The Impossibility of Restricting Tradeable Priorities in School Assignment

Transcription

1 The Impossibility of Restricting Tradeable Priorities in School Assignment Umut Mert Dur North Carolina State University Thayer Morrill North Carolina State University February 2016 Abstract Ecient school assignments are made by allowing students to trade their priorities. However, a school board typically gives the highest priority at a school to students who have a sibling attending the school, and a school board might not want to allow a student to trade such a school-specic priority. This paper addresses the following question: is it possible to design a strategy-proof trading mechanism where the designer can specify priorities that are not allowed to be traded? We demonstrate that it is impossible to restrict which priorities are traded and maintain minimal eciency properties. Specically, we dene a mechanism to be perfect if each agent is assigned her top choice whenever such an assignment is feasible. We show that even this minimal level of eciency is incompatible with making some of the priorities untradeable. We also show that an agent's improvement in priorities may be punished by any non-wasteful, individually rational and mutually best mechanism which allows agents to trade only the tradeable priorities. JEL Classication: C78, D61, H75, I28 Key Words: Matching Theory, Market Design, School Choice Problem Address: 2801 Founders Drive 4102 Nelson Hall, Raleigh, NC, 27695; umutdur@gmail.com; web page: Address: 2801 Founders Drive 4102 Nelson Hall, Raleigh, NC, 27695; thayermorrill@gmail.com; web page: 1

2 1 Introduction A school board implementing a centralized assignment mechanism typically faces a choice between the Deferred Acceptance algorithm, hereafter DA, (Gale and Shapley, 1962) and Top Trading Cycles, hereafter TTC, (Shapley and Scarf, 1974). is strategy-proof and fair but is not Pareto ecient. TTC is strategy-proof and ecient but is not fair. Since virtually every school district has chosen DA over TTC, it is tempting to conclude that school boards value fairness over eciency. 1 DA However, Boston, the rst district to choose between the two algorithms, gave a dierent reason for choosing DA. In his memo to the Boston School Committee discussing the choice of algorithm, Superintendent Payzant writes: 2 Another algorithm we have considered, Top Trading Cycles Mechanism, presents the opportunity for the priority for one student at a given school to be traded for the priority of a student at another school, assuming each student has listed the others school as a higher choice than the one to which he/she would have been assigned. There may be advantages to this approach, particularly if two lesser choices can be traded for two higher choices. It may be argued, however, that certain priorities e.g., sibling priority apply only to students for particular schools and should not be traded away. This motivates the following question: is it possible to design a strategy-proof trading mechanism where the designer can specify priorities that are not allowed to be traded? In this paper, we demonstrate that it is impossible to restrict which priorities are traded and maintain minimal eciency properties. Specically, we dene a mechanism to be perfect if each agent is assigned her top choice whenever such an assignment is feasible. We demonstrate that even this minimal level of eciency is incompatible with making some of the priorities untradeable and some of the priorities tradeable. In order to make our result as broad as possible, we intentionally do not dene a specic trading procedure or ways of restricting priorities. Rather, we consider weak properties that are satised by any such procedure and show that they are fundamentally incongruent. A minimal fairness restriction on a mechanism is as follows: if school a has capacity for q a students, then the q a highest ranked student should be assigned to a or 1 In 2012, New Orleans school district adopted TTC. 2 This memo can be found at Accessed August 6,

3 a school that they prefer to a. What if a school board gives student i highest priority at a because i's sibling attends a, but the intention of the school board is that i may only use this priority to be assigned to a? We interpret this as follows. If i is assigned to a school other than a, and she did not use her priority at a, then the next highest ranked student will inherit i's priority. Specically, if a school a has capacity for q students and r of the students with restricted priorities are not assigned to a, then the top q + r ranked students at a must be assigned to a or a school they prefer to a. 3 We say a mechanism limits trades if it satises this property. We show that any Pareto ecient mechanism cannot limit trades (Proposition 1). We next show that this result is robust to alternate properties of a trading rule. In a trading rule, which trades an agent is able to make is independent of preferences. These are determined by the ownership structure. Preferences only determine which trades an agent wants to make. We dene consistent trading as follows. Suppose agents submit some set of preferences, and student i is assigned to school a. If i were to change her preferences to rank school a rst, would there be any impact on any of the trades that were previously made? For a reasonable trading rule, we argue no. The trades i is able to make have not changed, and the trades i chooses to make should not change either as choosing the same set of trades results in i receiving her most preferred object. Since i makes the exact same trades and otherwise the problem is identical, then the mechanism should make the same trades. We prove that no mechanism is perfect, makes consistent trades, and limits trades (Proposition 4). Next we consider an alternative to perfection. If a student is not able to trade her restricted priorities, is she able to trade the priorities that are not restricted? Surprisingly, the answer is no. We dene a mechanism to eciently limit trades if it never makes an assignment that is Pareto dominated by an alternative assignment that also limits trades. We prove that there does not exist a mechanism that makes consistent trades and eciently limits trades (Proposition 5). Finally, we consider strategy-proofness. We show there is no mechanism that makes consistent trades, limits trades, is strategyproof, non-wasteful, and individually rational (Proposition 6). Making consistent trades is a stronger condition than nonbossiness. Similarly, limiting trades is a strictly weaker fairness notion than eliminating justied envy. It is intriguing that these basic properties are incompatible. Moreover, we show 3 Implicit in this denition we assume that only the top q priorities may be restricted. This is consistent with a school board restricting the trade of sibling or walk zone priorities as these are typically the highest priorities at a school. 3

4 that any hierarchical exchange rule, introduced by Pápai (2000), fails to limit trades (Corollary 1). The intuition for these results is as follows. Trading mechanisms, in the tradition of Papai's hierarchical exchanges, work by having an ownership and inheritance structure. Each object is owned by some agent, and when an agent makes a trade, the objects she owns but did not use are inherited by some other agent. For a standard trading procedure, we can always nd a trade that the agents want and are able to make. In the terminology of Top Trading Cycles, there always exists a cycle. However, if we limit trading, then eventually an object will be owned by an agent who is not allowed to trade the object. Now, we can always nd a trade that agents want to make, but we can no longer be sure that they are able to make it. It may be that every cycle involves an agent trading a priority at which she is restricted. In this case, we are stuck. Any way that we proceed will either give agents the incentive to misreport their preferences, induce ineciencies, or keep students from being assigned schools they should be guaranteed admission to. Our nal result is to demonstrate that if we use a trading mechanism but do not allow agents to trade their restricted priorities, then an agent may be harmed by even having this priority. Here, we formally dened what it means to be a trading mechanism rather than using properties satised by a trading mechanism. Specically, we say an assignment mechanism is a trading mechanism if at each step it selects and processes a cycle and this cycle is not aected by the preferences reported by the other students. We do not restrict how ownership is determined or how objects are inherited. We show a surprising result. If agents are not allowed to trade restricted priorities, then for every mutual best and individually rational trading rule, there exists an assignment problem where a student is better o having no priority at an object (being ineligible for that object) rather than having a priority but one that is restricted (Theorem 1). We conclude by using simulations to estimate how frequently restricted priorities are traded under TTC. The simulations show that when the correlation between student preferences increase or students' tendency to be assigned to their siblings' schools increases, the number of restricted priority traded under TTC mechanism decreases. Moreover, under realistic environments we observe that very few of the restricted priorities are traded under TTC. Our paper contributes to the growing literature on the ecient assignment of agents to indivisible objects when no object is owned by any of the agents. This topic was pioneered by Pápai (2000) who introduced the version of TTC that we study here. TTC 4

5 is part of a broader class of mechanisms introduced by Pápai (2000) called hierarchical exchange rules. Pycia and Ünver (2011a) introduce a class of trading mechanisms called trading cycles that extend hierarchical exchange rules. Kesten (2004) introduced an alternative trading algorithm called Equitable Top Trading Cycles. Morril (2014) introduces an alternative called Clinch and Trade. Both Equitable Top Trading Cycles and Clinch and Trade are designed to make an ecient assignment with fewer instances of justied envy than TTC. This problem is important because of its applicability to assigning students to public schools. This problem was rst considered by Balinski and Sönmez (1999) and then by Abdulkadiro lu and Sönmez (2003). Abdulkadiro lu, Pathak, Roth, and Sönmez (2005) discusses the market design considerations of applying DA and TTC to the school assignment problem. The organization of the rest of the paper is as follows: in the next section we describe the model and the axioms we use in our analysis. In Section 3 we demonstrate the impossibility results. In Section 4, we introduce a class of trading mechanism and show that students might be punished when their priorities are improved. In Section 5, using simulations we measure the performance of the TTC mechanism in terms of the number of restricted priorities traded. 2 Model A school choice problem is a list [I, S, q, P,, e] where I is the set of students, S is the set of schools, q = (q s ) s S is the quota vector where q s is the number of available seats in school s, P = (P i ) i I is the preference prole where P i is the strict preference of student i over the schools, = ( s ) s S is the priority prole where s is the strict priority relation of school s over I, 5

6 e = (e s ) s S is the restricted priority prole where e s is the set of students with restricted priority for school s. We assume that the only priorities a school board chooses to restrict are among the q s highest priorities at a given school s. Let s be the no-school (being unassigned) option and s S where q s =. Without loss of generality e s =. For a given priority prole, we denote the rank of student i in s with r s (i). That is, if i s j then r s (i) < r s (j). For a given problem, we denote the set of students with restricted priority at least one school in S with I R and the set of students without restricted priority with I U. That is, I R = s S e s and I U = I \ I R. In the rest of the paper we x I,S, q,, and e and represent a problem with P. Let R i be the at-least-as-good-as relation associated with P i for all i I. A matching µ : I S is a function which assigns a school to each student such that no school is assigned to more students than the number of available seats in that school. We denote the set of matchings with M. In matching µ, the assignment of student i is denoted by µ i and the set of students assigned to s is denoted by µ s. Let µ s be the set of students not assigned to school s, i.e., µ s = I \ µ s. A mechanism is a procedure which selects a matching for each problem. The matching selected by mechanism φ in problem P is denoted by φ(p ). Let φ i (P ) be the assignment of student i I in matching φ(p ). A matching µ is perfect if all students in I are assigned to their most preferred school in S. Formally, µ is perfect if µ i P i x for all x S \ {µ i } and i I. For every problem, the existence of a perfect matching is not guaranteed. To see this consider a problem in which all students rank school s at the top of their preference lists and the number of available seats in school s is less than the number of students in I. A mechanism φ is perfect if it selects the perfect matching whenever it exists. A matching µ Pareto dominates another matching ν if each student i I weakly prefers her assignment in µ to her assignment in ν and there exists at least one student j who strictly prefers her assignment in µ to her assignment in ν. A matching µ is Pareto ecient if there does not exist another matching ν which Pareto dominates µ. A mechanism φ is Pareto ecient if φ(p ) is Pareto ecient for any problem P. A matching µ is individually rational if each student i I weakly prefers his assignment to being unassigned option, i.e. µ i R i s for all i I. A mechanism φ is individually rational if it selects individually rational matching for all problems. A matching µ is non-wasteful if whenever a student i prefers another school s to his assignment then all the available seats of s are lled. A mechanism φ is non-wasteful 6

7 if it selects non-wasteful matching in all problems. It is easy to see that perfection implies Pareto eciency, and Pareto eciency implies individual rationality and non-wastefulness. Hence, if a matching is wasteful or individually irrational or Pareto inecient then it cannot be perfect. A matching µ is fair if there does not exist a student school pair (i, s) where s P i µ i and i s j for some j µ s. A mechanism φ is fair if for any problem P its outcome φ(p ) is fair. A matching µ is mutually best if there does not exist a student school pair (i, s) such that s P i x for all x S \{s}, r s (i) q s and µ(i) s. A mechanism φ is mutually best if for any problem P its outcome φ(p ) is mutually best. A mechanism φ is strategy-proof if there does not exist a student i and a preference relation P such that φ i (P, P i ) P i φ i (P i ). A mechanism φ makes consistent trades if a student cannot change the matching selected by moving his match to the top of his preference list. That is, φ makes consistent trades if for all preferences P and all agents i, φ(p ) = φ(p i, P i ) where P i is any preference prole for i such that φ i (P ) P i s for all s φ i (P ). A mechanism φ is nonbossy if a student cannot change the assignment of the other students without changing his own assignment by submitting dierent preference list. That is, φ is nonbossy if for any P and P i φ i (P i, P i ) = φ i (P ) implies φ(p i, P i ) = φ(p ). 3 Results In this section we consider whether or not it is possible to design a trading mechanism that allows students to trade some but not all of their priorities. Our notion of trading mechanism is a generalization of Papai's hierarchical exchanges (Pápai, 2000). Seats at a school are assigned to an owner. An owner is allowed to keep one of her seats or make a trade, and the seats she does not use are inherited by other agents. Papai provides a tight characterization of hierarchical exchanges when objects have capacity for only one agent. In the school assignment problem where one object may be assigned to many agents there is enough exibility to allow for a variety of strategy-proof and ecient trading mechanisms. The variation of Top Trading Cycles introduced by Abdulkadiro lu and Sönmez (2003) is perhaps the most intuitive, but Equitable Top Trading Cycles (Kesten, 2004), Clinch and Trade (Morril, 2014), and Prioritized Trading Cycles (Morrill, 2014) 7

8 are alternative trading mechanisms. Unfortunately we nd that restricting trades is incompatible with even the most basic notions of eciency. So that this result is as general as possible, we deliberately do not dene a specic trading mechanism but rather consider properties that we would expect to hold in any trading mechanism. Similarly, we do not dene what it means to limit the trading of priorities, but instead we consider properties that would hold in a trading mechanism that limited trading. The rst condition that we consider is limiting trades. Suppose a school a has capacity for q a students but a school board has deemed k of the q a priorities schoolspecic and therefore untradeable. Intuitively, if any of the k agents that are not allowed to trade their priority at a are not assigned to a, then some other student will inherit her priority. We assume that the student with highest priority will inherit. Specically, a mechanism limits trades if whenever k k of the students with restricted priority at a are assigned to a school other than a, then the q a +k students with highest priority at a are assigned to either a or a school they prefer to a. Also notice that any violation of this condition would be an instance of justied envy, so this is a strictly weaker condition than eliminating justied envy (fairness). Denition 1 A matching µ limits trades if there does not exist a student school pair (i, s) where sp i µ i and r s (i) q s + e s µ s. A mechanism φ limits trades if φ(p ) limits trades for any P. First, we demonstrate that there is no mechanism that limits trades and is Pareto ecient. Since any fair mechanism also limits trades, this impossibility result is stronger than the impossibility between fairness and Pareto eciency (Balinski and Sönmez, 1999). Proposition 1 There does not exist a mechanism which limits trades and is Pareto ecient. Proof. Suppose φ is Pareto ecient and limits trades. Consider the following problem. Let S = {a, b, c, s }, I = {i, j, k}, q = (1, 1, 1, ), e a = {i} and e s = for all s S \ {a}. The preferences and priorities are: a b c P i P j P k i k j c a a k a c b 8 b b c

9 ( i j k There is a unique matching which limits trades: λ = a c b ) ( ). But λ is not Pareto ecient because it is Pareto dominated by µ = i j k c a b. Next, we relax Pareto eciency. The restriction on trading mechanisms that we consider, making consistent trades, is closely related to non-bossiness. Specically, if a mechanism makes consistent trades then it is also non-bossy (Proposition 2) and if a strategy-proof mechanism is non-bossy then it also makes consistent trades (Proposition 3). Proposition 2 Any mechanism which makes consistent trades is non-bossy. Proof. Suppose not. Let φ make consistent trades but be bossy. Then there exists a problem and a student i such that φ i (P ) = φ i (P, P i ) = s and φ(p ) φ(p, P i ). Let s P x for all x S \ {s}. Since φ makes consistent trades φ( P, P i ) = φ(p ) and φ( P, P i ) = φ(p, P i ). Then φ( P, P i ) = φ(p ) = φ(p, P i ). This is a contradiction. Proposition 3 Any strategy-proof mechanism which is non-bossy makes consistent trades. Proof. Suppose under P, i is assigned to a and let P i be any preference in which i ranks a at the top of the list. Then i must be assigned to a under φ(p i, P i ) as otherwise she could protably misreport her preferences as P i. If φ is nonbossy, then since changing i's preferences did not change i's assignment, it cannot change the assignment of any other agents, and it must be that φ(p ) = φ(p i, P i ). Therefore, φ makes consistent trades. Since TTC is strategy-proof and non-bossy, Proposition 3 implies that TTC makes consistent trades. Perfection is a signicantly weaker requirement than Pareto eciency. However, when a trading mechanism makes consistent trades, even this level of eciency is incompatible with limiting trades. Proposition 4 There does not exist a mechanism which makes consistent trades, limits trades and is perfect. Proof. Suppose φ satises all these axioms. Consider the following problem. Let S = {a, b, c, s }, I = {i, j, k}, q = (1, 1, 1, ), e a = {i} and e s = for all s S \ {a}. We x the following preferences and priorities: 9

10 a b c P i P j i k j c a k a c Here, we do not need any restriction on P k. We dene two matchings µ and λ as follows: ( ) i j k µ = c a b b b ( ) i j k λ = a c b Limiting trades implies that regardless of the preferences submitted by the other students, when i submits P i, he will be assigned to either a or c as i must not receive a school worse than a. Similarly, when j submits P j, he will be assigned to either a or c as j must not receive a school worse than c. Therefore, k is assigned to b no matter what preferences she submits. If k ranks a rst, then by limiting trades, j cannot be assigned to a. Therefore, φ must make assignment λ when i reports P i, j reports P j and k ranks a at the top of his list. Since k is assigned to b and φ makes consistent trades, if she ranks b rst then φ must continue to make assignment λ. However, when k ranks b rst, µ is a perfect assignment, a contradiction. These results demonstrate that there are signicant unintended consequences to designating certain priorities as untradeable. The eect of eliminating some types of trades is to impose a limit on all trades that may occur. Here we consider another way of signicantly weakening Pareto eciency. We show that it is not even possible to eliminate trades so that the outcome is undominated by alternatives that also limit trades. We call this eciently limiting trades. Denition 2 A matching µ eciently limits trades if there does not exist any other matching ν which limits trades and Pareto dominates µ. limits trades if φ(p ) eciently limits trades for any P. A mechanism φ eciently Proposition 5 There does not exist a mechanism which makes consistent trades and eciently limits trades. Proof. Suppose φ satises all these axioms. We use the same problem used in the proof of Proposition 4. 10

11 As in the proof of Proposition 4 limiting trades implies that regardless of the preferences submitted by the other students, when i submits P i and j submits P j, φ assigns i and j to either a or c and k is assigned to b no matter what preferences she submits. If k ranks a rst, then by limiting trades, j cannot be assigned to a. Therefore, φ must make assignment λ. Since k is assigned to b, if she ranks b rst then since φ makes consistent trades, φ must continue to make assignment λ. However, when k ranks b rst, µ limits trades and Pareto dominates λ, a contradiction. Next we consider the strategic implications of limiting trades, and again we nd a negative result. When we try to limit trades, we create situations where an algorithm gets stuck. Specically, in a trading algorithm we can no longer be sure that a trading cycle always exists. Whichever way we get unstuck will enable agents to strategically misrepresent their preferences. Proposition 6 There does not exist a mechanism which makes consistent trades, limits trades and is strategy-proof and non-wasteful. Proof. Suppose φ satises all these axioms. We use the same problem used in the proof of Proposition 4. As in the proof of Proposition 4 limiting trades implies that regardless of the preferences submitted by the other students, when i submits P i and j submits P j, φ assigns i and j to either a or c and k is assigned to b no matter what preferences she submits. If k ranks a rst, then by limiting trades, j cannot be assigned to a. Therefore, φ must make assignment λ. Since k is assigned to b, if she ranks b rst then since φ makes consistent trades, φ must continue to make assignment λ. Let b P k x for all x S \ {b}. We have shown that φ(p i, P j, P k ) = λ. Consider the preference prole (P i, P j, P k ) where P i : c P i s P i bp i a. In problem (P i, P j, P k ) due to strategy-proofness and limiting trades i will be assigned to s, φ i (P i, P j, P k ) = s. Moreover, in φ(p i, P j, P k ) k will be assigned to b and j will be assigned to either c or a. Otherwise, φ cannot satisfy limiting trades. If φ j (P i, P j, P k ) = c then the available seat in a is wasted, i.e. a P j φ j (P i, P j, P k ) and φ a (P i, P j, P k ) < q a. If φ j (P i, P j, P k ) = a then the available seat in c is wasted, i.e. c P i φ j (P i, P j, P k ) and φ c (P i, P j, P k ) < q c. This contradicts with the fact that φ is non-wasteful. Pápai (2000) shows that combination of strategy-proofness and non-bossiness is equivalent to group strategy-proofness. Hence, Proposition 2 and Proposition 6 imply that there does not exist a group strategy-proof and non-wasteful mechanism that limits trades. 11

12 Proposition 7 There does not exist a mechanism which is group strategy-proof, nonwasteful and limits trades. Proof. Proposition 2, Proposition 3 and Pápai (2000) imply that the set of strategyproof, non-bossy and non-wasteful mechanisms, the set of group strategy-proof and nonwasteful mechanisms, and the set of strategy-proof, non-wasteful mechanisms making consistent trades are equivalent. In Proposition 6, we have shown that any mechanism belongs to the set of strategy-proof, non-wasteful mechanisms making consistent trades does not limit trades. By the equivalency, any group strategy-proof and non-wasteful mechanism does not limit trades. Pápai (2000) demonstrates that a mechanism is group-strategyproof, Pareto ecient, and reallocation proof if and only if it is a hierarchical exchange rule. Hence, as a direct corollary of Proposition 7, any hierarchical exchange rule does not limit trades. Corollary 1 There does not exist a hierarchical exchange rule which limits trades. 4 Trading Mechanisms In this section we focus on the class of trading mechanisms which includes TTC (Abdulkadiro lu and Sönmez, 2003; Pápai, 2000). In particular, we investigate whether there exists a trading mechanism which satises desired features while not allowing the trade of restricted priorities. Before starting our analysis we dene the class of trading mechanisms. We say a mechanism φ belongs to the class of trading mechanisms if the followings are true: 1. For any problem P, φ selects cycles 4 recursively and each student i x is assigned to s x for all x {1,..., n} where i n+1 = i 1. Denote the rst cycle selected in problem P by φ c (P ). 2. If there is no perfect matching for problem P then the cycle selected in a problem P is not aected by the preference prole of the students who are not in the cycle, i.e. φ c (P ) = φ c (P I c, P I c) where I c is the set of students in φ c (P ). A cycle (i 1, s 1, i 2, s 2,..., i n, s n ) respects the restricted priorities if i x does not have restricted priority for s x 1 for all x {1,..., n}. A trading mechanism φ respects 4 A cycle (i 1, s 1, i 2, s 2,..., i n, s n ) is a list of students and schools in which student i x points to s x and s x points to i x+1 for all x {1,..., n} where i n+1 = i 1. 12

13 the restricted priorities if for any problem P the cycle selected by φ, φ c (P ), respects the restricted priorities. We now demonstrate that an attempt to limit what priorities are tradeable may lead to severe unintended consequences under trading mechanisms. We show that if a school does not allow some of the priorities to be traded, then a student can be harmed by having one of these restricted priorities. For example, if we do not allow students to trade sibling priorities, then a student may prefer to be ranked last by a school then to have the highest, but restricted sibling priority. For convenience, we present the argument as a comparison between having the highest, but restricted priority versus the lowest, but unrestricted priority. However, the same argument implies that a student can be better o by being declared unacceptable by a school than by having the highest priority at the same school. 5 We consider this an unintended consequence because surely the intention of restricting a priority such as sibling priority is not to harm the students that have an older sibling attending a school they are not interested in attending. Theorem 1 Let φ be any mutually best, individually rational, non-wasteful trading mechanism which respects the restricted priorities. Under φ, a student can be worse o having the highest priority at a school if this priority is restricted than having the lowest (but unrestricted) priority. Proof. Consider the following problem. S = {a, b, s }, I = {i, j, k} and q = (1, 1, ). Suppose i has restricted priority at a. a b P i P j P k i j b a a j i a b s k k There is no perfect matching for this problem. We are focusing on mutual best, individually rational and non-wasteful mechanisms. Suppose ψ is a trading mechanism and satises all. By non-wastefulness all seats will be lled. We have the following observations. ψ cannot select a trading cycle in which i or k is assigned to b and j is not a part of the trading cycle: Suppose for contradiction ψ selects a cycle, not including j, that 5 We do not frame it this way since we have not allowed schools to declare students unacceptable in our model. However, all algorithms could be easily modied to allow this and the same result would hold. 13

14 assigns b. By denition, the same cycle is selected regardless of the preferences j submits. However, if j ranks b at the top of his preference list, then by mutually best j must be assigned to b. Therefore, the cycle is not preserved, a contradiction. ψ cannot select a trading cycle in which j or k is assigned to a and i is not a part of the trading cycle: This is similar to the above argument. If not, then i's preferences do not change the cycle selected. However, if i ranks a at the top of his preference list, then ψ cannot be mutually best and select the same cycle. The remaining cycles are: 1. i a i, 2. j b j, 3. i b j a i, 4. i a j b i. By assumption, i has restricted priority at a. Therefore, (3) is not the cycle selected. Since (3) is the only case in which i is assigned to b, ψ must assign i to a. Now consider the following priority structure in which i has the lowest, but unrestricted priority at a. a b P i P j P k j j b a a k i a b s i k We can interpret this priority prole as follows: In the rst problem i has sibling priority but in the second one i does not have sibling priority. In this second problem any mutually best and individually rational mechanism assigns j to a and k to. And in order not to waste the seat in b, i will be assigned to b: i becomes better o by losing his sibling priority at school a. An alternative way of presenting Theorem 1 is to use a similar concept to the respecting improvement in the test scores introduced by Balinski and Sönmez (1999). Since the priority structure in the school choice environment does not only depend on the test scores we use an axiom called respecting improvement in the priorities. We say that is an improvement in the priorities for agent i I if: (1) i s j = i s j for all s S, 14

15 (2) there exists at least one agent j and school s such that j s i s j, and (3) k s l k s l for all s S and l, k I\{i}. A mechanism φ respects improvements in the priorities if is an improvement in the priorities for agent i I, then φ i (, P )R i φ i (, P ). That is, a mechanism respects improvements in the priorities if an agent is not punished for having higher priorities for some schools. Now we can re-state our result in Theorem 1 by using respecting improvements in the priorities. Corollary 2 There does not exist a mutually best, individually rational, non-wasteful trading mechanism which respects the restricted priorities and improvements in the priorities. Proof. We refer to the proof of Theorem 1. 5 Simulations Since it is impossible to completely eliminate the trade of the restricted priorities without inducing eciency lose, a natural question is how often are the priorities we would like to restricted actually traded in TTC. We estimate the number of trades using computer simulations. We develop our setup by taking some important aspects of the school choice problem into account. For instance, a school may become more desirable to a student if his elder sibling is already attending that school. Hence, a student with sibling priority may prefer to be assigned to his sibling's school if there is another school that he slightly prefers to his sibling's school. We expect to there to be correlation in the preferences. Specically, we expect there to be good schools that all of the students are more likely to vote highly than bad schools. We incorporate these points in our denition of the preferences of the agents over the schools. Let U i,s be the utility of student i I for school s S. It is dened as: U i,s = β (α Z(s) + (1 α) Z(i, s)) + (1 β) sibling(i) siblingpriority(i, s) where α, β [0, 1]. 6 The correlation in the agent preferences is captured by α. Parameter β captures the tendency of the students to be assigned to their elder sibling's 6 Erdil and Ergin (2008) dene utilities in their simulations similarly. 15

16 current school. As β decreases being assigned to the elder sibling's school becomes more preferable. Z(s) is an i.i.d standard uniformly distributed random variable and represents the common tastes of students on school s. Z(i, s) is also an i.i.d standard uniformly distributed random variable and represents the tastes of student i on school s. Student i's value for attending his sibling's school is denoted by sibling(i) and its value is drawn from a standard uniform distribution. The last term in the utility equation, siblingprioritiy(i, s), is equal to 1 if i has sibling attending to s and 0 otherwise. It is worth mentioning that a student may not rank his elder sibling's school for higher (1 β), since the random variable sibling also plays role in his taste on his elder sibling's school. In our simulations, we set the number of students equal to the number of available seats and each school has the same number of seats. Then, given the number of students and the schools we index the students by i = 1, 2,..., n and schools by s = 1, 2,...m where n = m q. We rst randomly decide which students have sibling priority in which schools. Next, we determine the priority structure from the sibling prole and an ordering that is drawn randomly. We generate random variables Z(s), Z(i, s) and sibling(i), and by using these random variables and predetermined α and β we determine the utilities. For each setup we run the TTC mechanism and calculate the number of restricted priority traded. By keeping the sibling prole and utilities the same, we run the TTC mechanism 10,000 times by using dierent random orderings of the students. Figure 1: Simulations with 5 Schools with 20 Seats 16

17 Figure 2: Simulations with 10 Schools with 10 Seats We set the number of students to 100 and the number of students with sibling priority to 40. We consider two scenarios. In the rst one we set the number of schools to 5 and in the second one we set the number of schools to 10. Under both cases the number of traded restricted priorities decreases as β decreases and α increases. That is to say, if students mostly agree on the relative qualities of the schools and a school becomes more desirable when an elder sibling is attending to that school, then TTC almost eliminates the trades of restricted priorities. 6 Conclusion A school board might reasonably object to students being able to trade school-specic priorities such as sibling attendance or being within walking distance to the school. This paper demonstrates that it is impossible to design a trading mechanism that makes a subset of the priorities untradeable without sacricing even the most basic eciency properties. This suggests that if the trading of some priorities is completely unacceptable to a school board, then they should use the DA mechanism instead of the TTC mechanism. However, if the trading of these priorities is something the board wishes to avoid but is not necessarily a dealbreaker, then our design objective should be modifying TTC to minimize, not eliminate, the trading of such priorities. 17

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