The Impossibility of Restricting Tradeable Priorities in School Assignment Umut Mert Dur∗
Thayer Morrill †
North Carolina State University
North Carolina State University
April 2015
Abstract E cient 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-speci c 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 e ciency properties. Speci cally, we de ne 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 e ciency 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.
1
Introduction
A school board implementing a centralized assignment mechanism typically faces a choice between the Deferred Acceptance algorithm, hereafter DA, (Gale and Shap-
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1
ley, 1962) and Top Trading Cycles, hereafter TTC, (Shapley and Scarf, 1974). is strategy-proof and fair but is not Pareto e cient. cient but is not fair.
DA
TTC is strategy-proof and e -
Since virtually every school district has chosen DA over TTC,
1
it is tempting to conclude that school boards value fairness over e ciency.
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 e ciency properties. Speci cally, we de ne 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 e ciency 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 de ne a speci c trading procedure or ways of restricting priorities.
Rather, we consider weak
properties that are satis ed by any such procedure and show that they are fundamentally
a has capacity for qa students, then the qa highest ranked student should be assigned to a or 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
incongruent. A minimal fairness restriction on a mechanism is as follows: if school
1 In
2012, New Orleans school district adopted TTC. memo can be found at http://www.iipsc.org/resources/tpayzant-memo-05.25.2005.pdf. Accessed August 6, 2013. 2 This
2
will inherit
i's
priority. Speci cally, if a school
a
has capacity for
the students with restricted priorities are not assigned to students at
a
limits trades
must be assigned to
a
a,
q
students and
then the top
or a school they prefer to
a.
q+r
r
of
ranked
We say a mechanism
if it satis es this property. We show that any Pareto e cient 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 de ne
consistent trading
some set of preferences, and student preferences to rank school
a
i
as follows. Suppose agents submit
is assigned to school
a.
If
i
were to change her
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 to make have not changed, and the trades choosing the same set of trades results in
i
is able
i chooses to make should not change either as 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 de ne a mechanism to
e ciently 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 e ciently 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 justi ed envy. It is intriguing that these basic properties are incompatible. Moreover, we show 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
3
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 ine ciencies, 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 de ned what it means to be a trading mechanism rather than using properties satis ed by a trading mechanism. Speci cally, 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 e cient 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 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 e cient assignment with fewer instances of justi ed envy than TTC.
4
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
is the set of students,
• S
is the set of schools,
• q = (qs )s∈S s, • P = (Pi )i∈I
[I, S, q, P, , e]
is the quota vector where
qs
where
is the number of available seats in school
is the preference pro le where
Pi
is the strict preference of student
i
over the schools,
• = ( s )s∈S s over I ,
is the priority pro le where
• e = (es )s∈S
is the restricted priority pro le where
restricted priority for school
s
is the strict priority relation of school
es
is the set of students with
s.
s∅ ∈ S where qs∅ = ∞. Without loss of generality es∅ = ∅. For a given priority pro le , we denote the rank of student i in s with rs (i). That is, if i s j then rs (i) < rs (j). For a given problem, we denote R the set of students with restricted priority at least one school in S with I and the set U R U R of students without restricted priority with I . That is, I = ∪s∈S es and I = I \ I .
Let
s∅
be the no-school (being unassigned) option and
5
In the rest of the paper we x
Ri
I , S , q , ,
e
and
and represent a problem with
be the at-least-as-good-as relation associated with A
matching µ : I → S
Pi
for all
P.
Let
i ∈ I.
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
student
i
µ−s
is denoted by
µi
M.
In matching
µ,
the assignment of
s is denoted by µs . Let = I \ µs . A mechanism
and the set of students assigned to
be the set of students not assigned to school
s,
i.e.,
µ−s
is a procedure which selects a matching for each problem.
The matching selected by
φ in problem P is denoted by φ(P ). Let φi (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 µi Pi x for all x ∈ S \ {µi }. For every problem, mechanism
the existence of a perfect matching is not guaranteed. To see this consider a problem in which all students rank school of available seats in school
s
s
at the top of their preference lists and the number
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
prefers her assignment in
j
µ
to her assignment in
who strictly prefers her assignment in
µ
ν
φ
A matching
is Pareto e cient if
µ
is
φ(P )
i∈I
µ
is
which Pareto dominates
µ.
to her assignment in
ν
ν.
A matching
is Pareto e cient for any problem
individually rational
assignment to being unassigned option, i.e.
weakly
and there exists at least one student
Pareto e cient if there does not exist another matching A mechanism
if each student
if each student
µi Ri s∅
for all
i ∈ I i ∈ I.
P.
weakly prefers his 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
if it selects non-wasteful matching in all problems. It is easy to see that perfection implies Pareto e ciency, and Pareto e ciency implies individual rationality and non-wastefulness.
Hence, if a matching is wasteful or
individually irrational or Pareto ine cient then it cannot be perfect.
fair if there does not exist a student school pair (i, s) where s Pi µi and i s j for some j ∈ µs . A mechanism φ is fair if for any problem P its outcome A matching
φ(P )
µ
is
µ
is
is fair.
A matching such that
mutually best
if there does not exist a student school pair
s Pi x for all x ∈ S \{s}, rs (i) ≤ qs 6
and
µ(i) 6= s.
A mechanism
(i, s)
φ is mutually
best
if for any problem
P
its outcome
φ(P )
is mutually best.
φ is strategy-proof if there does not exist a student i and a preference 0 relation P such that φi (P , P−i ) Pi φi (Pi ). 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(Pi0 , P−i ) where Pi0 is any 0 preference pro le for i such that φi (P ) Pi s for all s 6= φi (P ). A mechanism φ is nonbossy if a student cannot change the assignment of the other A mechanism
0
students without changing his own assignment by submitting dierent preference list. That is,
φ
is
nonbossy
if for any
P
and
Pi0 φi (Pi0 , P−i ) = φi (P )
implies
φ(Pi0 , 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 e cient 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) are alternative trading mechanisms. Unfortunately we nd that restricting trades is incompatible with even the most basic notions of e ciency. So that this result is as general as possible, we deliberately do not de ne a speci c trading mechanism but rather consider properties that we would expect to hold in any trading mechanism. Similarly, we do not de ne 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 capacity for
qa
limiting trades
students but a school board has deemed
7
k
.
Suppose a school
of the
qa
a
has
priorities school-
speci c and therefore untradeable. Intuitively, if any of the to trade their priority at
a
are not assigned to
a,
k agents that are not allowed
then some other student will inherit
her priority. We assume that the student with highest priority will inherit. Speci cally, a mechanism limits trades if whenever
k0 ≤ k
of the students with restricted priority at
a are assigned to a school other than a, then the qa + k 0 students with highest priority at a are assigned to either a or a school they prefer to a. This is a generalization of mutual best.
3
Also notice that any violation of this condition would be an instance of justi ed
envy, so this is a strictly weaker condition than eliminating justi ed envy (fairness).
limits trades
De nition 1 A matching µ if there does not exist a student school pair (i, s) where sPi µi and rs (i) ≤ qs + |es ∩ 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 e cient. Since any fair mechanism also limits trades, this impossibility result is stronger than the impossibility between fairness and Pareto e ciency (Balinski and Sönmez, 1999).
Proposition 1 There does not exist a mechanism which limits trades and is Pareto e cient. Proof.
problem. Let
s ∈ S \ {a}.
φ is Pareto e cient and limits trades. Consider the following S = {a, b, c, s∅ }, I = {i, j, k}, q = (1, 1, 1, ∞), ea = {i} and es = ∅ for all
Suppose
The preferences and priorities are:
a i k
b k
c j
Pi c a b
Pj a c b
Pk a b c
i j k There is a unique matching which limits trades: λ = a c b ! i j k e cient because it is Pareto dominated by µ = . c a b 3A
! . But
λ is not Pareto
matching µ is mutually best if whenever there exists student school pair (i, s) such that s Pi µi then rs (i) > qs . 8
Next, we relax Pareto e ciency.
The restriction on trading mechanisms that we
consider, making consistent trades, is closely related to non-bossiness. Speci cally, 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
i such that φi (P ) = φi (P 0 , P−i ) = s and φ(P ) 6= φ(P 0 , P−i ). ¯ x for all x ∈ S \ {s}. Since φ makes consistent trades φ(P¯ , P−i ) = φ(P ) and Let sP φ(P¯ , P−i ) = φ(P 0 , P−i ). Then φ(P¯ , P−i ) = φ(P ) = φ(P 0 , P−i ). This is a contradiction. a problem and a student
Proposition 3 Any strategy-proof mechanism which is non-bossy makes consistent trades. Proof.
Pi0 be any preference in which i 0 ranks a at the top of the list. Then i must be assigned to a under φ(Pi , P−i ) as otherwise she could pro tably misreport her preferences as Pi . If φ is nonbossy, then since changing i's preferences did not change i's assignment, it cannot change the assignment of any 0 other agents, and it must be that φ(P ) = φ(Pi , P−i ). Therefore, φ makes consistent Suppose under
P, i
is assigned to
a
and let
trades. Since TTC is strategy-proof and non-bossy, Proposition 3 implies that TTC makes consistent trades. Perfection is a signi cantly weaker requirement than Pareto e ciency.
However,
when a trading mechanism makes consistent trades, even this level of e ciency is incompatible with limiting trades.
Proposition 4 There does not exist a mechanism which makes consistent trades, limits trades and is perfect. Proof. φ satis es all these axioms. Consider the following problem. Let S = {a, b, c, s∅ }, I = {i, j, k}, q = (1, 1, 1, ∞), ea = {i} and es = ∅ for all s ∈ S \ {a}. We x Suppose
the following preferences and priorities:
a i k
b k
c j
9
Pi c a b
Pj a c b
Pk .
Here, we do not need any restriction on
µ
We de ne two matchings
and
λ
as
follows:
i j k c a b
µ=
!
i j k a c b
λ=
!
Limiting trades implies that regardless of the preferences submitted by the other students, when
i
submits
a school worse than
c
as
j
a.
Pi ,
a
he will be assigned to either
Similarly, when
j
submits
c.
must not receive a school worse than
Pj ,
or
c
as
i
must not receive
he will be assigned to either
Therefore,
k
b
is assigned to
a
or
no matter
what preferences she submits. If
k
ranks
a
must make assignment his list. Since
φ
k
j
rst, then by limiting trades,
λ
when
is assigned to
b
i
reports
and
φ
must continue to make assignment
Pi , j
a. k ranks a
cannot be assigned to reports
Pj
and
Therefore,
at the top of
makes consistent trades, if she ranks
λ.
However, when
k
ranks
b
rst,
φ
µ
b
rst then
is a perfect
assignment, a contradiction. These results demonstrate that there are signi cant 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 signi cantly weakening Pareto e ciency. 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 e ciently limiting trades.
e ciently limits trades
De nition 2 A matching µ if there does not exist any other matching ν which limits trades and Pareto dominates µ. A mechanism φ e ciently limits trades if φ(P ) e ciently limits trades for any P . Proposition 5 There does not exist a mechanism which makes consistent trades and e ciently limits trades. Proof.
Suppose
φ
satis es 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
i
no matter what preferences she submits.
and If
j k
to either ranks
a
a
or
c
and
k
is assigned to
b
rst, then by limiting trades,
must make assignment
λ.
Since
k
submits
j
is assigned to
10
Pi
and
j
submits
cannot be assigned to
a.
Pj
,
φ
assigns
Therefore,
φ
b, if she ranks b rst then since φ makes
consistent trades, rst,
µ
φ
must continue to make assignment
limits trades and Pareto dominates
λ,
λ.
k
However, when
ranks
b
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. Speci cally, 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
satis es 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
i
no matter what preferences she submits.
and If
j k
to either ranks
a
a
or
c
and
k
is assigned to
b
rst, then by limiting trades, Since
k
j
Pi
and
j
submits
cannot be assigned to
a.
Pj
,
φ
assigns
Therefore,
φ
b, if she ranks b rst then since φ makes consistent trades, φ must continue to make assignment λ. Let b Pk x for all x ∈ S \ {b}. We have shown that φ(Pi , Pj , Pk ) = λ. Consider the 0 0 0 0 0 0 preference pro le (Pi , Pj , P k ) where Pi : c Pi s∅ Pi bPi a. In problem (Pi , Pj , P k ) due 0 to strategy-proofness and limiting trades i will be assigned to s∅ , φi (Pi , Pj , P k ) = s∅ . 0 Moreover, in φ(Pi , Pj , P k ) k will be assigned to b and j will be assigned to either c or a. 0 Otherwise, φ cannot satisfy limiting trades. If φj (Pi , Pj , P k ) = c then the available seat 0 0 0 in a is wasted, i.e. a Pj φj (Pi , Pj , P k ) and |φa (Pi , Pj , P k )| < qa . If φj (Pi , Pj , P k ) = a 0 0 then the available seat in c is wasted, i.e. c Pi φj (Pi , Pj , P k ) and |φc (Pi , Pj , P k )| < qc . This contradicts with the fact that φ is non-wasteful.
must make assignment
λ.
submits
is assigned to
Pápai (2000) shows that combination of strategy-proofness and non-bossiness is equivalent to group strategy-proofness.
Hence, Proposition 2 and Proposition 6 im-
ply that there does not exist a group strategy-proof and non-wasteful mechanism that limits trades.
Proposition 7 There does not exist a mechanism which is group strategy-proof, nonwasteful and limits trades.
11
Proof.
Proposition 2, Proposition 3 and Pápai (2000) imply that the set of strategy-
proof, 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 e cient, 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 satis es desired features while not allowing the trade of restricted priorities. Before starting our analysis we de ne the class of trading mechanisms. We say a mechanism
φ
belongs to the class of trading mechanisms if the followings
are true:
P , φ selects cycles4 recursively and each student ix x ∈ {1, ..., n} where in+1 = i1 . Denote the rst cycle selected in
1. For any problem
sx for all c by φ (P ). to
2. If there is no perfect matching for problem
P
is assigned problem
P
then the cycle selected in a problem
P is not aected by the preference pro le of the students who are not in the cycle, i.e. φc (P ) = φc (PI c , P˜−I c ) where I c is the set of students in φc (P ). A cycle (i1 , s1 , i2 , s2 , ..., in , sn ) respects the restricted priorities if ix does not have restricted priority for sx−1 for all x ∈ {1, ..., n}. A trading mechanism φ respects c the restricted priorities if for any problem P the cycle selected by φ, φ (P ), respects the restricted priorities.
4A
cycle (i1 , s1 , i2 , s2 , ..., in , sn ) is a list of students and schools in which student ix points to sx and sx points to ix+1 for all x ∈ {1, ..., n} where in+1 = i1 .
12
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
5
priority at the same school.
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 then she would be by having the lowest (but unrestricted) priority. Proof.
Consider the following problem.
S = {a, b, s∅ }, I = {i, j, k}
and
q = (1, 1, ∞).
a i j k
b j i k
Pi b a
Pj a b
Suppose
i
has restricted priority at
Pk a s∅
There is no perfect matching for this problem.
We are focusing on mutual best,
individually rational and non-wasteful mechanisms. Suppose and satis es all.
a.
ψ
By non-wastefulness all seats will be lled.
is a trading mechanism 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 assigns b. By de nition, the same cycle is selected regardless of the preferences j submits. 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 modi ed to allow this and the same result would hold. 13
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. However, if
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
Since (3) is the only case in which
i
a.
Therefore, (3) is not the cycle selected.
is assigned to
b, ψ
must assign
Now consider the following priority structure in which stricted priority at
i
i
to
a.
has the lowest, but unre-
a. a j k i
b j i k
Pi b a
Pj a b
Pk a s∅
We can interpret this priority pro le as follows: In the rst problem priority but in the second one
i
i
has sibling
does not have sibling priority.
In this second problem any mutually best and individually rational mechanism assigns
i
j
to
a
and
k
to
∅.
And in order not to waste the seat in
becomes better o by losing his sibling priority at school
b, i
will be assigned to
b:
a.
An alternative way of presenting Theorem 1 is to use a similar concept to the 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
˜
(1)
is an improvement in the priorities for agent
˜ sj i s j =⇒ i
for all
i∈I
if:
s ∈ S,
(2) there exists at least one agent
j
and school
14
s0
such that
˜ s0 j, j s0 i
and
˜ s l for all s ∈ S and l, k ∈ I\{i}. k s l ⇐⇒ k ˜ is an improvement A mechanism φ respects improvements in the priorities if ˜ P )Ri φi ( , P ). That is, a mechanism respects in the priorities for agent i ∈ I, then φi ( , (3)
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.
5
We refer to the proof of Theorem 1.
Simulations
Since it is impossible to completely eliminate the trade of the restricted priorities without inducing e ciency 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. Speci cally, 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 de nition of the preferences of the agents over the schools. Let
Ui,s
be the utility of student
i∈I
for school
s ∈ S.
It is de ned as:
Ui,s = β × (α × Z(s) + (1 − α) × Z(i, s)) + (1 − β) × sibling(i) × siblingpriority(i, s) 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 current school. As β decreases being assigned to the elder sibling's school becomes where
6 Erdil
and Ergin (2008) de ne utilities in their simulations similarly. 15
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
and its value
sibling(i)
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 where
n = m × q.
i = 1, 2, ..., n
and schools by
s = 1, 2, ...m
We rst randomly decide which students have sibling priority in
which schools. Next, we determine the priority structure from the sibling pro le and
Z(s), Z(i, s) and sibling(i), and by using these random variables and predetermined α and β we determine
an ordering that is drawn randomly. We generate random variables
the utilities. For each setup we run the TTC mechanism and calculate the number of restricted priority traded. By keeping the sibling pro le 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
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-speci c 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 sacri cing even the most basic e ciency 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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