D = d′ where d ̸= d′ , from our preferential suppositions, we have Xj0 Pi Xi0 . This, along with the fairness of ψ A , implies that j ≻d i, showing the fairness of X 1 for this case. Let us now consider the case where (Xj1 )D = (Xi1 )D = d. That is, they are assigned to the same department. By the definition of MCOP, they are assigned to the same department d at X 0 as well. If Xj0 Pi Xi0 , then by the fairness of ψ A , we have j ≻d i. Let us assume that Xi0 Ri Xj0 and Xj1 Pi Xi1 . This implies that there is either (or both) a TA or a RA shortage at department d at X 0 , and either (or both) student i or student j changes his contract in the course of the second stage of MCOP. Assume for a contradiction that i ≻d j. We have the following cases to consider. Case 1. Xi1 = Xi0 and Xj1 ̸ = Xj0 . In words, student i manages to keep his contract while student j changes his contract in the course of the second stage of MCOP. By the definition of MCOP, it implies that student j signs a contract-type which falls short of its need at X 0 . This in turn implies that Xi0 Ri Xj1 . Therefore, as Xi0 = Xi1 , we have Xi1 Ri Xj1 . Case 2. Let us assume that Xi1 ̸ = Xi0 . That is, because of TA or RA (or both) shortages of department d at X 0 , student i changes his contract in the course of the second stage of the MCOP. In this case, as student j has a worse ranking than student i (by our supposition), by the definition of MCOP, he is forced to sign a T type of a R-type contract of department d. Therefore, each of them 33 Due to the stability of the MCOP, student i does not envy any student without a contract. Hence, student j has to sign a contract at X 1 .

receives either a T -type or a R-type contract at X 1 . However, as, by our supposition, Xj1 Pi Xi1 , they sign different contracts at X 1 . Let us assume that student i and student j sign T -type and Rtype contracts, respectively. By our assumption, student i prefers R-type to T -type. However, this preference of student i is not respected in the second stage of MCOP. This is because whenever its his turn to pick a contract in the second stage of MCOP, the remaining TA shortage is no less than the total number of remaining students, including student i, who yet to pick their contracts within department d. In this case, however, student j, who has a worse ranking than student i, has to sign T -type contract as well, contradicting our supposition. The proof for the other case where student i and student j respectively sign R-type and T -type contracts follow from the same arguments. This contradicts our starting supposition, hence completing the proof. □ Appendix G Proof of Theorem 4. (i) Let us consider a problem consisting of only one student, say i, and only one department d. Let qFd = 1 and qT = qTd = 1, with all other capacities are equal to zero. Student d i’s preferences is such that he prefers the F -type to the T -type, and he would rather sign a contract than have no contract at all. Student i is admissible to department d. There are two stable allocations here. At the MCOP outcome, student i receives a T -type contract. At the other stable allocation, he signs a F -type contract. Let ψ be a mechanism that gives the latter allocation at this problem and coincides with the MCOP outcome at every other problem instance. It is immediate to see that ψ is a stable mechanism that dominates MCOP. (ii) Let φ be a stable mechanism that limitedly respects the TA/RA needs. Let us consider a problem P and write MCOP(P) = X , φ (P) = X ′ , and ψ A (P) = X ′′ . We will show that X ′ cannot dominate X . We first claim that if there is a student i such that (Xi )D = d and (Xi′ )D = d′ where d ̸ = d′ , then X cannot be dominated by X ′ . To see this, let us construct an artificial problem as follows. For each department d ∈ D, we introduce two different slot-types and denote them by dm and dn . The first one department d’s with financial support slots while the latter is the same department’s without financial support slots. The capacities are such that qdm = qFd + qTd + qRd and qdn = qNd . Let us define the students’ preferences P ′′ as follows: For any student i, (i) (ii) (iii) (iv ) (v )

for for for for for

any any any any any

dm and dˆ m , dm Pi′′ dˆ m if and only if (d, F ) Pi (dˆ , F ). dm and dˆ n , dm Pi′′ dˆ n if and only if (d, F ) Pi (dˆ , N). dn and dˆ n , dn Pi′′ dˆ n if and only if (d, N) Pi (dˆ , N). dm , dm Pi′′ ∅ if and only if (d, F ) Pi ∅. dn , dn Pi′′ ∅ if and only if (d, N) Pi ∅.

The slots’ ranking lists over the students, say ≻′′ , directly come from the associated department’s ranking lists. That is, for any dm and dn , ≻′′dm =≻′′dn =≻d . Let P ′′ denote this artificial problem. Given an allocation X˜ at P, let us define the associated matching µ(X˜ ) at P ′′ as follows: For any student i, (i) µi (X˜ ) = dm iff X˜ i = (d, s) for some s ∈ {F , T , R}. (ii) µi (X˜ ) = dn iff X˜ i = (d, N). (iii) µi (X˜ ) = ∅ iff i ∈ / X˜ I . We now claim that for any stable allocation X˜ at P, the associated matching µ(X˜ ) is stable at P ′′ . Assume for a contradiction that µ(X˜ ) is not stable. First, by construction, for no student i ∈ I, ∅ Pi′′ µi (X˜ ). Moreover, for any k ∈ {m, n} and any i ∈ µdk (X˜ ), i ≻′′dk ∅. Hence, the lack of stability implies that there exists a student i who blocks the matching with another slot. Note that if he is assigned to dn for some d, then he cannot block with dm .

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Because, otherwise, X˜ could not have been stable at P (by the definition of the Aw FS choices and the preferential suppositions). Moreover, because we treat all of the financial support options the same by considering only one slot for them, it implies that he blocks with another department’s slot. That is, there exists a slot of another department which has an excess capacity or someone assigned to that slot has a worse ranking than student i. But then, by the definition of the Aw FS choices and our assumptions on the students’ preferences, it implies that X˜ cannot be stable at P, which finishes the proof of this claim. Next, let us observe that the DA outcome at P ′′ associates with the ψ A outcome at P. That is, for any problem P, µ(ψ A (P)) = DA(P ′′ ). To see this, let us consider our earlier artificial problem construction in Appendix C. For any given matching σ there, we define the associated matching µ(σ ) in the above artificial problem as follows: µi (σ ) = dm iff σi = ds for some s ∈ {F , T , R}, µi (σ ) = dn iff σi = dN , and µi (σ ) = ∅ iff σi = ∅. Then, it is immediate to see that the DA outcome at the artificial problem in Appendix C, say P ′ , associates with the DA outcome at P ′′ (this is because at P ′′ , all different slots that are associated with different financial support options of each department are considered as only one slot whose capacity is the sum of their individual capacities. Moreover, the students’ preferences and their ranking lists at the departments are all preserved). As the DA outcome at P ′ associates with the ψ A outcome at P, it in turn implies that the DA outcome at P ′′ associates to the ψ A outcome at P. Note that as each slot with a financial support is treated as the same in the artificial problem, it also implies that for each problem P, MCOP(P) associates with the DA outcome in the artificial problem P ′′ . Let us now consider X , X ′ , and X ′′ . By our initial supposition, for some student i, (Xi )D = d and (Xi′ )D = d′ where d ̸ = d′ (here, either d or d′ can be ∅, representing having no contract). Both µ(X ′′ ) and µ(X ′ ) are stable at P ′′ , and moreover, the former is the DA outcome at P ′′ , which is the unanimously preferred stable outcome by the students at P ′′ . This, along with the fact that student i is assigned to different departments at those matchings and (Xi )D = (Xi′′ )D , implies that Xi Pi Xi′ (here, we also invoke our preferential suppositions). Hence, X ′ cannot dominate X . Let us now assume that every student is assigned to the same department at both X ′ and X (note that in this case, their associated matchings at P ′′ may be the same). Let us consider department d, along with the best ranked student assigned to itself, say i. We claim that Xi Ri Xi′ . Note that as no student’s department changes in the second stage of MCOP, each student’s assigned department under X and X ′ is also the same as that under X ′′ . In the second stage of MCOP, student i picks his favorite department d contract subject to the TA/RA needs. More explicitly, given the better-ranked student choices, student i may be forced to receive his non-favorable contract type only when otherwise those shortages would be exacerbated. As no other stable allocation respects the TA/RA needs more than X ′ , and the departments have the same set of students at both allocations, it implies that Xi Ri Xi′ . If it is strict, then we are done. Otherwise, Xi = Xi′ . In the latter case, we repeat the same arguments for the second bestranked student who is assigned to the department d, and so on. From here, we eventually conclude that either they are the same allocation or there exists at least one student who prefers X to X ′ , which finishes the proof. □

writing, let φ = MCOP. Consider a problem P and student i with φi (P) = (d, s). Assume for a contradiction that student i benefits from reporting a false preference Pi′ . That is, φ (Pi′ , P−i ) Pi φ (P). Let P ′ = (Pi′ , P−i ) and φi (P ′ ) = (d′ , s′ ). We first claim that d′ = d. Assume for a contradiction that d ̸ = d′ (here, by abusing notation, d can be ∅, representing having no contract. Because of the stability of φ , d′ cannot be ∅).

Appendix H

Proof. For ease of writing, let us write φ for MCOP. Assume for a contradiction that at some problem P and for student i, φ (P) Pi ψ A (P). Let φi (P) = (d, s′ ) and ψiA (P) = (d, s) where s ̸= s′ (recall that the students’ assigned departments do not change in the second stage of φ ). This implies that s′ ̸ = N and student i’s

Proof of Theorem 5. First of all, from Kominers and Sönmez (2016) and the fact that the Aw FS choices fall into their class of choice functions, we know that ψ A is strategy-proof. For ease of

Because the students’ assigned departments remain the same in the course of the second stage of MCOP, we have (ψiA (P ′ ))D = d′ and (ψiA (P))D = d. If s′ = N, then it means that at P ′ , student i’s contract-type does not change in the second stage of φ . Hence, ψiA (P ′ ) = (d′ , N). This, along with our preferential suppositions and the initial supposition that (d′ , N) Pi (d, s), implies that ψiA (P ′ ) Pi ψiA (P), contradicting the strategy-proofness of ψ A . On the other hand, if s′ ̸ = N, then (ψiA (P ′ ))S ∈ {F , T , R}. Then, similar to the previous case, from our preferential suppositions, we have ψiA (P ′ ) Pi ψiA (P), contradicting the strategy-proofness of ψ A . Hence, d′ = d. Above shows that in MCOP, by misreporting his preferences, student i cannot be better off through being assigned to a different department. Yet another possibility is that student i can continue to be assigned to the same department but with a more favorable financial support term. To study this case, let φi (P ′ ) = (d, s′ ) where s′ ̸ = s and (d, s′ ) Pi (d, s) (recall that φi (P) = (d, s)). We now claim that the same set of students are assigned to department d at both φ (P ′ ) and φ (P). As no student’s department is changed in the second stage of φ , student i is assigned to department d under both ψ A (P) and ψ A (P ′ ). For ease of writing, let ψ A (P) = X and ψ A (P ′ ) = X ′ . Let us now revisit the artificial matching problem formulation in Appendix G. Let us write P ′′ for the artificial problem that is associated to problem P. Because of our construction there, µ(X ′ ), the associated matching of X ′ in that artificial problem P ′′ , is stable at P ′′ . Likewise, µ(X ) is also stable at P ′′ . Hence, by the rural hospitals theorem, the same number of students are assigned to the departments at both X and X ′ . Moreover, because ψ A is strategy-proof, we have ψ A (P) Ri ψ A (P ′ ). This in turn implies that department d has to admit TA or RA shortage (or both) at either (or both) of allocations X and X ′ . This, along with our above observation, implies that at both µ(X ) and µ(X ′ ), slot dm , whose total capacity is qdm = qFd + qTd + qRd , has excess capacity. Then, by invoking the rural hospitals theorem once more, we conclude that the same set of students is assigned to dm at both µ(X ) and µ(X ′ ), which in turn implies that those students are assigned to department d at both φ (P) and φ (P ′ ). That is, the same set of students is assigned to department d at φ (P) and φ (P ′ ). In the second stage of MCOP, student i cannot affect the contract assignments of the students who are ranked better than himself. This implies that each student j ∈ Xd (note that Xd = Xd′ ) with j ≻d i, we have φj (P) = φj (P ′ ). This in turn means that in the second stage of MCOP at both problems P and P ′ , either student i is forced to sign the same contract or he picks the more favorable one between T and R-type contracts of department d. But then, it means that student i cannot benefit from reporting P ′ under φ , completing the proof. □ We will use the following lemma in the proof of Theorem 6. Lemma 1. For any problem P and student i ∈ I, ψ A (P) Ri MCOP(P).

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contract is changed in the course of the second stage of φ . This in turn means that department d has a shortage at ψ A (P). First, let us observe that student i cannot obtain a F -type contract of department d in the second stage of φ . That is, s′ ̸ = F . Assume for a contradiction that s′ = F . As (d, s′ ) Pi (d, s), in the course of ψ A , student i offers F -type contract to department d before he offers (d, s), and the former is rejected. This means that department d exhausts all of its F -type contracts for the better-ranked students in ψ A . In the second stage of φ , if those better-ranked students continue to receive the F -type contracts, then student i cannot obtain a F -type contract. Otherwise, they may be forced to sign a T -type or R-type contract because of the remaining shortages. But then, as student i has a worse ranking than all of those, he is forced to sign either of T -type and R-type contract as well, showing that he cannot obtain a F -type contract of department d at φ (P). That is, s′ ̸ = F . Therefore, s′ ∈ {T , R}. Without loss of generality, assume that φi (P) is a T -type contract. This, along with our supposition that φ (P) Pi ψ A (P), implies that department d’s T -type contracts are fully exhausted at ψ A (P). And moreover, as ψ A is fair, any student who receives a T -type contract of department d at ψ A (P) has a better ranking than student i (with respect to ≻d ). This implies that department d has only a RA shortage at ψ A (P). This in turn implies that each student who receives a T -type contract prefers T -type to R-type contract of department d. Given this as well as the fact that each such student has a better ranking than student i, student i cannot receive a T -type contract in the second stage of φ , contradicting our starting supposition. The symmetric arguments show the case where student i signs R-type contract at φ (P). □ Proof of Theorem 6. Kominers and Sönmez (2016) show that COP respects improvements under their slot-specific priority choices. Their improvement notion coincides with ours. Moreover, we already show that the Aw FS choices fall into their class of choice functions. Hence, we can immediately conclude that ψ A respects improvements. In what follows, we show that the same is true for MCOP as well. We now add the student rankings ≻ to the problem notation. Let us consider (P , ≻) and (P , ≻′ ) such that ≻′ is an improvement over ≻ for student i. For ease of exposition, let ψiA (P , ≻) = (d, s), ψiA (P , ≻′ ) = (d′ , s′ ), and φ = MCOP. As ψ A respects improvements, we have (d′ , s′ ) Ri (d, s). Assume for a contradiction that φ does not respect improvements and student i prefers φi (P , ≻ ) to φi (P , ≻′ ). From (d′ , s′ ) Ri (d, s) and our assumptions on the preferences, we have d = d′ . By Lemma 1, we know that ψ A (P , ≻) Ri φ (P , ≻). Hence, if we sum up our findings so far, we have (d, s′ ) Ri (d, s) Ri φi (P , ≻) where ψiA (P , ≻′ ) = (d, s′ ) and ψiA (P , ≻) = (d, s). Therefore, the only way for student i to be worse off at ≻′ is that department d has either a TA shortage or a RA shortage (or both) and his contract-type is changed in the course of the second stage of φ . The rest of the proof follows from the same arguments we have in the proof of Theorem 5. We now claim that the same set of students is assigned to department d at both (P , ≻) and (P , ≻′ ). Let us consider the artificial problem, introduced in Appendix G, associated with problem (P , ≻). Let us write ψ A (P , ≻) = X and ψ A (P , ≻′ ) = X ′ . We know that µ(X ) is stable in that artificial problem. Moreover, it is easy to see that µ(X ′ ) is also stable in the same artificial problem. Moreover, as department d has a shortage at X ′ , by the rural hospitals theorem, the same set of students is assigned to department d slots at the artificial problem at both µ(X ) and µ(X ′ ). This in turn implies that the same set of students is assigned to department d at both X and X ′ . In the second stage of MCOP, student i cannot affect the contract assignments of the students who are ranked better than

himself. This implies that each student j ∈ Xd (note that Xd = Xd′ ) with j ≻′d i, we have φj (P , ≻) = φj (P , ≻′ ). This in turn means that in the second stage of MCOP if student i is forced to sign a T -type of R-type contract at (P , ≻′ ), then so is he at (P , ≻). On the other hand, as his ranking is at least weakly increased at ≻′d , if he picks his favorite one among those types at (P , ≻), then so does he at (P , ≻′ ). The converse, however, may not be true. That is, he may choose his favorite among T -type and R-type contracts of department d at (P , ≻′ ), but not at (P , ≻). These altogether shows that φi (P , ≻′ ) Ri φi (P , ≻), which finishes the proof. □ Appendix I Proof of Theorem 7. ‘‘If’’ Part. We have already proven that MCOP is stable, strategy-proof, and that limitedly respects the TA/RA needs. ‘‘Only If’’ Part. Let ψ be mechanism such that it is stable, strategy-proof, and that limitedly respects the TA/RA needs. For ease of writing, let us write φ for MCOP. We first claim that for each problem P and department d, [ψd (P)]I = [φd (P)]I . That is, each department admits the same set of students at both outcomes. Assume for a contradiction that for some student i, [ψi (P)]D = d and [φi (P)]D = d′ where d ̸ = d′ (either of them can be ∅ as well). For ease of writing, let ψ (P) = X and φ (P) = X ′ . Let us first observe that φi (P) Pi ψi (P). To see this, let us revisit the artificial problem construction in Appendix G. There, we observe that φ (P) associates with the DA outcome in the artificial problem P ′′ . By using the notations in Appendix G, we have µ(X ′ ) = DA(P ′′ ). Moreover, as we find in that appendix, µ(X ) defines a stable matching in P ′′ . Note that as d ̸= d′ , we have µi (X ) ̸ = µi (X ′ ). As DA produces the unanimously preferred stable matchings in the standard matching setting, we have µi (X ′ ) Pi′′ µi (X ). This, along with our preference suppositions, implies that Xi′ Pi Xi . Let Pi′ : Xi′ , ∅; P ′ = (Pi′ , P−i ), and ψ (P ′ ) = X ′′ . It is immediate to observe that X ′ is stable in P ′ . Hence, both µ(X ′′ ) and µ(X ′ ) define a stable matching in the artificial problem that is associated with P ′ . Hence, by the rural hospitals theorem, we have Xi′′ ̸ = ∅. This, along with the individual rationality of ψ , implies that ψi (P ′ ) = Xi′ . This shows that under ψ , student i profitably misreports in problem P, contradicting the strategy-profess of ψ . Hence, we conclude that for each student i, [ψi (P)]D = [φi (P)]D . That is, under both ψ and φ , each department admits the same set of students at each problem. For a department d, let us now consider (Xd )I , the set of students who receive a department d contract under X . Note that from above, (Xd )I = (Xd′ )I . Let us order those students on the basis of their rankings at department d, and from best to worst, we write (Xd )I = {i1 , . . . , ik }. Let us now consider student i1 . First of all, Xi′ Ri1 Xi1 . This is 1 because, by the definition of MCOP, student i1 has freedom to pick his more preferred contract-type as long as the TA/RA needs can be satisfied with the rest. That is, if he cannot obtain his favorite contract-type of department d under X ′ , then it is because of the TA/RA needs. This, along with the facts that ψ limitedly respects the TA/RA needs and (Xd )I = (Xd′ )I , implies that he cannot obtain a more favorable contract-type under X , implying that Xi′ Ri1 Xi1 . 1 Let us now assume that Xi′ Pi1 Xi1 . That is, Xi′ ̸ = Xi . Let us 1 consider Pi′ : Xi′ , ∅; and P ′ = (Pi′ , P−i ). By the strategy-proofness of both φ and ψ , we have φi1 (P ′ ) = Xi′ and ψi1 (P ′ ) = ∅. 1 At problem P ′ , we have two stable allocations ψ (P ′ ) and φ (P ′ ); however, department d admits different sets of students. This contradicts our first finding above. This in turn implies that Xi′ = 1 Xi1 . If repeat the same arguments one by one for the later students

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in the ordering, then we easily obtain that Xi = Xi′ for each i ∈ (Xd )I , implying that Xd = Xd′ . As this is true for each department, we have X = X ′ . This in turn shows that ψ is nothing but MCOP, completing the proof. □ The Independence of the Axioms COP under the Aw FS choices, the first stage of MCOP, is both stable and strategy-proof. However, it is immediate to see that it fails to limitedly respect the TA/RA needs. Let us consider a department-ordering D = {d1 , . . . , dn }. Then, one by one following that ordering, each department chooses its best-ranked admissible student group up to its total TA/RA needs. Let Id be the set of students that department d admits at the end. Then, for each department d, let the best student group in Id (with respect to ≻d ) of size min{qT , |Id |} receive T -type contracts of d department d, and the rest (if any) receive R-type contract of the same department. Let us denote that mechanism by ψ . Then, it is easy to see that ψ is strategy-proof and it limitedly respects the TA/RA needs. However, it fails to be stable (for instance, a student may be chosen by department d while he finds any department d contract unacceptable). Lastly, let us consider a problem where {i, j} ⊆ I and d ∈ D. Let qFd = qT = qR = 1. Let the preferences be such that Pi = d d Pj : (d, T ), (d, F ), (d, R), ∅; and all the other students (if any) find every contract unacceptable. Department d’s rank order is such that ≻d : i, j, . . . , ∅. Let us consider a mechanism ψ such that at problem P, ψi (P) = (d, R), ψj (P) = (d, T ), and every body else is unassigned. At each other problem, let ψ produce the same outcome as MCOP. Then, it is immediate to see that ψ is stable and that limitedly respects the TA/RA needs. It is, however, not strategy-proof. To see this, let Pi′ : (d, F ), (d, T ), (d, R), ∅; and P ′ = (Pi′ , P−i ). Then, ψi (P ′ ) = (d, T ), ψi (P ′ ) = (d, R), and every body else is unassigned, hence student i benefits from misreporting. References Abdulkadiroğlu, A., Sönmez, T., 2003. School choice: A mechanism design approach. Amer. Econ. Rev. 93 (3), 729–747. Abizada, A., 2016. Stability and incentives for college admissions with budget constraints. Theor. Econ. 11 (2), 735–756. Abizada, A., Dur, U.M., 2017. College admissions with complementarities. In: mimeo. Afacan, M.O., 2013. Application fee manipulations in matching markets. J. Math. Econom. 49 (6), 446–453.

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