Constrained school choice: an experimental QRE analysis

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 1 3 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 1 3 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). 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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