A Characterization of the Top Trading Cycles Mechanism for the School Choice Problem

Munich Personal RePEc Archive

A Characterization of the Top Trading

Cycles Mechanism for the School Choice

Problem

Dur, Umut

University of Texas at Austin

15 September 2012

Online at

https://mpra.ub.uni-muenchen.de/41366/

for the Shool Choie Problem

∗

Umut Mert Dur

†

University of Texas at Austin

September 14, 2012

Abstrat

This paper haraterizes the top trading yles mehanism for the shool hoie problem.

Shools may have multiple available seatsto be assigned to students. For eah shoola strit

priorityordering of students isdetermined by the shooldistrit. Eah student hasstrit

pref-erene over the shools. We rst dene weaker forms of fairness, onsisteny and resoure

monotoniity. Weshowthatthetop tradingylesmehanismistheuniqueParetoeient and

strategy-proof mehanism that satises the weaker forms of fairness, onsisteny and resoure

monotoniity. To our knowledge this is the rst axiomati approah to the top trading yles

mehanisminthe shoolhoieproblem where shools have aapaitygreater than one.

Key Words: Top Trading CylesMehanism, ShoolChoieProblem

JEL Classiation: C78, D61,D78, I20

1 Introdution

Intheir seminalpaper, Abdulkadiro§luand Sönmez[2003℄ introdue the shoolhoie problem.

Be-fore that paper, in some of the major ities students were assigned to publi shools via deient

mehanismswhihgivehighinentivestothe studentstomisreporttheirtruepreferenesinorderto

get better alloations. To eliminate the gaming,they propose two ompeting strategy-proof

meh-anisms: the Top Trading Cyle (TTC) mehanism and the Deferred Aeptane (DA) mehanism.

∗

IoweveryspeialthankstoThomasWiseman,UtkuÜnver,OnurKestenandThayerMorrillfordetailedomments

andsuggestions.IwouldliketoalsothankTayfunSönmez,MarinPeskiandAy³eKabukçuo§luforusefuldisussions.

†

Address: TheUniversityofTexasatAustin,DepartmentofEonomis ;e-mail: umutdurgmail.om;webpage:

TheTTCmehanismisnotonlystrategy-proofbut alsoParetoeient. However, itfailstobefair .

On the other hand, the DA mehanism satises fairness but fails to be Pareto eient. When the

poliy makers deided to adopt one of the two strategy-proof mehanisms, the DA mehanism was

seleted due toitsbetter features interms of respeting shool distritpriorities. 2

However, in 2012

New OrleansReovery ShoolDistrit beame the rst shool distritto adoptTTC.

Adoption of the TTC by New Orleans shool distrit shows us that some shool distrits may

value eieny over fairness. If Pareto eieny and strategy-proofness are the main objetives of

theshooldistritsthenTTC anbeonsidered oneofthe andidates. However, itisnot the unique

Pareto eient and strategy-proof mehanism. For instane, the serial ditatorship mehanism also

satises these two axioms. 3

In this paper, we try to help the poliy makers who are willing to

adopt a Pareto eient and strategy-proof mehanism by providingthe full haraterization of the

TTCmehanism. Our haraterizationisbasedonParetoeieny, strategy-proofness, mutualbest

along with two axioms that we introdue: resoure monotoniity for top-ranked students and weak

onsisteny. We show that TTC mehanism is the unique mehanism satisfying Pareto eieny,

strategy-proofness,mutualbest,weakonsistenyandresouremonotoniityfortop-rankedstudents.

Mutual best

4

requires that a student be assigned to the shool at the top of his preferene

whenever he has the highest priority at that shool. A mehanism is resoure monotoni for

top-ranked students if the assignment of the top-ranked student for ashoolis not worsened whenthe

number of available seats in that shool inreases. A mehanism issaid tobeweakly onsistent if

theremovalofasetofagentswiththeirassignmentsdoesnotaettheassignmentsoftheremaining

agentsaslongaseahagentisthetop-rankedstudentforoneoftheassignmentoftheremovedagent.

Mutual best, weak onsisteny and resoure monotoniity for top-ranked students are weaker

formsof fairness, onsisteny 5

and resoure monotoniity 6

, respetively. TTC mehanism does not

satisfy fairness, onsisteny and resoure monotoniity. In partiular, there does not exist a

meh-anism that is fair, strategy-proof and onsistent. 7

Moreover Pareto eieny and fairness are

in-1

Fairnessis thenaturalounterpartofthestabilityin theshoolhoieontext[BalinskiandSönmez, 1999℄. An

alloationisfairiftheredoesnotexistastudentwhoprefersanothershooltohisassignmentandthatshooladmitted

astudentwithlowerpriority. 2

Shooldistritsin Boston,NewYorkCityandDenverhaveadoptedversionsof theDA mehanism. 3

Pyia andÜnver[2011b℄providealassofmehanismssatisfyingstrategy-proofnessandParetoeieny inthe

shoolhoieproblem. 4

Morrill[2012℄usesthesameaxiomin theharaterizationofTTC in ashoolhoieproblem whereeahshool

hasonlyoneavailable seat. 5

A mehanismis onsistentifwheneverasetofagentsareremovedwiththeirassignmentsthenalltheremaining

agentswillbeassignedtotheirinitialassignmentwhenwerunthemehanismonlyonsideringtheremainingagents

andremainingopiesoftheobjets. 6

Resouremonotoniityrequiresthatifthenumberofavailableobjetsinreasesthenallagentsshouldbeaeted

inthesamediretion[ChunandThomson,1988℄. 7

AlaldeandBarbera[1994℄showthatDAmehanismistheuniquestrategy-proofandfairmehanismbutitfails

ompatible. Therefore, we annot have a mehanism satisfying all of the axioms. Kesten [2006℄

shows thatTTCsatisesfairness,onsistenyandresouremonotoniityifthepriorityordersatises

strong ayliity ondition. In this paper, we show that TTC is not totally unsuessful in these

threedimensionsand noneof theParetoeient andstrategy-proofmehanismsan performbetter

thanTTC in allthe three dimensions.

Amehanismwhihfailstosatisfymutualbest,resouremonotoniityfortop-rankedstudentsand

onsisteny maynot meetthe demands of both students(families)and shooldistrits. Weonsider

mutual best as a must fairness requirement in the shool hoie ontext. For instane, most shool

distrits give highest priority at a shool to a student whose elder sibling is already attending that

shoolandmostofthefamilieshaveprefereneoverkeepingtheirhildreninthesameshool[Pathak,

2011℄. Therefore, both parents and shool distrits benet from the mutually best mehanisms .

Similarly,resouremonotoniityfortop-rankedstudentsisamustresouremonotoniityrequirement.

We modify this requirement in two ways. When publi goods are alloated, we should not have a

derease in the welfare of any of the agents. Otherwise, providing less and less publi goods will

be a lear solution for the poliy makers. Therefore, we restrit our attention to the mehanisms

underwhihthe welfareofagentsweaklyinreases whenthe numberofavailableobjetsinreases. 10

We also modify the resoure monotoniity axiom by only requiring not to have a redution in the

welfareofthe top-ranked student forthe shool whosenumberof seats has inreased. Therefore any

resouremonotonimehanismunderwhihwelfareoftheagentsweaklyinreasewithaninreasein

thenumberofavailableobjetssatisesresoure monotoniityfortop-ranked students. Consisteny

isadesiredproperty inthe shoolhoie ontextwherethe assignmentproess for dierent types of

shoolsare doneseparately. Forinstane, inNew YorkCitytheassignmentofexamandmainstream

shoolsaredoneseparately[Abdulkadiro§luetal.,2009℄. Therefore, runningaonsistentmehanism

willpreventthe requestofremainingagentsforanotherrun whentheotheragentsare removed with

theirassignments.

Although,mutual best and resoure monotoniity for top-ranked students axioms are enoughto

proveour uniqueness result,the TTC mehanismsatisesstronger forms ofthese two axioms. TTC

respets thepriorityofstudent

i

forshool

s

if thenumberofstudentswith higherpriorityforshool

s

islessthanthenumberofavailableseatsinthatshool. Moreover,if thepoliymakers andfamilies

are only sensitive to priority violation in the upper priority groups then TTC an be onsidered to

haveagoodperformane intermsof respetingpriorities. UnderTTC mehanism,thestudents who

areranked atthe top

q

of thepriorityorderof shool

s

annotbemadeworse odue totheinrease inthe numberof availableseats from

q

to

q

′

.

8

BalinskiandSönmez[1999℄showthattheredoesnotexistfairandParetoeientmehanism. 9

Serialditatorshipmehanismsatisesfourofthem. Itfails tobefair. 10

shoolmayhavemorethanoneavailableseat. Abdulkadiro§luandChe[2010℄andMorrill[2012℄

pro-videalternativeharaterizations of TTCmehanisminthe shool hoieontextwhere eahshool

isrestrited to have only one available seat. Abdulkadirogluand Che show that TTC mehanismis

the only mehanism that is Pareto eient, strategy-proof and reursively respets top priorities. 11

Morill haraterizes the TTC mehanism in two dierent ways. He rst shows that TTC is the

uniquemehanismwhihisstrategy-proof,Paretoeient, andindependentof irrelevantrankings 12

andsatisesmutualbest. HealsodemonstratesthatTTCisthe uniquemehanismsatisfyingPareto

eieny,independene of irrelevantrankings, weakMaskin monotoniity and mutualbest. Results

of these two papers do not hold in the shool hoie problem where shools may have more than

one available seat [Morrill, 2012℄. Sönmez and Ünver [2010℄ provide the haraterization of the you

requestmy house-Iget your turn (YRMH-IGYT)mehanismsinthe housealloationproblems with

existing tenants [Abdulkadiro§lu and Sönmez, 1999℄. They show that YRMH-IGYT mehanism is

the unique mehanism satisfying Pareto eieny, strategy-proofness, individual rationality, weak

neutrality 13

and onsisteny. 14

Pyia and Ünver [2011a℄introdue a lass of mehanism alled

trad-ing yles mehanisms and show that in the house alloation problem a mehanism is individually

rational,Pareto eient,groupstrategy-proof if anonlyif itisatrading yles mehanism. 15

Pyia

and Ünver [2011b℄ also analyze trading yles mehanism in the shool hoie environment where

eah shool may have more than one available seat and show that trading yles mehanisms are

Pareto eient and strategy-proof.

Therest of thepaperisorganized asfollows: InSetion2we introduethe modelandproperties

ofmehanisms. InSetion3wedesribetheTTCmehanism. WepresentourmainresultsinSetion

4. In Setion5 we show the independene of axioms used in our main results. A brief onlusion is

given in the nal setion.

2 Model

Ashoolhoieproblem isa list

[

I, S, q, P,

≻

]

where

•

I

is the set of students,

11

A mehanismrespets topprioritiesif anagentisassignedan objet, thentheagentthat istop-rankedby that

objetshouldnotbeassignedtoaworseobjetthanthat objet. 12

A mehanism is independent of irrelevant rankings if wheneverthe ranking of an agent at an objet's priority

orderdoesnotaettheassignmentofthatagentthenitdoesnotaettheassignmentofalltheotheragents. 13

Ifamehanismsatisesweakneutralitythentheoutomeofthatmehanismwillnotdependonthenamesofthe

unoupiedobjets. 14

Sönmez and Ünver [2010℄ also onsider a weaker version of onsisteny in the house alloation problem with

existingtenants. 15

•

S

is the set of shools,

•

q

= (

q

s

)

s

∈

S

is the quota vetor where

q

s

isthe numberof available seats inshool

s

,

•

P

= (

P

i

)

i

∈

I

is the preferene prole where

P

i

is the strit preferene of student

i

over the

shoolsinludingno-shooloption,

• ≻

= (

≻

s

)

s

∈

S

isthe priority prolewhere

≻

s

isthe priority relationof shool

s

over

I

.

We denote the no-shool option with

s

∅

and

q

s

∅

=

∞

. Let

R

i

be the at-least-as-good-as relation assoiated with the strit preferene order

P

i

and for all

s, s

′

_∈

_S

_∪

_s

∅

sR

i

s

′

if and only if

s

=

s

′

or

sP

i

s

′

. Weassume that there are no ties inthe priority proles of shools. 16

A mathing is a funtion

µ

:

I

→

S

∪

s

∅

suh that

µ

(

i

) =

s

and

µ

(

i

) =

s

′

if only if

s

=

s

′

_.

If

µ

(

i

) =

s

∅

thenstudent

i

isunassigned. Inamathing

µ

,the numberof studentsassignedtoashool

s

annot exeed the total number of available seats in shool

s

. Let

M

be the set of all possible

mathings.

A mehanism is a proedure whih selets a mathing for eah problem. That is, a mehanism

ϕ

takes the preferene prole of the students, the priority order of students for shools, the quota

vetor,thenseletsamathingforeveryproblem. Themathingseletedbymehanism

ϕ

inproblem

[

I, S, q, P,

≻

]

is denoted by

ϕ

[

I, S, q, P,

≻

]

. Let

ϕ

[

I, S, q, P,

≻

] (

i

)

denote the assignment of student

i

∈

I

by mehanism

ϕ

for problem

[

I, S, q, P,

≻

]

.

Student

i

stritlyprefers mathing

µ

tomathing

µ

′

if hestritlyprefers

µ

(

i

)

to

µ

′

₍

_i

₎

_{, µ}

₍

_i

₎

_P

i

µ

′

(

i

)

.

A mathing

µ

is Pareto eient if there does not exist a mathing

µ

′

_{∈ M}

in whih eah student

is not worse o and at least one student is stritly better o. More formally, mathing

µ

is Pareto eientif there doesnot exist a mathing

µ

′

_{∈ M}

where

µ

′

₍

_i

₎

_R

i

µ

(

i

)

for eah

i

∈

I

and

µ

′

₍

_j

₎

_P

j

µ

(

j

)

atleastforone

j

∈

I.

Amehanism

ϕ

isParetoeient ifforallproblemsitseletsaParetoeient mathing.

A mehanism

ϕ

is strategy-proof if it is (weakly) dominantstrategy for allstudents to tell their preferenes truthfully. Formally, a mehanism

ϕ

is strategy-proof if for every preferene prole

P

and

P

′

i

ϕ

[

I, S, q, P,

≻

] (

i

)

R

i

ϕ

[

I, S, q,

(

P

i

′

, P

−

i

)

,

≻

] (

i

)

for all student

i

∈

I.

Here,

P

−

i

represents the true preferene prole of studentsexept

i.

Let

t

≻

i

be the set of shools ranking

i

over all other students under priority prole

≻

. Formally,

t

≻

_i

=

{

s

∈

S

|

i

≻

s

j

∀

j

∈

I

\

i

}

. A mehanism

φ

is mutually best if whenever there exists

s

∈

t

≻

i

suh that

sP

i

s

′

forall

s

′

_∈

_S

_{\ {}

_s

_}

then

φ

[

I, S, q, P,

≻

] (

i

) =

s

for all

i

∈

I

. 17

A mehanism

φ

is resoure monotoni if for all

s

∈

S

, all

q

′

s

≤

q

s

either for all

i

∈

I

,

φ

[

I, S, q, P,

≻

] (

i

)

R

i

φ

[

I, S,

(

q

s

′

, q

−

s

)

, P,

≻

] (

i

)

orforall

i

∈

I φ

[

I, S,

(

q

′

s

, q

−

s

)

, P,

≻

] (

i

)

R

i

φ

[

I, S, q, P,

≻

] (

i

)

. 18

16

Shooldistritsmostlyuserandomtiebreakingrules. 17

Morrill[2012℄denesmutualbestsimilarly. 18

I use a dierent version of resoure monotoniity. Intuitively, if student

i

has the highest priority for shool

s

then his welfare should not be worsened when the number of seats in shool

s

in-reases. I formally dene resoure monotoniity for top-ranked students as follows: A mehanism

φ

is resoure monotoni for top-ranked students if for all

i

∈

I

and all

q

′

s

≥

q

s

>

0

where

s

∈

t

≻

i

φ

[

I, S,

(

q

_s

′

, q

−

s

)

, P,

≻

] (

i

)

R

i

φ

[

I, S, q, P,

≻

] (

i

)

.

Before introduingour onsisteny axiomwe need additionalnotation.

Forany shool

s

∈

S

, priority order

≻

s

, and a set of students

J

⊂

I

,let

≻

J

s

bethe restrition of priority order

≻

s

to students in

J

. Let

≻

J

_{= (}

_≻

J

s

)

s

∈

S

and

≻

−

J

_{= (}

_≻

I

\

J

s

)

s

∈

S

.

Given a problem

[

I, S, q, P,

≻

]

, a set of students

J

⊂

I

, and a quota prole

q

˜

≤

q

we say

J, S,

q, P

˜

−

J

,

≻

J

istherestrition oftheproblem

[

I, S, q, P,

≻

]

tostudentsin

J

andquotaprole

q

˜

. 19

Amehanismisonsistent ifwhenever aset ofstudents are removed withtheir assignmentsthen

all the remaining students willbe assigned to their initial assignment when we run the mehanism

onlyonsidering the remaining students and objets. 20

Formally,a mehanism

φ

is onsistent if for any problem

[

I, S, q, P,

≻

]

, when we remove a set of students

J

⊂

I

together with their assignments

φ

[

I, S, q, P,

≻

](

J

)

,then for any

i

∈

I

\

J

φ

[

I

\

J, S,

q, P

˜

−

J

,

≻

−

J

](

i

) =

φ

[

I, J, q, P,

≻

](

i

)

where

q

˜

s

is the numberof availableseats remainingin shool

s

.

In this paper, we introdue a weaker version of the onsisteny axiom. 21

A mehanism satises

weakonsistenyifwheneverweremoveasetofstudentswiththeirassignmentsuhthatthestudent

withthehighestpriorityforoneoftheremovedstudent'sassignmentisalsoanotherremoved student

thenthe assignments of the remainingstudents do not hange.

A mehanism

φ

is weakly onsistent if for any problem

[

I, S, q, P,

≻

]

, when we remove a set of students

J

⊂

I

togetherwiththeirassignments

φ

[

I, S, q, P,

≻

](

J

)

satisfying

|

t

≻

j

∩

φ

[

I, S, q, P,

≻

](

J

)

|

=

1

for eah

j

∈

J

,then forany

i

∈

I

\

J

φ

[

I

\

J, S,

q, P

˜

−

J

,

≻

−

J

](

i

) =

φ

[

I, J, q, P,

≻

](

i

)

.

Our restrition on the set of students and seats removed is simple. It is easy to see that any

mehanismwhihisonsistentbased onthe traditionaldenition satisestheweakerformof itthat

wedene here.

19

SimilarnotationisusedinSönmezandÜnver[2010℄. 20

SeeThomson[1990℄andErgin[2000℄forrelatedresults. 21

Sönmez andÜnver[2010℄alsomodiesthedenition oftheonsisteny axiom. Inthat paper,theyharaterize

YRMH-IGYTinthehousealloationproblemwithexistingtenants. YRMH-IGYTalsofailstosatisfytheonsisteny

Intheshoolhoieontext,theTTCmehanismwasrstintroduedbyAbdulkadiro§luandSönmez

[2003℄. It was based on the Gale's top trading yles algorithm [Shapley and Sarf, 1974℄. It is a

diretmehanism and forany given problem

[

I, S, q, P,

≻

]

it worksiteratively ina number of steps:

Top Trading Cyles Mehanism (TTC):

Step 1: Assign a ounter to eah shool and set it to the quota of eah shool. Eah student

pointstohismostpreferredshool. Eahshoolpointstothetop-ranked studentinitspriorityorder.

Shool

s

∅

points to all students pointing to it. Due to the niteness there is at least one yle. 22

Assign every student in the yles to the shool he points to and remove him. The ounter of eah

shoolin ayle isredued by one and if it redues tozero, the shool is alsoremoved.

In general,

Step k: Eah student points to his most preferred shool among the remaining ones. Eah

remainingshool points to the student with the highest priority among the remainingones. Shool

s

∅

pointstoallstudentspointingtoit. Thereisatleastone yle. Assigneverystudentintheyles

tothe shoolhe points to and remove him. The ounter of eah shool ina yle isredued by one

and if itredues tozero, the shoolis alsoremoved.

The algorithmterminates when allstudents are assigned.

We illustratethe dynamis of TTC mehanismin the followingexample.

Example 1 Let

S

=

{

s

1

, s

2

, s

3

, s

4

}

,

I

=

{

i

1

, i

2

, i

3

, i

4

, i

5

}

and

q

= (1

,

1

,

1

,

2)

. The preferenes of

students and priorities are as follows:

i

1

:

s

1

P

_i

1

s

2

P

i

1

s

3

P

i

1

s

4

s

1

:

i

5

≻

s

1

i

3

≻

s

1

i

4

≻

s

1

i

2

≻

s

1

i

1

i

2

:

s

2

P

_i

2

s

1

P

i

2

s

4

P

i

2

s

3

s

2

:

i

3

≻

s

2

i

1

≻

s

2

i

4

≻

s

2

i

2

≻

s

2

i

1

i

3

:

s

1

P

_i

3

s

3

P

i

3

s

4

P

i

3

s

2

s

3

:

i

3

≻

s

3

i

2

≻

s

3

i

4

≻

s

3

i

1

≻

s

3

i

5

i

4

:

s

3

P

_i

4

s

4

P

i

4

s

1

P

i

4

s

2

s

4

:

i

1

≻

s

4

i

3

≻

s

4

i

2

≻

s

4

i

5

≻

s

4

i

4

i

5

:

s

4

P

_i

5

s

1

P

i

5

s

2

P

i

5

s

3

Step 1: Eah students points to his most preferred shool and eah shools points to the student

withthe highest priority. There is only one yle:

(

s

1

, i

5

, s

4

, i

1

)

. We assign eah student in the yle tothe shoolhe points to and remove him:

µ

(

i

1

) =

s

1

and

µ

(

i

5

) =

s

4

. We also redue the ounterof

eah shool in the yle and removeonly

s

1

sine its ounter redues to zero.

Step2: Eahremainingstudentspoints tohismostpreferredremainingshoolandeahremaining

shools points to the student with the highest priority among the remaining ones. There is only

one yle:

(

s

3

, i

3

)

. We assign the student in the yle to the shool he points to and remove him:

22

µ

(

i

3

) =

s

1

. We alsoredue the ounter of the shoolin the yle and remove it,

s

3

, sine its ounter

redues to zero.

Step3: Eahremainingstudentspoints tohismostpreferredremainingshoolandeahremaining

shools points to the student with the highest priority among the remaining ones. There is only one

yle:

(

s

2

, i

4

, s

4

, i

2

)

. We assigneah students in the yle to theshoolhe points to and remove him:

µ

(

i

2

) =

s

2

and

µ

(

i

4

) =

s

4

. We also redue the ounter of eah shool in the yle and remove only

both of them sine their ounter redue to zero.

The mehanism terminates sine allstudents are assigned.

4 Results

In the following theorem, we show that TTC is Pareto eient, strategy-proof, weakly onsistent,

resoure monotoni for top-ranked students and mutually best. Moreover, there does not exist

anothermehanism satisfyingall these axioms. We prove it inthe Appendix.

Theorem 1 In shool hoie problem TTC is the unique mehanism satisfying

•

Pareto eieny

•

Strategy-proofness

•

Weak onsisteny

•

Resoure monotoniity for top-ranked students

•

Mutual best.

In the next setion, we show that there always exist another mehanism satisfying only four of

the ve axioms.

Mutualbestanbeonsideredasaveryweakfairnessrequirementandsatisfyingitmaynotmake

a mehanism more desirable. In the following proposition, we show that TTC mehanism satises

muhstronger fairness requirement.

Proposition 1 Under TTC mehanism, eah student weakly prefers his assignment to eah shool

s

for whih he isranked at the top

q

s

portion of thatshool's priority order.

Proof. Suppose not. Let student

i

's rankfor shool

s

be

r < q

s

and he be assigned to shool

s

′

suhthat

sP

i

s

′

s

will keep pointing

i

until he is removed. Therefore,

i

will be assigned to

s

whenever he points to

that shool. Now onsider the ase that

i

is assigned before

s

points to him. In this ase,

i

should beassigned toa better shool and he never pointsto

s

.

We an also show that TTC mehanism satises a general form of resoure monotoniity for

top-ranked student.

Proposition 2 When the number of available seats in shool

s

is inreased from

q

s

to

q

˜

s

, keeping everything else the same, then TTC mehanism assigns top

q

s

students in shool

s

's priority order to weakly better shools.

Proof. We refertothe proofof Theorem1. Thepart thatwe prove TTCmehanismisresoure

monotonifortop-ranked studentsan beextended for top

q

s

students. Itfollows fromthefat that therst

q

≤

q

s

seats of shool

s

annotbelled beforetop

q

studentsinshool

s

'spriorityorderare removed.

So far, we show that TTC mehanism outperforms other strategy-proof and Pareto eient

mehanisms. Someshooldistritsonsiderfairnessasthemostimportantonernandthesedistrits

selet DA mehanism instead of the TTC mehanism. In the rest of this setion, we fous on the

fairness and the performane of the TTC interms of respeting priorities.

Inthemostoftheshooldistrits,prioritystrutureisdeterminedbasedonsomeexogenousrules.

Forinstane,Bostonshooldistritgivesthehighestpriorityforashooltothestudentslivinginthe

samewalkzoneandhavingasiblingattendingthatshool. 23

Theseondpriorityisgiventostudents

havingasiblingattending that shoolbut livingoutsidethe walk zoneof that shool. Students who

are only livingin the same walk zone have the third priority and the fourth priority is given tothe

remainingstudents. Ties between students in the same priority group is broken by randomlottery.

That is, the priority struture,

≻

, in any problem is determined based on the priority groups and

randomdraw. Publipoliymakersand familiesmightgivemoreimportanerespetingprioritiesin

the upper priority groups [Abdulkadiro§lu,2011℄. InProposition 3,we show that TTCis suessful

atrespetingprioritiesintheupperprioritygroupsundersomerealistionditions. Beforepresenting

our resultsweneed some notation.

Suppose thereare

n

prioritygroupsandrespetingprioritiesinthe rst

n

∗

prioritygroupismore

important. Let

G

i

:

S

→ N

be a funtion and

G

i

(

s

)

be the priority group that student

i

belongs tofor shool

s

. We say student

i

's preferene

P

i

is perfetly orrelated with the priority groups if the following ondition holds: if

G

i

(

s

)

< n

∗

and

G

i

(

s

)

< G

i

(

s

′

₎

then

sP

i

s

′

. A preferene prole

23

P

= (

P

i

)

i

∈

I

is perfetly orrelated with the priority groups if eah student's preferene is perfetly

orrelated with the priority groups. As an example, suppose the rst priority group (sibling-walk

zone)inBostonisgivenmoreimportanethantheothers. Thenthepreferene proleofthestudents

is perfetly orrelated with the priority groups if eah student having sibling-walk zone priority in

some shool ranks one of the shools for whih he has sibling-walk zone priority at the top of his

preferene list.

Now we are ready to present our result on the performane of the TTC mehanism in terms of

respeting priorities.

Proposition 3 Let

π

be the outome of TTC mehanism in problem

[

I, S, q, P,

≻

]

. There does not

exist a student and shool pair

(

i, s

)

suh that

G

i

(

s

)

< n

∗

,

sP

i

π

(

i

)

, there exists another student

j

assignedto

s

and

i

≻

s

j

if any one of the followingonditions holds:

(a) Thetotalnumber of students in therst

n

∗

prioritylass ofeahshool

s

isless thanor equal to

q

s

.

(b)Preferene prole

P

isperfetly orrelated with the priority groups.

5 Independene of Axioms

Belowwe show the independene of axioms mentioned in Theorem 1.

•

Strategy-proof, weakly onsistent, resoure monotoni for top-ranked students, and mutually

best, but not Pareto eient: Consider the following problem. Two shools

S

=

{

a, b

}

with

one availableseat and two students

I

=

{

1

,

2

}

. Letthe preferene prole

P

and priority order

≻

be

P

1

P

2

b

a

a

b

s

∅

s

∅

≻

a

≻

b

1

2

2

1

Let mehanism

ψ

assign

2

to

b

and

1

to

a.

Let

ψ

selet the same assignment in the above problem independent of preferenes. For all other problems,

ψ

selets the same mathing as TTC mehanism. Mehanism

ψ

fails tobe Pareto eient and satises other 4 properties.

•

Strategy-proof, weakly onsistent, resoure monotoni for top-ranked students, and Pareto

e-ient, but not mutually best: Serial ditatorship mehanism isstrategy-proof, (weakly)

onsis-tent,and Pareto-eient. Moreover, whenthe numberof available seatsin ashoolisinrease

all students' welfare weakly improve. That is, it satises more generalized version of the

•

Strategy-proof,weaklyonsistent,Paretoeient, andmutualbestmehanism, butnotresoure

monotoni for top-ranked students: Consider the following problem: Two shools

S

=

{

a, b

}

with one available seat and three students

I

=

{

1

,

2

,

3

}

. Let the preferene prole

P

and priority order

≻

be

P

1

P

2

P

3

b

b

a

a

a

b

s

∅

s

∅

s

∅

≻

a

≻

b

1

3

2

1

3

2

Let mehanism

ψ

assign

3

to

a

and

1

to

b

in this problem. If the number of available seats in shool

a

is inreased to 2 then

ψ

assigns

1

and

3

to

a

and

2

to

b

. Let

ψ

selet the same assignment in the above problemwhere

a

has two available seats and

1

ranks

a

above

s

∅

and assign

1

to

s

∅

if he ranks

a

below

s

∅

. For all other problems

ψ

selets the same mathing as TTC mehanism. Mehanism

ψ

fails to be resoure monotoni for top-ranked students and satises other 4 properties.

•

Strategy-proof, Pareto eient, mutually best mehanism, resoure monotoni for top-ranked

students but notweaklyonsistent: Consider thefollowingproblem. Threeshools

S

=

{

a, b, c

}

with one available seat and three students

I

=

{

1

,

2

,

3

}

. Let the preferene prole

P

and priority order

≻

be

P

1

P

2

P

3

c

a

a

a

b

b

b

c

c

≻

a

≻

b

≻

c

1

1

1

2

2

2

3

3

3

Let mehanism

ψ

assign

1

to

c

and

2

to

b

and

3

to

a

in this problem. Let

ψ

selet the same mathing as long as

1

and

3

submit the same preferenes and

2

ranks

b

over

s

∅

. If we remove

1

with his assignmentthen

2

is assignedto

a

and

3

is assignedto

b

. Forall other problems

ψ

seletsthesamemathingasTTCmehanism. Mehanism

ψ

failstobeonsistentandsatises other 4 properties.

•

Pareto eient, mutually best mehanism, resoure monotoni for top priority students and

onsistentbutnotstrategy-proof: TheBostonmehanismisParetoeient,resouremonotoni

and onsistent[Kojimaand Ünver, 2010℄. Moreover, inthe rst step of theBoston mehanism

when astudent appliestohismost popularshoolfor whih he hasthe highestpriorityhe will

be assignedto that shool. Therefore it satises mutual best. The Boston mehanism fails to

TTC mehanism has been studied extensively in the market design literature. It and its variants

havebeenproposedasoneofthebestalternativesinmanymathingmarketsinludingpublishool

hoie systems, on-ampus housing and the kidney exhange programs. However, TTC mehanism

hasneverbeen haraterizedfortheases whereobjetshaveaapaity greaterthanone, i.e. shool

hoie problem. In this paper, we provide the rst haraterization of the TTC mehanism in the

shool hoie problem. Our haraterization willhelp the shool distrits hoose between

strategy-proof and Pareto eient mehanisms. In partiular, TTC mehanism is the unique strategy-proof

and Pareto eient mehanism satisfyingmutual best, weak onsisteny and resoure monotoniity

fortop-ranked students.

We also fous on the performane of the TTC mehanism in terms of respeting priorities. We

show that TTC mehanism respets priorities in the upper priority lasses. If the poliy makers

and families are only sensitive for the priority violations in the upper priority lasses then TTC

mehanism willmeet their needs.

Appendix

Proof of Theorem 1.

Werst showthattheTTC mehanismsatisesallofthe axiomsinthetheorem. Then,weshow

thatitistheuniquemehanismsatisfyingallof theaxioms. Paretoeieny and strategy-proofness

of TTCfollows fromAbdulkadiro§luand Sönmez[2003℄.

Mutual Best: Suppose TTC does not satisfy mutual best. Then, there exists a student shool

pair,

(

i, s

)

,suh thatstudent

i

has the highestpriority forshool

s

and prefers shool

s

toany other shool and

i

is not assigned to

s

by TTC. In the rst step of the TTC,

s

will point to

i

and

i

will pointto

s

. They willformayle and

i

willbeassignedto

s

. Therefore, TTCsatisesmutualbest. Resoure Monotoniity: To show that TTC is resoure monotoni for top-ranked students

take a student shool pair

(

i, s

)

suh that

s

∈

t

≻

i

and

q

s

>

0

. Denote the assignment of TTC in problem

[

I, S, q, P,

≻

]

with

µ

. Now onsider the problem

[

I, S,

(˜

q

s

, q

−

s

)

, P,

≻

]

where

q

˜

s

> q

s

. We onsidera variantofthe TTC mehanism inwhih onlyoneyle is removed ineahstep.

24

Fixthe

yleseletionrule. Inpartiular,let

Cy

(

k

)

bethe ylethat isseleted inthe

k

th

step ofthevariant

of the TTC mehanism when we onsider the problem

[

I, S, q, P,

≻

]

. Let

s

be removed in step

k

of TTCwhen we onsider problem

[

I, S, q, P,

≻

]

. Wewill alsoselet

Cy

(˜

k

)

in step

˜

k < k

if we observe thatyle when we run the variant of TTCfor the problem

[

I, S,

(˜

q

s

, q

−

s

)

, P,

≻

]

.

24

Shool

s

annot be removed before student

i

is assigned to a shool in problem

[

I, S, q, P,

≻

]

. Therefore,

i

is assigned in step

k

′

_≤

_k

in the problem

[

I, S, q, P,

≻

]

. To see this reall that in the TTCmehanism,

s

willpointto

i

until

i

isremoved. Therefore,noneoftheseatsof

s

willbeassigned toany student before

i

is removed. Also note that allthe yles seleted in step

k

′′

_{< k}

′

inproblem

[

I, S, q, P,

≻

]

will be observed in step

k

′′

of TTC when we onsider the problem

[

I, S,

(˜

q

s

, q

−

s

)

, P,

≻

]

beause none of them inludes a student pointing to

s

and an inrease in the number of available seats in

s

willnot aet their assignments. As aresult the set of remainingshoolsin step

k

′

of the

TTCmehanismin both problemwillbe the sameand we willobserve the yle inluding

i

inboth problems.

Weak Consisteny: We again onsider the variant of the TTC that is dened above. Let

J

be the set of students and let

µ

(

J

)

be their assignments. Due to the requirement in the denition of the weak onsisteny we only hek the ase in whih eah student in

J

has the highest priority for one of the shools in

µ

(

J

)

. Suppose none of the students in

J

belongs to a

Cy

(

k

)

where

k <

k

˜

. Then, it is lear that the assignment of students in

Cy

(

k

)

where

k <

k

˜

willnot be aeted by the removal of students in

J

with their assignments. Suppose

i

∈

Cy

(˜

k

)

. Let

µ

(

i

)

be his assignment. Therefore,

i

1

who isthe top-ranked student inthe priority orderof

µ

(

i

)

should bein

J

. This is also true for the top-ranked student of the shool that

i

1

is assigned. Due to the niteness we should have a yle. That is,

Cy

(˜

k

)

⊆

J

and

µ

(

Cy

(˜

k

))

⊆

µ

(

J

)

. Therefore, removing these students before runningtheTTCmehanismorremovingthemwithinthe mehanismwillnotaettheassignments

ofthe remainingstudents.

Uniqueness: Suppose there exists another mehanism

φ

satisfying all these 5 properties and there exists a problem

[

I, S, q, P,

≻

]

in whih

φ

and TTC selet dierent mathings. We will onsider the version of TTC mehanism in whih only one yle is removed in a step and if there

are more than 1 yle the one whih willbe removed is seleted based on some exogenous rule, i.e.

the yle with the shool having the lowest index . Then suppose that eahstudent removed before

step

k

≥

1

of the TTC mehanism is assigned to the same shool under

φ

and TTC. Denote these students with set

J

. Let

i

be the student who is removed in the step

k

of TTC and assigned to a dierentshoolby

φ

. Ifwe removestudents assignedintherst step ofTTCwith theirassignments thenassignmentsof the remainingstudents inthe outome of both mehanismswillnot hange due

to the weak onsisteny. We an ontinue removing all students in

J

with their assignments and still remaining students will be assigned to the same shools.

25

Denote the redued problem with

h

˜

I,

S,

˜

q,

˜

P ,

˜

≻

˜

i

. Here,

I

˜

=

I

\

J

,

S

˜

=

S

,

q

˜

s

=

−

P

i

∈

J

1(

φ

[

q, P,

≻

](

i

) =

s

) +

q

s

,

P

˜

=

P

˜

I

and

≻

˜

=

≻

˜

I

. In

this redued problem student

i

will be removed in the rst step of the TTC mehanism. Let

s

be the shool pointing student

i

in the rst step of TTC mehanism in the redued problem. By the

25

denition of the TTC mehanism student

i

should be the top-ranked student in

≻

˜

s

. We onsider two ases. In the rst ase student

i

is assignedto

s

and in the seond ase

i

isassigned toanother shoolby TTC.

Case 1: Student

i

pointsto the shool

s

in the rst step of TTC. Shool

s

should be the most preferred shool in

P

i

among the ones having available seats. Suppose

i

reports

P

′

i

:

sP

i

′

s

∅

. Due to the strategy-proofness TTC will assign

i

to

s

and

φ

will assign to

s

∅

. Any mutual best mehanism shouldassign

i

to

s

inthe redued problem. Therefore,

φ

fails tosatisfy mutual best.

Case 2: In this ase

i

is assigned to

s

′

₆

₌

_s

and there is anotherstudent

j

assigned to shool

s

. Nowsuppose student

i

reports

s

′

_P

′

i

sP

i

′

s

∅

. TTC will seletthe same mathing. Due to the strategy-proofness

φ

will assign

i

to either

s

where he is top-ranked or

s

∅

.

26

First onsider the latter ase

inwhih

i

is assigned to

s

∅

by

φ

when he submits

s

′

_P

′

i

sP

i

′

s

∅

. Now onsider the ase that

i

submits

sP

_i

′′

s

∅

and keeping everything same. Due to the strategy-proofness he will be assigned to

s

∅

by

φ

.

However this will violate mutual best. Therefore the latter ase is not possible. Therefore, when

i

submits

P

′

i

he will be assigned to

s

by

φ

. Now onsider the ase where

i

submits

P

′

i

and

q

˜

s

= 1

. Sine

φ

is resoure monotoni for top-ranked students,

i

annot be assigned to his top hoie

s

′

by

φ

. Then he willbe assigned to

s

or

s

∅

. Due tothe aforementioned reasonshe willbeassigned to

s

.

Therefore student

j

who is assignedto

s

by TTC willbe assigned toan other shool by

φ

. Given

s

isthe top hoie of

j

amongthe shoolswith availableseats

j

prefers hisassignment underTTC to

φ

.

Notethatsinestudent

j

isassignedtoashoolbyTTCinthe rststep thereshouldbeanother shool

s

′′

where

j

is the top-ranked student. Ifwe repeat the same steps for student

j

then we will show that when

q

˜

s

′′

= 1

and

j

submits

sP

′

j

s

′′

P

j

′

s

∅

he will be assigned to

s

′′

by

φ

. We an keep ontinue and show that

φ

will assign all the students who are assigned in the rst step of TTC to one of the shools pointing to them in the rst step of TTC in the redued problem. Therefore

they will be assigned to stritly worse shool by

φ

and no other student will be assigned to those shools sine all shools quota will be equalized to 1 when we keep repeating. Therefore a trade

between these students willinrease the welfarewithout worseningany other student and

φ

failsto bePareto-eient.

Proof of Proposition 3. Part (a) of the proposition is a diret result of Proposition 2. We

prove Part (b) by using the denition of the TTC mehanism. In partiular, we use the variant of

TTCmehanisminwhihonlyone yle isremoved ineahstep (see Proofof Theorem1). Consider

the students who are ranked atthe top of the priority orderof shools. Then among these students

26

Here, it is possible that

i

an be also assigned to another shoolthat he doesn't inlude to his preferene list.

However,weanprovethatthiswill violateeitherstrategy-proofnessofmutual best asasimilarwaythatwefollow

nd the students who are pointed by shools that they belong to the

k

th

priority group and there

does not exist a student pointed by a shool that he belongs to the

l

th

priority group where

l < k

. If

k > n

∗

then we are done. If

k

≤

n

∗

then among these students, selet the one who is favored

in the random draw and denote him by

i

1

. We laim that in this step,

i

1

is pointed by his most popularshool. Suppose not. Thenhe ispointedby anothershool

s

and hismost popularshool

s

′

is pointing another student

i

′

. Given

s

′

_P

i

1

s

then

G

i

1

(

s

)

≥

G

i

1

(

s

′

₎

. Moreover,

G

i

1

(

s

′

_{) =}

_{k > G}

i

′

(

s

′

)

sine

i

1

is the most favored student in the random draw and

i

′

_≻

s

′

i

1

. This ontradits with the

fatthat theredoes not existastudent pointed by ashoolthat he belongs tothe

l

th

priority group

where

l < k

. Then student

i

1

's priority is not violated in any shool beause he is assigned to his most popularshool.

WeshowinTheorem1thatTTCmehanismsatisesweakonsisteny. Thatis,whenweremove

i

1

withhisassignmenttheremainingstudentswillbeassignedtothesameshoolbyTTCmehanism

in the updated problem. Therefore, we an onsider the redued problem as a new problem and

repeat the steps above and showthat there does not exist astudent and shoolpair

(

i, s

)

suh that

G

i

(

s

)

< n

∗

,

sP

i

π

(

i

)

, there exists another student

j

assignedto

s

and

i

≻

s

j

.

Referenes

AtilaAbdulkadiro§lu. Generalized mathing for shoolhoie. Working paper, 2011.

Atila Abdulkadiro§lu and Yeon-Koo Che. The role of priorities in assigning indivisible objets: A

haraterization of top trading yles. Working paper, 2010.

Atila Abdulkadiro§lu and Tayfun Sönmez. House alloation with existing tenants. Journal of

Eo-nomi Theory,88:233260, 1999.

AtilaAbdulkadiro§luand Tayfun Sönmez. Shoolhoie: A mehanismdesignapproah. Amerian

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