Vol.:(0123456789)
Social Choice and Wel a e (2023) 61:587–624
h ps://doi.o g/10.1007/s00355-023-01458-2
1 3
ORIGINAL PAPER
Cons ained school choice: anexpe imen al QRE analysis
Jo geAlcalde‑Unzu1 · FlipKlijn2 · Ma cVo sa z3
Recei ed: 28 Sep embe 2021 / Accep ed: 23 Ma ch 2023 / Published online: 18 May 2023
© The Au ho (s) 2023
Abs ac
The heo e ical li e a u e on public school choice p oposes cen alized mechanisms ha
assign child en o schools on he basis o pa en s’ p e e ences and he p io i ies child en
ha e o di e en schools. The ela ed expe imen al li e a u e analyzes in de ail how
a ious mechanisms a e in e ms o wel a e and s abili y o he esul ing ma chings,
ye o en p o ides only agg ega e s a is ics o he indi idual beha io ha leads o hese
ou comes (i.e., he deg ee o which subjec s ell he u h in he induced simul aneous
mo e game). In his pape , we show ha he quan al esponse equilib ium (QRE) ade-
qua ely desc ibes indi idual beha io and he esul ing ma ching in h ee cons ained
p oblems o which he immedia e accep ance mechanism and he s uden -op imal s a-
ble mechanism coincide. Speci ically, he compa a i e s a ics o he logi -QRE wi h
isk-neu al and expec ed-payo -maximizing agen s cap u e he di ec ional changes
o subjec beha io and he p e alence o he di e en s able ma chings when ca dinal
payo s (i.e., ela i e p e e ence in ensi ies) a e modi ied in he expe imen .
1 In oduc ion
1.1 Mo i a ion
In many public school choice p og ams, cen alized mechanisms such as he s u-
den -op imal s able mechanism and he immedia e accep ance mechanism a e used
We hank an associa e edi o and wo anonymous e iewe s o hei aluable commen s and
sugges ions which imp o ed he pape . The au ho s g a e ully acknowledge inancial suppo
om Fundación Ramón A eces. J. Alcalde-Unzu g a e ully acknowledges inancial suppo om
Minis e io de Ciencia, Inno ación y Uni e sidades (PGC2018-093542-B-I00 and PID2021-
127119NB-I00). F. Klijn g a e ully acknowledges inancial suppo om AGAUR–Gene ali a
de Ca alunya (2017-SGR-1359 and 2021-SGR-00416) and he Spanish Agencia Es a al de
In es igación (AEI) h ough g an s ECO2017-88130-P and PID2020-114251GB-I00 and he Se e o
Ochoa P og amme o Cen es o Excellence in R&D (Ba celona School o Economics CEX2019-
000915-S). M. Vo sa z g a e ully acknowledges inancial suppo om Minis e io de Ciencia,
Inno ación y Uni e sidades (PGC2018-096977-B-I00 and PID2021-122919NB-I00).
* Flip Klijn
lip.kli[email p o ec ed]
Ex ended au ho in o ma ion a ailable on he las page o he a icle
588
J.Alcalde-Unzu e al.
1 3
o assign child en o schools on he basis o pa en s’ p e e ences and he p io i ies o
child en o di e en schools (based on, e.g., walking dis ance, siblings, e c.).1 In a
cons ained se ing whe e pa en s can only ank a limi ed numbe o schools, which
happens in many eal-li e applica ions,2 he e may be incen i es o beha e s a egi-
cally. Speci ically, a e y common si ua ion is ha he pa en s p e e hei child o
go o school x ins ead o school y bu he child has highe p io i y a school y han a
school x. The pa en s hen ace an impo an decision. Will hey epo hei p e e -
ences on he pai (x,y) u h ully o will hey mis ep esen hei p e e ences by ank-
ing school y abo e school x? Ou pape con ibu es o he expe imen al li e a u e
ha analyzes he s uc u e o manipula ions in school choice p oblems by s udying
how ca dinal payo s a ec subjec beha io .
Fo he cons ained se ing, labo a o y expe imen s ha e shown ha subjec s may
ail o coo dina e on a Nash equilib ium.3 In ac , o he bes o ou knowledge, he e
a e no s udies ha gi e clea -cu desc ip ions o subjec s’ beha io and esul ing
ma chings. We conside h ee cons ained p oblems o which he immedia e accep -
ance mechanism and he s uden -op imal s able mechanism coincide and show ha
quan al esponse heo y in oduced by McKel ey and Pal ey (1995) is capable o
desc ibing he main beha io al pa e ns.4 F om a logi -QRE pe spec i e, i he pay-
o o school x is much highe han he payo o school y, he e a e incen i es o
play a s a egy in which school x is anked abo e school y. On he o he hand, i he
payo s o he wo schools a e close o each o he , s a egies ha ank school y abo e
school x ha e he ad an age ha no much payo is o egone and ha he isk o ge -
ing an e en wo se ou come han school y diminishes. We nex de ail he expe imen-
al design and ou main indings o cla i y hese poin s u he .
1.2 Expe imen andcon ibu ion
We conside a se ing wi h h ee s uden s and wo schools (
s1
and
s2
) such ha each
s uden can only apply o one school. The p e e ences o he s uden s and he p io i-
ies s uden s ha e a he schools emain he same h oughou he expe imen . Two
pa ame e s a y in ou s udy. Fi s , he numbe o sea s schools o e changes each
six ounds o an expe imen al session. School
s1
o e s one and school
s2
o e s wo
sea s in he i s six ounds o a session (we e e o his si ua ion as p oblem
P1
),
bo h schools o e one sea in he nex six ounds (p oblem
P2
), and in he inal six
1 The s uden -op imal mechanism is he mechanism based on he de e ed accep ance algo i hm (Gale
and Shapley 1962). The immedia e accep ance mechanism is also known as he “Bos on mechanism.”
Fo u he de ails we e e o Abdulkadi oğlu and Sönmez (2003) who ini ia ed he ma ke design li e a-
u e on school choice.
2 See Du (2019), Du and Mo ill (2020), and Kojima and Ün e (2014) o de ails.
3 See he subsec ion “Rela ed li e a u e” a he end o he In oduc ion o e e ences.
4 The quan al esponse equilib ium has been applied as an equilib ium no ion in many expe imen al se -
ings including all-pay auc ions (Ande son e al. 1998), he a ele ’s dilemma (Cap a e al. 1999), ju y
decision ules (Gua naschelli e al. 2000), al e na ing o e ba gaining games (Goe ee and Hol 2000),
coo dina ion games (Ande son e al. 2001), i s p ice auc ions (Goe ee e al. 2002), gene alized ma ch-
ing pennies games (Goe ee e al. 2003), capaci y alloca ion games (Chen e al. 2012), and wo-sided
ma ching (Echenique e al. 2016).
589
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Cons ained school choice: anexpe imen al QRE analysis
ounds
s1
o e s wo and
s2
o e s one sea (p oblem
P3
). Ou ea men a iable is
he ma e ial payo subjec s ecei e om hei ma ches. A subjec always ecei es
1 mone a y uni i she is assigned o he school she likes mos and no hing i she
emains unma ched. In ea men L (low payo ), a subjec ecei es 0.3 mone a y
uni s i she is ma ched o he school she likes leas . This payo equals 0.7 mone a y
uni s in ea men H (high payo ). In each o
P1
and
P3
, he e a e wo s able ma ch-
ings: he s uden -op imal s able ma ching
𝜇I
and he school-op imal s able ma ching
𝜇S
. All s uden s weakly p e e
𝜇I
o
𝜇S
. The e is a unique s able ma ching in
P2
.5
Ou expe imen al p edic ions a e based on he logi e sion o he quan al
esponse equilib ium (logi -QRE) wi h isk-neu al and expec ed-payo -maximizing
agen s, which is pa ame ized by a “ a ionali y pa ame e ”
𝜆≥0
.6 I is well-known
ha o la ge
𝜆
, he logi -QRE ends owa ds a Nash equilib ium in mixed s a egies.
Fo bo h
P1
and
P3
, he esul ing ma ching unde his limi ing logi -QRE is
𝜇I
in
ea men L bu
𝜇S
in ea men H. The unde lying in ui ion is ha a he pu e s a -
egy Nash equilib ium ha yields
𝜇I
some s uden s ake he isk o being unma ched.
This isk comes wi h he po en ial bene i o being ma ched o he mos p e e ed
school. Fo i being wo hwhile o ake his isk (ins ead o applying o he wo s
school, which ensu es a sea , bu comes wi h a low payo ), he payo di e ence
be ween he wo schools mus be su icien ly high, which is he case in ea men L
bu no in ea men H. A simila a gumen applies in he gene al (non-limi ing) case
as well. In ac , we es ablish ha independen ly o he ac ual a ionali y pa ame e s
in ea men s L and H,
𝜇I
is mo e likely o be ob ained han
𝜇S
in ea men L bu
𝜇S
is mo e p e alen han
𝜇I
in ea men H. Ou da a suppo s his p edic ion, which
we conside o be ou main inding. In
P1
o ea men L,
𝜇I
is ob ained in 30.2%
and
𝜇S
in 27.2% o he cases ( he equency o an uns able ma ching is hus 42.6%).
On he o he hand, in
P1
o ea men H,
𝜇I
is ob ained in 19.7% and
𝜇S
in 45.1% o
he cases. The equencies o
P3
a e 43.2% o
𝜇I
and 37.0% o
𝜇S
in ea men L,
bu 24.1% o
𝜇I
and 59.9% o
𝜇S
in ea men H. In each o he wo p oblems, he
equency di e ence be ween
𝜇I
and
𝜇S
is signi ican a he 5-pe cen le el o ea -
men H bu no o ea men L.
The logi -QRE also pe mi s us o de i e hypo heses ega ding s uden beha -
io . Poin p edic ions indica e o a gi en p oblem
P∈{P1,P2,P3}
and o a gi en
ea men whe he a s uden is mo e likely o apply o school
s1
o o school
s2
. And
ea men compa isons e e o how he p obabili y wi h which a s uden applies o
each o he wo schools changes be ween he wo ea men s. The expe imen al da a
is in almos all ins ances consis en wi h hese p edic ions. In pa icula , o
P1
and
P3
, 4 ou o a o al o 5 s a is ical es s ega ding ea men compa isons yield a one-
sided p alue below 0.05. Wi h espec o he poin p edic ions, he e is mo e noise
5 P oblem
P2
has been added o he design mainly o comple eness easons. We hus concen a e in he
in oduc ion on ou wo main p oblems
P1
and
P3
. A comple e se o p edic ions and he co esponding
da a analysis o p oblem
P2
is p esen ed alongside he o he wo p oblems in Sec s. 2 and 3, espec i ely.
6 We concen a e on he logi -QRE because Haile e al. (2008) show ha in i s mos gene al o m, he
quan al esponse equilib ium is no alsi iable in any game (any beha io can be a ionalized). See also
Goe ee e al. (2005).
590
J.Alcalde-Unzu e al.
1 3
in he da a because “only” 3 ou o a o al o 12 compa isons a e signi ican a he
5-pe cen le el.
We also es ima e he logi -QRE o he pooled da a a he p oblem le el ia maxi-
mum likelihood. The es ima ed
𝜆
is la ges in
P2
, which is a guably he simples
se ing because i has a unique Nash equilib ium ou come (suppo ed by an in ini e
numbe o Nash equilib ia in mixed s a egies). In each o
P1
and
P3
, he es ima ed
𝜆
is smalle in ea men L han in ea men H. Finally, we ind ha his disc epancy
in he es ima ion o
𝜆
is educed i agen s a e isk-a e se bu no i hey ha e p e -
e ences o he s uden -op imal s able ma ching ( he expec ed u ili y depends posi-
i ely on he p obabili y ha his ma ching is eached) o he expec ed g oup payo .
1.3 Rela ed li e a u e
The e is a s eadily inc easing numbe o expe imen al s udies ha complemen he
heo e ical li e a u e on school choice (see Hakimo and Küble 2020 o an excel-
len o e iew). I is by now well unde s ood ha some subjec s ail o epo hei
ue p e e ences unde he uncons ained s uden -op imal s able mechanism. Unde
he immedia e accep ance mechanism, subjec beha io is e en less in line wi h he
ue p e e ences (see, e.g., Calsamiglia e al. 2010; Chen and Kes en 2019; Chen and
Sönmez 2006; Fea he s one and Niede le 2016; Pais and Pin é 2008). The kind o
de ia ions subjec s employ unde he wo mechanisms a e e y simila . In pa icula ,
he o de o wo schools a he op o hei p e e ence anking migh be swi ched i
s uden s ha e a highe p io i y in he less desi ed o he wo schools.
The exis ing expe imen s on school choice mainly compa e a ious mechanisms
in e ms o hei ou come (s abili y and wel a e o he esul ing ma ching), bu only
o e a less de ailed o e iew o he beha io ha leads o hese ou comes (i.e., o en
he ocus is a compa ison o u h- elling a es ac oss mechanisms). I seems he e-
o e necessa y o de elop al e na i e models ha cap u e he obse ed beha io al
pa e ns. The logi -QRE is a na u al s a ing poin because i akes ela i e p e e -
ence in ensi ies explici ly in o accoun .7 Apa om ou s udy, only Echenique e al.
(2016) and D ey uss e al. (2021) es ima e a logi -QRE in a wo-sided ma ching
ma ke . The main di e ence be ween ou s udy and Echenique e al. (2016) is ha
we speci ically design he expe imen in such a way ha he p edic ions o he logi -
QRE depend on he ea men condi ion. Fu he mo e, Echenique e al. (2016) do
no de i e es able hypo heses om he logi -QRE.
Using he da a o Li (2017) on he (s a egy-p oo ) andom se ial dic a o ship
mechanism, D ey uss e al. (2021) sugges ha expec a ion-based loss a e sion can
explain indi iduals’ de ia ions om u h- elling. When he unde lying mechanism
is s a egy-p oo , he logi -QRE wi h expec ed-payo -maximizing agen s con e ges
owa ds u h ul p e e ence e ela ion, which implies ha de ia ions o a ional sub-
jec s a e likely o be caused by non-s anda d p e e ences.
7 Abdulkadi oğlu e  al. (2011) highligh in a es ic ed en i onmen ha he immedia e accep ance
mechanism allows pa en s o exp ess hei ela i e p e e ence in ensi ies.
591
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Cons ained school choice: anexpe imen al QRE analysis
1.4 Remainde
We p oceed as ollows. In Sec .2, we de ail he expe imen al design and p ocedu es
and we de i e p edic ions om he logi -QRE. In Sec .3, we p esen all expe imen-
al esul s. Sec ion4 concludes. The appendix con ains he expe imen al ins uc-
ions, o mal p oo s, addi ional da a analysis, and heo e ical p edic ions o lis s o
leng h 2 (i.e., he uncons ained mechanisms).
2 Labo a o y expe imen
2.1 Design andp ocedu es
Ou expe imen is designed o analyze beha io in h ee school choice p oblems in
which h ee s uden s (labeled 1, 2, and 3) seek o ob ain a sea a schools
s1
and
s2
.
The p e e ences o he s uden s and he p io i ies o he schools a e p esen ed in
Table1 and a e he same in all h ee p oblems. The numbe o sea s schools o e ,
howe e , changes o e he cou se o an expe imen al session and is gi en in Table2.
Du ing he expe imen , subjec s assume he ole o s uden s. Schools a e no s a-
egic playe s. The in o ma ion in Table1 is common knowledge. The numbe o
sea s schools o e is made public a he beginning o each ound o he expe imen .
Gi en his in o ma ion, he subjec s’ ask is o submi a single applica ion o a school
(no necessa ily o he mos p e e ed school) o be used by a cen al clea inghouse
o assign s uden s o schools. Speci ically, we ocus on he cons ained immedia e
accep ance o , equi alen ly, he cons ained de e ed accep ance mechanism ( he
equi alence is due o he ac ha lis s a e o leng h 1):
Table 1 P e e ences o s uden s
o e schools and p io i ies o
schools o e s uden s
P e e ences P io i ies
123
s1
s2
Bes ma ch:
s1
s1
s2
3 2
Second bes :
s2
s2
s1
1 1
Thi d bes : 2 3
Table 2 Numbe o sea s a
schools P oblem
#
sea s
s1
s2
P1
1 2
P2
1 1
P3
2 1

592
J.Alcalde-Unzu e al.
1 3
Cons ained Immedia e/De e ed Accep ance Mechanism
S ep 1. Each s uden sends an applica ion o exac ly one school.
S ep 2. Each school ha has ecei ed a leas one applica ion accep s he applica-
ion om he s uden wi h he highes p io i y (among all ecei ed appli-
ca ions). I a school wi h wo sea s has ecei ed a leas wo applica ions,
hen i also accep s he applica ion om he s uden wi h he second high-
es p io i y (among all ecei ed applica ions). All o he applica ions (i
any) a e ejec ed.
Each s uden is assigned o he school ha she applied o p o ided ha
he school accep ed he applica ion. I he applica ion o a s uden was
ejec ed, hen he s uden emains wi hou a sea .
The expe imen was p og ammed wi hin he z-T ee oolbox p o ided by Fisch-
bache (2007) and ca ied ou a Lineex (www. lineex. es) hos ed a he Uni e si y o
Valencia. A he beginning o a session, subjec s ecei ed w i en ins uc ions ha
we e ead aloud by an ins uc o (see Appendix1). Pa icipan s we e in o med in
pa icula ha he expe imen would ake a o al o 18 ounds and ha he numbe
o sea s schools o e would change each six ounds. In all sessions,
P1
was always
played i s ( ounds 1–6),
P2
always played second ( ounds 7–12), and
P3
always
played hi d (13–18).8 Then, he compu e so wa e s a ed. The p og am ma ched
pa icipan s anonymously in o g oups o h ee. Wi hin each g oup, one subjec
was assigned he ole o s uden 1, ano he subjec he ole o s uden 2, and a hi d
subjec he ole o s uden 3. G oups and oles did no change o e he cou se o
he expe imen . Be o e ound 1, subjec s i s wen indi idually o e an illus a i e
example ( o ge used o he mechanism) and hen played a ial ound ha was no
aken in o accoun o paymen ( o become amilia wi h he compu e so wa e). A
he beginning o each ound, he compu e sc een p esen ed he p e e ences o he
h ee g oup membe s, he p io i ies o he wo schools, and he numbe o sea s a
each school. Subjec s ook hen hei espec i e decisions. A he end o each ound,
each subjec was in o med o he esul ing ma ch.
Ou ea men a iable is he payo subjec s ecei e. In each ound, a subjec
ecei ed 1ECU i she ended up a he mos p e e ed school and 0ECU i he appli-
ca ion was ejec ed (in which case she ended up unma ched). In ea men L (low),
a subjec ecei ed 0.3ECU i she ended up a he second mos p e e ed (o equi a-
len ly, he leas p e e ed) school. In ea men H (high), his payo was 0.7ECU.
Each ECU was wo h 1€.
8 The e is no need o coun e balance he o de in which p oblems a e played because we do no compa e
ou comes be ween p oblems.
593
1 3
Cons ained school choice: anexpe imen al QRE analysis
We an wo sessions (one wi h 42 and ano he one wi h 39 subjec s) pe ea -
men . In o al, 162 unde g adua es om a ious disciplines pa icipa ed in he
expe imen . Each session las ed abou 90min. Apa om he payo subjec s accu-
mula ed du ing he session, hey also ecei ed a show-up ee o 3€. Subjec s ea ned
on a e age 13.73€ in ea men L and 14.60€ in ea men H.
2.2 P edic ions
In his subsec ion, we de i e ou expe imen al p edic ions. A (school choice)
p oblem is a i e- uple
P=⟨I,S,q,PI,PS⟩
whe e
•
I={1, 2, …,n}
is a ini e se o s uden s (indi iduals);
• S is a ini e se o schools;
•
q≡(qs)s∈S
whe e o each
s∈S
,
qs
is he numbe o a ailable sea s a school
s;
•
PI≡(Pi)i∈I
is a p o ile o s ic p e e ence ela ions o he s uden s, whe e
o each
i∈I
,
Pi
is a comple e, i e lexi e, and ansi i e bina y ela ion o e
S∪{i}
; and
•
PS≡(Ps)s∈S
is a p o ile o s ic p io i y ela ions o he schools, whe e o
each
s∈S
,
Ps
is a comple e, i e lexi e, and ansi i e bina y ela ion o e I.
We assume ha o each s uden i, each school is p e e ed o he ou side op ion
(being unma ched), which is deno ed by i. A ma ching o s uden s o schools is a
unc ion
𝜇∶I
→
S
such ha o each
i∈I
and o each
s∈S
,
•
𝜇(i)∈S∪{i}
and
•
|
𝜇
−1(
s
)|
≤q
s
.
We will o en w i e a ma ching
𝜇
as he ec o
(𝜇(1),𝜇(2),…,𝜇(n))
, whe e o
each i,
𝜇(i)
is called s uden i’s ma ch. Whene e
𝜇(i)=i
, s uden i is unma ched,
i.e., emains wi hou sea . A pai
(i,s)∈I×S
blocks ma ching
𝜇
i
sP
i𝜇(i)
and
•
|
𝜇
−1(
s
)|
<q
s
o
•
|
𝜇
−1(
s
)|=
q
s
and he e exis s
k∈𝜇−1(s)
such ha
iP
sk
.
A ma ching is s able i no pai (i,s) blocks i . The se o s able ma chings is
non-emp y (Gale and Shapley 1962). Mo eo e , he e exis s a s uden -op imal
s able ma ching
𝜇I
which is weakly p e e ed by all s uden s o all o he s able
ma chings. Simila ly, he e exis s a school-op imal s able ma ching
𝜇S
which is
Table 3 Se o all s able
ma chings P oblem
𝜇I
𝜇S
P1
(s1,s2,s2)
≠
(s2,s2,s1)
P2
(1, s2,s1)
=
(1, s2,s1)
P3
(
s
1,
s
1,
s
2)
≠
(
s
1,
s
2,
s
1)
594
J.Alcalde-Unzu e al.
1 3
s uden -pessimal, i.e., all s uden s weakly p e e any o he s able ma ching o
𝜇S
.
Table3 shows all s able ma chings in ou h ee p oblems. In p oblems
P1
and
P3
,
he e a e wo s able ma chings (
𝜇I
and
𝜇S
). Since he e is only one s able ma ch-
ing in p oblem
P2
, he side-op imal s able ma chings
𝜇I
and
𝜇S
coincide in his
p oblem.
We conside he s a egic game induced by he school choice mechanism. Le
I={1, 2, 3}
be he se o playe s. Each playe
i∈I
has a se o pu e s a egies
Ai={ai1,ai2}
, whe e
aij
is playe i’s s a egy o sending an applica ion o school
sj
. Le
A≡×i∈IAi
deno e he se o s a egy-p o iles
a=(a1,a2,a3)
.
Le
P∈{P1,P2,P3}
. Fo each
a∈A
and each
i∈I
, le
𝛽i(a,P)
deno e he
ma ch o playe i when playe s send applica ions a and schools ha e p io i ies and
sea a ailabili y as gi en by
P
. Fix a p oblem
P∈{P1,P2,P3}
and a ea men
T∈{L,H}
. We assume ha playe i has a u ili y unc ion
ui∶A
→
ℝ
ha e lec s
he possible pe - ound payo s in he expe imen , i.e.,
He e, good (bad) ma ch e e s o playe i’s mos (leas ) p e e ed school a
P
.
A s a egy-p o ile
a∈A
is a Nash equilib ium (in pu e s a egies) i o each
i∈I
and each
ai∈Ai
,
u
i
(a)≥u
i
(a
i
,(a
j
)j
≠
i)
. Fo each p oblem, he se o Nash
equilib ium ou comes o he simul aneous-mo e game induced by he cons ained
immedia e accep ance mechanism coincides wi h he se o s able ma chings
(Hae inge and Klijn 2009). Hence, while he e is a unique s able ma ching and
he e o e a unique Nash equilib ium ou come in p oblem
P2
, he e a e wo Nash
equilib ium ou comes in each o he o he wo p oblems.
Le
Δi
deno e he se o mixed s a egies o playe i. Mo e speci ically, an ele-
men o
Δi
is a p obabili y dis ibu ion
pi∶Ai
→
ℝ
, i.e.,
pi(ai1)+pi(ai2)=1
and
o each
aij ∈Ai
,
pi(aij)
≥
0
. Le
Δ
≡
×i∈IΔi
. The domain o he u ili y unc ion u
is ex ended om A o
Δ
by de ining o each
p∈Δ
,
Since
pi(ai1)+pi(ai2)=1
, we can deno e
pi(ai1)
by
pi
and
pi(ai2)
by
1−pi
, and hus
in e changeably e e o an elemen o
Δi
by
(pi,1−pi)
o simply by
pi
. Fo each
p=(p1,p2,p3)∈Δ
and each
i∈I
, we le
p
−i
≡(p
j
)j
≠
i
.
A s a egy-p o ile
p∈Δ
is a Nash equilib ium in mixed s a egies i o each
i∈I
and each
pi∈Δ
i
,
ui(p)≥ui(pi,p−i)
. The se o Nash equilib ia in mixed
s a egies is compu ed and depic ed o each p oblem in Fig.4 o Appendix2.
The esul ing p obabili y dis ibu ions o e ma chings a e p esen ed in Table4.
u
i(a)≡
⎧
⎪
⎨
⎪
⎩
0 i
𝛽
i(a,P)=i, i.e., playe iis unma ched;
0.3 i 𝛽i(a,P)is playe i�s bad ma ch and T=L;
0.7 i 𝛽i(a,P)is playe i�s bad ma ch and T=H
;
1 i 𝛽
i
(a,P)is playe i�s good ma ch.
u
i(p)≡
∑
a=(a1,a2,a3)∈A[∏
i∈I
pi(ai)
]
⋅ui(a)
.
595
1 3
Cons ained school choice: anexpe imen al QRE analysis
Table4 highligh s ha he e a e mul iple Nash equilib ium ou comes in p ob-
lems
P1
and
P3
. The no ion o quan al esponse equilib ium o no mal o m
games, in oduced by McKel ey and Pal ey (1995), can be used as an equilib-
ium selec ion c i e ion. Quan al esponse equilib ia in i s logi o m a e de ined
by means o a non-nega i e pa ame e
𝜆
ha is in e sely ela ed o he playe s’
e o le el. Gi en
𝜆≥0
, a s a egy-p o ile
p∗∈Δ
is a logi quan al esponse
equilib ium (logi -QRE) i o each
i∈I
and each
aij ∈Ai
,
I
𝜆=0
, hen playe s choose uni o mly a andom. By Theo em2 in McKel ey and
Pal ey (1995), i
𝜆→∞
, hen
p∗
con e ges o a Nash equilib ium in mixed s a e-
gies. We e e o his Nash equilib ium as he limi ing logi -QRE.
In each p oblem, since each o he h ee playe s has wo pu e s a egies, he sys-
em o equa ions (1) educes o h ee equa ions wi h h ee unknowns, i.e.,
p∗
1
,
p∗
2
,
and
p∗
3
( he p obabili ies wi h which he s uden s apply o school
s1
). Gi en
𝜆≥0
,
we indica e his logi -QRE p obabili y o s uden i in p oblem
P∈{P1,P2,P3}
o
ea men
T∈{L,H}
by
p∗
i(𝜆|P,T)
. Figu e1 depic s he logi -QRE p obabili ies
o all ins ances (combina ions o p oblems and ea men s) o ou expe imen .
Mos impo an ly, in each o he p oblems
P1
and
P3
, he limi ing logi -QRE
depends on he ea men . In pa icula , panel1 o Fig.1 e eals ha in
P1
(
q1=1
and
q2=2
) o ea men L, as
𝜆
g ows la ge, he p obabili y wi h which s uden 1
applies o school
s1
and s uden s 2 and 3 apply o school
s2
app oaches 1. So, all
s uden s a e accep ed wi h p obabili y 1 o la ge
𝜆
and he esul ing ma ching is
(s1,s2,s2)
, which co esponds o he s uden -op imal s able ma ching o
P1
. Howe e ,
(1)
p∗
i(aij)= e
𝜆
⋅
u
i
(a
ij
,p∗
−i
)
e
𝜆⋅ui(ai1,p∗
−i)
+e
𝜆⋅ui(ai2,p∗
−i)
.
Table 4 P obabili y dis ibu ion
o e ma chings induced by Nash
equilib ia in mixed s a egies
(p1,p2,p3)
x=0.3
in ea men L and
x=0.7
in ea men H. Recall ha in
P2
,
𝜇I=𝜇S
P oblem T ea men
(p1,p2,p3)
𝜇I
𝜇S
Uns able
P1
L,H(1,0,0) 1 0 0
L,H(0,0,1) 0 1 0
L(0.3,0,0.7) 0.09 0.49 0.42
L(0,0.3,0.7) 0 0.49 0.51
H(0.7,0,0.3) 0.49 0.09 0.42
H(0,0.7,0.3) 0 0.09 0.91
P2
L,H
(p1,0,1)
o each
p1∈[0, 1]
110
P3
L,H(1,1,0) 1 0 0
L,H(1,0,1) 0 1 0
L(1,0.3,0.7) 0.09 0.49 0.42
H(1,0.7,0.3) 0.49 0.09 0.42
602
J.Alcalde-Unzu e al.
1 3
E idence on Hypo hesis 1:
We show ha he expe imen al da a in Table5 conside ably suppo s Hypo hesis 1.
Conside i s
P1
. I is expec ed ha in ea men L, s uden 1 is mo e and s uden s
2 and 3 a e less likely o apply o
s1
han o
s2
. Also, in ea men H, s uden 3 is
expec ed o be mo e and s uden s 1 a e 2 a e expec ed o be less likely o apply o
s1
han o
s2
. Only he da a o s uden 1 in ea men L is ou igh agains his p edic-
ion: he expe imen al equency wi h which he s uden applies o
s1
is 49%, ye
his equency is expec ed o be g ea e han 50%. F om a s a is ical poin o iew,
only he compa ison o s uden 2 in ea men H is signi ican a he 5% le el ( he
one-sided p alue is 0.039). Wi h espec o he ea men e ec s, s uden s 1 and 2
a e expec ed o be mo e and s uden 3 is expec ed o be less likely o apply o
s1
in
ea men L han in ea men H. The expe imen al da a goes in he co ec di ec ion.
The one-sided p alues o he co esponding Mann–Whi ney U es s a e 0.062 o
s uden 1, 0.031 o s uden 2, and 0.024 o s uden 3.
Conside nex
P2
. I is expec ed ha in bo h ea men s, s uden 3 is mo e and
s uden 2 is less likely o apply o
s1
han o
s2
. I is e iden om Table5 ha he
expe imen al da a is comple ely in line wi h hese p edic ions e en hough he
compa ison o s uden 3 in ea men L is only signi ican a a one-sided p alue
o 0.098. Wi h espec o s uden 1, o whom he poin p edic ion was exp essed
in a weak sense, he hypo hesis ha his s uden applies o school
s1
wi h p ob-
abili y
1
2
canno be ejec ed in ei he o he wo ea men s. The expe imen al da a
is also consis en wi h he ambiguous ea men e ec s in Hypo hesis1: S uden s
1 and 2 a e mo e and s uden 3 is less likely o apply o
s1
in ea men L han in
ea men H. The one-sided p alues o he co esponding Mann–Whi ney U es s
a e 0.019 o s uden 1, 0.069 o s uden 2, and 0.067 o s uden 3.
Conside inally
P3
. I is expec ed ha in ea men L, s uden s 1 and 2 a e mo e
and s uden 3 is less likely o apply o
s1
han o
s2
. Also, in ea men H, s uden s
1 and 3 a e expec ed o be mo e and s uden 2 is expec ed o be less likely o apply
s1
han o
s2
. The expe imen al da a is again consis en wi h hese p edic ions. The
compa isons o s uden 1 a e in bo h ea men s signi ican a he 1% le el. Finally,
s uden 2 (s uden 3) is supposed o apply o
s1
mo e (less) o en in ea men L han
in ea men H. The da a con i ms hese p edic ions: he one-sided p alues o he
Mann-Whi ney U es s a e 0.040 o s uden 2 and 0.025 o s uden 3.
◻
A i s obse a ion om Table6 is ha e en hough ou se ing is highly s ylized,
i is no s aigh o wa d o subjec s o coo dina e in
{P1,P3}
on a Nash equilib ium
in pu e s a egies. A Nash equilib ium in pu e s a egies leads o a s able ma ch-
ing, howe e an uns able ma ching is eached in
P1
in mo e han 35% and in
P3
in
mo e han 16% o he cases. Mo eo e , he p obabili y dis ibu ion o e ma chings
in hese wo p oblems does no coincide wi h he one induced by he non-degen-
e a e mixed s a egy Nash equilib ia in Table4. To see his, we employ Wilcoxon
signed- ank es s a he g oup le el o analyze whe he he empi ical dis ibu ion
o e ma chings is di e en om he one ha a ises om he non-degene a e Nash
equilib ia in mixed s a egies. The ( uly independen ) obse a ion o a g oup is he
p obabili y wi h which a ma ching (
𝜇I
,
𝜇S
, o uns able) is eached o e he cou se o
he six ounds in which he p oblem is played. We ind ha in ea men L o
P1
,
pS

603
1 3
Cons ained school choice: anexpe imen al QRE analysis
is signi ican ly di e en om 0.49 ( wo-sided
p=0.0052
). Also, in ea men H o
P1
,
pS
is signi ican ly g ea e han 0.09 (one-sided
p≤0.0001
). And he p obabili y
o an uns able ma ching is signi ican ly di e en om 0.42 in ea men L ( wo-sided
p<0.0001
) and ea men H ( wo-sided
p=0.0052
) o
P3
. Finally, in
P2
he unique
s able ou come is “only” ob ained in 78% o he cases in ea men L and in 90% o
he cases in ea men H. This hin s a misplays, ye he pe - ound da a a he bo -
om o Fig.5 in Appendix3 p o ides e idence o lea ning and coo dina ion e ec s
because uns able ma chings occu less o en in la e ounds o a p oblem.
E idence on Hypo hesis 2:
Hypo hesis 2 s a es ha o each p oblem
P∈{P1,P3}
, i is mo e likely o ob ain
𝜇I
han
𝜇S
in ea men L and i is less likely o ob ain
𝜇I
han
𝜇S
in ea men H. We ind
o
P1
ha he p obabili y di e ence be ween
𝜇I
and
𝜇S
is
0.302 −0.272 =0.030
in ea men L and
0.197 −0.451 =−0.254
in ea men H. The one-sided p alues
o he Wilcoxon signed- ank es s a he g oup le el a e 0.3528 in ea men L and
0.0078 in ea men H. Simila obse a ions hold o p oblem
P3
: he p obabili y
di e ence be ween
𝜇I
and
𝜇S
is
0.432 −0.370 =0.062
in ea men L (one-sided
p=0.3625) and
0.241 −0.599 =−0.358
in ea men H (one-sided p=0.0033). We
conclude ha he expe imen al da a is suppo i e o Hypo hesis 2.
◻
In he inal pa o his sec ion, we es ima e he logi -QRE ia maximum likeli-
hood. Suppose ha a o al o K subjec s pa icipa e in a gi en ea men
T∈{L,H}
in
each s uden ole i. Le
x
l,i
be a pa icula obse a ion o subjec l in ole i o ound
∈{1, …,6}
. We de ine
x
l,i
=1
i s uden l in ole i applies in ound o school
s1
. O h-
e wise,
x
l,i
=0
. Unde he logi -QRE,
p∗
i(𝜆)
is he p obabili y ha
x
l,i
=1
and
1−p∗
i(𝜆)
is he p obabili y ha
x
l,i
=0
. The join likelihood o obse ing he da a is hen
The da a can be pooled i i is assumed ha subjec beha io is ime-independen ,
i.e., o all
, �∈{1, …,6}
,
x
l,i
=x
�
l,i
≡x
l,i
.12 Then,
Le p
i
=
∑K
l=1
x
l,i
∕
K
be he empi ical p obabili y om he expe imen ha subjec s
in s uden ole i apply o school
s1
. We inally ob ain ha
L
(𝜆)=
3
∏
i=1
K
∏
l=1
6
∏
=1
p∗
i(𝜆)x
l,i⋅(1−p∗
i(𝜆))1−x
l,i
.
ln
L(𝜆)=6⋅
3
∑
i=1
K
∑
l=1
xl,i⋅ln(p∗
i(𝜆))+(1−xl,i)⋅ln(1−p∗
i(𝜆))
.
ln
L(𝜆)=6⋅K⋅
3
∑
i=1
pi⋅ln(p∗
i(𝜆)) + (1−pi)⋅ln(1−p∗
i(𝜆))
.
12 Time-independence is a es ic i e condi ion, bu we do no ha e su icien da a poin s o pe o m an
es ima ion o each ound.
604
J.Alcalde-Unzu e al.
1 3
In o de o maximize his unc ion, we calcula e he equilib ium p obabili ies o he
logi -QRE nume ically on a ine g id— he unique model pa ame e
𝜆
is a ied in
s eps o 0.01 be ween 0 and 10, which means ha 1000 di e en alues o
𝜆
a e
conside ed—and e alua e he objec i e unc ion a hese equilib ium alues. Fo
each o he 1000 es ima ions o
𝜆
we use a andom sample ha consis s o 60% o
he a ailable da a o each s uden .13 This yields a dis ibu ion o es ima es o
𝜆
and
allows us o analyze ea men e ec s. We deno e by

𝜆T
j
he mean o he es ima ed
dis ibu ion o p oblem
Pj
o ea men T.
Table7 p esen s he es ima ion esul s o h ee di e en models (u ili y unc-
ions). We i s concen a e on he case when he subjec s a e isk-neu al expec ed-
payo maximize s (Model I) and he only pa ame e o be es ima ed is
𝜆
. The
in ui ion o he logi -QRE is ha la ge alues o
𝜆
imply ha choices a e close o
Nash equilib ium beha io . In his sense,
𝜆
measu es he a ionali y o he obse ed
beha io . In ou expe imen ,
P2
is a guably he simples o he h ee p oblems. In
his p oblem, he con inuum o mixed s a egy Nash equilib ia leads o he same
(s able) ma ching. In he o he wo p oblems, he e a e wo s able ma chings and a
coo dina ion p oblem a ises because di e en Nash equilib ia induce di e en s able
(o e en uns able) ma chings. Ou es ima ion esul s in Table7 suppo his in e -
p e a ion since in Model I,

𝜆
is la ges in
P2
( o bo h ea men s).14 Fu he mo e,
Table 7 Maximum likelihood es ima ion esul s
Means and s anda d de ia ions (in pa en hesis) a e ob ained ia 1000 andom samples wi h 60% o he
obse a ions each. Fo each p oblem, ea men e ec s a e signi ican a wo-sided
p<0.0001
(Mann–
Whi ney U es )
P1
P2
P3
H L H L H L
Model I
𝜆
2.663 0.699 4.347 8.127 2.586 1.808
(0.089) (0.173) (0.255) (0.261) (0.073) (0.078)
Model II
𝜆
1.309 0.404 0.917 0.989 2.566 1.794
(0.024) (0.107) (0.048) (0.007) (0.071) (0.078)
1.000 1.000 0.979 0.993 0.001 0.001
(0.000) (0.000) (0.025) (0.008) (0.000) (0.000)
Model III
𝜆
4.764 1.390 4.346 2.280 11.372 1.798
(0.097) (0.093) (0.258) (0.104) (2.597) (0.078)
c0.678 0.000 1.000 0.000 0.252 1.000
(0.030) (0.000) (0.000) (0.000) (0.025) (0.000)
13 The log-likelihood unc ion is s ic ly conca e and we ind an in e io solu ion on he conside ed g id.
Hence, e en hough he logi -QRE is de ined o all
𝜆>0
, he e is no need o widen he g id.
14 I migh be su p ising ha

𝜆
L
2
=
8.1266
is subs an ially g ea e han

𝜆
H
2
=
4.3436
e en hough he e a e
only mino ea men di e ences in Table5. The logi -QRE ajec o ies in Fig.1 p o ide an explana ion
o his. Acco ding o he empi ical da a, he e is a high likelihood ha s uden 3 applies o school
s1
. Ye ,
a high
p∗
3
equi es a subs an ially g ea e
𝜆
in ea men L han in ea men H.
605
1 3
Cons ained school choice: anexpe imen al QRE analysis
i is wo h no ing ha in each o he p oblems
P1
and
P3
,

𝜆
is la ge in ea men H
han in ea men L. Fo a possible explana ion, ecall ha o la ge
𝜆
, he logi -QRE
con e ges o
𝜇S
in ea men H and o
𝜇I
in ea men L. A he pu e s a egy Nash
equilib ium ha induces
𝜇S
, s uden s apply o hei “sa e y schools,” i.e., hey maxi-
mize he p obabili y o being accep ed. Ye , a he pu e s a egy Nash equilib ium
ha yields
𝜇I
some s uden s ake he isk o emaining unma ched (i some o he
o he s uden s de ia es). This isk comes wi h he po en ial bene i o being ma ched
o he good school. I he payo o he (bad) sa e y school is close o he payo o
he good school, which is he case in ea men H, hen a subjec may p e e o send
an applica ion o he (bad) sa e y school, which would esul in a ela i ely la ge
𝜆
.
On he o he hand, i he e is a big payo di e ence be ween he good and he (bad)
sa e y school, which is he case in ea men L, he logi -QRE demands o make he
isky choice bu some subjec s migh s ill apply o hei (bad) sa e y school due o
isk p e e ences (see, e.g., Klijn e al. 201315), which would esul in a ela i ely
small
𝜆
. As a consequence, one could expec o ob ain la ge es ima es o
𝜆
in ea -
men H han in ea men L o bo h
P1
and
P3
.
We add ess in Model II he ques ion whe he subjec s’ isk a e sion indeed
a ec s he es ima ion o
𝜆
by assuming a CARA u ili y unc ion on expec ed pay-
o s.16 Equa ion (1) hen becomes
whe e is he A ow-P a measu e o isk a e sion. I
=0
, subjec s a e isk-neu-
al; and i
→1
, hen
ui(aij,p∗
−i) ∕(1− )
ends o
ln(ui(aij,p∗
−i))
.
Table7 shows ha

is in all p oblems consis en ac oss ea men s, ye he e a e
impo an di e ences be ween p oblems. Risk a e sion is maximal in
P1
and
P2
, while
 →0
in
P3
. I is wo h no ing ha playe 1 has he dominan s a egy
p1=1
in
P3
,
which implies ha he op imal beha io o his playe is independen o he isk a e -
sion. I one compa es speci ica ions unde he p emise ha
𝜆
is exogenous and is “sup-
posed o be” cons an ac oss ea men s, Model II imp o es upon Model I. In Model I,
he a io
|
𝜆H
j
∕

𝜆L
j|
is 3.81 in
P1
, 0.53 in
P2
, and 1.43 in
P3
. The a ios o Model II a e
close o 1 han he a ios o Model I: 3.24 in
P1
, 0.92 in
P2
, and 1.43 in
P3
.
Finally, we s udy in Model III whe he he sys ema ic di e ences o he be ween-
ea men es ima es o
𝜆
in Model I a e explained by subjec s ying o coo dina e on
he s uden -op imal s able ma ching. Fo ha i is assumed ha he u ili y unc ion
is a linea combina ion o he expec ed payo s and he p obabili y ha he s uden -
op imal s able ma ching is eached. In pa icula ,
p∗
i(aij)= e
𝜆
⋅
[
ui
(
aij,p
∗
−i
) ∕(
1
−
)]
e
𝜆⋅[ui(ai1,p∗
−
i) ∕(1− )]
+e
𝜆⋅[ui(ai2,p∗
−
i) ∕(1− )]
,
15 Klijn e al. (2013) show expe imen ally ha he s uden -op imal s able mechanism is mo e obus o
changes in he ca dinal p e e ence s uc u e han he immedia e accep ance mechanism and ha subjec s
wi h a highe deg ee o isk a e sion (measu ed h ough a Hol -Lau y lo e y ask) a e mo e likely o
play a “p o ec i e s a egy” unde he s uden -op imal s able mechanism bu no unde he immedia e
accep ance mechanism.
16 We a e e y g a e ul o an anonymous e e ee o sugges ing us o analyze he impac o isk p e e -
ences and p e e ences o he s uden -op imal s able ma ching on he es ima es o
𝜆
.
606
J.Alcalde-Unzu e al.
1 3
whe e
c∈[0, 1]
. We ind ha he es ima ion ou come o his model is wo se han
ha o Model II. Fi s ,
c
a ies subs an ially be ween ea men s and be ween p ob-
lems (and no only be ween p oblems as in Model II). And second, he a io
|
𝜆H
j
∕

𝜆L
j|
is 3.42 in
P1
, 1.90 in
P2
, and 6.32 in
P3
. Fo all p oblems, hese a ios a e u he
away om 1 han he a ios o Model II. One na u al al e na i e o Model III is o
assume ha subjec s ha e p e e ences o he expec ed g oup payo ins ead o p e -
e ences o he s uden -op imal s able ma ching. We also es ima e his al e na i e
model and, pe haps su p isingly, no e idence o his ype o p e e ences o e i-
ciency is ound. In all cases,
c=1
.
One impo an conclusion ha can be d awn om he es ima ion esul s in Table7
is ha including he isk pa ame e educes he a io
|
𝜆H
j
∕

𝜆L
j|
in each o he h ee
p oblems. We explo e his poin u he by e-es ima ing Model I o a ious non-ze o
le els o . In ac , one ca ea o Model II is ha

is 1 o close o 1 in p oblems
P1
and
P2
, ye

is 0 in p oblem
P3
. This s ands in con as wi h he idea ha a subjec ’s isk
a e sion is an ex e nal cha ac e is ic ha is cons an h oughou he expe imen .
Table8 shows ha

𝜆
dec eases as inc eases. The las column o Table8 cal-
cula es o each le el o isk a e sion he a iance o

𝜆
o e all 6 numbe s in he
same ow. In his calcula ion, he
𝜆
es ima es a e no malized o he uni in e al. The
a iance is lowes o
=0.6
, which means ha in e media e alues o seem o be
mo e app op ia e i he deg ee o isk a e sion is conside ed an ex e nal cha ac e is-
ic and one aims a minimizing he a iance o

𝜆
.
p∗
i(aij)= e
𝜆
⋅[c⋅ui(aij,p
∗
−i)+(1−c)⋅p(
𝜇I
)]
e
𝜆⋅[c⋅ui(ai1,p∗
−i)+(1−c)⋅p(𝜇I)]
+e
𝜆⋅[c⋅ui(ai2,p∗
−i)+(1−c)⋅p(𝜇I)]
,
Table 8 Maximum likelihood es ima ion esul s o
𝜆
Means and s anda d de ia ions (in pa en hesis) a e ob ained ia 1000 andom samples wi h 60% o he
obse a ions each
P1
P2
P3
Va (
𝜆)
H L H L H L
Model I
=0
2.663 0.699 4.347 8.127 2.586 1.808 0.124
(0.089) (0.173) (0.255) (0.261) (0.073) (0.078)
=0.2
2.319 0.621 3.351 5.727 2.232 1.558 0.119
(0.070) (0.154) (0.180) (0.159) (0.056) (0.070)
=0.4
2.016 0.554 2.521 3.931 1.921 1.338 0.114
(0.055) (0.139) (0.119) (0.110) (0.041) (0.062)
=0.6
1.749 0.496 1.842 2.598 1.648 1.144 0.113
(0.043) (0.125) (0.068) (0.066) (0.029) (0.055)
=0.8
1.515 0.446 1.300 1.604 1.408 0.974 0.139
(0.032) (0.114) (0.038) (0.035) (0.020) (0.048)
→1
1.309 0.404 0.875 0.976 1.199 0.825 0.124
(0.024) (0.107) (0.010) (0.021) (0.011) (0.042)
607
1 3
Cons ained school choice: anexpe imen al QRE analysis
4 Concluding ema ks
An impo an pa o he expe imen al li e a u e on school choice ocuses on com-
pa ing di e en cen alized assignmen mechanisms. The heo e ical li e a u e is
some imes able o p o ide e y sha p p edic ions. Fo example, in an uncons ained
se ing in which subjec s can ank all schools, bo h he s uden -op imal s able mecha-
nism and he op ading cycles mechanism a e s a egy-p oo , ha is, in he induced
simul aneous-mo e games subjec s ha e an incen i e o e eal hei p e e ences
u h ully. One can hen use he u h- elling a es obse ed in a labo a o y expe -
imen o make in e ences abou he quali y o he decisions. Since he immedia e
accep ance mechanism is manipulable, he compa ison o subjec beha io be ween
he s uden -op imal s able mechanism and he immedia e accep ance mechanism
is no as s aigh o wa d, e en in an uncons ained se ing. Thus, compa ing u h-
elling a es be ween mechanisms is only use ul in ce ain clea ly de ined ins ances.
In gene al, he e is a need o speci y a beha io al model ha pe mi s he de i a ion
o es able p edic ions and compa e he quali y o he decisions be ween ea men s.
The quan al esponse equilib ium is a na u al ool in his espec .
Wi h he help o a labo a o y expe imen we ha e analyzed he e ec s o changes
in he ca dinal payo s (i.e., ela i e p e e ence in ensi ies) on subjec beha io o
h ee s ylized school choice p oblems. Ou main inding is ha he logi -QRE co -
ec ly cap u es he quali a i e changes o bo h subjec beha io and he esul ing
ma ching. We no e ha he logi -QRE can also be used o de i e p edic ions ac oss
mechanisms, which is a ecu ing opic in he expe imen al li e a u e. Gi en he
po en ial in e es o u u e esea ch, we b ie ly discuss an example. Conside he
p oblem wi h h ee s uden s (1, 2, and 3) and h ee schools (
s1
,
s2
, and
s3
) in Table9.
Each school has 1 sea and gi es a s ic ly posi i e payo . Also assume ha s uden s
submi lis s o size3.
The e a e h ee s able ma chings. In he s uden -op imal s able ma ching, each
s uden is assigned o he bes ma ch. In he “median” s able ma ching, each s uden
is assigned o he second bes ma ch. And in he school-op imal s able ma ching,
each s uden is assigned o he wo s ma ch. The uncons ained s uden -op imal s a-
ble mechanism is s a egy-p oo , which implies ha agen s epo hei p e e ences
u h ully in he limi ing logi -QRE. Since s uden s canno emain unma ched unde
he immedia e accep ance mechanism, he wo s school mus be anked las (no
doing so is a weakly domina ed s a egy). Whe he s uden s ank hei bes school o
hei second bes school i s in he limi ing logi -QRE unde he immedia e accep -
ance mechanism is a unc ion o he payo s uc u e. I he payo o he bes and he
Table 9 P e e ences o s uden s
o e schools and p io i ies o
schools o e s uden s
P e e ences P io i ies
123
s1
s2
s3
Bes ma ch
s1
s2
s3
2 3 1
Second bes
s2
s3
s1
3 1 2
Thi d bes
s3
s1
s2
1 2 3

608
J.Alcalde-Unzu e al.
1 3
second bes school a e “close enough” o each o he , he limi ing logi -QRE is equal
o he median s able ma ching. And i he e is a “su icien ” payo di e ence o
hese wo schools, he s uden -op imal s able ma ching is ob ained.
Finally, pa o he ecen expe imen al li e a u e on school choice s udies mecha-
nisms ha inco po a e conce ns o a i ma i e ac ion (i.e., Klijn e al. 2016; Kawagoe
e al. 2018), allow o in o ma ion acquisi ion (Chen and He 2021), o employ a dynamic
assignmen p ocedu e (i.e., Klijn e al. 2019; Bó and Hakimo 2020; Du e al. 2021).
Also, he e a e inno a i e expe imen al designs ha analyze he oo s o sub-op imal
beha io in expe imen al ma ching ma ke s (i.e., Guillen and Hakimo 2017; Du e al.
2018; Ding and Scho e 2019; Guillen and Vesz eg 2021). I seems wo hwhile o mo e
closely s udy he p edic i e powe o he quan al esponse equilib ium in hese se ings.
Appendix1: Ins uc ions ( ansla ed omSpanish)
Gene al ins uc ions
Dea pa icipan , hank you o aking pa in his expe imen . The pu pose o his ses-
sion is o s udy how people make decisions. The session will las abou 90min. In addi-
ion o he 3 Eu o show-up ee you can—depending on you decisions—ea n some mo e
money. In o de o ensu e ha he expe imen akes place in an op imal se ing, we would
like o ask you o abide o he ollowing ules du ing he whole expe imen :
• Please, do no communica e wi h o he pa icipan s!
• Do no o ge o swi ch o you mobile phone!
• Read he ins uc ions ca e ully. I some hing is unclea o i you ha e any ques ion
now o a any ime du ing he expe imen , please ask one o he expe imen e s. How-
e e , do no ask ou loud, aise you hand ins ead. We will answe ques ions p i a ely.
I you do no obey he ules, he da a becomes useless o us and in his case we will
ha e o exclude you om his expe imen and you will no ecei e any mone a y
compensa ion. Payo s du ing he expe imen a e exp essed in ECU (expe imen al
cu ency uni s). A he end o he session you will ecei e 1 Eu o o each ECU
ob ained in he cou se o he expe imen .
Desc ip ion
The basic decision en i onmen in he expe imen is as ollows. The e a e h ee s u-
den s—le us call hem
E1
,
E2
, and
E3
— ha can be assigned o a school. The e a e
wo schools—deno ed
C1
and
C2
—and each school can ha e 1 o 2 a ailable sea s
(we will speci y his la e on in hese ins uc ions).
Since he schools di e in hei loca ion and quali y, s uden s ha e di e en opinions
ega ding which school hey would like o a end. The desi abili y o schools in e ms o
loca ion and quali y is exp essed in a able such as Table10. No e: his is an illus a i e
example and hence any able you will see la e in he expe imen migh be di e en .
609
1 3
Cons ained school choice: anexpe imen al QRE analysis
Each column gi es he p e e ences o a pa icula s uden . Conside he column
ha is ma ked
E3
. This column gi es he p e e ences o s uden
E3
and ells us ha
he/she would mos o all like o ob ain a sea in school
C1
. The e o e, he leas p e-
e ed school o s uden
E3
is
C2
. Finally, no ob aining a sea a any o he schools is
he wo s possible ou come. The columns o
E1
and
E2
ha e simila in e p e a ions.
Some s uden s ha e al eady a b o he o sis e a ending one o he schools. Also,
he s uden s di e in walking dis ance o he schools. The au ho i ies use hese and
o he ac o s o de e mine he schools’ p io i ies o e he s uden s. Each school has a
p io i y o de ing whe e all s uden s a e anked. The p io i y o de ings o he schools
can be summa ized in a able such as Table11. No e: his is an illus a i e example
and hence any able you will see la e in he expe imen migh be di e en .
Each column gi es he p io i y o de ing o a pa icula school. Conside he col-
umn ha is ma ked
C2
. This column gi es he p io i y o de ing o school
C2
and ells
us ha his school gi es he highes p io i y o ecei ing s uden
E2
. I his is no
possible, hen school
C2
gi es p io i y o s uden
E3
o be en olled. The lowes p io -
i y s uden o school
C2
is s uden
E1
. The column o
C1
has simila in e p e a ions.
The ma ching p ocedu e
To decide i and how s uden s a e assigned o schools, he ollowing p ocedu e is
ollowed. I consis s o wo phases.
Phase 1. S uden s a e asked o simul aneously and independen ly send an applica-
ion o one school. Fo ins ance, i can happen ha he s uden s apply o schools as
desc ibed by Table12. He e each column shows he applica ion o a s uden .
No e: his is an illus a i e example o h ee applica ions. Each s uden is ee o
apply o he school ha he/she hinks is app op ia e. The applica ion does no neces-
sa ily ha e o coincide wi h he mos p e e ed. In ac , in ou example s uden s
E1
and
E3
apply o hei mos p e e ed school, bu his is no he case o s uden
E2
.
Table 10 P e e ences o
s uden s
E1
,
E2
and
E3
o e
schools
E1
E2
E3
Mos p e e ed school
C2
C1
C1
Leas p e e ed school
C1
C2
C2
Table 11 P io i y o de ings o
schools
C1
and
C2
C1
C2
Highes p io i y s uden
E1
E2
Second highes p io i y school
E2
E3
Lowes p io i y school
E3
E1
Table 12 Applica ions o
s uden
E1
,
E2
, and
E3
E1
E2
E3
School applied o
C2
C2
C1
610
J.Alcalde-Unzu e al.
1 3
Phase 2. The s uden s’ applica ions oge he wi h he schools’ p io i y o de ings
de e mine an assignmen o s uden s o schools in he ollowing way.
• S ep 1: Each school ha has ecei ed a leas one applica ion accep s he applica-
ion om he s uden wi h he highes p io i y (among all ecei ed applica ions). I a
school wi h wo sea s has ecei ed a leas wo applica ions, hen i also accep s he
applica ion om he s uden wi h he second highes p io i y (among all ecei ed
applica ions). All o he applica ions (i any) a e ejec ed.
• S ep 2: Each s uden is assigned o he school ha she applied o p o ided ha he
school accep ed he applica ion. I he applica ion o a s uden was ejec ed, hen he
s uden emains wi hou a sea .
The expe imen
In he beginning o he expe imen , he compu e andomly di ides he pa icipan s in o
g oups o 3. The assignmen p ocess is andom and anonymous, so no pa icipan will
know who is in which g oup. Then, each pa icipan in a g oup ge s andomly assigned
he ole o a s uden in such a way ha one g oup membe will be in he ole o s uden
E1
, ano he g oup membe will be in he ole o s uden
E2
, and a hi d membe will be
in he ole o s uden
E3
.
You will play he basic decision si ua ion explained abo e 18 imes in o al. The
composi ion o he g oup and he oles o he pa icipan s wi hin each g oup do no
change o e he cou se o he expe imen ( o example, i you a e assigned he ole o
s uden
E2
, hen his will be you ole un il he end o he expe imen ; also, you will
always be playing wi h he same pa icipan in ole
E1
and wi h he same pa icipan in
ole
E3
). E e y 6 ounds he numbe s o sea s school o e change. The i s able wi h
s uden p e e ences and he second able wi h p io i y o de ings o schools emain he
same in all 18 ounds o he expe imen .
In each o he 18 ounds, payo s a e such ha you ecei e 1 ECU i you end up
a he school you p e e mos ,
x
ECU i you a e assigned o you second mos p e-
e ed school, and 0 ECU i you end up unassigned. A he end o he expe imen , we
will sum up you payo s o e he 18 ounds. You inal payo will be equi alen o he
sum o he pe - ound ECUs and he 3 Eu o show-up ee.
The i s hing you will see when he compu e p og am s a s is an illus a i e
example. Then, he e will be one ial ound ha does no coun o you inal payo
so ha you can amilia ize you sel wi h he compu e p og am. A e wa ds, he i s o
he 18 ounds ha coun o paymen s a s.
No e: in he expe imen al sessions o ea men H, he pa ame e x ook he alue 0.7;
in he expe imen al sessions o ea men L, he pa ame e x ook he alue 0.3.
Appendix2: Nash equilib ia
Le x be he pe - ound payo o ob aining a sea a he second mos p e e ed
school, i.e.,
x=0.3
in ea men L and
x=0.7
in ea men H. Recall ha a mixed
s a egy o playe /s uden i is comple ely desc ibed by he p obabili y
pi∈[0, 1]
611
1 3
Cons ained school choice: anexpe imen al QRE analysis
wi h which he s uden applies o school
s1
(so,
1−pi
is he p obabili y wi h which
s uden i applies o school
s2
).
The se o NE inmixed s a egies o p oblem
P1
One can easily e i y ha he no mal- o m game is gi en by Tables13 and 14. Thus,
he bes esponse co espondences o he h ee s uden s a e as ollows17:
We compu e he se o Nash equilib ia by checking he h ee Cases I, II, and III
below. Le
(p1,p2,p3)
be a Nash equilib ium.
I:
1−p3>x
. I ollows om
b 1
ha
p1=1
. Since
p1=1, (1−p1)(1−p3)
=0
<
x
. The e o e,
p2=0
by
b 2
. Hence,
x<1=p1+p2−p1p2
. So,
p3=0
by
b 3
. Now one easily e i ies ha he s a egy-p o ile
(p1,p2,p3)=(1, 0, 0)
is indeed a Nash equilib ium.
II:
1−p3<x
. I ollows om
b 1
ha
p1=0
. Since
p1=0
,
(1−p1)(1−p3)
=1
−
p3
<
x
. The e o e,
p2=0
by
b 2
. Hence,
x>0=p1+p2−p1p2
.
b
1(p2,p3)=
⎧
⎪
⎨
⎪
⎩
1 i 1 −p3>x
[0,1] i 1 −p3=x
0 i 1 −p3<x
b
2(p1,p3)=
⎧
⎪
⎨
⎪
⎩
1 i (1−p1)(1−p3)>
x
[0,1] i (1−p1)(1−p3)=
x
0 i (1−p1)(1−p3)<
x
b
3(p1,p2)=
⎧
⎪
⎨
⎪
⎩
1 i x>p1+p2−p1p2
[0,1] i x=p1+p2−p1p2
0 i x<p1+p2−p1p2
Table 13 Playe 3 plays
s1
1∖2
s1
s2
s1
0, 0, x
0,
x
,x
s2
x, 0, x
x,x,x
Table 14 Playe 3 plays
s2
1∖2
s1
s2
s1
1, 0, x
1,
x
,1
s2
x,1,1
x,x,0
17 Fo ins ance, gi en s a egies
(p2,p3)
o s uden s 2 and 3, he expec ed u ili y o s uden 1 om apply-
ing o
s1
is
1×(1−p3)
because
s1
has only one sea and only s uden 3 has highe p io i y o he school
han s uden 1. Simila ly, applying o
s2
yields he expec ed u ili y
x×1
, because
s2
has wo sea s and
s uden 1 has he second highes p io i y o he school (so, applying o he school gua an ees en ance).
618
J.Alcalde-Unzu e al.
1 3
We compu e he se o Nash equilib ia by checking he wo cases I and II below. Le
(p1,p2,p3)
be a Nash equilib ium.
I:
p3≠1
. F om
b 1
,
p1=1
. Then, om
b 2
and
b 3
,
p2∈[0, 1]
and
p3=0
,
espec i ely. One easily e i ies ha o each
p2∈[0, 1]
, he s a egy-p o ile
(p1,p2,p3)=(
1, p
2
,0
)
is indeed a Nash equilib ium.
II:
p3=1
. Then, om
b 3
,
p1=p2=0
. One easily e i ies ha he s a egy-
p o ile
(p1,p2,p3)=(
0, 0, 1
)
is indeed a Nash equilib ium.
The se o NE inmixed s a egies o p oblems
PIA
2
One can easily e i y ha he no mal- o m game is gi en by Tables23 and 24. Thus,
he bes esponse co espondences a e as ollows:
b
1(p2,p3)=
{
1 i p3≠1
[0,1] i p3=1
b
2(p1,p3)={1 i p1≠1 and p3≠
1
[0,1] i p1=1 o p3=1
b
3(p1,p2)=
{
0 i p1≠0 o p2≠0
[0,1] i p
1
=0 and p
2
=
0
Table 21 Playe 3 plays
s1,s2
1∖2
s1,s2
s2,s1
s1,s2
x,x,x
x,x,x
s2,s1
x,x,x
x,x,x
Table 22 Playe 3 plays
s2,s1
1∖2
s1,s2
s2,s1
s1,s2
1,
x
,1
1,
x
,1
s2,s1
x,1,1
x,x,x
Table 23 Playe 3 plays
s1,s2
1∖2
s1,s2
s2,s1
s1,s2
0,
x
,x
0,
x
,x
s2,s1
x
, 0, x
0,
x
,x

619
1 3
Cons ained school choice: anexpe imen al QRE analysis
We compu e he se o Nash equilib ia by checking he h ee Cases I, II, and III
below. Le
(p1,p2,p3)
be a Nash equilib ium.
I:
1−p3>xp2
. Since
p2≥0
and
x>0
,
p3<1
. F om
b 1
,
p1=1
. Since
(x+1)p1p3=(x+1)p3<x+p3=x−1+p1+p3
, i ollows om
b 2
ha
p2=0
. Then, om
b 3
,
p3=1
, which con adic s
p3<1
. Hence, he e is no
equilib ium in his case.
II:
1−p3<xp2
. Since
p3≤1
,
p2>0
. F om
b 1
,
p1=0
. F om
b 3
,
p3=1
. Since
(x+1)p1p3=0<x=x−1+p1+p3
, i ollows om
b 2
ha
p2=0
, which
con adic s
p2>0
. Hence, he e is no equilib ium in his case.
III:
1−p3=xp2
. We dis inguish among h ee subcases.
• Subcase
p2=0
. Since
1−p3=xp2
,
p3=1
. One easily e i ies ha o each
p1∈[0, 1]
, he s a egy-p o ile
(p1,p2,p3)=(p1,0,1)
is indeed a Nash equi-
lib ium.
• Subcase
p2=1
. Since
1−p3=xp2
,
p3∈(0, 1)
. Hence, om
b 3
,
x(p1+p2)=(1+x)p1p2
, i.e.,
p1=x∈(0, 1)
. Then, om
b 2
,
(x+1)p1p3=x−1+p1+p3
. Thus,
(x+1)xp3=p3−1
. In he case ha
x=0.3
, we ob ain
p3>1
, and in he case ha
x=0.7
, we ob ain
p3<0
. In
ei he case, his yields a con adic ion wi h
p3∈(0, 1)
. Hence, he e is no
equilib ium in his subcase.
• Subcase
p2∈(0, 1)
. Since
1−p3=xp2
,
p3∈(0, 1)
. F om
b 3
,
x(p1+p2)=(1+x)p1p2
. F om
b 2
,
(x+1)p1p3=x−1+p1+p3
. Using he
la e equali y, subs i u ing
p3=1−xp2
, applying s aigh o wa d simpli i-
ca ions, and inally subs i u ing
x(p1+p2)=(1+x)p1p2
yields p
1
p
2=x
1−x
2 .
The e o e,
b
1(p2,p3)=
⎧
⎪
⎨
⎪
⎩
1 i 1 −p3>xp2
[0,1] i 1 −p3=xp2
0 i 1 −p3<xp2
b
2(p1,p3)=
⎧
⎪
⎨
⎪
⎩
1 i (x+1)p1p3>x−1+p1+p
3
[0,1] i (x+1)p1p3=x−1+p1+p
3
0 i (x+1)p1p3<x−1+p1+p
3
b
3(p1,p2)=
⎧
⎪
⎨
⎪
⎩
1 i x(p1+p2)>(1+x)p1p2
[0,1] i x(p1+p2)=(1+x)p1p2
0 i x(p1+p2)<(1+x)p1p2
Table 24 Playe 3 plays
s2,s1
1∖2
s1,s2
s2,s1
s1,s2
1, 0, 1
1,
x
,0
s2,s1
x,1,0
0, x,x
620
J.Alcalde-Unzu e al.
1 3
Hence, p1+p2=
x
(1−x)
. Finally, i is easy o check ha o each o
x=0.3
and
x=0.7
, he unique solu ion
(p1,p2)
o he sys em p
1
p
2=x
1−x
2 and
p1
+
p2
=x
(1−x)
sa is ies
p2>1
which con adic s
p2∈(0, 1)
. Hence, he e is
no equilib ium in his subcase.
The se o NE inmixed s a egies o p oblem
PDA
2
One can easily e i y ha each p o ile o pu e s a egies induces payo s (0,x,x).
The e o e, each p o ile o mixed s a egies is a Nash equilib ium.
The se o NE inmixed s a egies o p oblems
PIA
3
and
PDA
3
One can easily e i y ha he no mal- o m game is gi en by Tables25 and 26. No e
ha epo ing u h ully is a weakly dominan s a egy. The bes esponse co e-
spondences a e as ollows:
We compu e he se o Nash equilib ia by checking he wo cases I and II below. Le
(p1,p2,p3)
be a Nash equilib ium.
x
(p1+p2)=(1+x)p1p2=(1+x)
x
1−x
2=(1+x)
x
(1−x)(1+x)
=
x
(1−x)
.
b
1(p2,p3)=
{
1 i p2≠0
[0,1] i p2=0
b
2(p1,p3)={1 i p1≠1 o p3≠1
[0,1] i p1=1 and p3=
1
b
3(p1,p2)=
{
0 i p1≠0 and p2≠
0
[0,1] i p
1
=0 o p
2
=0
Table 25 Playe 3 plays
s1,s2
1∖2
s1,s2
s2,s1
s1,s2
1,
x
,x
1,
x
,x
s2,s1
x,1,x
1,x,x
Table 26 Playe 3 plays
s2,s1
1∖2
s1,s2
s2,s1
s1,s2
1,1,1
1,x,x
s2,s1
x,1,x
1,x,x
621
1 3
Cons ained school choice: anexpe imen al QRE analysis
Table 27 Lis s o leng h 2
P obabili y dis ibu ion o e ma chings induced by Nash equilib ia in mixed s a egies
(
p
1,
p
2,
p
3)
. Recall
ha in
P2
,
𝜇I=𝜇S
P oblem T ea men
(p1,p2,p3)
𝜇I
𝜇S
Uns able
PIA
1
,P
DA
1
L,H
(1, p2,0)
o each
p2∈[0, 1]
1 0 0
L,H(1,0,1) 0 1 0
PIA
2
L,H
(p1,0,1)
o each
p1∈[0, 1]
1 1 0
PDA
2
L,H
(p1,p2,p3)
o each
p1,p2,p3∈[0, 1]
1 1 0
PIA
3
,P
DA
3
L,H(1,1,0) 1 0 0
L,H(1,0,1) 0 1 0
Panel 5: P oblem 3 − T ea men L Panel 6: P oblem 3 − T ea men H
Panel 3: P oblem 2 − T ea men L Panel 4: P oblem 2 − T ea men H
Panel 1: P oblem 1 − T ea men L Panel 2: P oblem 1 − T ea men H
15913172125291591317212
52
9
0.1
0.3
0.5
0.7
0.9
0.1
0.3
0.5
0.7
0.9
0.1
0.3
0.5
0.7
0.9
λ
pi
∗ (λ|Pj, T)
Fig. 7 Lis s o leng h 2. Logi -QRE p obabili ies unde he immedia e accep ance mechanism. Colo
scheme:
(p∗
1,p∗
2,p∗
3)→
(black, da k-g ay, ligh -g ay)
622
J.Alcalde-Unzu e al.
1 3
I:
p2≠0
. F om
b 1
,
p1=1
. Then, om
b 3
,
p3=0
. Thus, om
b 2
,
p2=1
. One
easily e i ies ha he s a egy-p o ile
(p1,p2,p3)=(1, 1, 0)
is indeed a Nash
equilib ium.
II:
p2=0
. Then, om
b 2
,
p1=p3=1
. One easily e i ies ha he s a egy-
p o ile
(p1,p2,p3)=(1, 0, 1)
is indeed a Nash equilib ium.
Funding Open Access unding p o ided hanks o he CRUE-CSIC ag eemen wi h Sp inge Na u e. The
au ho s g a e ully acknowledge inancial suppo om Fundación Ramón A eces. J. Alcalde-Unzu g a e-
ully acknowledges inancial suppo om Minis e io de Ciencia, Inno ación y Uni e sidades (PGC2018-
093542-B-I00 and PID2021-127119NB-I00). F. Klijn g a e ully acknowledges inancial suppo om
AGAUR–Gene ali a de Ca alunya (2017-SGR-1359 and 2021-SGR-00416) and he Spanish Agencia
Es a al de In es igación (AEI) h ough g an s ECO2017-88130-P and PID2020-114251GB-I00 and he
Se e o Ochoa P og amme o Cen es o Excellence in R&D (Ba celona School o Economics CEX2019-
000915-S). M. Vo sa z g a e ully acknowledges inancial suppo om Minis e io de Ciencia, Inno ación
y Uni e sidades (PGC2018-096977-B-I00 and PID2021-122919NB-I00).
Decla a ions
Con lic o in e es The au ho s ha e no ele an inancial o non- inancial in e es s o disclose.
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License,
which pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as
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Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps
and ins i u ional a ilia ions.
Au ho s and A ilia ions
Jo geAlcalde‑Unzu1 · FlipKlijn2 · Ma cVo sa z3
Jo ge Alcalde-Unzu
jo ge.alcalde@una a a.es
Ma c Vo sa z
m [email p o ec ed]
1 Depa men o Economics andINARBE, Public Uni e si y o Na a e, Campus A osadia,
31006Pamplona, Spain
2 Ins i u e o Economic Analysis (CSIC) andBa celona School o Economics, Campus UAB,
08193Bella e a(Ba celona), Spain
3 Depa amen o de Análisis Económico, Uni e sidad Nacional de Educación aDis ancia
(UNED), Paseo Senda del Rey 11, 28040Mad id, Spain