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Corrigendum to “Affirmative action in school choice: A new solution” [Math. Social Sci. 92 (2018) 1–9]
Mathematical Social Sciences 97 (2019) 61–64
Contents lists available at ScienceDirect
Mathematical Social Sciences journal homepage: www.elsevier.com/locate/mss
Corrigendum
Corrigendum to ‘‘Affirmative action in school choice: A new solution’’ [Math. Social Sci. 92 (2018) 1–9] Fangli Ding a , Shihuang Hong a , Zhenhua Jiao b, *, Xinggang Luo c a b c
School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China School of Business, Shanghai University of International Business and Economics, Shanghai, 200433, China Management School, Hangzhou Dianzi University, Hangzhou, 310018, China
highlights • This note investigates the responsiveness of the minority reserves type affirmative action in school choice. • We provide a counterexample which shows that a mechanism in Ju et al. (2018) is not responsive to the minority reserves type affirmative action. • For the responsiveness of Dogan’s mechanism, it is crucial to iteratively remove only the last-step minority interrupters.
article
info
Article history: Received 25 January 2018 Received in revised form 11 June 2018 Accepted 18 July 2018 Available online 22 September 2018
a b s t r a c t In (Ju et al., 2018) Ju et al. investigate the responsiveness of affirmative action in school choice. For the minority reserves type affirmative action they introduce a new matching mechanism and claim that their mechanism is responsive to the minority reserves policy (Theorem 1 in Ju et al., 2018). However, we construct a counterexample, which shows that it is not the case. © 2018 Elsevier B.V. All rights reserved.
1. Introduction School choice programs aim to give students the option to choose their school. At the same time, underrepresented students should be favored to close the opportunity gap. Currently, many school districts in the United States have affirmative action policies to favor minority students and help them attend more preferred schools. The majority quota is a traditional type of affirmative action. This type of policy gives minority students higher chances to attend more preferred schools by limiting the number of admitted majority students at schools. The majority quota affirmative action policy, however, may hurt all the minority students and result in avoidable inefficiency. To circumvent these shortcomings caused by majority quota, Hafalir et al. (2013) propose another type of affirmative action called ‘‘minority reserves’’ affirmative action. This type of policy is to reserve some seats at schools for the minority students, and to require that a reserved seat at a school can be assigned to a majority student only if no minority student prefers that school to her assignment. DOI of original article: https://doi.org/10.1016/j.mathsocsci.2017.12.002. author. * Corresponding E-mail addresses: [email protected] (S. Hong), [email protected] (Z. Jiao). https://doi.org/10.1016/j.mathsocsci.2018.07.002 0165-4896/© 2018 Elsevier B.V. All rights reserved.
Since the affirmative action policies in school choice aim to improve the welfare of the minority students, a satisfactory assignment mechanism should have the following property: running such a mechanism, a stronger affirmative action policy should not hurt a minority student without benefiting anyone of the minority students. If an assignment mechanism satisfies this property, then we say that it is responsive to the affirmative action. Unfortunately, for the responsiveness of affirmative actions, much more impossibility results than positive results have been obtained in the literature. Kojima (2012) shows that neither DA (studentproposing deferred acceptance algorithm) nor TTC (top trading cycles algorithm) is responsive to the majority quota affirmative action. For the responsiveness of Boston mechanism to majority quota policy, Afacan and Salman (2016) also obtain negative result. For the minority reserves affirmative action, Afacan and Salman (2016) show that the Boston mechanism is responsive. Hafalir et al. (2013) introduce the deferred acceptance algorithm with minority reserves (DAm ) for school choice with the minority reserves affirmative action. Although DAm satisfies desirable efficiency property, it is not responsive to the minority reserves affirmative action. More recently, Ju et al. (2018) propose an efficiency-improved deferred acceptance mechanism with minority reserves (EIDAm ) and claim that EIDAm is responsive to the minority reserves affirmative action. This note provides a counterexample to show that it is not the case.
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F. Ding, S. Hong, Z. Jiao et al. / Mathematical Social Sciences 97 (2019) 61–64
2. Model We follow the model in Ju et al. (2018). Let I and S be finite and disjoint sets of students and schools, respectively. There are two types of students: minority students and majority students. Let I m and I M denote the sets of minority and majority students, respectively. They are nonempty sets such that I m ∪ I M = I and I m ∩ I M = ∅. For each student i ∈ I, ≻i is a strict (i.e., complete, transitive, and anti-symmetric) preference relation over S ∪{i}, where i denotes i’s outside option. School s is acceptable to student i if s≻i i. The list of preferences for a group of students I ′ is denoted by ≻I ′ = (≻i )i∈I ′ . For each school s ∈ S, ≻s is a strict priority order over I. The list of priorities for a group of schools S ′ is denoted by ≻S ′ = (≻s )s∈S ′ . For agents a, b in S (resp. I) and c in I (resp. S), a⪰c b denotes either a≻c b or a = b. For each s ∈ S, qs is the capacity of s or the number of seats in s. Let q = (qs )s∈S be the capacity profile. Hafalir et al. (2013) introduce the following type of affirmative action: Each school s ∈ S has a minority reserve parameter rs and gives priority to minority students up to the minority reserve rs . For school s, if the number of minority students admitted to it is less than rs , then any minority applicant is given priority over any majority applicant at s. If there are not enough minority students to fill up the reserves, majority students can still be assigned to the reserved seats. This type of affirmative action is called ‘‘minority reserves’’. Let r = (rs )s∈S be the minority reserves profile. A school choice problem or simply a problem is a 6-tuple G = (I , S , ≻I , ≻S , q, r). For two minority reserves profiles r = (rs )s∈S and r ′ = (rs′ )s∈S , if rs ≤ rs′ for each school s ∈ S and rs′ < r ′ s′ for at least one school s′ ∈ S , then G′ = (I , S , ≻I , ≻S , q, r ′ ) is said to have stronger minority reserves than problem G = (I , S , ≻I , ≻S , q, r). A matching is an assignment of students to schools such that each student is assigned to a school or to her outside option, and no more students are assigned to a school than its capacity. Formally, a matching µ is a mapping from I ∪ S to the subsets of I ∪ S such that (1) µ(i) ⊆ S ∪ {i} and |µ(i)| = 1 for every i ∈ I,1 (2) µ(s) ⊆ I and |µ(s)| ≤ qs for every s ∈ S, (3) µ(i) = {s} if and only if i ∈ µ(s) for every i ∈ I and s ∈ S. A matching µ′ Pareto dominates µ for the minority students if (i) µ′ (i)⪰i µ(i) for every i ∈ I m ; and (ii) µ′ (i′ )≻i′ µ(i′ ) for some i′ ∈ I m . A mechanism φ chooses a matching φ (G) for each problem G. Definition 1. A matching mechanism φ is said to be responsive to the minority reserves affirmative action if there are no problems G = (I , S , ≻I , ≻S , q, r) and G′ = (I , S , ≻I , ≻S , q, r ′ ) such that G′ has stronger minority reserves than G, and φ (G) Pareto dominates φ (G′ ) for the minority students. In other words, running a matching mechanism φ , if increasing the level of affirmative action never results in a Pareto inferior assignment for the minority students, then φ is responsive to the minority reserves affirmative action. 3. Two matching mechanisms For completeness, in this section we provide the deferred acceptance mechanism with minority reserves (DAm ) proposed by Hafalir et al. (2013) and the efficiency improved deferred acceptance mechanism with minority reserves (EIDAm ) proposed by Ju et al. (2018). 1 Since µ(i) is a one-element set, if µ(i) = {s} we also write it as µ(i) = s.
The DAm is implemented by the deferred acceptance algorithm with minority reserves. For a problem G, the deferred acceptance algorithm with minority reserves runs as follows: Step 1: Start with the matching in which no student is matched. Each student i applies to her first choice school. Each school s first accepts as many as rs minority applicants with the highest priorities if there are enough minority applicants. Then it accepts applicants with the highest priorities from the remaining applicants until its capacity is filled or the applicants are exhausted. The rest of the applicants, if any remains, are rejected by s. In general, at Step k: Start with the tentative matching obtained at the end of Step k − 1. Each student i who got rejected at Step k − 1 applies to her first choice school (call it s) among all schools that have not rejected i before. The school s considers the new applicants and students admitted tentatively at Step k − 1. Among these students, school s first accepts as many as rs minority students with the highest priorities if there are enough minority students. Then it accepts students with the highest priorities from the remaining students. The rest of the students, if any remains, are rejected by s. If there are no rejections, then the algorithm stops. The algorithm terminates when no rejection occurs and the tentative matching at that step is finalized. Since no student reapplies to a school that has rejected her, the algorithm stops in finite time. In order to introduce the EIDAm mechanism, we first specify the concept of under-demanded school proposed by Kesten and Kurino (2016). Definition 2. Given a matching µ for a problem G, a school s is said to be under-demanded in matching µ if all students weakly prefer their assignments under µ to school s, that is, µ(i)⪰i s for each student i. The EIDAm is implemented by the efficiency-improved deferred acceptance algorithm with minority reserves, which operates as follows: Round 0. Given a problem G, run DAm for G to obtain the matching µ0 . Find all the under-demanded schools in µ0 , and remove all these schools together with the students that are matched to them. The assignments between the schools and the students that are removed are finalized. There may be students who are matched to themselves, then they are also removed and assigned no school finally. If there are no more agents left over after the removal, then stop. Otherwise, go to the next round. Round k (k ≥ 1). Denote the problem that consists of the remaining schools and students by Gk , re-run DAm m for Gk to obtain matching µk . Find all the under-demanded schools in µk , and remove all these schools together with the students that are matched to them. The assignments between the schools and the students that are removed are finalized. If there are no more agents left over after the removal, then stop. Otherwise, go to the next round. The algorithm terminates when no more agents are left over. Ju et al. (2018) show that this algorithm is well-defined. 4. Result Ju et al. (2018) claim that EIDAm is responsive to the minority reserves affirmative action. However, we find that it is not the case. Specifically, we have the following result. Theorem 1. EIDAm is not responsive to the minority reserves affirmative action. Proof. We slightly modify an example in Jiao and Tian (2018). Let S = {s1 , s2 , s3 , s4 }, I M = {i1 , i3 , i4 } and I m = {i2 }. All schools have a capacity of 1: q = (1, 1, 1, 1). The minority reserves profile is
F. Ding, S. Hong, Z. Jiao et al. / Mathematical Social Sciences 97 (2019) 61–64
r = (0, 0, 0, 0) Students’ preferences and schools’ priorities are given by the following table: ≻i 1
≻i 2
≻i3
≻i 4
≻s 1
≻s 2
≻s 3
≻s 4
s1 s4
s1 s2
s2 s4
s3 s1
i4 i1
i3 i2
i2 i4
i1 i3
. . .
. . .
. . .
. . .
. . .
. . .
s3 s4
i2 i3
Then the procedure of the DAm is as follows: step
s1
1
i1 ,i2
s2
s3
i3
i4
s4
i2 , i3
2 3
i2 , i4 i1 , i4
4 5
i4
i3
i2
i1
One can see that s4 is the only under-demanded school in the DAm matching µ0 . We remove s4 together with i1 and re-run DAm for the remaining students and schools. Then the matching µ1 is step 1
s1
s2
s3
i2
i3
i4
One can see that all schools are under-demanded in matching µ1 . We remove all of them and all students. Then the final EIDAm matching with respect to G = (I , S , ≻I , ≻S , q, r) is
µ=
(
s1 i2
s2 i3
s3 i4
s4 i1
)
.
Let r ′ = (0, 1, 0, 0). Then G′ = (I , S , ≻I , ≻S , q, r ′ ) has stronger minority reserves than G = (I , S , ≻I , ≻S , q, r). For problem G′ = (I , S , ≻I , ≻S , q, r ′ ), the procedure of the DAm is as follows: step
s1
1
s2
s3
i3
i4
i1 ,i2
2
s4
i2 ,i3
3
i1
i2
i4
i3
One can see that s3 and s4 are the only under-demanded schools in the above matching. We remove s3 and s4 together with i3 and i4 , and re-run DAm for the remaining students and schools. Then we get the following matching step
s1
1
i1 ,i2
2
i1
s2
i2
One can see that s2 is under-demanded in the above matching. We remove s2 together with i2 . In the next round we remove s1 together with i1 . Then the final EIDAm matching with respect to G′ = (I , S , ≻I , ≻S , q, r ′ ) is
µ = ′
(
s1 i1
s2 i2
s3 i4
s4 i3
)
.
One can see that, running EIDAm , student i2 under G′ is worse than under G. Then, under this mechanism, the stronger minority reserves affirmative action makes all minority students (only one— i2 ) worse off. Thus, EIDAm is not responsive to the minority reserves affirmative action. □ Remark 1. For school choice problems, Kesten (2010) first introduces the efficiency-improved deferred acceptance algorithm that
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iteratively removes ‘‘last-step interrupting students’’. Tang and Yu (2014) obtain that iteratively removing ‘‘last-step interrupting students’’ and iteratively removing ‘‘under-demanded schools with students matched to them’’ produce the same matching outcome. Thus, the EIDAm can be approximately regarded as an alternative mechanism which iteratively removes ‘‘last-step interrupting students’’. Based on such an observation, we illustrate the intuition of our counterexample as follows. Suppose that s ∈ S and s′ ∈ S m compete a seat at school c, and s has higher priority than s′ at c. Running the DAm algorithm, if school c temporarily accepts s and rejects s′ , then maybe it happens that, for a weaker affirmative action problem there exists a rejection cycle starting from s and s finally is an interrupter at school c,2 while for a stronger affirmative action problem there does not exist any rejection cycle or interrupter. Thus, under the EIDAm mechanism, for the weaker affirmative action problem we remove the interrupter s, student s′ be admitted by c and gets better off. However, for the stronger reserve-based affirmative action problem, there exists no interrupter, then the DA mechanism coincides the EIDAm mechanism. We suppose S m = {s′ }. Then we obtain that, under the EIDAm mechanism, increasing the level of affirmative action results in a Pareto inferior assignment for the minority. Therefore, the EIDAm mechanism is not responsive to the minority reserves affirmative action. Remark 2. Doğan (2016) introduces a mechanism which iteratively removes ‘‘last-step interrupting minority students’’ and shows that his mechanism is responsive. We note that the intuition behind our counterexample does not apply to Doğan’s mechanism. Specifically, according to the assumption S m = {s′ } in Remark 1, we can see that, for the weaker affirmative action problem, s is an interrupter, but she is not a minority student. Since Doğan’s mechanism iteratively removes ‘‘last-step interrupting minority students’’, under Doğan’s mechanism s will not be removed. Thus, running Doğan’s mechanism, for the weaker affirmative action problem we do not remove s, student s′ does not get better off. Therefore, the matching outcome comparison for student s′ described in Remark 1 cannot appear under Doğan’s mechanism. Remark 3. For the matching problem described in Remark 1, if we assume that s is a minority student, we give a further argument for the Doğan’s mechanism. Running the DAm algorithm, for a weaker affirmative action problem, if s is a last-step interrupting minority student at school c, then under the Doğan’s mechanism we remove s and student s′ becomes better off. For a stronger affirmative action problem, if s is no longer a last-step interrupting minority student at school c, then s will not be removed under the Doğan’s mechanism. Thus, s becomes better off under the stronger affirmative action, and consequently, the Doğan’s mechanism is responsive, as increasing the level of affirmative action does not result in a Pareto inferior assignment for the minority students. From the analysis above, we can see that, for the responsiveness of the Doğan’s mechanism, it is crucial to iteratively remove only the last-step minority interrupters. Acknowledgments The authors acknowledge financial support from the Zhejiang Provincial Natural Science Foundation of China (No. LY16G010008), the China Postdoctoral Science Foundation (Nos. 2016M600302, 2018T110380) and the National Natural Science Foundation of China (No. 71771068). 2 Formally, in the DA procedure of a school choice problem, if student s is tentatively accepted by school c at some step t and is later rejected by school c at some step t ′ > t, and at least one other student is rejected by school c at some step r such that t ≤ r < t ′ , then student s is an interrupter at school c.
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References Afacan, M.O., Salman, U., 2016. Affirmative actions: The Boston mechanism case. Econ. Lett. 141, 95–97. Doğan, B., 2016. Responsive affirmative action in school choice. J. Econ. Theory 165, 69–105. Hafalir, I.E., Yenmez, M.B., Yildirim, M.A., 2013. Effective affirmative action in school choice. Theor. Econ. 8, 325–363. Jiao, Z., Tian, G., 2018. Two further impossibility results on responsive affirmative action in school choice. Econ. Lett. 166, 60–62.
Ju, Y., Lin, D., Wang, D., 2018. Affirmative action in school choice: A new solution. Math. Soc. Sci. 92, 1–9. Kesten, O., 2010. School choice with consent. Quart. J. Econ. 125, 1297–1348. Kesten, O., Kurino, M., 2016. Do Outside Options Matter in Matching: A New Perspective on the Trade-Offs in Student Assignment. Working Paper. Kojima, F., 2012. School choice: Impossibilities for affirmative action. Games Econ. Behav. 75 (2), 685–693. Tang, Q., Yu, J., 2014. A new perspective on Kesten’s school choice with consent idea. J. Econ. Theory 154, 543–561.
Contents lists available at ScienceDirect
Mathematical Social Sciences journal homepage: www.elsevier.com/locate/mss
Corrigendum
Corrigendum to ‘‘Affirmative action in school choice: A new solution’’ [Math. Social Sci. 92 (2018) 1–9] Fangli Ding a , Shihuang Hong a , Zhenhua Jiao b, *, Xinggang Luo c a b c
School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China School of Business, Shanghai University of International Business and Economics, Shanghai, 200433, China Management School, Hangzhou Dianzi University, Hangzhou, 310018, China
highlights • This note investigates the responsiveness of the minority reserves type affirmative action in school choice. • We provide a counterexample which shows that a mechanism in Ju et al. (2018) is not responsive to the minority reserves type affirmative action. • For the responsiveness of Dogan’s mechanism, it is crucial to iteratively remove only the last-step minority interrupters.
article
info
Article history: Received 25 January 2018 Received in revised form 11 June 2018 Accepted 18 July 2018 Available online 22 September 2018
a b s t r a c t In (Ju et al., 2018) Ju et al. investigate the responsiveness of affirmative action in school choice. For the minority reserves type affirmative action they introduce a new matching mechanism and claim that their mechanism is responsive to the minority reserves policy (Theorem 1 in Ju et al., 2018). However, we construct a counterexample, which shows that it is not the case. © 2018 Elsevier B.V. All rights reserved.
1. Introduction School choice programs aim to give students the option to choose their school. At the same time, underrepresented students should be favored to close the opportunity gap. Currently, many school districts in the United States have affirmative action policies to favor minority students and help them attend more preferred schools. The majority quota is a traditional type of affirmative action. This type of policy gives minority students higher chances to attend more preferred schools by limiting the number of admitted majority students at schools. The majority quota affirmative action policy, however, may hurt all the minority students and result in avoidable inefficiency. To circumvent these shortcomings caused by majority quota, Hafalir et al. (2013) propose another type of affirmative action called ‘‘minority reserves’’ affirmative action. This type of policy is to reserve some seats at schools for the minority students, and to require that a reserved seat at a school can be assigned to a majority student only if no minority student prefers that school to her assignment. DOI of original article: https://doi.org/10.1016/j.mathsocsci.2017.12.002. author. * Corresponding E-mail addresses: [email protected] (S. Hong), [email protected] (Z. Jiao). https://doi.org/10.1016/j.mathsocsci.2018.07.002 0165-4896/© 2018 Elsevier B.V. All rights reserved.
Since the affirmative action policies in school choice aim to improve the welfare of the minority students, a satisfactory assignment mechanism should have the following property: running such a mechanism, a stronger affirmative action policy should not hurt a minority student without benefiting anyone of the minority students. If an assignment mechanism satisfies this property, then we say that it is responsive to the affirmative action. Unfortunately, for the responsiveness of affirmative actions, much more impossibility results than positive results have been obtained in the literature. Kojima (2012) shows that neither DA (studentproposing deferred acceptance algorithm) nor TTC (top trading cycles algorithm) is responsive to the majority quota affirmative action. For the responsiveness of Boston mechanism to majority quota policy, Afacan and Salman (2016) also obtain negative result. For the minority reserves affirmative action, Afacan and Salman (2016) show that the Boston mechanism is responsive. Hafalir et al. (2013) introduce the deferred acceptance algorithm with minority reserves (DAm ) for school choice with the minority reserves affirmative action. Although DAm satisfies desirable efficiency property, it is not responsive to the minority reserves affirmative action. More recently, Ju et al. (2018) propose an efficiency-improved deferred acceptance mechanism with minority reserves (EIDAm ) and claim that EIDAm is responsive to the minority reserves affirmative action. This note provides a counterexample to show that it is not the case.
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F. Ding, S. Hong, Z. Jiao et al. / Mathematical Social Sciences 97 (2019) 61–64
2. Model We follow the model in Ju et al. (2018). Let I and S be finite and disjoint sets of students and schools, respectively. There are two types of students: minority students and majority students. Let I m and I M denote the sets of minority and majority students, respectively. They are nonempty sets such that I m ∪ I M = I and I m ∩ I M = ∅. For each student i ∈ I, ≻i is a strict (i.e., complete, transitive, and anti-symmetric) preference relation over S ∪{i}, where i denotes i’s outside option. School s is acceptable to student i if s≻i i. The list of preferences for a group of students I ′ is denoted by ≻I ′ = (≻i )i∈I ′ . For each school s ∈ S, ≻s is a strict priority order over I. The list of priorities for a group of schools S ′ is denoted by ≻S ′ = (≻s )s∈S ′ . For agents a, b in S (resp. I) and c in I (resp. S), a⪰c b denotes either a≻c b or a = b. For each s ∈ S, qs is the capacity of s or the number of seats in s. Let q = (qs )s∈S be the capacity profile. Hafalir et al. (2013) introduce the following type of affirmative action: Each school s ∈ S has a minority reserve parameter rs and gives priority to minority students up to the minority reserve rs . For school s, if the number of minority students admitted to it is less than rs , then any minority applicant is given priority over any majority applicant at s. If there are not enough minority students to fill up the reserves, majority students can still be assigned to the reserved seats. This type of affirmative action is called ‘‘minority reserves’’. Let r = (rs )s∈S be the minority reserves profile. A school choice problem or simply a problem is a 6-tuple G = (I , S , ≻I , ≻S , q, r). For two minority reserves profiles r = (rs )s∈S and r ′ = (rs′ )s∈S , if rs ≤ rs′ for each school s ∈ S and rs′ < r ′ s′ for at least one school s′ ∈ S , then G′ = (I , S , ≻I , ≻S , q, r ′ ) is said to have stronger minority reserves than problem G = (I , S , ≻I , ≻S , q, r). A matching is an assignment of students to schools such that each student is assigned to a school or to her outside option, and no more students are assigned to a school than its capacity. Formally, a matching µ is a mapping from I ∪ S to the subsets of I ∪ S such that (1) µ(i) ⊆ S ∪ {i} and |µ(i)| = 1 for every i ∈ I,1 (2) µ(s) ⊆ I and |µ(s)| ≤ qs for every s ∈ S, (3) µ(i) = {s} if and only if i ∈ µ(s) for every i ∈ I and s ∈ S. A matching µ′ Pareto dominates µ for the minority students if (i) µ′ (i)⪰i µ(i) for every i ∈ I m ; and (ii) µ′ (i′ )≻i′ µ(i′ ) for some i′ ∈ I m . A mechanism φ chooses a matching φ (G) for each problem G. Definition 1. A matching mechanism φ is said to be responsive to the minority reserves affirmative action if there are no problems G = (I , S , ≻I , ≻S , q, r) and G′ = (I , S , ≻I , ≻S , q, r ′ ) such that G′ has stronger minority reserves than G, and φ (G) Pareto dominates φ (G′ ) for the minority students. In other words, running a matching mechanism φ , if increasing the level of affirmative action never results in a Pareto inferior assignment for the minority students, then φ is responsive to the minority reserves affirmative action. 3. Two matching mechanisms For completeness, in this section we provide the deferred acceptance mechanism with minority reserves (DAm ) proposed by Hafalir et al. (2013) and the efficiency improved deferred acceptance mechanism with minority reserves (EIDAm ) proposed by Ju et al. (2018). 1 Since µ(i) is a one-element set, if µ(i) = {s} we also write it as µ(i) = s.
The DAm is implemented by the deferred acceptance algorithm with minority reserves. For a problem G, the deferred acceptance algorithm with minority reserves runs as follows: Step 1: Start with the matching in which no student is matched. Each student i applies to her first choice school. Each school s first accepts as many as rs minority applicants with the highest priorities if there are enough minority applicants. Then it accepts applicants with the highest priorities from the remaining applicants until its capacity is filled or the applicants are exhausted. The rest of the applicants, if any remains, are rejected by s. In general, at Step k: Start with the tentative matching obtained at the end of Step k − 1. Each student i who got rejected at Step k − 1 applies to her first choice school (call it s) among all schools that have not rejected i before. The school s considers the new applicants and students admitted tentatively at Step k − 1. Among these students, school s first accepts as many as rs minority students with the highest priorities if there are enough minority students. Then it accepts students with the highest priorities from the remaining students. The rest of the students, if any remains, are rejected by s. If there are no rejections, then the algorithm stops. The algorithm terminates when no rejection occurs and the tentative matching at that step is finalized. Since no student reapplies to a school that has rejected her, the algorithm stops in finite time. In order to introduce the EIDAm mechanism, we first specify the concept of under-demanded school proposed by Kesten and Kurino (2016). Definition 2. Given a matching µ for a problem G, a school s is said to be under-demanded in matching µ if all students weakly prefer their assignments under µ to school s, that is, µ(i)⪰i s for each student i. The EIDAm is implemented by the efficiency-improved deferred acceptance algorithm with minority reserves, which operates as follows: Round 0. Given a problem G, run DAm for G to obtain the matching µ0 . Find all the under-demanded schools in µ0 , and remove all these schools together with the students that are matched to them. The assignments between the schools and the students that are removed are finalized. There may be students who are matched to themselves, then they are also removed and assigned no school finally. If there are no more agents left over after the removal, then stop. Otherwise, go to the next round. Round k (k ≥ 1). Denote the problem that consists of the remaining schools and students by Gk , re-run DAm m for Gk to obtain matching µk . Find all the under-demanded schools in µk , and remove all these schools together with the students that are matched to them. The assignments between the schools and the students that are removed are finalized. If there are no more agents left over after the removal, then stop. Otherwise, go to the next round. The algorithm terminates when no more agents are left over. Ju et al. (2018) show that this algorithm is well-defined. 4. Result Ju et al. (2018) claim that EIDAm is responsive to the minority reserves affirmative action. However, we find that it is not the case. Specifically, we have the following result. Theorem 1. EIDAm is not responsive to the minority reserves affirmative action. Proof. We slightly modify an example in Jiao and Tian (2018). Let S = {s1 , s2 , s3 , s4 }, I M = {i1 , i3 , i4 } and I m = {i2 }. All schools have a capacity of 1: q = (1, 1, 1, 1). The minority reserves profile is
F. Ding, S. Hong, Z. Jiao et al. / Mathematical Social Sciences 97 (2019) 61–64
r = (0, 0, 0, 0) Students’ preferences and schools’ priorities are given by the following table: ≻i 1
≻i 2
≻i3
≻i 4
≻s 1
≻s 2
≻s 3
≻s 4
s1 s4
s1 s2
s2 s4
s3 s1
i4 i1
i3 i2
i2 i4
i1 i3
. . .
. . .
. . .
. . .
. . .
. . .
s3 s4
i2 i3
Then the procedure of the DAm is as follows: step
s1
1
i1 ,i2
s2
s3
i3
i4
s4
i2 , i3
2 3
i2 , i4 i1 , i4
4 5
i4
i3
i2
i1
One can see that s4 is the only under-demanded school in the DAm matching µ0 . We remove s4 together with i1 and re-run DAm for the remaining students and schools. Then the matching µ1 is step 1
s1
s2
s3
i2
i3
i4
One can see that all schools are under-demanded in matching µ1 . We remove all of them and all students. Then the final EIDAm matching with respect to G = (I , S , ≻I , ≻S , q, r) is
µ=
(
s1 i2
s2 i3
s3 i4
s4 i1
)
.
Let r ′ = (0, 1, 0, 0). Then G′ = (I , S , ≻I , ≻S , q, r ′ ) has stronger minority reserves than G = (I , S , ≻I , ≻S , q, r). For problem G′ = (I , S , ≻I , ≻S , q, r ′ ), the procedure of the DAm is as follows: step
s1
1
s2
s3
i3
i4
i1 ,i2
2
s4
i2 ,i3
3
i1
i2
i4
i3
One can see that s3 and s4 are the only under-demanded schools in the above matching. We remove s3 and s4 together with i3 and i4 , and re-run DAm for the remaining students and schools. Then we get the following matching step
s1
1
i1 ,i2
2
i1
s2
i2
One can see that s2 is under-demanded in the above matching. We remove s2 together with i2 . In the next round we remove s1 together with i1 . Then the final EIDAm matching with respect to G′ = (I , S , ≻I , ≻S , q, r ′ ) is
µ = ′
(
s1 i1
s2 i2
s3 i4
s4 i3
)
.
One can see that, running EIDAm , student i2 under G′ is worse than under G. Then, under this mechanism, the stronger minority reserves affirmative action makes all minority students (only one— i2 ) worse off. Thus, EIDAm is not responsive to the minority reserves affirmative action. □ Remark 1. For school choice problems, Kesten (2010) first introduces the efficiency-improved deferred acceptance algorithm that
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iteratively removes ‘‘last-step interrupting students’’. Tang and Yu (2014) obtain that iteratively removing ‘‘last-step interrupting students’’ and iteratively removing ‘‘under-demanded schools with students matched to them’’ produce the same matching outcome. Thus, the EIDAm can be approximately regarded as an alternative mechanism which iteratively removes ‘‘last-step interrupting students’’. Based on such an observation, we illustrate the intuition of our counterexample as follows. Suppose that s ∈ S and s′ ∈ S m compete a seat at school c, and s has higher priority than s′ at c. Running the DAm algorithm, if school c temporarily accepts s and rejects s′ , then maybe it happens that, for a weaker affirmative action problem there exists a rejection cycle starting from s and s finally is an interrupter at school c,2 while for a stronger affirmative action problem there does not exist any rejection cycle or interrupter. Thus, under the EIDAm mechanism, for the weaker affirmative action problem we remove the interrupter s, student s′ be admitted by c and gets better off. However, for the stronger reserve-based affirmative action problem, there exists no interrupter, then the DA mechanism coincides the EIDAm mechanism. We suppose S m = {s′ }. Then we obtain that, under the EIDAm mechanism, increasing the level of affirmative action results in a Pareto inferior assignment for the minority. Therefore, the EIDAm mechanism is not responsive to the minority reserves affirmative action. Remark 2. Doğan (2016) introduces a mechanism which iteratively removes ‘‘last-step interrupting minority students’’ and shows that his mechanism is responsive. We note that the intuition behind our counterexample does not apply to Doğan’s mechanism. Specifically, according to the assumption S m = {s′ } in Remark 1, we can see that, for the weaker affirmative action problem, s is an interrupter, but she is not a minority student. Since Doğan’s mechanism iteratively removes ‘‘last-step interrupting minority students’’, under Doğan’s mechanism s will not be removed. Thus, running Doğan’s mechanism, for the weaker affirmative action problem we do not remove s, student s′ does not get better off. Therefore, the matching outcome comparison for student s′ described in Remark 1 cannot appear under Doğan’s mechanism. Remark 3. For the matching problem described in Remark 1, if we assume that s is a minority student, we give a further argument for the Doğan’s mechanism. Running the DAm algorithm, for a weaker affirmative action problem, if s is a last-step interrupting minority student at school c, then under the Doğan’s mechanism we remove s and student s′ becomes better off. For a stronger affirmative action problem, if s is no longer a last-step interrupting minority student at school c, then s will not be removed under the Doğan’s mechanism. Thus, s becomes better off under the stronger affirmative action, and consequently, the Doğan’s mechanism is responsive, as increasing the level of affirmative action does not result in a Pareto inferior assignment for the minority students. From the analysis above, we can see that, for the responsiveness of the Doğan’s mechanism, it is crucial to iteratively remove only the last-step minority interrupters. Acknowledgments The authors acknowledge financial support from the Zhejiang Provincial Natural Science Foundation of China (No. LY16G010008), the China Postdoctoral Science Foundation (Nos. 2016M600302, 2018T110380) and the National Natural Science Foundation of China (No. 71771068). 2 Formally, in the DA procedure of a school choice problem, if student s is tentatively accepted by school c at some step t and is later rejected by school c at some step t ′ > t, and at least one other student is rejected by school c at some step r such that t ≤ r < t ′ , then student s is an interrupter at school c.
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