The college admission problem with entrance criterion∗ Azar Abizada† and Siwei Chen‡ October 12, 2011

Abstract We study the college admission problem with entrance criterion. Each student receives a score from a central exam. The students are divided into two groups: those who are eligible to apply to colleges, and those who are not. Eligibility respects the students’ scores. Each college has a strict preference relation over the students and each student has a strict preference relation over the colleges. We extend the model studied by Perach and Rothblum (2010) to a general case where the students may have the same scores from the central exam. We study ways of assigning students to colleges that respect eligibility criteria. We define several notions of stability. We define three new rules based on the McVitie-Wilson algorithm that satisfy different notions of stability. JEL Classification: C78, D63 Keywords: eligibility, quasi-stability, multi-stability, strategy-proofness.

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Introduction In countries like China, France, South Korea, Turkey, Greece, Azerbaijan, Albania etc.

college admission is processed through a central education system. The central authority ∗

We would like to thank William Thomson for his guidance, and invaluable comments. We also would

like to thank Youngsub Chun for his suggestions. † Department of Economics, University of Rochester. [email protected] ‡ Department of Economics, University of Rochester. [email protected]

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administers a cental exam. All students must take this exam. Each college has several departments. In some countries like France, Azerbaijan, etc., each student takes different central exams according to the specializations he/she chooses (For example, a student who wants to be an engineer takes only science exam, and a student who wants to be a historian takes a social science exam). Therefore, they can apply only to a limited number of departments. In some other countries like Turkey, Albania etc., there are no restrictions and each student can apply to any department. Based on students’ scores on this exam, the central authority declares some students eligible to apply to college and assigns them seats. After the assignments are made, if some colleges have empty seats, the central authority may hold a second round to fill those seats. In China, this process continues for at most three level. In Turkey, for a long time, the central exam had two levels. Using the results of the first level, the central authority decided which students were eligible to apply to college. The second level was administered to assign students to specific colleges. In this level, students were tested on different subjects. In each college, each department considered scores only from certain subjects (For example, an engineering department only considered student’s mathematics and science scores. On the other hand, business administration and economics only considered student’s mathematics and social science scores.). Therefore, each department may have different preferences over students. In South Korea, after the first level of the exam, which identifies eligible students, some colleges may conduct college-specific exams. Preferences over students are based on students’ scores from this exam. As we can see, entrance (eligibility) criterion is used in college admissions in many countries. We model the entrance (eligibility) criterion in college admission problems. Each college has a certain number of seats. Each student has the option of not attending college. Similarly, each college has the option of having an empty seat. Each student has strict preferences over the colleges and the option of not attending college, and each college has strict preferences over the students and having an empty seat. The central education system conducts a central exam. All students have to take this exam. Then, the central education system decides which students are eligible to apply to college. From a fairness perspective, eligibility criteria should respect the students’ scores. It would be unfair to declare a student eligible if he has a lower score than some other student who is declared ineligible. We study the ways of allocating students to colleges that respect eligibility criterion. The college admission problem is first studied by Gale and Shapley (1962) in a seminal

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paper in which they proposed the now well-known deferred-acceptance algorithm. Many variants and extensions of the original model with useful applications have been studied (Knuth (1976), Roth (1984), Gustfield and Irving (1989), Roth and Sotomayor (1990), Roth (2002), Abdulkadiroˇglu and S¨onmez (2003), Abdulkadiroˇglu (2005a, b), Roth et al. (2007), Roth (2008), and references therein.). It is only recently that the college admission problem with an entrance criterion has been studied (Perach, Polak and Rothblum (2007), Perach and Rothblum (2010)). They design an algorithm that respects eligibility criteria and produces a weakly stable outcome. They also study incentive compatibility properties. One of the assumptions in Perach et al. (2008) and Perach and Rothblum (2010) is that students have distinct scores from the exam. In China, every year approximately 10 million students take college admission exam. In Turkey, this number is 500 thousand every year. It is highly likely that some students will obtain the same score. Therefore, we should not ignore the possibility. We generalize the previously studied model to allow students to have the same score. In most of the literature on two-sided matching problems, it is assumed that agents have strict preferences over the agents on the opposite side. Allowing indifferences in preferences result in many complications. Therefore, most researchers assume strictness of preferences. Allowing students to have the same score in exam results in similar type of difficulties. To overcome these difficulties, we first modify the eligibility criterion. We define several notions. Each notion respects eligibility. The first notion is defined in the following way: no pair of a college and an eligible student prefer each other to their assignments. Another notion respects not only eligibility but also score rankings: no pair of a college and an eligible student are such that the student prefers the college to his assignment, and the college is assigned either a student with a lower score or some other student with the same score whom it prefers. We are also interested in strategy-proofness, the requirement that no student can ever benefit by misreporting his preference. We look for rules to assign students to colleges that respect the eligibility criteria and are stable and strategy-proof. We define three new rules. All three respect the eligibility criterion. Our first rule satisfies the first notion of stability, but fails to be strategy-proof. Our second rule only satisfies a weaker version of the stability notion, but it is strategy-proof. Our last rule satisfies this weaker stability notion, and it too is strategy-proof. We discuss other properties of our rules and the results of previous literature that extend to our model.

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The rest of the paper is organized as follows: in Section 2, we define the model. In Section 3, we define axioms. In Section 4, we provide rules. In Section 5 we state and prove our main results.

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Model There are a finite set of students S = {s1 , s2 , . . . , sl } and a finite set of colleges

C = {c1 , c2 , . . . , cn }. Each student has the option of not attending college. We denote this option ∅c . Similarly, each college has the option of having an empty seat. We denote this option ∅s . Each student s ∈ S has a strict preference relation Rs over the colleges and ∪ staying at home, i.e. Rs is a strict order on C {∅c }. Let RS ∈ R S be the preference profile of the students. Each college c ∈ C has a strict preference relation Rc over the students and ∪ having an empty seat, i.e. Rc is a strict order on S {∅s }. Let RC ∈ R C be the preference ∪ profile of the colleges. Let N ≡ S C. Let R ≡ (Ri )i∈N ∈ R N be a preference profile of N . We use R and (RS , RC ) interchangeably. Each student s ∈ S has a score ms ∈ R+ from a central exam. Let m ≡ (ms1 , ms2 , . . . , msl ) be the list of scores. Each college c ∈ C has a capacity qc ∈ Z+ . Let q ≡ (qc1 , qc2 , . . . , qcn ) be the list of capacities. We assume that ∅c has an infinite capacity, i.e. q∅c = ∞. A problem is a list π ≡ (S, C, R, m, q). Let Π be the set of all problems. An assignment ∪ for π is a mapping µ : S → C {∅c } such that for each c ∈ C, the number of students assigned to it does not exceed its capacity, i.e. |µ−1 (c)| ≤ qc . Let I ⊆ S. We refer to students who are not in I as eligible, and to students in I as ineligible. An outcome for π is a pair (µ, I). Let M (µ) ≡ {s ∈ S : µ(s) ∈ C} be the set of students assigned to colleges at µ. Let U (µ) ≡ S \ M (µ) be the set of students who are not attending college at µ. Let R(µ, I) ≡ (S \ I) \ M (µ) be the set of rejected students at (µ, I). Let O(π) be the set of all outcomes for π. A rule associates each problem with an outcome for it. Formally, it is a ∪ mapping φ : Π → O(π) such that for each π ∈ Π, φ(π) ∈ O(π). π∈Π

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Axioms Let π = (S, C, R, m, q) be a problem. Let (µ, I) be an outcome for π. Let µ′ be an

assignment for π. 4

Our first requirement is that eligibility should respect students’ scores and it should not waste any seat. Outcome (µ, I) is (weakly) plausible for π if 1. for each s ∈ I and each s′ ∈ / I, ms < (≤) ms′ ; 2. if I ̸= ∅, then for each c ∈ C, |µ−1 (c)| = qc .

College c is acceptable to student s if c Rs ∅c . Let Cs be the set of acceptable colleges for s. Student s is acceptable to college c if s Rc ∅s . Let Sc be the set of acceptable students for c. Student s and college c are mutually acceptable if (s, c) ∈ Sc × Cs . A pair (s, c) ∈ S × C blocks µ for π if 1. c Ps µ(s); 2. either s ∈ Sc and |µ−1 (c)| < qc , or there is s′ ∈ µ−1 (c) such that s Pc s′ .

Assignment µ is stable for π if 1. for each pair (s, µ(s)) ∈ S × C, s and µ(s) are mutually acceptable; 2. there is no pair (s, c) ∈ S × C that blocks µ for π. Let St(π) be the set of all stable assignments for π. Assignment µ ∈ St(π) is student-optimal in St(π) if for each µ′ ∈ St(π), and each s ∈ S, µ(s) Rs µ′ (s). Outcome (µ, I) is (weakly) quasi-stable for π if 1. (µ, I) is (weakly) plausible for π; 2. for each pair (s, µ(s)) ∈ S × C, s and µ(s) are mutually acceptable; 5

3. there is no pair (s, c) ∈ (S \ I) × C that blocks µ for π. Let QSt(π) be the set of all quasi-stable outcomes for π. Let wQSt(π) be the set of all weakly quasi-stable outcomes for π. Assignment µ is justified-envy-free for π if for each s ∈ S and each c ∈ C such that c Ps µ(s), and for each s′ ∈ µ(c), we have either ms′ > ms , or ms = ms′ and s′ Pc s. Assignment µ is multi-stable for π if 1. for each pair (s, µ(s)) ∈ S × C, s and µ(s) are mutually acceptable; 2. it is justified-envy-free for π. Let M St(π) be the set of all multi-stable outcomes for π. An assignment µ (weakly) eligibility-dominates µ′ if for each s ∈ M (µ) \ M (µ′ ) and each s′ ∈ M (µ′ ) \ M (µ), we have ms > (≥) ms′ . Next, we introduce the axioms. Let φ be a rule. • The rule should always select a quasi-stable outcome. Quasi-stability: For each π ∈ Π, φ(π) ∈ QSt(π). • The rule should always select a weakly quasi-stable outcome. Weak quasi-stability: For each (π) ∈ Π, φ(π) ∈ wQSt(π). • The rule should always select a multi-stable outcome. Multi-stability: For each π ∈ Π, φ(π) ∈ M St(π). • No student should ever benefit from misrepresenting his preference. This is a strong incentive compatibility property. Strategy-proofness: For each (S, C, R, m, q) ∈ Π, each s ∈ S, and each Rs′ ∈ Rs , φ(S, C, R, m, q) Rs φ(S, C, (Rs′ , R−s , RC ), m, q). 6

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Rules In this section we introduce three new rules. All rules are based on the McVitie-Wilson

algorithm. We first recall the algorithm (McVitie and Wilson (1970), Perach and Rothblum (2010)). At each step of the algorithm we randomly select a student, and let the student apply to his most preferred college. If either the college has an empty seat and the student is acceptable to it, or the college is full but the student is preferred by the college to a previously accepted student for it, then the student is accepted by the college. Acceptance at each step is tentative and may be withdrawn at the following steps. For each unassigned student, if there is a college that is acceptable to the student and hasn’t rejected him, then we can select this student at some later step. The algorithm stops when each of the unassigned students has been rejected by all his acceptable colleges, or no unassigned students are left. Let R ∈ R N . The McVitie-Wilson rule, M W : ∪ For each t ∈ Z+ and each c ∈ C {∅c }, let Mtc be the set of students assigned to college c at Step t. For each t ∈ Z+ , let Ut be the set of unassigned students at Step t. Step 0: For each c ∈ C

∪ {∅c }, let M0c ≡ ∅. Let U0 ≡ S.

Step 1: Select a student from U0 ; call him s1 . Let s1 apply to his most preferred college; call it c1 . (i) If s1 ∈ Sc1 and |M0c1 | < qc1 , then s1 is assigned to c1 . Let U1 ≡ U0 \ {s1 }, M1c1 ≡ {s1 }, and for each c ̸= c1 , M1c ≡ ∅. (ii) If either s1 ∈ / Sc1 or |M0c1 | = qc1 , then s1 is rejected by c1 . Let U1 ≡ U0 and for each ∪ c ∈ C {∅c }, M1c ≡ ∅. Step 2: Select a student from U1 ; call him s2 . [Note that s2 can be the same as s1 .] Let s2 apply to his most preferred college; call it c2 . (i) If s2 ∈ Sc2 and |M1c2 | < qc2 , then s2 is assigned to c2 . Let U2 ≡ U1 \ {s2 }, M2c2 ≡

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M1c2

∪

{s2 }, and for each c ̸= c2 , M2c ≡ M1c .

(ii) If s2 ∈ Sc2 , |M1c2 | = qc2 , and there is s ∈ M1c2 such that s2 Pc2 s, then s2 is assigned ∪ ∪ to c2 and s is rejected by c2 . Let U2 ≡ (U1 \ {s2 }) {s}, M2c2 ≡ (M1c2 {s2 }) \ {s} and for each c ̸= c2 , M2c ≡ M1c . (iii) If either s2 ∈ / Sc2 , or |M1c2 | = qc2 and for each s ∈ M1c2 , s Pc2 s2 , then s2 is rejected ∪ by c2 . Let U2 ≡ U1 , and for each c ∈ C {∅c }, M2c ≡ M1c . Step t = 3, 4 . . .: Select a student from Ut−1 ; call him st . [Note that st can be the same as sk for any k < t.] Let st apply to his most preferred college; call it ct . ct (i) If st ∈ Sct and |Mt−1 | < qct , then st is assigned to ct . Let Ut ≡ Ut−1 \ {st }, Mtct ≡ ∪ ct c Mt−1 {st }, and for each c ̸= ct , Mtc ≡ Mt−1 . ct ct (ii) If st ∈ Sct , |Mt−1 | = qct , and there is s ∈ Mt−1 such that st Pct s, then st is assigned ∪ ct ∪ to ct and s is rejected by ct . Let Ut ≡ (Ut−1 \ {st }) {s}, Mtct ≡ (Mt−1 {st }) \ {s} and for c each c ̸= ct , Mtc ≡ Mt−1 . ct ct (iii) If either st ∈ / Sct , or |Mt−1 | = qct and there for each s ∈ Mt−1 , s Pct st , then st is ∪ c rejected by ct . Let Ut ≡ Ut−1 , and for each c ∈ C {∅c }, Mtc ≡ Mt−1 .

Let t∗ be such that Ut∗ −1 ̸= ∅ and Ut∗ = ∅. The algorithm stops at Step t∗ . ∪ For each c ∈ C {∅c } and each s ∈ Mtc∗ −1 , let µ(s) ≡ c. Then, M W (S, C, R, q) ≡ µ. The next proposition states the equivalence between the McVitie-Wilson rule and the well known Deferred-Acceptance rule (McVitie and Wilson (1971), Roth and Sotomayor (1990)). Proposition 1 The McVitie-Wilson rule coincides with the student-proposing DeferredAcceptance rule. Remark 1 The McVitie-Wilson rule is stable and strategy-proof. Our first rule is based on an algorithm constructed in the following way. We initially declare all the students ineligible. At Step 1, we select the ineligible students whose scores are the highest. These students become eligible. We apply the M-W algorithm to the problem involving only these students. Step 1 ends when the M-W algorithm stops. If some ineligible students are left unassigned and some seats are left empty, we cancel the assignments made 8

at this step and move to the next step. At Step 2 we select the ineligible students whose scores are the highest. [Note that overall, these students are the ones with the second highest score.] These students become eligible. We apply the M-W algorithm to the problem involving these students and the ones selected at Step 1. If at the end of the step, some ineligible students are left unassigned and some seats are left empty, we cancel the assignments made at this step and move to the next step. We repeat the process. The algorithm stops when either no ineligible student is left unassigned, or no seat is left empty. The High-to-Low rule with deferred acceptances, HtLDA: For each t ∈ Z+ , let It be the set of ineligible students at Step t, and Et the set of eligible students at Step t. Step 0: Let I0 ≡ S and E0 ≡ ∅. Step 1: Let S1 ≡ {s ∈ I0 | ∀s′ ∈ I0 , ms ≥ ms′ }. [Note that S1 may be a singleton.] Let I1 ≡ I0 \ S1 and E1 ≡ S1 . Let µ1 ≡ M W (E1 , C, (RE1 , RC ), q). If either for each c ∈ C, |µ−1 1 (c)| = qc , or I1 = ∅, STOP. Otherwise proceed to Step 2. Step 2: Let S2 ≡ {s ∈ I1 | ∀s′ ∈ I1 , ms ≥ ms′ }. Let I2 ≡ I1 \ S2 and E2 ≡ E1

∪

S2 .

Let µ2 ≡ M W (E2 , C, (RE2 , RC ), q). If either for each c ∈ C, |µ−1 2 (c)| = qc , or I2 = ∅, STOP. Otherwise proceed to Step 3. Step t= 3, 4,...: Let St ≡ {s ∈ It−1 | ∀s′ ∈ It−1 , ms ≥ ms′ }. Let It ≡ It−1 \ St and ∪ Et ≡ Et−1 St . Let µt = M W (Et , C, (REt , RC ), q). If either for each c ∈ C, |µ−1 t (c)| = qc , or It = ∅, STOP. Otherwise proceed to Step t + 1. Let t∗ be the step at which the algorithm stops. Let µt∗ ≡ M W (Et∗ , C, (REt∗ , RC ), q). Then, HtLDA(S, C, R, m, q) = (µt∗ , It∗ ). Our second rule is similar to the HtLDA rule. The only difference is that in the algo-

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rithm through which the rule is defined, we select the students one by one. The algorithm is constructed in the following way. We initially declare all students ineligible. At Step 1, if there is only one ineligible student whose score is the highest, we select him. If there are more than one ineligible student whose score is the highest, we randomly select one of them. This student becomes eligible. We apply the M-W algorithm to the problem involving only this student. Step 1 ends when the M-W algorithm stops. If some ineligible students are left unassigned and some seats are left empty, we cancel the assignments made at this step and move to the next step. At Step 2, we select an ineligible student whose score is the highest. Once again, if there are more than one ineligible student whose score is the highest, we randomly select one of them. This student become eligible. We apply the M-W algorithm to the problem involving this student and the one selected at Step 1. If at the end of the step, some ineligible students are left unassigned and some seats are left empty, we cancel the assignments made at this step and move to the next step. We repeat the process. The algorithm stops when either no ineligible student is left unassigned, or no seat is left empty. The Modified High-to-Low rule with deferred acceptances, M HtLDA: Let R ∈ R N . For each t ∈ Z+ , let It be the set of ineligible students at Step t, and Et the set of eligible students at Step t (Note that Et ≡ S\It ). Step 0: Let I0 ≡ S and E0 ≡ ∅. Step 1: Let S1 ≡ {s ∈ I0 | ∀s′ ∈ I0 , ms ≥ ms′ } [Note that S1 may be a singleton]. Select a student in S1 at random; call him s1 . Let I1 ≡ I0 \ {s1 } and E1 ≡ {s1 }. Let µ1 ≡ M W (E1 , C, (RE1 , RC ), q). If either for each c ∈ C, |µ−1 1 (c)| = qc , or I1 = ∅, STOP. Otherwise proceed to Step 2. Step 2: Let S2 ≡ {s ∈ I1 | ∀s′ ∈ I1 , ms ≥ ms′ }. Select a student in S2 at random; call him s2 . Let I2 ≡ I1 \ {s2 } and E2 ≡ E1 ∪ {s2 }. Let µ2 ≡ M W (E2 , C, (RE2 , RC ), q). If either for each c ∈ C, |µ−1 2 (c)| = qc , or I2 = ∅, STOP. Otherwise proceed to Step 3. Step t=3, 4, ...: Let St ≡ {s ∈ It−1 | ∀s′ ∈ It−1 , ms ≥ ms′ }. Select a student in St 10

at random; call him st . Let It ≡ It−1 \ {st } and Et ≡ Et−1 ∪ {st }. Let µt = M W (Et , C, (REt , RC ), q). If either for each c ∈ C, |µ−1 t (c)| = qc , or It = ∅, STOP. Otherwise proceed to Step t + 1. Let t∗ be the step at which the algorithm stops. Let µt∗ ≡ M W (Et∗ , C, (REt∗ , RC ), q). Then, M HtLDA(S, C, R, m, q) = (µt∗ , It∗ ). Our next rule is used in college admissions in many countries (China, South Korea, Azerbaijan until two years ago etc.). It is defined through an algorithm, which we construct in the following way. We initially declare all students ineligible. At Step 1, we select the ineligible students whose scores are the highest. These students become eligible. We apply the M-W algorithm to the problem involving only these students. Step 1 ends when the M-W algorithm stops. Assignments made by the M-W algorithm at this step are permanent. If some ineligible students are left unassigned and some seats are left empty, we move to the next step. At Step 2, we select the ineligible students whose scores are the highest. [Note that overall, these students are the ones with the second highest score.] These students become eligible. We apply the M-W algorithm to the problem involving only these students. Step 2 ends when the M-W algorithm stops. Assignments made by the M-W algorithm at this step are permanent. If some ineligible students are left unassigned and some seats are left empty, we move to the next step. We repeat the process. The algorithm stops when either no ineligible student is left unassigned, or no seat is left empty. The High-to-Low rule with immediate acceptances, HtLIA: For each t ∈ Z+ , let It be the set of ineligible students at Step t, and Et the set of eligible students at Step t. Step 0: Let I0 ≡ S, and E0 ≡ ∅. Step 1: Let E1 ≡ {s ∈ I0 | ∀s′ ∈ I0 , ms ≥ ms′ }. [Note that S1 may be a singleton.] Let I1 ≡ I0 \ E1 . For each c ∈ C, let qc1 ≡ qc . Let q 1 ≡ (qc11 , qc12 , . . . , qc1n ). Let µ1 ≡ M W (E1 , C, (RE1 , RC ), q 1 ). 1 If either for each c ∈ C, |µ−1 1 (c)| = qc , or I1 = ∅, STOP. Otherwise proceed to Step 2.

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Step 2: Let E2 ≡ {s ∈ I1 | ∀s′ ∈ I1 , ms ≥ ms′ }. Let I2 ≡ I1 \ E2 . For each c ∈ C 2 2 2 2 let qc2 ≡ qc1 − |µ−1 1 (c)|. Let q ≡ (qc1 , qc2 , . . . , qcn ).

Let µ2 ≡ M W (E2 , C, (RE2 , RC ), q 2 ). 2 If either for each c ∈ C, |µ−1 2 (c)| = qc , or I2 = ∅, STOP. Otherwise proceed to Step 3.

Step t= 3, 4,...: Let Et ≡ {s ∈ It−1 | ∀s′ ∈ It−1 , ms ≥ ms′ }. Let It ≡ It−1 \ Et . For t t t t each c ∈ C let qct ≡ qct−1 − |µ−1 t−1 (c)|. Let q ≡ (qc1 , qc2 , . . . , qcn ).

Let µt = M W (Et , C, (REt , RC ), q t ). t If either for each c ∈ C, |µ−1 t (c)| = qc , or It = ∅, STOP. Otherwise proceed to Step t + 1.

∗

Let t∗ be the step at which the algorithm stops. Let µt∗ ≡ M W (Et∗ , C, (REt∗ , RC ), q t ). Then, HtLIA(S, C, R, m, q) = (µt∗ , It∗ ). Next, we discuss the relation between HtLDA and HtLIA rules. Proposition 2 For each π ∈ Π, if (µ, I) ≡ HtLDA(π) and (µ′ , I ′ ) ≡ HtLIA(π) then if for each s, s′ ∈ S \ I, we have ms = ms′ , then HtLDA(π) = HtLIA(π).

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Main Results Before stating our main results, we introduce two notions: the maximal set of ineligi-

ble students at quasi-stable outcomes, I ∗ (π) ≡ {s ∈ S : there is some (µ, I) ∈ QSt(π) such that s ∈ I}, and the minimal set of rejected students at quasi-stable outcomes, Let R∗ (π) ≡ {s ∈ S : for each (µ, I) ∈ QSt(π), s ∈ R(µ, I)}.

Theorem 1 The High-to-Low rule with deferred acceptances is quasi-stable. Proof. Let π = (S, C, R, m, q) ∈ Π. Let (µ, I) ≡ HtLDA(S, C, R, m, q). By the definition of HtLDA, M W (S \ I, C, (RS\I , RC ), q) = µ. Since M W = DA and DA is stable, µ ∈ St(S \ I, C, (RS\I , RC ), m, q). Because the students are selected in a decreasing order of scores in the I-M-W algorithm, the first condition of plausibility is satisfied. The I-M-W 12

algorithm stops when either there are no ineligible students left, or there are no empty seats left. This implies the second condition of plausibility. Thus, HtLDA is quasi-stable. Our next theorem states some properties of the HtLDA rule. Theorem 2 For each π = (S, C, R, m, q) ∈ Π, if (µ, I) ≡ HtLDA(π), then 1. I = I ∗ (π); 2. R(µ, I) = R∗ (π); 3. For each (µ′ , I ′ ) ∈ QSt(π), µ eligibility-dominates µ′ ; 4. For each (µ′ , I ′ ) ∈ QSt(π) and each s ∈ S \ I, we have µ(s) Rs µ′ (s). Proof. Statement 1. By the definition of I ∗ (π), I ⊆ I ∗ (π). Suppose by contradiction that there are π = (S, C, R, m, q) ∈ Π, (µ′ , I ′ ) ∈ QSt(π), and s ∈ S such that s ∈ I ′ and s ∈ / I. The case I = ∅ is trivial. Let I ̸= ∅. Let µ′′ ≡ M W (S \ I, C, (RS\I , RC ), q). Let b ≡ max{ms : s ∈ I}. Let b′ ≡ max{ms : s ∈ I ′ }. Step 1. Since the I-M-W algorithm terminates at b < b′ for π, there is c∗ ∈ C such that |µ′′−1 (c∗ )| < qc∗ . Since (µ′ , I ′ ) ∈ QSt(π), then for each c ∈ C, |µ′−1 (c)| = qc . Thus, there are s1 ∈ S and c1 ∈ C such that µ′ (s1 ) = c1 and µ′′ (s1 ) = ∅c . Step 2. Since c1 Ps1 ∅c and µ′′ ∈ St(S\I ′ , C, (RS\I ′ , RC ), mS\I ′ , q), we have |µ′′−1 (c1 )}| = qc1 , and for each s ∈ µ′′−1 (c1 ), s Pc1 s1 . Then, µ′ ∈ QSt(π) implies that there are s2 ∈ S and c2 ∈ C such that µ′ (s2 ) = c2 , µ′′ (s2 ) = c1 , and c2 Ps2 c1 . Step 3. Since c2 Ps2 c1 and µ′′ ∈ St(S\I ′ , C, (RS\I ′ , RC ), mS\I ′ , q), we have |µ′′−1 (c2 )| = qc2 , and for each s ∈ µ′′−1 (c2 ), s Pc2 s2 . Then, µ′ ∈ QSt(π) implies that there are s3 ∈ S and c3 ∈ C such that µ′ (s3 ) = c3 , µ′′ (s3 ) = c2 , and c3 Ps3 c2 . Step 4. · · · · · · Repeating the argument, we obtain that there are s ∈ S and c, c ∈ C such that µ′ (s) = c, µ′′ (s) = c, c Ps c, and |µ′′−1 (c)| < qc . Thus, the pair (s, c) blocks µ′′ for (S \ I ′ , C, (RS\I ′ , RC ), mS\I ′ , q). This contradicts the fact that µ′′ ∈ St(S \ I ′ , C, (RS\I ′ , RC ), mS\I ′ , q). Statement 2. Suppose by contradiction that there are π = (S, C, R, m, q) ∈ Π, (µ′ , I ′ ) ∈ 13

QSt(π), and s1 ∈ S such that s1 ∈ R(µ, I), but s1 ∈ / R(µ′ , I ′ ). Let b ≡ max{ms : s ∈ I}. Let b′ = max{ms : s ∈ I ′ }. Step 1. By part 1, b′ ≤ b. Since s1 ∈ / R(µ′ , I ′ ), there is c1 ∈ C such that µ′ (s1 ) = c1 . Since (µ, I) ∈ QSt(π) and c1 Ps1 ∅c , we have |µ−1 (c)}| = qc , and for each s ∈ µ−1 (c), s Pc1 s1 . Step 2. Since(µ′ , I ′ ) ∈ QSt(π), there are s2 ∈ S and c2 ∈ C such that µ(s2 ) = c1 , µ′ (s2 ) = c2 , and c2 Ps2 c1 . Since (µ, I) ∈ QSt(π), we have |µ−1 (c2 )| = qc2 , and for each s ∈ µ−1 (c2 ), s Pc2 s2 . Step 3. Since(µ′ , I ′ ) ∈ QSt(π), there are s3 ∈ S and c3 ∈ C such that µ(s3 ) = c2 , µ′ (s3 ) = c3 , and c3 Ps3 c2 . Since (µ, I) ∈ QSt(π), we have |µ−1 (c3 )| = qc3 , and for each s ∈ µ−1 (c3 ), s Pc3 s3 . Step 4. · · · · · · Repeating the argument, we end up with one of the following two cases: Case 1: There are s ∈ S and c ∈ C such that µ′ (s) = c, |µ−1 (c)| = qc , and for each s ∈ µ−1 (c) and each c ∈ C, we have s Pc s and c Rs c. Thus, there is s ∈ S such that µ(s) = c and µ′ (s) = c ̸= c. Since c Ps c and s Pc s, the pair (s, c) blocks µ′ for π. This contradicts the fact that (µ′ , I ′ ) ∈ QSt(π). Case 2: There are s, s ∈ S and c, c ∈ C such that µ(s) = c, µ′ (s) = c, c Ps c, µ(s) = c, and µ′ (s) = ∅c . Since (µ, I) ∈ QSt(π), we have s Pc s. Since c Ps ∅c , the pair (s, c) blocks µ′ for π. This contradicts the fact that (µ′ , I ′ ) ∈ QSt(π). Statement 3. Let (µ′ , I ′ ) ∈ QSt(π) and (µ′ , I ′ ) ̸= (µ, I). Let b ≡ max{ms : s ∈ I}. We need to show that for each s ∈ M (µ) \ M (µ′ ) and each s′ ∈ M (µ′ ) \ M (µ), ms > b ≥ ms′ . By plausibility, for each s ∈ M , ms > b. Suppose by contradiction that there is s′ ∈ M (µ′ )\M (µ) such that ms′ > b. This implies that s′ ∈ S \ I. Since s′ ∈ / M (µ), we have s′ ∈ R(µ, I). By part 2, R(µ, I) ⊆ R(µ′ , I ′ ). Thus, s′ ∈ R(µ′ , I ′ ). Thus, s′ ∈ / M (µ′ ). This contradicts the assumption that s′ ∈ M (µ′ ) \ M (µ). Statement 4. Let (µ′ , I ′ ) ∈ QSt(π). Let µ′′ ≡ M W (S \ I ′ , C, (RS\I ′ , RC ), mS\I ′ , q). By Part 1, I ′ ⊆ I. Since µ = M W (S \ I, C, RS\I , RC , mS\I , q), then for each s ∈ S \ I, we have µ(s) Rs µ′ (s). Since µ′ , µ′′ ∈ St(S \ I ′ , C, (RS\I ′ , RC ), mS\I ′ , q) and µ′′ is student-optimal in St(S \ I ′ , C, (RS\I ′ , RC ), mS\I ′ , q), then for each s ∈ S \ I ′ , we have µ′′ (s) Rs µ′ (s). Thus, for each s ∈ S \ I, we have µ(s) Rs µ′ (s). 14

The only statement in Theorem 1 of Perach and Rothblum (2010) that doesn’t hold in this model is: for each s ∈ S, if all colleges are acceptable for s, and s is acceptable for each c ∈ C, then s is not rejected at (µ, I). This is shown by the following example. Example 1: Let S = {s1 , s2 , s3 } with m = {6, 4, 4}, and C = {c1 , c2 } with q = {1, 1}. Preferences are as follows: Rs1

Rs2

Rs3

Rc1

Rc2

c1

c2

c2

s1

s1

c2

c1

c1

s2

s

∅c

∅c

∅c

s3

s3

∅s

∅s

Let I0 ≡ S and E0 ≡ ∅. Step 1. Let S1 ≡ {s1 }, I1 ≡ {s2 , s3 }, and E1 ≡ {s1 }. Apply the M-W algorithm to (E1 , C, RC , RE1 , q). Student s1 is assigned to college c1 . Since there is an empty seat in c2 and there are students in I1 , we proceed to Step 2. Step 2. Let S2 ≡ {s2 , s3 }, I2 ≡ ∅, and E2 ≡ {s1 , s2 , s3 }. Apply the M-W algorithm to (E2 , C, RC , RE2 , q). Student s1 is assigned to college c1 , student s2 is assigned to college c2 , and student s3 is rejected by both colleges. Since I2 = ∅, the algorithm stops. Let µ be such that µ(s1 ) = c1 , µ(s2 ) = c2 , and µ(s3 ) = s3 . The outcome is (µ, ∅). ∩ Note that Cs3 = {c1 , c2 } and s3 ∈ Sc1 Sc2 , but s3 ∈ R(µ, ∅). This is a violation of the property above. The How-to-Low rule with deferred acceptances is not strategy-proof. This is shown by the following example. Example 2: Let S = {s1 , s2 , s3 , s4 } with m = {6, 6, 4, 4}, and C = {c1 , c2 } with q = {1, 1}. Preferences are as follows:

15

Rs1

Rs2

Rs3

Rs4

Rc1

Rc2

Rs′ 2

c1

c1

c1

c2

s3

s4

c2

∅c

c2

c2

c1

s2

c1

c2

∅c

∅c

∅c

s1 .. .

s2 .. .

∅c

Let µ be such that µ(s1 ) = ∅c , µ(s2 ) = ∅c , µ(s3 ) = c1 , and µ(s4 ) = c2 .

Then,

HtLDA(S, C, R, m, q) = (µ, ∅). Let µ′ be such that µ(s1 ) = c1 , µ(s2 ) = c2 , µ(s3 ) = ∅c , and µ(s4 ) = ∅c . Then, HtLDA(S, C, (Rs′ 2 , , R−s2 , RC ), m, q) = (µ′ , {s3 , s4 }). Suppose that Rs2 is the true preference of student s2 . Since c2 Ps2 c1 , student s2 is better off by misreporting his preference to be Rs′ 2 . Thus, HtLDA is not strategy-proof. Since the outcome of M HtLDA rule is not plausible, therefore M HtLDA rule is not quasi-stable. But it is weakly quasi-stable. The proof is omitted, because it is very similar to the proof of Theorem 1. Theorem 3 The Modified High-to-Low rule with deferred acceptances is weakly quasi-stable. Next, we show that M HtLDA is strategy-proof. We need to introduce two lemmas. Lemma 1 For each (S, C, R, m, q) ∈ Π, each s ∈ S, and each Rs′ ∈ Rs , if (µ, I) ≡ M HtLDA(S, C, R, m, q), (µ′ , I ′ ) ≡ M HtLDA(S, C, (Rs′ , R−s , RC ), m, q), and I ′ ⊆ I, then µ(s) Rs µ′ (s). Proof. By the definition of M HtLDA, µ = M W (S \ I, C, (RS\I , RC ), mS\I , q), and µ′ = M W (S \ I ′ , C, (Rs′ , RS\(I ′ ∪{s}) , RC ), mS\I ′ , q). Let µ′′ ≡ M W (S \ I ′ , C, (RS\I ′ , RC ), mS\I ′ , q). Since M W is strategy-proof, then µ′′ (s) Rs µ′ (s). Since I ′ ⊆ I, then µ(s) Rs µ′′ (s). Thus, µ(s) Rs µ′ (s).

Lemma 2 For each (S, C, R, m, q) ∈ Π, each s ∈ M , and each Rs′ ∈ Rs , if (µ, I) ≡ M HtLDA(S, C, R, m, q), (µ′ , I ′ ) = M HtLDA(S, C, (Rs′ , R−s , RC ), m, q), and for each c ∈ C, µ(s) Rs′ c, then 16

1. I ⊆ I ′ . 2. µ′ (s) = µ(s). 3. for each s′ ∈ S \ I ′ , we have µ′ (s′ ) Rs′ µ(s′ ). Proof. Let I ̸= ∅. By plausibility, for each c ∈ C, |µ−1 (c)| = qc . By the definition of M HtLDA, µ = M W (S\I, C, (RS\I , RC ), mS\I , q) and µ′ = M W (S\I ′ , C, (Rs′ , RS\(I ′ ∪{s}) , RC ), mS\I ′ , q). Let µ′′ ≡ M W (S\I, C, (Rs′ , RS\(I∪{s}) , RC ), mS\I , q). Since µ ∈ St(S\I, C, (RS\I , RC ), mS\I , q) and for each c ∈ C, µ(s) Rs′ c, we have µ ∈ St(S \ I, C, (Rs′ , RS\(I∪{s}) , RC ), mS\I , q). Since µ′′ is student-optimal in St(S\I, C, (Rs′ , RS\(I∪{s}) , RC ), mS\I , q), then for each s′ ∈ S\I, we have µ′′ (s′ ) Rs′ µ(s′ ). Since I ̸= ∅, then for each c ∈ C, |µ−1 (c)}| = qc . Thus, for each c ∈ C, |µ′′−1 (c)| = qc . This implies that the MHtLDA algorithm stops at b′ = max{ms : s ∈ I ′ } ≥ b = max{ms : s ∈ I} for (S \ I, C, Rs′ , (RS\(I∪{s}) , RC ), mS\I , q). Thus, I ⊆ I ′ . Thus, for each s′ ∈ S \ I ′ , we have µ′ (s′ ) Rs′ µ′′ (s′ ) Rs′ µ(s′ ). Since µ′ (s) Rs′ µ(s) and for each c ∈ C, µ(s) Rs′ c, we have µ′ (s) = µ(s). The case where I = ∅ follows the same logic, where b′ ≥ b holds trivially.

Theorem 4 The Modified High-to-Low rule with deferred acceptances is strategy-proof. Proof.

For each (S, C, R, m, q) ∈ Π, each s ∈ S, and each Rs′ ∈ Rs , let (µ, I) ≡

M HtLDA(S, C, R, m, q) and (µ′ , I ′ ) ≡ M HtLDA(S, C, (Rs′ , R−s , RC ), m, q). If s ∈ I, then (µ′ , I ′ ) = (µ, I). If s ∈ S \ I and I ′ ⊆ I, then by Lemma 3, µ(s) Rs µ′ (s). The only case left is s ∈ S \ I and I ⊂ I ′ . Suppose by contradiction that µ′ (s) Ps µ(s). Let c ≡ µ′ (s). By Lemma 4, we can assume that for each c′ ∈ C, c Rs′ c′ . Let the students in S become eligible one-by-one in such a way that in the end, the set of students who are still ineligible is I ′ , and s is the last student to become eligible. There are two cases: Case 1: All colleges are already full when s becomes eligible. Consider the execution of the M HtLDA algorithm when the preference profile is (RS , RC ). When s becomes eligible, the assignment from the previous step is the same as the one when the preference profile is (Rs′ , R−s , RC ). Thus, all the colleges are already full. The new assignment after s becoming eligible does not create empty seats. Thus, at the end of this 17

step, all the colleges are still full. Since no additional students can become eligible, then I = I ′ . This contradicts the assumption that I ⊂ I ′ . Case 2: There is exactly one empty seat when s becomes eligible. Let cl be the college that has one empty seat. Consider the execution of the M HtLDA algorithm when the preference profile is (Rs′ , R−s , RC ). When s enters, he applies to c and he is tentatively admitted. [It is possible that c = cl .] Let s1 be the student who is rejected by c to be replaced by s. Then, he applies to c1 and is tentatively admitted. Let s2 be the student who is rejected by c1 to be replaced by s1 . He then applies to c2 and is tentatively admitted · · · · · · This process ends when the empty seat in cl is filled by some student sl . We obtain {s0 ≡ s, s1 , ... , sl } and {c0 ≡ c, c1 , ... , cl }, where for each 0 ≤ j ≤ l, sj is rejected by cj−1 and he is tentatively admitted to cj . Consider the execution of the M HtLDA algorithm when the preference profile is (RS , RC ). When s becomes eligible, he applies to the colleges according to Rs until he is tentatively admitted to some college c′ . Let s′ be the student who is rejected by c′ to be replaced by s. He applies to c′′ and is tentatively admitted. Let s′′ be the students who is rejected by c′′ to be replaced by s′ . He applies to c′′′ and is tentatively admitted · · · · · · If there is sj ∈ {s1 , ... , sl } such that sj is rejected before the process reaches the step where s applies to c, then the process will consist of {sj , sj+1 , ... , sl } and {cj , cj+1 , ... , cl }, where for each j ≤ k ≤ l, sk is rejected by sk−1 and he is tentatively admitted to ck . If not, then after s applies to c and he is tentatively admitted, the process will be the same as the one when the preference profile is (Rs′ , R−s , RC ). Since the empty seat in cl is filled in both cases, no additional students can become eligible. Thus, I = I ′ . This contradicts the assumption that I ⊂ I ′ . Next we will discuss the properties of HtLIA. Although it is plausible, it is easy to show that it is not quasi-stable. But it turns out that HtLIA rule is multi-stable and strategy-proof.

Theorem 5 The High-to-Low rule with immediate acceptances is multi-stable. Proof. All we need to show is for each π ∈ Π, HtLIA rule is justified-envy-free for π. Let (µ, I) ≡ HtLIA(π). We need to show that for each s ∈ S and each c ∈ C such that c Ps µ(s), then for each s′ ∈ µ(c), either 18

(1) ms′ > ms , or (2) ms = ms′ and s′ Pc s. Part (1) follows from the definition of the rule: since a student with higher score becomes eligible earlier than the one with lower score, and since assignments are permanent, part (1) is satisfied. Part (2) follows from the proof of Theorem 1. Theorem 6 The High-to-Low rule with immediate acceptances is strategy-proof. Proof. At each step, rule only considers students with the same score for the available seats. The McWitie-Wilson rule used at each step is strategy-proof. Since assignment of a student is decided at the step he becomes eligible, and this assignment is permanent, no student has incentive to misrepresent.

6

Conclusion Our first rule treats the students with equal scores equally when determining eligibility.

Either they are all eligible, or they are all ineligible. But this equal treatment of equals results in a loss of strategy-proofness. Our second rule determines the students’ eligibility one-by-one. There could be at most one group of students with the same score such that some of the students are eligible and some of them are ineligible. We regain strategy-proofness (Theorem 4).

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[3] Abdulkadiroˇglu A., and S¨onmez T. (2003), ”School choice: a mechanism design approach”, American Economic Review, Vol.93, 729-747. [4] Gale, D. and L. Shapley (1962), ”College admissions and the stability of marriage”, American Mathematical Monthly, Vol.69, 9-15. [5] Gustfield D, and Irving RW (1989) The stable marriage problem: structure and algorithms. The MIT Press, Cambridge. [6] Knuth D.E. (1976) Marriages stables. Les Presses de L’Universite Montreal, Montreal. [7] McVitie D.G. and Wilson L.B. (1970), ”Stable marriage assignment for unequal sets”, BIT, Vol.10, 295-209. [8] Perach N., Polak J., and Rothblum U.G. (2007), ”A stable matching model with an entrance criterion applied to the assignment of students to dormitories at the technion”, International Journal of Game Theory, Vol.36, 519-535. [9] Perach N, and Rothblum U.G. (2010), ”Incentive compatibility for the stable matching model with an entrance criterion”, International Journal of Game Theory, Vol.39, 657667. [10] Roth A.E. (1984), ”The evolution of the labor market for medical interns and residents: a case study in game theory”, Journal of Political Economy, Vol.92, 991-1016. [11] Roth A.E. (2002), ”The economist as engineer: game theory, experimentation, and computation as tools for design economics”, Econometrica, Vol.70, 1341-1378. [12] Roth A.E. (2008), ”Deferred acceptance algorithms: history, theory, practice, and open questions”, International Journal of Game Theory, Vol.36, 537-569 [13] Roth, A. and M. Sotomayor (1990), Two-Sided Matching. New York: Cambridge University Press. [14] Thomson (2010), Strategy-proof resource allocation rules. Lecture Notes.

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