College Admissions with Entrance Exams: Centralized versus Decentralized

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1 Is E. Hflir Rustmdjn Hkimov Dorothe Kübler Morimitsu Kurino College Admissions with Entrnce Exms: Centrlized versus Decentrlized Discussion Pper SP II October 2014 (WZB Berlin Socil Science Center Reserch Are Mrkets nd Choice Reserch Unit Mrket Behvior

2 Wissenschftszentrum Berlin für Sozilforschung ggmbh Reichpietschufer Berlin Germny Copyright remins with the uthor(s. Discussion ppers of the WZB serve to disseminte the reserch results of work in progress prior to publiction to encourge the exchnge of ides nd cdemic debte. Inclusion of pper in the discussion pper series does not constitute publiction nd should not limit publiction in ny other venue. The discussion ppers published by the WZB represent the views of the respective uthor(s nd not of the institute s whole. Is E. Hflir, Rustmdjn Hkimov, Dorothe Kübler, Morimitsu Kurino College Admissions with Entrnce Exms: Centrlized versus Decentrlized Affilition of the uthors: Is E. Hflir Crnegie Mellon University, Pittsburgh Rustmdjn Hkimov WZB Berlin Socil Science Center Dorothe Kübler WZB Berlin Socil Science Center nd Technicl University Berlin Morimitsu Kurino University of Tsukub, Jpn

3 Wissenschftszentrum Berlin für Sozilforschung ggmbh Reichpietschufer Berlin Germny Abstrct College Admissions with Entrnce Exms: Centrlized versus Decentrlized by Is E. Hflir, Rustmdjn Hkimov, Dorothe Kübler nd Morimitsu Kurino * We theoreticlly nd experimentlly study college dmissions problem in which colleges ccept students by rnking students efforts in entrnce exms. Students hold privte informtion regrding their bility level tht ffects the cost of their efforts. We ssume tht student preferences re homogeneous over colleges. By modeling college dmissions s contests, we solve nd compre the equilibri of centrlized college dmissions (CCA in which students pply to ll colleges, nd decentrlized college dmissions (DCA in which students cn only pply to one college. We show tht lower bility students prefer DCA wheres higher bility students prefer CCA. The min qulittive predictions of the theory re supported by the experimentl dt, yet we find number of behviorl differences between the mechnisms tht render DCA less ttrctive thn CCA compred to the equilibrium benchmrk. Keywords: College dmissions, incomplete informtion, student welfre, contests, ll-py uctions, experiment JEL clssifiction: C78; D47; D78; I21 * E-mil: [email protected], [email protected], [email protected], [email protected].

4 1 Introduction Throughout the world nd every yer, millions of prospective university students pply for dmission to colleges or universities during their lst yer of high school. Admission mechnisms vry from country to country, yet in most countries there re government gencies or independent orgniztions tht offer stndrdized dmission exms to id the college dmission process. Students invest lot of time nd effort to do well in these dmission exms, nd they re heterogeneous in terms of their bility to do so. In some countries, the ppliction nd dmission process is centrlized. For instnce, in Turkey university ssignment is solely determined by ntionl exmintion clled YGS/LYS. After lerning their scores, students cn pply to number of colleges. Applictions re lmost costless s ll students need only to submit their rnk-order of colleges to the centrl uthority. 1 On the other hnd, Jpn hs centrlized Ntionl Center test, too, but ll public universities including most prestigious universities require the cndidte to tke nother, institution-specific secondry exm which tkes plce on the sme dy. This effectively prevents the students from pplying to more thn one public university. 2 The dmissions mechnism in Jpn is decentrlized, in the sense tht colleges decide on their dmissions independent of ech other. In the United Sttes, students tke both centrlized exms like the Scholstic Aptitude Test (SAT, nd lso complete college-specific requirements such s college dmission essys. Students cn pply to more thn one college, but since the ppliction process is costly, students typiclly send only few pplictions (the mjority being between two to six pplictions, see Chde, Lewis, nd Smith, Hence, the United Sttes college dmissions mechnism flls in between the two extreme cses. In this pper, we compre the institutionl effects of different college dmission mechnisms on the equilibrium efforts of students nd student welfre. To do this, we model college dmissions with dmission exms s contests (or ll-py uctions in which the cost of effort represents the pyment mde by the students. We focus on two extreme cses: in the centrlized model (s in the Turkish mechnism students cn freely pply to ll colleges, wheres in the decentrlized model (s in the Jpnese mechnism for public colleges students cn only pply to one college. For simplicity, in our min model we consider two colleges tht differ in qulity nd ssume tht 1 Greece, Chin, South Kore, nd Tiwn hve similr ntionl exms tht re the min criterion for the centrlized mechnism of college dmissions. In Hungry, the centrlized dmission mechnism is bsed on score tht combines grdes from school with n entrnce exm (Biro, There re ctully two stges where the structure of ech stge is s explined in Section 4. The difference between the stges is tht the cpcities in the first stge re much greter thn those in the second stge. Those who do not get dmission to ny college spend one yer prepring for the next yer s exm. Moreover, the Jpnese high school dmissions uthorities hve dopted similr mechnisms in locl districts. Although the mechnism dopted vries cross prefectures nd is chnging yer by yer, its bsic structure is tht ech student chooses one mong specified set of public schools nd then tkes n entrnce exm t his or her chosen school. The exms re held on the sme dy. Finlly, institution-specific exms tht prevent students from pplying to ll colleges hve lso been used nd debted in the United Kingdom, notbly between the University of Cmbridge nd the University of Oxford. We thnk Ken Binmore for pointing this out. 2

5 students hve homogeneous preferences for ttending these colleges. 3 More specificlly, ech of the n students gets utility of v 1 by ttending college 1 (which cn ccommodte q 1 students nd gets utility of v 2 by ttending college 2 (which cn ccommodte q 2 students. We suppose 0 < v 1 < v 2, nd hence college 2 is the better nd college 1 is worse of the two colleges. Students utility from not being ssigned to ny college is normlized to 0. Following the mjority of the literture on contests with incomplete informtion, we suppose tht n bility level in the intervl [0, 1], is drwn i.i.d. from the common distribution function, nd the cost of exerting n effort e for student with bility level is given by e. Thus, given n effort level, the higher the bility the lower the cost of exerting the effort. In the centrlized college dmissions problem (CCA, ll students rnk college 2 over college 1. Hence, the students with the highest q 2 efforts get into college 2, students with the next highest q 1 efforts get into college 1, nd students with the lowest n q 1 q 2 efforts re not ssigned to ny college. In the decentrlized college dmissions problem (DCA, students need to simultneously choose which college to pply to nd how much effort to exert. Then, for ech college i {1, 2}, students with the highest q i efforts mong the pplicnts to college i get into college i. It turns out tht the equilibrium of CCA cn be solved by stndrd techniques, such s in Moldovnu, Sel, nd Shi (2012. In this monotone equilibrium, higher bility students exert higher efforts, nd therefore the students with the highest q 2 bility levels get dmitted to the good college (college 2, nd students with bility rnkings between q 2 +1 nd q 1 +q 2 get dmitted to the bd college (college 1 (Proposition 1. Finding the equilibrium of DCA is not strightforwrd. It turns out tht in equilibrium, there is cutoff bility level tht we denote by c. All higher bility students (with bilities in (c, 1] pply to the good college, wheres lower bility students (with bility levels in [0, c] use mixed strtegy when choosing between the good nd the bd college. Students effort functions re continuous nd monotone in bility levels (Theorem 1. Our pper therefore contributes to the ll-py contests literture. To the best of our knowledge, ours is the first pper to model nd solve competing contests where the plyers hve privte informtion regrding their bilities nd sort themselves into different contests. After solving for the equilibrium of CCA nd DCA, we compre the equilibri in terms of students interim expected utilities. We show tht students with lower bilities prefer DCA to CCA when the number of sets is smller thn the number of students (Proposition 2. The min intuition for this result is tht students with very low bilities hve lmost no chnce of getting set in CCA, wheres their probbility of getting set in DCA is bounded wy from zero due to the fewer number of pplictions thn the cpcity. Moreover, we show tht students with higher bilities prefer CCA to DCA (Proposition 3. 4 The min intuition for this result is tht 3 In Section 6, we discuss the cse with three or more colleges. 4 More specificlly we obtin single crossing condition: if student who pplies to college 2 in the decentrlized mechnism prefers the centrlized mechnism to the decentrlized mechnism, then ll higher bility students lso 3

6 high-bility students (i cn only get set in the good school in DCA, wheres they cn get sets in both the good nd the bd school in CCA, nd (ii their equilibrium probbility of getting set in the good school is the sme cross the two mechnisms. We test the theory with the help of lb experiments. We implement five mrkets for the college dmissions gme tht re designed to cpture different levels of competition (in terms of the supply of sets, the demnd rtio, nd the qulity difference between the two colleges. We compre the two college dmission mechnisms nd find tht in most (but not ll mrkets, the comprisons of the students ex-nte expected utilities, their effort levels, nd the students preferences regrding the two college dmission mechnisms re well orgnized by the theory. However, the experimentl subjects exert higher effort thn predicted. The overexertion of effort is prticulrly pronounced in DCA, which mkes it reltively less ttrctive for the pplicnts compred to CCA. The rest of the pper is orgnized s follows. The introduction (Section 1 ends with discussion of the relted literture. Section 2 introduces the model nd preliminry nottion. In sections 3 nd 4 we solve the model for the Byesin Nsh equilibri of the centrlized nd decentrlized college dmission mechnisms, respectively. Section 5 offers comprisons of the equilibri of the two mechnisms. Section 6 discusses the cse of three or more colleges. Section 7 presents our experimentl results. Finlly, section 8 concludes. Omitted proofs re given in the Appendix. 1.1 Relted literture College dmissions hve been studied extensively in the economics literture. Following the seminl pper by Gle nd Shpley (1962, the theory literture on two-sided mtching minly considers centrlized college dmissions nd investigtes stbility, incentives, nd the efficiency properties of vrious mechnisms, notbly the deferred-cceptnce nd the top trding cycles lgorithms. The student plcement nd school choice literture is motivted by the centrlized mechnisms of public school dmissions, rther thn by the decentrlized college dmissions mechnism in the US. This literture ws pioneered by Blinski nd Sönmez (1999 nd Abdulkdiroğlu nd Sönmez (2003. We refer the reder to Sönmez nd Ünver (2011 for recent comprehensive survey regrding centrlized college dmission models in the two-sided mtching literture. Recent work regrding centrlized college dmissions with entrnce exms include Abizd nd Chen (2011 nd Tung (2009. Abizd nd Chen (2011 model the entrnce (eligibility criterion in college dmissions problems nd extend models of Perch, Polk, nd Rothblum (2007 nd Perch nd Rothblum (2010 by llowing the students to hve the sme scores from the centrl exm. On the other hnd, by llowing students to submit their preferences fter they receive the test results, Tung (2009 djusts multi-ctegory seril dicttorship (MSD nlyzed by Blinski nd Sönmez (1999 in order to mke students better off. One crucil difference between the modelling in our pper nd the literture should be emhve the sme preference rnking. 4

7 phsized: In our pper student preferences ffect college rnkings over students through contests mong students, while student preferences nd college rnkings re typiclly independent in the two-sided mtching models nd school-choice models. The nlysis of decentrlized college dmissions in the literture is more recent. Chde, Lewis, nd Smith (2014 consider model where two colleges receive noisy signls bout the cliber of pplicnts. Students need to decide which colleges to pply to nd ppliction is costly. The two colleges choose dmissions stndrds tht ct like mrket-clering prices. The uthors show tht in equilibrium, college-student sorting my fil, nd they lso nlyze the effects of ffirmtive ction policies. In our model, the colleges re not strtegic plyers s in Chde, Lewis, nd Smith (2014. Another importnt difference is tht in our model the students do not only hve to decide which colleges to pply to, but lso how much effort to exert in order to do well in the entrnce exms. Che nd Koh (2013 study model in which two colleges mke dmission decisions subject to ggregte uncertinty bout student preferences nd liner costs for ny enrollment exceeding the cpcity. They find tht colleges dmission decisions become tool for strtegic yield mngement, nd in equilibrium, colleges try to reduce their enrollment uncertinty by strtegiclly trgeting students. In their model, s in Chde, Lewis, nd Smith (2014, students exm scores re costlessly obtined nd given exogenously. Avery nd Levin (2010, on the other hnd, nlyze model of erly dmission t selective colleges where erly dmission progrms give students n opportunity to signl their enthusism to the college they would like to ttend. In nother relted pper, Hickmn (2009 lso models college dmissions s Byesin gme where heterogeneous students compete for sets t colleges. He presents model in which there is n lloction mechnism mpping ech student s score into set t college. Hickmn (2009 is mostly interested in the effects of ffirmtive ction policies, nd the solution concept used is pproximte equilibrium in which the number of students is ssumed to be lrge so tht students pproximtely know their rnkings within the relized smple of privte costs. 5 In our pper, we do not require the number of students to be lrge. In nother recent pper by Slgdo-Torres (2013, students nd colleges prticipte in decentrlized mtching mechnism clled Costly Signling Mechnism (CSM in which students first choose costly observble score to signl their bilities, then ech college mkes n offer to student, nd finlly ech student chooses one of the vilble offers. Slgdo-Torres (2013 chrcterizes symmetric equilibrium of CSM which is proven to be ssertive, nd lso performs some comprtive sttics nlysis. CSM is decentrlized just like the decentrlized college dmissions model developed in this pper. However, CSM cnnot be used to model college dmission mechnisms (such s the ones used in Jpn tht require students to pply to only one college. Our pper is lso relted to the ll-py uction nd contests literture. Notbly, Bye, 5 In relted pper, Morgn, Sisk, nd Vrdy (2012 study competition for promotion in continuum economy. They show tht more meritocrtic profession lwys succeeds in ttrcting the highest bility types, wheres profession with superior promotion benefits ttrcts high types only under some ssumptions. 5

8 Kovenock, nd de Vries (1996 nd Siegel (2009 solve for ll-py uctions nd contests with complete informtion. We refer the reder to the survey by Konrd (2009 bout the vst literture on contests. Relted to our decentrlized mechnism, Amegshie nd Wu (2006 nd Konrd nd Kovenock (2012 both model competing contests in complete informtion setting. Amegshie nd Wu (2006 study model where one contest hs higher prize thn the other. They show tht sorting my fil in the sense tht the top contestnt my choose to prticipte in the contest with lower prize. In contrst, Konrd nd Kovenock (2012 study ll-py contests tht re run simultneously with multiple identicl prizes. They chrcterize set of pure strtegy equilibri, nd symmetric equilibrium tht involves mixed strtegies. In our decentrlized college dmissions model, the corresponding contest model is lso model of competing contests. The min difference in our model is tht we consider incomplete informtion s students do not know ech other s bility levels. A series of ppers by Moldovnu nd Sel (nd Shi studies contests with incomplete informtion, but they do not consider competing contests in which the prticiption in contests is endogenously determined. In Moldovnu nd Sel (2001, the contest designer s objective is to mximize expected effort. They show tht when cost functions re liner or concve in effort, it is optiml to llocte the entire prize sum to single first prize. Moldovnu nd Sel (2006 compre the performnce of dynmic sub-contests whose winners compete ginst ech other with sttic contests. They show tht with liner costs of effort, the expected totl effort is mximized with sttic contest, wheres the highest expected effort cn be higher with contests with two divisions. Moldovnu, Sel, nd Shi (2012 study optiml contest design where both wrds nd punishments cn be used. Under some conditions, they show tht punishing the bottom is more effective thn rewrding the top. This pper lso contributes to lrge experimentl literture on contests nd ll-py uctions, summrized in recent survey rticle by Dechenux, Kovenock, nd Sheremet (2012. Our setup in the centrlized mechnism with heterogeneous gents, two non-identicl prizes, nd incomplete informtion is closely relted to number of existing studies by Brut, Kovenock, nd Noussir (2002, Noussir nd Silver (2006, nd Müller nd Schotter (2010. These studies observe tht gents overbid on verge compred to the Nsh prediction. Moreover, they find n interesting bifurction, term introduced by Müller nd Schotter (2010, in tht low types underbid nd high types overbid. Regrding the optiml prize structure, it turns out tht if plyers re heterogeneous, multiple prizes cn be optiml to void the discourgement of wek plyers, see Müller nd Schotter (2010. Higher effort with multiple prizes thn with single prize ws lso found in setting with homogeneous plyers by Hrbring nd Irlenbusch (2003. We re not wre of ny previous experimentl work relted to our decentrlized dmissions mechnism where gents simultneously hve to choose n effort level nd decide whether to compete for the high or the low prize. The pper lso belongs to the experimentl literture on two-sided mtching mechnisms nd 6

9 school choice strting with Kgel nd Roth (2000 nd Chen nd Sönmez ( These studies s well s mny follow-up ppers in this strnd of the literture focus on the rnk-order lists submitted by students in the preference-reveltion gmes, but not on effort choice. Thus, the rnkings of students by the schools re exogenously given in these studies unlike in our setup where the colleges rnkings re endogenous. 2 The Model The college dmissions problem with entrnce exms, or simply the problem, is denoted by (S, C, (q 1, q 2, (v 1, v 2, F. There re 2 colleges college 1 nd college 2. We denote colleges by C. Ech college C C := {1, 2} hs cpcity q C which represents the mximum number of students tht cn be dmitted to college C, where q C 1. There re n students. We denote the set of ll students by S. Since we suppose homogeneous preferences of students, we ssume tht ech student hs the crdinl utility v C from college C {1, 2}, where v 2 > v 1 > 0. Thus we sometimes cll college 2 the good college nd college 1 the bd college. Ech student s utility from not being ssigned to ny college is normlized to be 0. We ssume tht q 1 + q 2 n. 7 Ech student is ssigned to one college or no set in ny college by the mechnisms nd the mechnisms tke the efforts into ccount while deciding on their dmissions. 8 Ech student s S mkes n effort e s. The students re heterogeneous in terms of their bilities, nd the bilities re their privte informtion. More specificlly, for ech s S, s [0, 1] denotes student s s bility. Abilities re drwn identiclly nd independently from the intervl [0, 1] ccording to continuous distribution function F tht is common knowledge. We ssume tht F hs continuous density f = df > 0. For student s with bility s, putting in n effort of e s results in disutility of e s s. Hence, the totl utility of student with bility from mking effort e is v C e/ if she is ssigned to college C, nd e/ otherwise. Before we move on to the nlysis of the equilibrium of centrlized nd decentrlized college dmission mechnisms, we introduce some necessry nottion. 2.1 Preliminry nottion First, for ny continuous distribution T with density t, for 1 k m, let T k,m denote the distribution of the k th (lowest order sttistics out of m independent rndom vribles tht re 6 For recent exmple for theory nd experiments in school choice literture, see Chen nd Kesten ( Mny college dmissions, including ones in Turkey nd Jpn, re competitive in the sense tht totl number of sets in colleges is smller thn the number of students who tke the exms. 8 In relity the performnce in the entrnce exms is only noisy function of efforts. For simplicity, we ssume tht efforts completely determine the performnce in the tests. 7

10 identiclly distributed ccording to T. Tht is, T k,m ( := m j=k Moreover, let t k,m ( denote T k,m ( s density: 0. t k,m (x := d d T k,m( = ( m T ( j (1 T ( m j. (1 j m! (k 1! (m k! T (k 1 (1 T ( m k t(. (2 For convenience, we let T 0,m be distribution with T 0,m ( = 1 for ll, nd t 0,m dt 0,m /d = Next, define the function p j,k : [0, 1] [0, 1] s follows: given j, k {0, 1,..., n}, for ech x [0, 1], define ( j + k p j,k (x := x j (1 x k. (3 j The function p j,k (x is interpreted s the probbility tht when there re (j+k students, j students re selected for one event with probbility x nd k students re selected for nother event with probbility (1 x. Suppose tht p 0,0 (x = 1 for ll x. Note tht with this definition, we cn write T k,m ( = m p j,m j (T (. (4 j=k 3 The Centrlized College Admissions Mechnism (CCA In the centrlized college dmissions gme, ech student s S simultneously mkes n effort e s. Students with the top q 2 efforts re ssigned to college 2 nd students with the efforts from the top (q to (q 1 + q 2 re ssigned to college 1. The rest of the students re not ssigned to ny colleges. 9 We now solve for the symmetric Byesin Nsh equilibrium of this gme. The following proposition is specil cse of the ll-py uction equilibrium which hs been studied by Moldovnu nd Sel (2001 nd Moldovnu, Sel, nd Shi (2012. Proposition 1. In CCA, there is unique symmetric equilibrium β C such tht for ech [0, 1], 9 In setup with homogeneous student preferences, this gme reflects how the Turkish college dmission mechnism works. In the centrlized test tht the students tke, since ll students would put college 2 s their top choice nd college 1 s their second top choice in their submitted preferences, the resulting ssignment would be the sme s the ssignment described bove. In school choice context, this cn be described s the following two-stge gme. In the first stge, there is one contest where ech student s simultneously mkes n effort e s. The resulting effort profile (e s s S is used to construct single priority profile such tht student with higher effort hs higher priority. In the second stge, students prticipte in the centrlized deferred cceptnce mechnism where colleges use the common priority. 8

11 ech student with bility chooses n effort β C ( ccording to β C ( = ˆ 0 x {f n q2,n 1(x v 2 + (f n q1 q 2,n 1(x f n q2,n 1(x v 1 } dx. where f k,m ( for k 1 is defined in Eqution (2 nd f 0,m (x is defined to be 0 for ll x. Proof. Suppose tht β C is symmetric equilibrium effort function tht is strictly incresing. Consider student with bility who chooses n effort s if her bility is. Her expected utility is v 2 F n q2,n 1( + v 1 (F n q1 q 2,n 1( F n q2,n 1( βc (. The first-order condition t = is v 2 f n q2,n 1( + v 1 (f n q1 q 2,n 1( f n q2,n 1( [βc (] Thus, by integrtion nd s the boundry condition is β C (0 = 0, we hve β C ( = ˆ 0 = 0. x {f n q2,n 1(x v 2 + (f n q1 q 2,n 1(x f n q2,n 1(x v 1 } dx. The bove strtegy is the unique symmetric equilibrium cndidte obtined vi the first-order pproch by requiring no benefit from locl devitions. Stndrd rguments show tht this is indeed n equilibrium by mking sure tht globl devitions re not profitble (for instnce, see Section 2.3 of Krishn ( The Decentrlized College Admissions Mechnism (DCA In the decentrlized college dmissions gme, ech student s chooses one college C s nd n effort e s simultneously. Given the college choices of students (C s s S nd efforts (e s s S, ech college C dmits students with the top q C effort levels mong its set of pplicnts ({s S C s = C}. 10 For this gme, we solve for symmetric Byesin Nsh equilibrium (γ(, β D ( ; c where c (0, 1 is cutoff, γ : [0, c] (0, 1 is the mixed strtegy tht represents the probbility of lower bility students pplying to college 1, nd β D : [0, 1] R is the continuous nd strictly incresing effort function. Ech student with type [0, c] chooses college 1 with probbility γ( (hence 10 In setup with homogeneous student preferences, this gme reflects how the Jpnese college dmissions mechnism works: ll public colleges hold their own tests nd ccept the top performers mong the students who tke their tests. In school choice context, this cn be described s the following two-stge gme. In the first stge, students simultneously choose which school to pply to, nd without knowing how mny other students hve pplied, they lso choose their effort level. For ech school C {1, 2}, the resulting effort profile (e s {s S Cs=C} is used to construct one priority profile C such tht student with higher effort hs higher priority. In the second stge, students prticipte in two seprte deferred cceptnce mechnisms where ech college C uses the priority C. 9

12 chooses college 2 with probbility 1 γ(, nd mkes effort β D (. (c, 1] chooses college 2 for sure, nd mkes effort β D (. 11 Ech student with type We now move on to the derivtion of symmetric Byesin Nsh equilibrium. Let symmetric strtegy profile (γ(, β( ; c be given. For this strtegy profile, the ex-nte probbility tht student pplies to college 1 is c γ(xf(xdx, while the probbility tht student pplies to 0 c college 2 is 1 γ(xf(xdx. Let us define function π : [0, c] [0, 1] tht represents the ex-nte 0 probbility tht student hs type less thn nd she pplies to college 1: π( := ˆ 0 γ(xf(xdx. (5 With this definition, the ex-nte probbility tht student pplies to college 1 is π(c, while the probbility tht student pplies to college 2 is 1 π(c. Moreover, p m,k (π(c is the probbility tht m students pply to college 1 nd k students pply to college 2 where p m,k ( is given in Eqution (3 nd π( is given in Eqution (5. Next, we define G( : [0, c] [0, 1], where G( is the probbility tht type is less thn or equl to, conditionl on the event tht she pplies to college 1. Tht is, G( := π( π(c. Moreover let g( denote G( s density. G k,m is the distribution of the k th order sttistics out of m independent rndom vribles tht re identiclly distributed ccording to G s in equtions (1 nd (4. Also, g k,m ( denotes G k,m ( s density. Similrly, let us define H( : [0, 1] [0, 1], where H( is the probbility tht type is less thn or equl to, conditionl on the event tht she pplies to college 2. Tht is, for [0, 1], H( = F ( π( 1 π(c if [0, c], F ( π(c 1 π(c if [c, 1]. Moreover, let h( denote H( s density. Note tht h is continuous but is not differentible t c. Let H k,m be the distribution of the k th order sttistics out of m independent rndom vribles distributed ccording to H s in equtions (1 nd (4. Also, h k,m ( denotes H k,m ( s density. 11 A nturl equilibrium cndidte is to hve cutoff c (0, 1, students with bilities in [0, c to pply to college 1, nd students with bilities in [c, 1] to pply to college 2. It turns out tht we cnnot hve n equilibrium of this kind. In such n equilibrium, (i type c hs to be indifferent between pplying to college 1 or college 2, (ii type c s effort is strictly positive in cse of pplying to college 1, nd 0 while pplying to college 2, hence there is discontinuity in the effort function. These two conditions together imply tht type c + ɛ student would benefit from mimicking type c ɛ student for smll enough ɛ. Forml rguments resulting in the nonexistence result re vilble from the uthors upon request. Therefore, we hve to hve some students using mixed strtegies while choosing which college to pply to. Derivtions show tht in equilibrium, lower bility students would use mixed strtegies, while the higher bility students re certin to pply to the better school. 10

13 We re now redy to stte the min result of this section, which chrcterizes the unique symmetric Byesin Nsh equilibrium 12 of the decentrlized college dmissions mechnism. The sketch of the proof follows the Theorem, wheres the more technicl prt of the proof is relegted to Appendix B. Theorem 1. In DCA, there is unique symmetric equilibrium (γ, β D ; c where student with type [0, c] chooses college 1 with probbility γ( nd mkes effort β D (; nd student with type [c, 1] chooses college 2 for sure nd mkes effort β D (. Specificlly, ˆ β D ( = v 2 x 0 n 1 m=q 2 p n m 1,m (π(ch m q2 +1,m(xdx. The equilibrium cutoff c nd the mixed strtegies γ( re determined by the following four requirements: (i π(c uniquely solves the following eqution for x q 1 1 v 1 m=0 q 2 1 p m,n m 1 (x = v 2 m=0 (ii Given π(c, c uniquely solves the following eqution for x q 2 1 v 1 = v 2 m=0 n 1 p n m 1,m (π(c + v 2 m=q 2 p n m 1,m (π(c p n m 1,m (x. m j=m q 2 +1 ( F (x π(c p j,m j. 1 π(c (iii Given π(c nd c, for ech [0, c, π( uniquely solves the following eqution for x( n 1 v 2 m=q 2 p n m 1,m (π(c m j=m q 2 +1 (iv Finlly, for ech [0, c], γ( is given by γ( = ( F ( x( n 1 p j,m j = v 1 p m,n m 1 (π(c 1 π(c m=q 1 π(cb( (1 π(ca( + π(cb( (0, 1, m j=m q 1 +1 ( x( p j,m j. π(c 12 More specificlly, we chrcterize the unique equilibrium in which (i students use mixed strtegy while deciding which college to pply to, nd (ii effort levels re independent of college choice nd monotone incresing in bilities. 11

14 where n 1 A( := v 1 n 1 B( := v 2 ( π( p m,n m 1 (π(c m p m q1,q 1 1, π(c m=q 1 ( F ( π( p n m 1,m (π(c m p m q2,q π(c m=q 2 Proof. Suppose tht ech student with type [0, 1] follows strictly incresing effort function β D nd type [0, c] chooses college 1 with probbility γ( (0, 1, nd type in (c, 1] chooses college 2 for sure. We first show how to obtin the equilibrium cutoff c nd the mixed strtegy function γ. A necessry condition for this to be n equilibrium is tht ech type [0, c] hs to be indifferent between pplying to college 1 or 2. Thus, for ll [0, c],. ( q1 1 v 1 m=0 ( q2 1 = v 2 p m,n m 1 (π(c + m=0 p n m 1,m (π(c + n 1 m=q 1 p m,n m 1 (π(cg m q1 +1,m( n 1 m=q 2 p n m 1,m (π(ch m q2 +1,m(. (6 The left-hnd side is the expected utility of pplying to college 1, while the right-hnd side is the expected utility of pplying to college 2. To see this, note tht q 1 1 m=0 p m,n m 1(π(c nd q 2 1 m=0 p n m 1,m(π(c re the probbilities tht there re no more thn (q 1 1 nd (q 2 1 pplicnts in colleges 1 nd 2, respectively. For m q 1, p m,n m 1 (π(cg m q1 +1,m( is the probbility of getting set in college 1 with effort when there re m other pplicnts in college 1. Similrly, for m q 2, p n m 1,m (π(ch m q2 +1,m( is the probbility of getting set in college 2 with effort, when there re m other pplicnts in college 2. Note tht we hve G m q1 +1,m( = m j=m q 1 +1 ( π( p j,m j π(c nd H m q2 +1,m( = m j=m q 2 +1 for ll [0, c]. The eqution (6 t = 0 nd = c cn hence be written s ( F ( π( p j,m j 1 π(c q 1 1 v 1 m=0 q 2 1 p m,n m 1 (π(c = v 2 m=0 q 2 1 n 1 v 1 = v 2 p n m 1,m (π(c + v 2 m=0 p n m 1,m (π(c, nd (7 m=q 2 p n m 1,m (π(c m j=m q 2 +1 ( F (c π(c p j,m j, (8 1 π(c 12

15 respectively. We show in Appendix B tht there is unique π(c tht stisfies Eqution (7, nd tht given π(c, the only unknown c vi F (c in Eqution (8 is uniquely determined. Moreover, using (7, we cn rewrite Eqution (6 s n 1 v 1 m=q 1 p m,n m 1 (π(c m j=m q 1 +1 ( π( n 1 p j,m j = v 2 p n m 1,m (π(c π(c m=q 2 m j=m q 2 +1 ( F ( π( p j,m j, 1 π(c for ll [0, c]. In Appendix B, we show tht given π(c nd c, for ech [0, c], there is unique π( tht stisfies Eqution (9 nd, moreover, tht we cn get the mixed strtegy function γ( by differentiting Eqution (9. Finlly we derive the unique symmetric effort function β D by tking first-order pproch in terms of G( nd H( which re determined by the equilibrium cutoff c nd the mixed strtegy function γ. Consider student with type [0, c]. A necessry condition for the strtegy to be n equilibrium is tht she does not wnt to mimic ny other type in [0, c]. Her utility mximiztion problem is given by mx [0,c] v 2 ( q2 1 m=0 p n m 1,m (π(c + n 1 m=q 2 p n m 1,m (π(ch m q2 +1,m( where the indifference condition (6 is used to clculte the expected utility. 13 βd (. (9 The first-order necessry condition requires the derivtive of the objective function to be 0 t =. Hence, n 1 v 2 m=q 2 p n m 1,m (π(ch m q2 +1,m( (βd ( Solving the differentil eqution with the boundry condition (which is β D (0 = 0, we obtin = 0. for ll [0, c] ˆ β D ( = v 2 x 0 n 1 m=q 2 p n m 1,m (π(ch m q2 +1,m(xdx 13 Equivlently, we cn write the mximiztion problem s mx [0,c] v 1 ( q1 1 m=0 p m,n m 1 (π(c + n 1 With the sme procedure, this gives the equivlent solution s m=q 1 p m,n m 1 (π(cg m q1+1,m( βd (, for ech [0, c]. ˆ β D ( = v 1 x 0 n 1 m=q 1 p m,n m 1 (π(cg m q1+1,m(xdx 13

16 Next, consider student with type [c, 1]. A necessry condition is tht she does not wnt to mimic ny other type in [c, 1]. Her utility mximiztion problem is then mx [c,1] v 2 ( q2 1 m=0 p n m 1,m (π(c + n 1 m=q 2 p n m 1,m (π(ch m q2 +1,m( βd (. Note tht lthough the objective function is the sme for types in [0, c] nd [c, 1], it is not differentible t the cutoff c. The first-order necessry condition requires the derivtive of the objective function to be 0 t =. Hence, n 1 v 2 m=q 2 p n m 1,m (π(ch m q2 +1,m( (βd ( Solving the differentil eqution with the boundry condition of continuity (which is β D (c = c v 2 x n 1 0 m=q 2 p n m 1,m (π(ch m q2 +1,m(xdx, we obtin for ech [c, 1]. ˆ β D ( = v 2 x 0 n 1 = 0. m=q 2 p n m 1,m (π(ch m q2 +1,m(xdx To complete the proof, we need to show tht not only locl devitions, but lso globl devitions cnnot be profitble. In Appendix B.2, we do tht nd hence show tht the uniquely derived symmetric strtegy (γ, β D ; c is indeed n equilibrium. 5 Comprisons As illustrted in sections 3 nd 4, the two mechnisms result in different equilibri. It is therefore nturl to sk how the two equilibri compre in terms of interim student welfre. We denote by EU C ( nd EU D ( the expected utility of student with bility under CCA nd DCA, respectively. Our first result concerns the preference of low-bility students. Proposition 2. Low-bility students prefer DCA to CCA if nd only if n > q 1 + q 2. Proof. First, let us consider the cse of n > q 1 + q 2. For this cse it is not difficult to see tht EU C (0 = 0 (becuse the probbility of being ssigned to ny college is zero, nd EU D (0 > 0 (becuse with positive probbility, type 0 will be ssigned to college. Since the utility functions re continuous, we cn then see tht there exists n ɛ > 0 such tht for ll x [0, ɛ], we hve EU D (x > EU C (x. Next, let us consider the cse of n = q 1 + q 2. For this cse, we hve EU C (0 = v 1. This is becuse with probbility 1, type 0 will be ssigned to college 1 by exerting 0 effort. Moreover, we 14

17 6 4 Effort Ability 6 4 EU Ability Effort in DCA Effort in CCA EU in DCA EU in CCA Figure 1: Efforts (left nd expected utility (right under CCA nd DCA Note: The figures were creted with the help of simultions for the following prmeters: n = 12, (q 1, q 2 = (5, 4, nd (v 1, v 2 = (5, 20. The equilibrium cutoff under DCA is clculted s c = hve EU D (0 < v 1. This is becuse type 0 should be indifferent between pplying to college 1 nd college 2, nd in the cse of pplying to college 1, the probbility of getting ssigned to college 1 is strictly smller thn 1. Since the utility functions re continuous, we cn then see tht there exists n ɛ > 0 such tht for ll x [0, ɛ], we hve EU C (x > EU D (x. Intuitively, when the sets re over-demnded (i.e., when n > q 1 +q 2, very low-bility students hve lmost no chnce of getting set in CCA, wheres their probbility of getting set in DCA is bounded wy from zero. Hence they prefer DCA. Although this result merely shows tht only students in the neighborhood of type 0 need to hve these kinds of preferences, explicit equilibrium clcultions for mny exmples (such s the mrkets we study in our experiments result in significnt proportion of low-bility students preferring DCA. We provide n explicit depiction of equilibrium effort levels nd interim expected utilities for specific exmple in Figure 1. Moreover, we estblish the reverse rnking for the high-bility students. Tht is, the highbility students prefer CCA in the following single-crossing sense: if student who pplies to college 2 in DCA prefers CCA to DCA, then ll higher bility students hve the sme preference rnking. Proposition 3. Let c be the equilibrium cutoff in DCA. We hve (i if EU C ( EU D ( for some > c, then EU C ( > EU D ( for ll >, nd (ii if EU C ( < EU D ( for some > c, then d d EU C ( > d d EU D (. 15

18 Proof. Let us define Then we hve K ( v 2 F n q2,n 1 (, L ( v 1 (F n q1 q 2,n 1 ( F n q2,n 1 (, M ( K ( + L (, ( q2 1 N ( = v 2 p n m 1,m (π (c + n 1 p n m 1,m (π (c H m q2 +1,m (. m=0 m=q 2 By integrtion by prts, we obtin EU C ( = M ( M (x xdx 0. EU C ( = 0 M (x dx. Similrly, nd by integrtion by prts, we obtin EU D ( = N ( N (x xdx 0, EU D ( = 0 N (x dx. Note tht, for > c, we hve N ( = K (. This is becuse students whose bility levels re greter thn c pply to college 2 in DCA, nd therefore set is grnted to student with bility level > c if nd only if the number of students with bility levels greter thn is not greter thn q 2. This is the sme condition in CCA, which is given by the expression K (!. (Also note tht we hve N ( K ( for < c, in fct we hve N ( > K (, but this is irrelevnt for wht follows. Now, for ny > c, we obtin d ( EU C ( = M ( d = K ( + L ( nd d ( EU D ( = N ( d = K (. 16

19 or Since L ( > 0, for ny > c, we hve d ( EU C ( > d ( EU D (, d d EU C ( + d d EU C ( > EU D ( + d d EU D (. This mens tht for ny > c, whenever EU C ( = EU D (, we hve d d EU C ( > d d EU D (. Then we cn conclude tht tht once EU C ( is higher thn EU D (, it cnnot cut through EU D ( from bove to below nd EU C ( lwys stys bove EU D (. To see this suppose EU C ( > EU D ( nd EU C ( < EU D ( for some > > c, then (since both EU C ( nd EU D ( re continuously differentible there exists (, such tht EU C ( = EU D ( nd d EU C ( < d EU D (, contrdiction. Hence (i is stisfied. Moreover, (ii is obviously d d stisfied since whenever EU C ( < EU D (, we hve to hve d EU C ( > d EU D (. d d Intuitively, since high-bility students (i cn only get set in the good college in DCA wheres they cn get set in both the good nd the bd college in CCA, nd (ii their equilibrium probbility of getting set in the good college is the sme cross the two mechnisms, they prefer CCA. One my lso wonder if there is generl ex nte utility rnking between DCA nd CCA. It turns out tht one cn find exmples where either DCA or CCA result in higher ex nte utility (or socil welfre The Cse of l Colleges Let us consider l colleges, 1,..., l, where ech college k hs the cpcity q k > 0 nd ech student gets the utility of v k from ttending college k (v l > v l 1 >... > v 2 > v 1 > 0. We conjecture 15 tht in the decentrlized mechnism there will be symmetric Byesin Nsh equilibrium ((γ k l k=1, βd, (c k l k=0 : (i c 0,..., c l re cutoffs such tht 0 = c 0 < c 1 <... < c l 1 < c l = 1; (ii β D is n effort function where ech student with bility mkes n effort level of β D (; (iii γ 1,..., γ l re mixed strtegies such tht for ech k {1,..., l 1}, ech student with bility [c k 1, c k pplies to college k with probbility γ k ( nd college k+1 with probbility 1 γ k (. Moreover, ech student with bility [c l 1, 1] pplies to college l, equivlently, γ l ( = 1. The equilibrium effort levels cn be identified s follows. Let k {1,..., l} be given. Let π k ( denote the ex-nte probbility tht student hs type less thn or equl to nd she pplies to college k. Then, π 1 ( = γ 0 1(xdF (x. For 14 Specific exmples re vilble from the uthors upon request. 15 As explined below, the strtegies re not formlly shown to be n equilibrium since we do not hve proof to show tht globl devitions re not profitble. 17

20 k {2,..., l} nd [c k 2, c k ], π k c ( = k 2 (1 γ k 1 (xdf (x if c k 1, ck 1 c k 2 (1 γ k 1 (xdf (x + c k 1 γ k (xdf (x if c k 1. We define H k to be the probbility tht type is less thn or equl to, conditionl on the event tht she pplies to college k: H k ( = πk ( π k (c k. In this equilibrium, ech student with bility [c k 1, c k ] exerts n effort of β D ( = β D (c k 1 + ˆ c k 1 x n 1 m=q k p m,n m 1 (π k (c k h k m q k +1,m(xdx where β D (0 = 0 nd h k m q k +1,m is the density of H k m q k +1,m. Similr to Theorem 1, it is possible to determine the formultion for cutoffs c 1,..., c l 1 nd mixed strtegies γ 1,..., γ l using the indifference conditions (See the Appendix C. This set of strtegies cn be shown to stisfy immunity for locl devitions, but prohibitively tedious rguments to check for immunity to globl devitions (s we hve done in the Appendix B prevent us from formlly proving tht it is indeed n equilibrium. By supposing n equilibrium of this kind, we cn ctully show tht propositions 2 nd 3 hold for l colleges. Proposition 2 trivilly holds, s students with the lowest bility levels get zero utility from CCA nd strictly positive utility from DCA. We cn lso rgue tht Proposition 3 holds since the students with bility levels [c l 1, 1] pply to college l only. This cn be observed by noting tht set is grnted to these students in college k if nd only if the number of students with bility levels greter thn is no greter thn q l, which is the sme condition in CCA. Hence, even in this more generl setup of l colleges, we cn rgue tht low-bility students prefer DCA wheres high-bility students prefer CCA. 7 The Experiment In this section, we present n experiment on college dmissions with entrnce exms. It is designed to test the results of the model nd generte further insights into the performnce of the centrlized (CCA nd the decentrlized college dmissions mechnism (DCA. In prticulr, we check which of the two mechnisms leds to higher student efforts nd welfre in the experiment. We investigte individul effort choices by the students in the two mechnisms s well s their choice of college in DCA. 18

21 7.1 Design of the experiment In the experiment, there re two colleges, college 1 (the bd college nd college 2 (the good college. There re 12 students who pply for positions, nd these students differ with respect to their bility. At the beginning, every student lerns her bility s. The bility of ech student is drwn from the uniform distribution over the intervl of 0 to 100. Students hve to choose n effort level e s tht determines their success in the ppliction process. The cost of effort is given by es s. In the centrlized college dmissions mechnism (CCA, ll students simultneously choose n effort level. Then the computer determines the mtching by dmitting the students with the highest effort levels to college 2 up to its cpcity q 2 nd the next best students, i.e., from rnk q to rnk q 1 + q 2, to college 1. All other students re unssigned. In the decentrlized college dmissions mechnism (DCA, the students simultneously decide not only on their effort level but lso on which of the colleges to pply to. The computer determines the mtching by ssigning the students with the highest effort mong those who hve pplied to college C, up to its cpcity q C. We implemented five different mrkets tht differ with respect to the totl number of open slots (q 1 + q 2, the number of slots t ech college (q 1 nd q 2 s well s the vlue of the colleges for the students (v 1 nd v 2. This llows us to investigte behvior under very different mrket conditions. Most relevnt from the point of view of the theoreticl predictions, we cn compre outcomes in mrkets where the number of students is equl to the number of sets (mrkets 1 nd 4 to mrkets with more students thn sets (mrkets 2, 3, nd 5. The prmeters in ech mrket were chosen so s to generte cler-cut predictions regrding the two min outcome vribles, effort nd the interim expected utility of ech student. In ech of the first four mrkets, one mechnism domintes the other in one of the two outcome vribles. The fifth mrket is designed to mke the two mechnisms s similr s possible. In order to provide vlid comprison of the observed verge effort nd utility levels in the mrkets where there is no dominnce reltionship, i.e., the cells in Tble 1 for which the predicted difference depends on the bility of the pplicnt, we compute the equilibrium effort nd utility levels for the reliztions of bilities in our experimentl mrkets. We then tke expected vlues given the relized bilities. Tble 1 provides n overview of the five mrkets together with the theoreticl predictions regrding the difference between CCA nd DCA. We employed between-subjects design. Students were rndomly ssigned either to the tretment with CCA or the tretment with DCA. In ech tretment, subjects plyed 15 rounds with one mrket per round. Ech of the five different mrkets ws plyed three times by every prticipnt, nd bilities were drwn rndomly for every round. These drws were independent, nd ech bility ws eqully likely. We employed the sme rndomly drwn bility profiles in both tretments in order to mke them s comprble s possible. Mrkets were plyed in blocks: first ll five 19

22 Tble 1: Overview of mrket chrcteristics Number of sets t [Vlue of] Predicted utility higher Predicted effort higher college 2 college 1 Mrket 1 6 [2000] 6 [1000] CCA depends; DCA in expecttion Mrket 2 2 [2000] 2 [1000] DCA no diff. in expecttion Mrket 3 2 [2000] 8 [1000] depends; DCA in expecttion CCA Mrket 4 3 [2000] 9 [1800] CCA DCA Mrket 5 9 [2000] 1 [1000] no diff. in expecttion no diff. in expecttion Notes: In columns 4 nd 5, one of the two mechnisms sometimes domintes the other for ll students, but the rnking of the mechnisms cn lso depend on the students bility. mrkets were plyed in rndom order once, then ll five mrkets were plyed in rndom order for second time, nd then gin rndomly ordered for the lst time. We chose this sequence of mrkets in order to ensure tht the level of experience does not vry cross mrkets. Prticipnts fced new sitution in every round s they never plyed the sme mrket with the sme bility twice. They received feedbck bout their lloction nd the points they erned fter every round. At the beginning of the experiment, students received n endowment of 2,200 points. At the end of the experiment, one of the 15 rounds ws rndomly selected for pyment. The points erned in this round plus the 2,200 endowment points were pid out in Euro with n exchnge rte of 0.5 cents per point. The experiment lsted 90 minutes, nd the verge ernings per subject were EUR The experiment ws run t the experimentl economics lb t the Technicl University Berlin. We recruited student subjects from our pool with the help of ORSEE by Greiner (2004. The experiments were progrmmed in z-tree, see Fischbcher (2007. For ech of the two tretments, CCA nd DCA, independent sessions were crried out. Ech session consisted of 24 prticipnts tht were split into two mtching groups of 12 for the entire session. In totl, six sessions were conducted, tht is, three sessions per tretment, with ech session consisting of two independent mtching groups of 12 prticipnts. Thus, we end up with six fully independent mtching groups nd 72 prticipnts per tretment. In the beginning of the experiment, printed instructions were given to the prticipnts (see Appendix D. Prticipnts were informed tht the experiment ws bout the study of decision mking, nd tht their pyoff depended on their own decisions nd the decisions of the other prticipnts. The instructions were identicl for ll prticipnts of tretment, explining in detil the experimentl setting. Questions were nswered in privte. After reding the instructions, ll individuls prticipted in quiz to mke sure tht everybody understood the min fetures of the experiment. 20

23 7.2 Experimentl results We first present the ggregte results in order to compre the two mechnisms. In second step, we study behvior in the two mechnisms seprtely to compre it to the point predictions nd to shed light on the resons for the ggregte findings. All results we report on re significnt t the 5% level Tretment comprisons: Aggregte results We compre the two college dmission mechnisms with respect to three properties, summrized in results 1 to 3. The first comprison concerns the expected utility of students in the two mechnisms, which is equl to the expected number of points erned, due to the ssumption of risk neutrlity. Second, we investigte whether one of the mechnisms leds to higher effort levels by the students thn the other mechnism. And the third spect we focus on is whether individuls of different bility prefer different mechnisms. Result 1 (Expected utility: In mrkets 1 nd 4, where n = q 1 + q 2, the verge utility of students in CCA is higher thn in DCA, s predicted by the theory. In mrkets 2 nd 3, the verge utility of students in DCA is not higher thn in CCA, in contrst to the theoreticl predictions. In mrket 5, there is no significnt difference both in theory nd in the dt. Support. Tble 2 presents the verge number of points or the verge utility of the prticipnts in the two different mechnisms in ll five mrkets. The third column displys the p-vlues for the two-sided Wilcoxon rnk-sum test for the equlity of distributions of equilibrium utilities nd efforts, bsed on the relized drw of bilities. Thus, in mrkets 1 to 4, we expect tht level of utility in the two mechnisms is significntly different. The lst column in the tble provides the p-vlues for the two-sided Wilcoxon rnk-sum test for the equlity of distributions of the observed number of points erned in the two mechnisms. Tble 2: Averge utility Utility higher Averge utility higher Averge utility Averge utility Observed utilities for ll students for relized types in CCA in DCA different in Mrket (predicted (predicted (observed (observed CCA nd DCA 1 CCA CCA, DCA DCA, depends; DCA in expecttion DCA, CCA CCA, no diff. in expecttion no diff., Notes: Columns 3 nd 6 show the p-vlues of the Wilcoxon rnk-sum test for equlity of the distributions. The equilibrium predictions for the comprison of utilities of students in mrkets 1 nd 4 re consistent with the experimentl dt, s the verge utility in CCA is significntly higher in both mrkets. Thus, with n equl number of pplicnts nd sets, CCA is preferble to DCA if 21

24 the gol is to mximize the utility of the students. This is due to the potentil miscoordintion of pplicnts in DCA. We fil to observe the superiority of DCA in both mrkets where this is predicted, nmely mrkets 2 nd 3. The reltionship is even reversed, with the verge utility being higher in CCA thn in DCA in both mrkets. Result 2 (Effort levels: In mrkets 1 nd 4, where n = q 1 + q 2, the verge effort level of students in DCA is higher thn in CCA. This is in line with the predictions. In mrket 3, the verge effort levels of students in CCA re not significntly higher thn in DCA, in contrst to the theoreticl prediction. In mrkets 2 nd 5, there is no difference in effort between the two mechnisms both in theory nd in the dt. Support. Tble 3 presents the verge effort levels of the prticipnts by different mechnisms nd mrkets. Anlogously to Tble 2, the third column shows the results of the Wilcoxon rnk sum test of the equilibrium efforts bsed on the relized drw of bilities. We expect effort to differ significntly between the two mechnisms only in mrkets 2 nd 3 (with mrginlly significnt difference in mrket 1. The lst column provides the p-vlues for the two-sided Wilcoxon rnksum test for the equlity of distributions of the observed effort levels in the two mechnisms. The equilibrium predictions regrding the comprison of efforts in mrkets 1 nd 4 re confirmed by the dt in tht effort is higher in DCA. In mrket 3 verge efforts re higher CCA thn in DCA s predicted, but the difference is not significnt. Tble 3: Averge effort Effort higher Averge effort higher Averge effort Averge effort Observed efforts for ll students for relized types in CCA in DCA different in Mrket (predicted (predicted (observed (observed CCA nd DCA 1 depends; DCA in expecttion DCA, no diff. in expecttion no diff., CCA CCA, DCA DCA, no diff. in expecttion no diff., Notes: Columns 3 nd 6 show the p-vlues of the Wilcoxon rnk-sum test for equlity of the distributions. Tking together results 1 nd 2, we observe tht in mrkets without shortge of sets (mrket 1 nd mrket 4 students re on verge better off in CCA where they exert less effort. In mrket 5 the results re lso in line with the theoreticl predictions with lmost identicl effort nd expected utility levels in both mechnisms. In the two remining mrkets with surplus of students over sets, mrkets 2 nd 3, the results contrdict the theory. Mrkets 2 nd 3 should led to higher verge utility of the students in DCA thn in CCA, which cnnot be observed in the lb. Therefore, the overll results suggest tht with respect to the utility of students, CCA performs better thn predicted reltive to DCA. Next we turn to the question whether students of different bilities prefer different mechnisms by providing n experimentl test of propositions 2 nd 3. According to Proposition 2 low-bility 22

25 students prefer DCA over CCA if there re more pplicnts thn sets in the mrket, s in our mrkets 2, 3, nd 5. Proposition 3 implies tht if ny student prefers CCA over DCA, then ll students with higher bility must lso prefer CCA. (Remember tht in mrkets 1 nd 4, ll students prefer CCA, nd we therefore do not consider these mrkets here. Result 3 (Expected utility of low- nd high-bility students: In mrkets 2 nd 3, the verge utilities of students with low bilities re higher in DCA, nd the verge utilities of students with high bilities re higher in CCA. There is no difference in the verge utilities of students in DCA nd CCA in mrket 5. Support: Tble 4 presents the regression results of the students utility on the 10% bility quntiles nd the dummies for the interction of ech quntile nd the DCA for mrket 2 nd mrket 3. The significnce of the dummy vribles for the interction of the DCA nd quntile reflects the significnce of the tretment difference for the corresponding 10% bility quntile. Coefficients for the interctions of the first to fourth quntiles (i.e., the students with the lowest bility nd the DCA re positive, nd two of them re significntly different from zero. Thus, the lowbility students hve on verge higher utility in DCA in mrkets 2 nd 3. Coefficients for the other quntiles re negtive, nd re significnt for the seventh nd tenth 10% quntiles. Thus, high-bility students hve, on verge, lower utility in DCA thn in CCA. This confirms the single-crossing property of Proposition 3. Overll, the results of mrkets 2 nd 3 lend support to propositions 2 nd 3. Note tht mrket 5 which we constructed s control to generte pproximtely the sme outcome for CCA nd DCA, does not yield significnt differences in the expected utility for high- nd low-bility students Point predictions regrding individul behvior Next we investigte the individul behvior of subjects in ech mechnism seprtely. In prticulr, we test the point predictions of the theory regrding the effort levels in CCA nd DCA s well s the choice between college 1 nd college 2 in DCA. This will help to understnd the results regrding the comprison of the two mechnisms, in prticulr the reltively poor performnce of DCA with respect to student utility. Figure 2 depicts the efforts of individuls, the kernel regression estimtion of efforts, nd the equilibrium predictions for ech of the mrkets nd mechnisms. All 10 pnels for the 10 mrkets show tht the kernel of effort increses in bility. Moroever, the observed effort levels often lie bove the predicted vlues. Result 4 (Individul efforts: Individul efforts in the experiments differ from the equilibrium efforts in eight out of 10 mrkets. In ll 10 mrkets verge efforts re greter thn verge equilibrium efforts. This overexertion of effort is significnt in ll five mrkets in DCA nd in three out of five mrkets in CCA. The observed effort levels differ from rndom behvior, nd equilibrium efforts hve predictive power for the observed effort levels in both mechnisms. 23

26 Tble 4: Utility differences cross bility quntiles Vrible Coefficient (Std. Err. 10% bility quntiles *** ( st quntile in DCA ( nd quntile in DCA *** ( rd quntile in DCA *** ( th quntile in DCA ( th quntile in DCA ( th quntile in DCA ( th quntile in DCA *** ( th quntile in DCA ( th quntile in DCA ( th quntile in DCA ** ( Intercept * ( N 864 R F (11, Notes: OLS estimtion of utility bsed on clustered robust stndrd errors t the subject level. *** denotes sttisticl significnce t the 1%-level, ** t the 5%-level, nd * t the 10%-level. 24

27 Figure 2: Individul efforts by bility 25

28 Tble 5: Individul efforts Averge Averge Averge observed equilibrium rndom p-vlue p-vlue efforts efforts efforts obs.=pred. obs.=rnd. (1 (2 (3 (4 (5 CCA Mrket Mrket Mrket Mrket Mrket DCA Mrket Mrket Mrket Mrket Mrket Support: In ll mrkets nd mechnisms, verge effort levels re higher thn predicted, s cn be tken from comprison of the first two columns in Tble 5. Column (4 provides the p-vlues of the Wilcoxon mtched-pirs signed-rnk test for the equlity of observed nd equilibrium efforts by mrkets nd mechnisms. In CCA the difference is significnt for three out of five mrkets (mrket 3, 4, nd 5 while in DCA it is significnt for ll five mrkets. Thus DCA leds to significnt overexertion in more mrkets thn CCA. One possible intution for this finding is tht the uncertinty is higher under DCA where students need to coordinte on the colleges, which leds to higher efforts. Next, we compre observed behvior to rndom choices. As the bility level of student determines her possible set of effort choices, rndom choices will differ for different bility types. Thus, we define the rndom choice s the choice of the effort in the middle of the intervl of ll fesible efforts of n pplicnt, see column (3. The behvior of subjects is significntly different from the rndom prediction in ll mrkets for both mechnisms s cn be tken from the p-vlues of the Wilcoxon mtched-pirs rnk-sum test for the equlity of observed nd rndom efforts in the lst column of Tble 5. We lso find tht in spite of the negtive results regrding the point predictions, the equilibrium effort levels hve significnt predictive power. This emerges from n OLS estimtion of observed efforts bsed on clustered robust stndrd errors t the level of mtching groups, presented in Tble 6. Moreover, there is no significnt difference with respect to the predictive power of the equilibrium in the different dmission systems (s the predictions for CCA nd the dummy vrible for CCA re both not significnt. As finl step, we investigte the choice of prticipnts to pply to college 1 or college 2 in DCA. Recll tht the symmetric Byesin Nsh equilibrium chrcterized in Theorem 1 hs the 26

29 Tble 6: Observed effort choices nd equilibrium predictions Vrible Coefficient (Std. Err. Equilibrium effort 0.741*** (0.047 Equilibrium effort in CCA (0.083 Dummy for CCA ( Intercept *** ( N 2160 R F (4, Notes: OLS estimtion of effort levels bsed on clustered robust stndrd errors t the level of mtching groups. *** denotes sttisticl significnce t the 1%-level, ** t the 5%- level, nd * t the 10%-level. property tht students with n bility bove the cutoff should lwys pply to the better college (college 2 wheres students with n bility below the cutoff should mix between the two colleges. Result 5 (Choice of college in DCA: In DCA, students bove the equilibrium bility cutoff choose the good college 2 more often thn students below the cutoff. Across ll mrkets nd controlling for bility, the equilibrium predictions regrding the probbility of choosing the good college hve predictive power for the subjects choices. Support: Tble 7 displys the cutoff bility for ech mrket in the first column. In the second column it provides the verge equilibrium probbility of choosing the good college 2 for students with bilities below the cutoff in the respective mrkets. The verge is clculted given the ctul reliztion of bilities in the experiment. This cn be compred to the observed frequency of choosing the good college in the experiment by these students in the next column. It emerges tht subjects choose the good college 2 more often thn predicted in ll five mrkets, but in some mrkets the predicted nd observed proportions re quite close. The next column (4 displys the proportion of subjects bove the cutoff pplying to college 2. Remember tht in equilibrium these types should pply to college 2 with certinty. Finlly, the lst column of Tble 7 presents the p-vlues for the test of equlity of the proportions of the choice of college 2 below nd bove the mrket-specific equilibrium cutoff. In ll mrkets the differences re significnt t 1% significnce level. Further evidence for the predictive power of the model is provided by Tble 8. It shows the results of probit model for the choice of the good college 2 in DCA. The coefficient for the 27

30 Tble 7: Proportion of choices of good college 2 Equ. prop. Obs. prop. Obs. prop. p-vlues for of choices of choices of choices equlity of Equilibrium of college of college of college proportions bility 2 below 2 below 2 bove bove nd cutoff the cutoff the cutoff the cutoff below the cutoff (1 (2 (3 (4 (5 Mrket % 33% 85% 0.00 Mrket % 51% 92% 0.00 Mrket % 27% 68% 0.00 Mrket % 17% 42% 0.00 Mrket % 64% 91% 0.00 equilibrium probbility of choosing the good college is significnt t the 1% significnce level. Tble 8: Choice of the good college 2 in DCA Vrible Coefficient (Std. Err. Equilibrium probbility of choosing the good college 1.684*** (0.106 Intercept -0.79*** (0.079 N 1080 Pseudo R Notes: Probit estimtion of dummy for the choice of the good college bsed on clustered robust stndrd errors t the subject level. *** denotes sttisticl significnce t the 1%-level, ** t the 5%-level, nd * t the 10%-level. Finlly, we investigte the ppliction decision of students by bility. Figure 3 presents the choices of subjects in DCA by mrkets nd bility quntiles, together with the equilibrium proportions. Students bove the equilibrium cutoff in mrket 1, mrket 2, nd mrket 5 choose the good college 2 lmost certinly, in line with the theory. The proportions of choices of students with low bility re lso close to the equilibrium mixing probbilities. The biggest difference between the observed nd the equilibrium proportions origintes from the students who re slightly below the cutoff. This finding is prticulrly evident in mrkets 1, 2, nd 4. To understnd this, remember tht the equilibrium is chrcterized by discontinuity of the probbility of the choice of college 2: students with bilities just bove the cutoff hve pure strtegy of choosing college 2, while students just below the cutoff choose college 1 with n lmost 100% probbility. Not surprisingly, in the experiment the choices of universities re smooth round the cutoff. Accordingly, we do not observe the predicted kink in the effort choices s is evident in Figure 2. This cn be due to the fct tht students with n bility level round the cutoff under- or overestimte the cutoff, which would result in the observed smoothing. 28

31 Figure 3: Choice of colleges by subjects in DCA 29

32 8 Conclusion In this pper, we study college dmissions exms which concern millions of students every yer throughout the world. Our model bstrcts wy from mny spects of rel-world college dmission gmes nd focuses on the following two importnt spects: (i colleges ccept students by considering student exm scores, (ii students hve differing bilities which re their privte informtion, nd the costs of getting redy for the exms re inversely relted to bility levels. Motivted by the Turkish nd the Jpnese college dmissions mechnisms, we focus on two extreme policies. In the centrlized model students cn freely nd costlessly pply to ll colleges wheres in the decentrlized mechnism, students cn only pply to one college. We consider model tht is s simple s possible by ssuming two colleges nd homogeneous student preferences over colleges in order to derive nlyticl results s Byesin Nsh solutions to the two mechnisms. 16 The solution of the centrlized dmissions mechnism follows from stndrd techniques in the contest literture. The solution to the decentrlized model, on the other hnd, hs interesting properties such s lower bility students using mixed strtegy when deciding which school to pply to. Our min result is tht low- nd high-bility students differ in terms of their preferences between the two mechnisms where high-bility students prefer the centrlized mechnism nd low-bility students the decentrlized mechnism. We employ experiments to test the theory nd to develop insights into the functioning of centrlized nd decentrlized mechnisms tht tke into ccount behviorl spects. We hve implemented five different mrkets chrcterized by the common vlues of the two colleges to the students s well s the cpcity of the two colleges. Overll, the min predictions of the theory re supported by the dt, in spite of few importnt differences. We find tht in mrkets with n equl number of sets nd pplicnts, the centrlized mechnism is better for ll pplicnts, s predicted by the theory. Agin in line with the theory we observe tht in mrkets with n overdemnd for sets, low-bility students prefer decentrlized dmissions mechnism wheres high-bility students prefer centrlized mechnism. However, in these mrkets we cnnot confirm the predicted superiority of the decentrlized mechnism for the students. This cn be scribed to one robust nd strk difference between theory nd observed behvior, nmely overexertion of effort, which is more pronounced in the decentrlized mechnism. For the evlution of the two mechnisms from welfre perspective, it mtters whether the effort spent prepring for the exm hs no benefits beyond improving the performnce in the exm or whether this effort is useful. If effort is purely cost, then welfre cn be mesured by the men utility of the students. In ll our mrkets, the centrlized mechnism t lest wekly outperforms the decentrlized mechnism with respect to this criterion. If the effort exerted by the students increses their productivity, then the decentrlized mechnism becomes reltively more ttrctive, where efforts re wekly higher thn in the centrlized mechnism. 16 We lso discuss the extension to more colleges in section 6. 30

33 A Appendix A.1 Preliminries The following lemmt re useful for the results given in the rest of the Appendix. Lemm 1. Let l, m be given integers. Then, d dx d dx ( l j=0 ( m j=l p j,m j (x p j,m j (x = m p l,m l 1 (x when 0 l < m, = m p l 1,m l (x when 0 < l m, d dx d dx ( l j=0 ( m j=l p m j,j (x p m j,j (x = m p m l 1,l (x when 0 l < m, = m p m l,l 1 (x when 0 < l m. Proof. We use the following eqution: ( m (m j + 1 = j 1 m! (j 1!(m j + 1! (m j + 1 = m! (j 1!(m j! = ( m j. (10 j The first formul: Suppose 0 = l. Then, l j=0 p j,m j(x = p 0,m (x = (1 x m. Its derivtive is m(1 x m 1 = m p 0,m 1 (x. Thus the formul holds. Consider nother cse where 0 < l. Then we hve d dx ( l j=0 p j,m j (x = d ( l ( m x j (1 x m j dx j j=0 l ( m = j x j 1 (1 x m j j j=1 l ( m = j x j 1 (1 x m j j j=1 l ( m = j x j 1 (1 x m j j j=1 l j=0 l+1 j=1 l+1 j=1 ( m (m jx j (1 x m j 1 j ( m (m j + 1x j 1 (1 x m j j 1 ( m j x j 1 (1 x m j (by (10 j Thus, 31

34 d dx ( l j=0 p j,m j (x = ( m (l + 1 x l (1 x m l 1 m! = l + 1 l!(m l 1! xl (1 x m l 1 (m 1! = m l!(m l 1! xl (1 x m l 1 = m p l,m l 1 (x. The second formul: Suppose l = m. Then, m j=l p j,m j(x = p m,0 (x = x m. Its derivtive is mx m 1 = mp m 1,0 (x. Thus the formul holds. Consider nother cse where l < m. Then we hve d dx ( m j=l p j,m j (x = d dx = = = ( m j=l m ( m j j=l m ( m j j=l m ( m j j=l ( m x j (1 x m j j j x j 1 (1 x m j j x j 1 (1 x m j j x j 1 (1 x m j ( m (m jx j (1 x m j 1 j m ( m (m j + 1x j 1 (1 x m j j 1 m ( m j x j 1 (1 x m j (by (10 j m 1 j=l j=l+1 j=l+1 Thus, d dx ( l j=0 p j,m j (x = ( m l x l 1 (1 x m l = l m! (l 1!(m l! xl 1 (1 x m l (m 1! = m (l 1!(m l! xl 1 (1 x m l = m p l 1,m l (x. The third formul: By the second formul, we hve d dx ( l j=0 The fourth formul: By the first formul, we hve d dx ( m j=l p m j,j (x = d ( m p j,m j (x = m p m l 1,l (x. dx j=m l p m j,j (x = d dx ( m l j=0 p j,m j (x = m p m l,l 1 (x. 32

35 B On Theorem 1 B.1 Derivtion of the symmetric equilibrium We show how to obtin the function γ : [0, c] (0, 1 nd the cutoff c from Eqution (6. Step 1: We show tht there is unique vlue π(c tht stisfies Eqution (7. Define function ϕ 1 : [0, 1] R: for ech x [0, 1], q 2 1 ϕ 1 (x = v 2 m=0 q 1 1 p n m 1,m (x v 1 m=0 Differentite ϕ 1 t ech x (0, 1: using Lemm 1, we hve p m,n m 1 (x. ϕ 1(x = v 2 (n 1 p (n 1 (q2 1 1,q 2 1(x + v 1 (n 1 p q1 1,(n 1 (q 1 1 1(x > 0. Thus, ϕ 1 is strictly incresing. Moreover, ϕ 1 (0 = v 1 < 0 nd ϕ 2 (1 = v 2 > 0. Thus, since ϕ 1 is continuous function on [0, 1], there is unique x (0, 1 such tht ϕ 1 (x = 0. Thus, since ϕ 1 (π(c = 0 by (7, there is unique π(c (0, 1 tht stisfies Eqution (7. Step 2: Given unique π(c, we now show tht there is unique cutoff c (0, 1. In Eqution (8, since π(c is known by Step 1, the the only unknown is c vi F (c. Define function ϕ 2 : [π(c, 1] R s follows: for ech x [π(c, 1], q 2 1 ϕ 2 (x = v 2 m=0 n 1 p n m 1,m (π(c + v 2 m=q 2 p n m 1,m (π(c m j=m q 2 +1 Differentite ϕ 2 t ech point x (π(c, 1: using Lemm 1, we hve ( x π(c p j,m j v 1. 1 π(c n 1 ϕ 2(x = v 2 ( p n m 1,m (π(c m=q π(c m p m q2,q 2 1 Thus, ϕ is strictly incresing. Moreover, ϕ 2 (1 = v 2 v 1 > 0 nd ( x π(c > 0. 1 π(c 33

36 q 2 1 n 1 ϕ 2 (π(c = v 2 p n m 1,m (π(c + v 2 m=0 q 2 1 = v 2 m=0 m=q 2 p n m 1,m (π(c m j=m q 2 +1 p j,m j (0 v 1 p n m 1,m (π(c v 1 ( p j,m j (0 = 0 for j m q q 1 1 = v 1 p m,n m 1 (π(c v 1 ( (7 < 0. m=0 Therefore, there is unique x (π(c, 1 such tht ϕ 2 (x = 0. Since ϕ 2 (F (c = 0, x = F (c. Thus, since F is strictly incresing, there is unique cutoff c (F 1 (π(c, 1 such tht c = F 1 (x. Step 3: From steps 1 nd 2, π(c nd c re uniquely determined. We now show tht for ech [0, c, there is unique π( (0, 1 tht stisfies (9. Fix [0, c. Define function ϕ 3 : [0, F (] R: n 1 ϕ 3 (x = v 2 m=q 2 p n m 1,m (π(c m j=m q 2 +1 ( F ( x n 1 p j,m j v 1 p m,n m 1 (π(c 1 π(c m=q 1 Let us differentite ϕ 3 t ech x (0, F ( by using Lemm 1: m j=m q 1 +1 ( x p j,m j. π(c n 1 ( ϕ 1 ( F ( x 3(x = v 2 p n m 1,m (π(c m p m q2,q 1 π(c π(c m=q 2 n 1 ( 1 ( x v 1 p m,n m 1 (π(c m p m q1,q π(c 1 1 < 0. π(c m=q 1 Thus, ϕ is strictly decresing. Moreover, n 1 ϕ 3 (0 = v 2 nd n 1 = v 2 m=q 2 p n m 1,m (π(c p n m 1,m (π(c m=q 2 > 0. m j=m q 2 +1 m j=m q 2 +1 ( F ( n 1 p j,m j v 1 p m,n m 1 (π(c 1 π(c m=q 1 ( F ( p j,m j ( p j,m j (0 = 0 1 π(c m j=m q 1 +1 p j,m j (0 34

37 n 1 ϕ 3 (F ( = v 2 n 1 = v 1 < 0. m=q 2 p n m 1,m (π(c m=q 1 p m,n m 1 (π(c m j=m q 2 +1 m j=m q 1 +1 n 1 p j,m j (0 v 1 p m,n m 1 (π(c m=q 1 ( F ( p j,m j ( p j,m j (0 = 0 π(c m j=m q 1 +1 ( F ( p j,m j π(c Thus, there is unique x (0, F ( such tht ϕ 3 (x = 0. Since ϕ 3 (π( = 0, x = π(. Hence, there is unique π( (0, 1 tht stisfies Eqution (9. Step 4: Finlly, we derive γ( for ech (0, c. Recll tht in (9, π( = γ(xf(xdx nd 0 π(c nd π( re known by previous steps. Differentite (9 with respect to by using Lemm 1: ( γ(f( p m,n m 1 (π(c π(c m=q 1 ( f( γ(f( p n m 1,m (π(c 1 π(c m=q 2 n 1 v 1 n 1 = v 2 Let us define the following functions: n 1 A( := v 1 m p m q1,q 1 1 ( π( π(c m p m q2,q 2 1 ( π( > 0, π(c ( F ( π( 1 π(c p m,n m 1 (π(c m p m q1,q 1 1 m=q 1 n 1 ( F ( π( B( := v 2 p n m 1,m (π(c m p m q2,q 2 1 > 0. 1 π(c m=q 2 Then, we cn write (11 s Solving for γ( in (12, we obtin γ(f( A( = π(c γ( =. (11 f((1 γ( B(. (12 1 π(c π(cb( (1 π(ca( + π(cb( (0, 1. By construction, function γ we hve derived stisfies Eqution (9. B.2 Verifiction: the cndidte is n equilibrium In this ppendix, we check for globl devitions nd confirm tht the unique symmetric equilibrium cndidte we hve derived in Theorem 1 is indeed n equilibrium. As preliminry nottion nd nlysis, let us clculte the probbility, denoted by P [1, b c, γ, β D ], tht student who mkes 35

38 effort e = β D (b nd pplies to college 1 ends up getting set in college 1: q1 1 P [1, b γ, β D m=0 ] = ˆp m,n m 1(c + n 1 m=q 1 ˆp m,n m 1 (cg m q1 +1,m(b if b [0, c] 1 if b c. Obviously, if the student chooses n effort more thn β(c, he will definitely get set in college 1. Otherwise, the first line represents the sums of the probbility of events in which e is one of the highest q 1 efforts mong the students who pply to college 1. Similrly, let us clculte the probbility, denoted by P [2, b β, γ], tht student who mkes effort e = β(b nd pplies to college 2 ends up getting set in college 2. q2 1 P [2, b γ, β D m=0 ] = ˆp n m 1,m(c + n 1 m=q 2 ˆp n m 1,m (ch m q2 +1,m(b if b [0, 1] 1 if b 1. Obviously, if the student chooses n effort more thn β(1, he will definitely get set in college Otherwise, the first line represents the sums of probbility of events in which e is one of the highest q 2 efforts mong the students who pply to college 2. Next, denote by U(r, b γ, β D, (or U(r, b for short the expected utility of type who chooses college 1 with probbility r nd mkes effort e = β D (b when ll of the other students follow the strtegy (γ, β D. We hve, U(r, b := rp [1, b γ, β D ]v 1 + (1 rp [2, b γ, β D ]v 2 e. We need to show tht for ech [0, 1], ech r [0, 1] nd ech b 0, Û( U(γ(, U(r, b. Fix [0, 1]. It is sufficient to show tht Û( U(0, b nd Û( U(1, b, s these two conditions together implies required no globl devition condition. Below, we show tht for ny [0, 1], nd for b 0, both Û( U(0, b nd Û( U(1, b hold. We consider two cses, one for lower bility students ( [0, c], one for higher bility students ( [c, 1]. As sub-cses, we nlyze b to be in the sme region (b is low for low, nd b is high for high, different region ( high, b low; nd low, b high, nd b being over 1. The no-devition results for the sme region is stndrd, wheres devitions cross regions need to be crefully nlyzed. Cse 1: Type [0, c] Cse 1-1: b [0, c]. Then, by our derivtion, we hve U(0, b = U(1, b nd lso Û( U(1, b cn be shown vi stndrd rguments (for instnce, see section nd Proposition 2.2 in Krishn, Hence, we cn conclude tht Û( U(1, e = U(0, e. Cse 1-2: b (c, 1]. We first show Û( U(1, b. 17 Of course, there is no type b with b > 1, if student chooses n effort e strictly greter thn β D (1, we represent him s mimicking type b > 1. 36

39 Next, we show Û( U(0, b. Û( U(1, c = v 1 βd (c v 1 βd (b ( β D (c β D (b. = U(1, b. Û( U(γ(c, c = P [2, c γ, β D ]v 2 βd (c ( = P [2, β D (c γ, β D ]v 2 βd (c + βd (c c c U(0, b c + βd (c βd (c ( c = P [2, b γ, β D ] βd (b + βd (b βd (b c = U(0, b + ( β D (b β D (c ( 1 1 c U(0, b ( β D (b β D (c, < c. Cse 1-3: b > 1 (or e > β D (1. βd (c = P [2, b γ, β D ]v 2 βd (b c + βd (c c = U(0, c c + βd (c c + βd (c c βd (c βd (c βd (c Moreover, Û( U(γ(c, c = v 1 βd (c > v 1 e ( β D (c β D (1 < e = U(1, b. Cse 2: Type [c, 1] Û( U(0, 1 (by Cse 1-2 = v 2 βd (1 > v 2 e ( e > β D (1 = U(0, b. Cse 2-1: b [0, c]. We first show Û( U(1, b. 37

40 Û( U(0, c = v 2 P [2, c γ, β D ] βd (c = U(γ(c, c c + βd (c c U(γ(b, b c + βd (c c = U(γ(b, b + βd (b βd (c βd (c βd (b = U(1, b + (β D (c β D (b c + βd (c c ( 1 c 1 U(1, b ( β D (c β D (b 0, c <. βd (c ( U(γ(b, b = U(1, b To obtin Û( U(0, b, note tht in the bove inequlities, if we use U(γ(b, b = U(0, b in the fourth line, we obtined the desired inequlity. Cse 2-2: b (c, 1]. First, by our derivtion, Û( U(0, e γ, βd, cn be shown vi stndrd rguments (for instnce, see section nd Proposition 2.2 in Krishn, Next, we show Û( U(1, b. Û( U(0, c = v 2 P [2, c γ, β D ] βd (c = v 1 βd (c Cse 2-3: b > 1 (or e > β D (1 nd v 1 βd (b ( v 2 P [2, c γ, β D ] = v 1 = U(1, b ( β D (c β D (b. Û( U(γ(c, c = U(1, c = v 1 βd (c v 1 e ( e > β D (1 > β D (c U(1, b. Û( U(0, 1 = v 2 βd (1 v 2 e ( e > β D (1 = U(0, b. 38

41 C Equilibrium Derivtion for l Colleges We show how to derive cutoffs, mixed strtegies, nd cost functions provided there exists n equilibrium s specified in section 6. The bsic procedure follows the one in Theorem 1. We first show how to obtin the equilibrium cutoffs c 1,..., c l 1 nd the mixed strtegy function γ 1,..., γ l 1. Let k {1,..., l 1}. A necessry condition for this to be n equilibrium is tht ech type [c k 1, c k ] hs to be indifferent between pplying to college 1 nd college 2. Thus, for ll [c k 1, c k ], ( q k 1 v k m=0 ( q k+1 1 = v k+1 n 1 p m,n m 1 (π k (c k + p m,n m 1 (π k (c k Hm q k k +1,m( m=q k m=0 p m,n m 1 (π k+1 (c k+1 + n 1 Step 1: Find π 1 (c 1,..., π l (c l. Eqution (13 cn be written s m=q k+1 p m,n m 1 (π k+1 (c k+1 H k+1 m q k+1 +1,m(. (13 q 1 1 v 1 m=0 q 2 1 p m,n m 1 (π 1 (c 1 = v 2 m=0 p m,n m 1 (π 2 (c 2, q k 1 v k 2 = v k p m,n m 1 (π k (c k for k {3,..., l}, (14 m=0 where the first eqution is Eqution (13 t = 0 under k = 1, which sys tht type = 0 is indifferent between college 1 nd 2; the second eqution follows from Eqution (13 t = c k under k 1 nd k, which sys tht type = c k 2 is indifferent between colleges k 2 nd k. Therefore, π 1 (c 1,..., π l (c l cn be obtined by solving Eqution (14. Step 2: Given π 1 (c 1,..., π l (c l, find cutoffs c 1,..., c l 1. We first show the following clim tht shows how to obtin π k (c k 1 from π 1 (c 1,..., π l (c l. π k (c k 1 = F (c k 1 k 1 j=1 πj (c j. Proof. For k = 2: Note tht π 1 (c 1 = c 1 γ 0 1(xdF (x. Thus π 2 (c 1 := c 1 (1 γ 0 1(xdF (x = F (c 1 π 1 (c 1. Suppose tht the clim is true up to k 1 where k 3. Then π k 1 (c k 1 := π k 1 (c k 2 + c k 1 c k 2 γ k 1 (xdf (x. Thus c k 1 c k 2 γ k 1 (xdf (x = π k 1 (c k 1 π k 1 (c k 2. Hence, by 39

42 the induction hypothesis, we hve π k (c k 1 : = ˆ ck 1 c k 2 (1 γ k 1 (xdf (x = F (c k 1 F (c k 2 ˆ ck 1 c k 2 γ k 1 (xdf (x = F (c k 1 F (c k 2 π k 1 (c k 1 + π k 1 (c k 2 k 2 = F (c k 1 F (c k 2 π k 1 (c k 1 + (F (c k 2 π j (c j k 1 = F (c k 1 π j (c j. j=1 j=1 Now Eqution (13 t = c k cn be rewritten s, for ech k {1,..., l 1}, q k+1 1 v k = v k+1 m=0 +v k+1 n 1 p m,n m 1 (π k+1 (c k+1 m=q k+1 p m,n m 1 (π k+1 (c k+1 where we use induction clim nd H k+1 m q k+1 +1,m(c k = m j=m q k+1 +1 m j=m q k+1 +1 ( F (ck (π 1 (c π k (c k p j,m j π k+1,(15 (c k+1 ( π k+1 (c k p j,m j. π k+1 (c k+1 Hence, given π 1 (c 1,..., π l (c l, we cn find c k by solving Eqution (15. Step 3: Given π 1 (c 1,..., π l (c l nd c 1,..., c l 1, for ech k {1,..., l 1} nd ech [c k 1, c k ], there is unique π k ( tht stisfies Eqution (16. Moreover, we cn get the mixed strtegy function γ k ( by differentiting Eqution (16. Eqution (13 t [c k 1, c k ] cn be rewritten s, for ech k {1,..., n 1}, q k 1 v k m=0 q k+1 1 = v k+1 n 1 p m,n m 1 (π k (c k + v k m=0 p m,n m 1 (π k+1 (c k+1 +v k+1 n 1 m=q k+1 p m,n m 1 (π k+1 (c k+1 m=q k p m,n m 1 (π k (c k m j=m q k+1 +1 m j=m q k +1 ( π k ( p j,m j π k (c k ( F ( F (ck 1 π k ( + π k (c k 1 p j,m j π k+1 (16, (c k+1 where we used the following eqution: for ech [c k 1, c k ], since π k ( := π k (c k 1 + c k 1 γ k (xdf (x, 40

43 π k+1 ( := ˆ c k 1 (1 γ k (xdf (x = F ( F (c k 1 π k ( + π k (c k 1. Differentite Eqution (16 with respect to by using Lemm 1: n 1 v k p m,n m 1 (π k (c k γ k(f( π k (c m=q k k mp m qk,q k 1 ( π k ( π k (c k n 1 = v k+1 p m,n m 1 (π k+1 (c k+1 f( γ k(f( π k+1 (c m=q k+1 k+1 Let us define the following functions: mp m qk+1,q k+1 1 ( π k+1 (. (17 π k+1 (c k+1 n 1 A k ( = v k n 1 B k ( = v k+1 Then we cn write (17 s m=q k p m,n m 1 (π k (c k mp m qk,q k 1 ( π k ( > 0 π k (c k m=q k+1 p m,n m 1 (π k+1 (c k+1 mp m qk+1,q k+1 1 γ k (f( π k (c k Solving for γ k ( in (18, we obtin γ k ( = ( π k+1 ( > 0. π k+1 (c k+1 A k ( = f((1 γ k( B k (. (18 π k+1 (c k+1 π k (c k B k ( π k+1 (c k+1 A k ( + π k (c k B k (. Step 4: We find the effort function β D. Consider student with type [c k 1, c k ]. A necessry condition is tht she does not wnt to mimic ny other type in [c k 1, c k ]. Her utility mximiztion problem is ( q k 1 mx v k p m,n m 1 (π k (c k + [c k 1,c k ] m=0 n 1 p m,n m 1 (π k (c k Hm q k k +1,m( m=q k βd (. The first-order necessry condition requires the derivtive of the objective function to be 0 t =. Hence 41

44 n 1 v k m=q k p m,n m 1 (π k (c k h m qk +1,m( (βd ( Solving the differentil eqution with the boundry condition t β D (c k 1, we obtin for ll [c k 1, c k ]. β D ( = β D (c k 1 + v k ˆ c k 1 x n 1 = 0. m=q k p m,n m 1 (π k (c k h k m q k +1,m(xdx D Experiment Instruction(not for publiction Welcome! This is n experiment bout decision mking. You nd the other prticipnts in the experiment will prticipte in sitution where you hve to mke number of choices. In this sitution, you cn ern money tht will be pid out to you in csh t the end of the experiment. How much you will ern depends on the decisions tht you nd the other prticipnts in the experiment mke. During the experiment you re not llowed to use ny electronic devices or to communicte with other prticipnts. Plese use exclusively the progrms nd functions tht re intended to be used in the experiment. These instructions describe the sitution in which you hve to mke decision. The instructions re identicl for ll prticipnts in the experiment. It is importnt tht you red the instructions crefully so tht you understnd the decision-mking problem well. If something is uncler to you while reding, or if you hve other questions, plese let us know by rising your hnd. We will then nswer your questions individully. Plese do not, under ny circumstnces, sk your question(s loud. You re not permitted to give informtion of ny kind to the other prticipnts. You re lso not permitted to spek to other prticipnts t ny time throughout the experiment. Whenever you hve question, plese rise your hnd nd we will come to you nd nswer it. If you brek these rules, we my hve to terminte the experiment. Once everyone hs red the instructions nd there re no further questions, we will conduct short quiz where ech of you will complete some tsks on your own. We will wlk round, look over your nswers, nd solve ny remining comprehension problems. The only purpose of the quiz is to ensure tht you thoroughly understnd the crucil detils of the decision-mking problem. Your nonymity nd the nonymity of the other prticipnts will be gurnteed throughout the entire experiment. You will neither lern bout the identity of the other prticipnts, nor will they lern bout your identity. 42

45 Generl description This experiment is bout students who try to enter the university. The 24 prticipnts in the room re grouped into two groups of 12 persons ech. These 12 prticipnts represent students competing for university sets. The experiment consists of 15 independent decisions (15 rounds, which represent different student dmission processes. At the end of ech round every student will receive t most one set in one of the universities or will remin unssigned. There re two universities tht differ in qulity. We refer to the best university s University 1. Admission to the best university (University 1 yields pyoff of 2,000 points for the students. Admission to University 2 yields smller pyoff for the students, which cn vry cross the rounds. Ech university hs certin number of sets to be filled, fctor which cn lso be different for ech of the rounds. Instructions for CCA The lloction procedure is implemented in the following wy: At the beginning of the ech round, every student lerns her bility. The bility of ech student is drwn uniformly from the intervl from 0 to 100. Thus every student hs n equl chnce of being ssigned every level of bility from the intervl. You will lern your own bility but not the bility of the other 11 students competing with you for the sets. The bility is drwn independently for ll prticipnts in every round. Admission to universities is centrlized nd is bsed on the mount of effort tht ech student puts into finl exm. In the experiment you cn choose level of effort. This effort is costly. The price of effort depends on your bility. The higher the bility the esier (cheper the effort. The higher the bility the esier (cheper the effort. The price of one unit of effort is determined s: 100 divided by the bility, 100/bility. On your screen you will see your bility for the round nd the corresponding price of one unit of effort. You will hve to decide on the mount of the effort tht you choose. In ech of the rounds you cn use the clcultor which will be on your screen. You cn use it to find out wht possible pyoffs prticulr effort in points cn yield. To gin better understnding of the experiment you cn insert different vlues. This will help you with your decision. In the beginning of ech round, every prticipnt receives 2,200 points tht cn either be used to exert effort or kept. After ech student hs decided how much effort to buy, these effort levels re sent to the centrlized clering house which then determines the ssignments to universities. The students who hve chosen the highest effort levels re ssigned to University 1 up to the cpcity of this university. They receive 2,000 points. The students with the next higher levels of effort re ssigned to University 2 up to its cpcity nd receive the corresponding mount of points. All other students who hve pplied remin unssigned nd will receive no points. Prticipnts tht 43

46 hve chosen the sme mount of effort will be rnked ccording to rndom drw. Ech prticipnt receives pyoff tht is determined s the sum of the non-invested endowment nd the pyoff f rom university dmission. Thus: Pyoff = Endowment price of effort units of effort + pyoff from ssignment Note tht your bility, the bility of the other prticipnts, nd the number of sets t University 1 nd University 2 vry in every round. Every point corresponds to 0.5 cents. Only one of the rounds will be relevnt for you ctul pyoff. This round will be selected rndomly by the computer t the end of the experiment. Exmple Let us consider n exmple with three hypotheticl persons: Juli, Peter, nd Simon. Imgine the following round: University 1 hs four sets, nd University 2 hs five sets. The dmission to University 1 yields 2,000 points nd the dmission to University 2 yields 1,000 points. Juli hs n bility of 25. Thus the cost of one unit of effort is 100/25 = 4 points for her. Her endowment is 2,200 points, which mens tht she cn buy mximum of 2,200/4 = 550 units of effort. Let us imgine tht Juli decided to buy 400 units of effort. Thus she hs to py 400*4 = 1,600 points nd keeps 600 points of her endowment. Peter hs n bility of 50. Thus the cost of effort for him is 100/50 = 2 points for one unit of effort. His endowment is 2,200 points. Thus he cn buy mximum of 2,200/2 = 1100 units of effort. Let us ssume tht Peter chose 600 units of effort. Thus he hs to py 600*2 = 1,200 points. Simon hs n bility of 80. Thus the cost of one unit of effort is 100/80 = 1.25 points for one unit of effort. His endowment is 2,200 points. Thus he cn buy mximum of 2,200/1.25 = 1760 units of effort. Let us imgine tht Simon decides to buy 500 units of effort. Thus he hs to py 500*1.25 = 625 points. Imgine tht the following effort levels were chosen by the other 9 prticipnts: 10, 70, 200, 250, 420, 450, 550, 700, 1,200. Thus, the four students with the highest effort levels re ssigned to University 1 nd receive pyoff of 2,000 points. These re the students with effort levels 1,200, 700, 600 (Peter, nd 550. Of the remining eight students, five students with the highest levels of efforts re ssigned to University 2 nd receive pyoff of 1,000 points. These re the students with the efforts levels 500 (Simon, 450, 420, 400 (Juli nd 250. The students with effort levels 10, 70, nd 200 remin unssigned. Thus, the pyoff for Juli is 2, 200 1, , 000 = 1, 600, for Peter 2, 200 1, , 000 = 3, 000 nd for Simon 2, , 000 = 2,

47 Instructions for DCA The lloction procedure is implemented s follows: At the beginning of the ech round, every student lerns her bility. The bility of ech student is drwn uniformly from the intervl from 0 to 100. Thus every student hs n equl chnce of being ssigned every level of bility from the intervl. You will lern your own bility but not the bility of the other 11 students competing with you for the sets. The bility is drwn independently for ll prticipnts in every round. The dmission to universities is decentrlized. Students first decide which university they wnt to pply to. Thus, you hve to choose one university you wnt to pply to. After the decision is mde, you will compete only with students who hve decided to pply to the sme university. The ssignment of sets t ech university is bsed on the mount of the effort tht ech student puts into finl test. In the experiment you cn choose level of effort. This effort is costly. The price of effort depends on your bility. The higher the bility the esier (cheper is the effort. The price of one unit of effort is determined s: 100 divided by the bility, 100/bility. On your screen you will see your bility for the round nd the corresponding price of one unit of effort. You will hve to decide on the mount of the effort tht you choose. In ech of the rounds you cn use the clcultor which will be on your screen. You cn use it to find out wht possible pyoffs prticulr effort in points cn yield. To gin better understnding for the experiment you cn insert different vlues. This will help you with your decision. In the beginning of ech round, every prticipnt receives 2,200 points tht cn be used to exert effort or kept. After ech student decides how much effort to buy, these efforts re used to determine the ssignments to universities. Among the students who pply to University 1, the students with the highest effort levels re ssigned to this university up to its cpcity nd receive 2,000 points. All other students who pplied to University 1 remin unssigned. Among those students who pply to University 2, the students with the highest effort levels re ssigned set up to the cpcity of University 2. They receive the corresponding mount of points. All other students who hve pplied to University 2 remin unssigned. Prticipnts tht hve chosen the sme mount of effort will be rnked ccording to rndom drw. Ech prticipnt receives pyoff tht is determined s the sum of the non-invested endowment nd the pyoff from university dmission. Thus: Pyoff = Endowment price of effort units of effort + pyoff from ssignment Note tht your bility, the bility of the other prticipnts, nd the number of sets t University 1 nd University 2 vry in every round. Every point corresponds to 0.5 cents. Only one of the rounds will be relevnt for you ctul pyoff. This round will be selected rndomly by the computer t the end of the experiment. 45

48 Exmple Let us consider n exmple with three hypotheticl persons: Juli, Peter, nd Simon. Imgine the following round: University 1 hs four sets, nd University 2 hs five sets. Juli hs n bility of 25 nd decides to pply to University 2. Thus the cost of one unit of effort is 100/25 = 4 points for her. Her endowment is 2,200 points, which mens tht she cn buy mximum of 2,200/4 = 550 units of effort. Let us imgine tht Juli decided to buy 400 units of effort. Thus she hs to py 400*4 = 1,600 points nd keeps 600 points of her endowment. Peter hs n bility of 50. He pplies to University 1. Thus the cost of effort for him is 100/50 = 2 points for one unit of effort. His endowment is 2,200 points. Thus he cn buy mximum of 2,200/2 = 1100 units of effort. Let us ssume tht Peter chose 600 units of effort. Thus he hs to py 600*2 = 1,200 points. Simon hs n bility of 80. He pplies to University 2. Thus the cost of one unit of effort is 100/80 = 1.25 points for one unit of effort. His endowment is 2,200 points. Thus he cn buy mximum of 2, 200/1.25 = 1, 760 units of effort. Let us imgine tht Simon decides to buy 500 units of effort. Thus he hs to py = 625 points. Imgine tht there re n dditionl four students who decide to pply to University 2 (competing with Juli nd Simon, nd five students who decide to pply to University 1 (competing with Peter. The following efforts were bought by the four prticipnts who pply to University 2, together with Juli: 10, 70, 450, 550. Thus, there re 6 contenders for 5 sets. All students, but one with the effort of 10, receive set t University 2 nd thus pyoff of 1,000 points. The following efforts were bought by the five other prticipnts who pply to University 1, together with Peter: 200, 250, 420, 700, 1,200. Thus, there re 6 contenders for 4 sets. The four students with the highest efforts re ssigned to University 1, including Peter, nd ll receive 2,000 points. The students with effort levels 200 nd 250 remin unssigned. Thus, the pyoff for Juli is, , 600+1, 000 = 1, 600, for Simon 2, , 000 = 2575 nd for Peter 2, 200 1, , 000 = References Abdulkdiroğlu, A., nd T. Sönmez (2003: School Choice: A Mechnism Design Approch, Americn Economic Review, 93, Abizd, A., nd S. Chen (2011: The College Admissions Problem with Entrnce Criterion, Working pper. Amegshie, J. A., nd X. Wu (2006: Adverse Selection in Competing All-py Auctions, mimeo. 46

49 Avery, C., nd J. Levin (2010: Erly Admissions t Selective Colleges, Americn Economic Review, 100, Blinski, M., nd T. Sönmez (1999: A Tle of Two Mechnisms: Student Plcement, Journl of Economic Theory, 84, Brut, Y., D. Kovenock, nd C. N. Noussir (2002: A Comprison of Multiple-Unit All- Py nd Winer-Py Auctions Under Incomplete Informtion, Interntionl Economic Review, 43, Bye, M. R., D. Kovenock, nd C. G. de Vries (1996: The All-py Auction with Complete Informtion, Economic Theory, 8, Biro, P. (2012: University Admission Prctices - Hungry, ccessed July 21, Chde, H., G. Lewis, nd L. Smith (2014: Student Portfolios nd the College Admissions Problem, Review of Economic Studies, 81(3, Che, Y.-K., nd Y. Koh (2013: Decentrlized College Admissions, Working pper. Chen, Y., nd O. Kesten (2013: From Boston to Chinese Prllel to Deferred Acceptnce: Theory nd Experiments on Fmily of School Choice Mechnisms., Working pper. Chen, Y., nd T. Sönmez (2006: School Choice: An Experimentl Study, Journl of Economic Theory, 127, Dechenux, E., D. Kovenock, nd R. M. Sheremet (2012: A Survey of Experimentl Reserch on Contests, Working pper. Fischbcher, U. (2007: z-tree: Zurich Toolbox for Redy-mde Economic Experiments, Experimentl Economics, 10, Gle, D., nd L. S. Shpley (1962: College Admissions nd the Stbility of Mrrige, Americn Mthemticl Monthly, 69, Greiner, B. (2004: An Online Recruitment System for Economic Experiments, in Forschung und wissenschftliches Rechnen, ed. by K. Kremer, nd V. Mcho, vol. 1A, pp Göttingen: Ges. für Wiss. Dtenverrbeitung. Hrbring, C., nd B. Irlenbusch (2003: An Experimentl Study on Tournment Design, Lbour Economics, 10, Hickmn, B. (2009: Effort, Rce Gps, nd Affirmtive Action: A Gme Theoric Anlysis of College Admissions, Working pper. 47

50 Kgel, J., nd A. E. Roth (2000: The Dynmics of Reorgniztion in Mtching Mrkets: A Lbortory Experiment motivted by Nturl Experiment, Qurterly Journl of Economics, 115, Konrd, K. A. (2009: Strtegy nd Dynmics in Contests. Oxford University Press, Oxford. Konrd, K. A., nd D. Kovenock (2012: The Lifebot Problem, Europen Economic Review, 27, Krishn, V. (2002: Auction Theory. Acdemic Press, New York. Moldovnu, B., nd A. Sel (2001: The Optiml Alloction of Prizes in Contests, Americn Economic Review, 91(3, (2006: Contest Architecture, Journl of Economic Theory, 126, Moldovnu, B., A. Sel, nd X. Shi (2012: Crrots nd Sticks: Prizes nd Punishments in Contests, Economic Inquiry, 50(2, Morgn, J., D. Sisk, nd F. Vrdy (2012: On the Merits of Meritocrcy, Working pper. Müller, W., nd A. Schotter (2010: Workholics nd Dropouts in Orgniztions, Journl of the Europen Economic Assocition, 8, Noussir, C. N., nd J. Silver (2006: Behvior in All-Py Auctions with Incomplete Informtion, Gmes nd Economic Behvior, 55, Perch, N., J. Polk, nd U. Rothblum (2007: A Stble Mtching Model with n Entrnce Criterion pplied to the Assignment of Students to Dormitories t the Technion, Interntionl Journl of Gme Theory, 36, Perch, N., nd U. Rothblum (2010: Incentive Comptibility for the Stble Mtching Model with n Entrnce Criterion, Interntionl Journl of Gme Theory, 39, Slgdo-Torres, A. (2013: Incomplete Informtion nd Costly Signling in College Admissions, Working pper. Siegel, R. (2009: All-py Contests, Econometric, 77, Sönmez, T., nd U. Ünver (2011: Mtching, Alloction, nd Exchnge of Discrete Resources, in Hndbook of Socil Economics, ed. by J. Benhbib, A. Bisin, nd M. O. Jckson, vol. 1A, pp North Hollnd. Tung, C. Y. (2009: College Admissions under Centrlized Entrnce Exm: A Compromised Solution, Working pper. 48

51 Discussion Ppers of the Reserch Are Mrkets nd Choice 2014 Reserch Unit: Mrket Behvior Hidekzu Anno, Morimitsu Kurino SP II Second-best incentive comptible lloction rules for multiple-type indivisible objects SPII Pblo Guillen, Rustmdjn Hkimov Monkey see, monkey do: truth-telling in mtching lgorithms nd the mnipultion of others SPII Roel vn Veldhuizen, Joep Sonnemns Nonrenewble resources, strtegic behvior nd the Hotelling rule: An experiment SPII Roel vn Veldhuizen, Hessel Oosterbeek, Joep Sonnemns Peers t work: From the field to the lb Sebstin Kodritsch On Time-Inconsistency in Brgining Dietmr Fehr, Juli Schmid Exclusion in the All-Py Auction: An Experimentl Investigtion Dvid Dnz The curse of knowledge increses self-selection into competition: Experimentl evidence Is E. Hflir, Rustmdjn Hkimov, Dorothe Kübler, Morimitsu Kurino College Admissions with Entrnce Exms: Centrlized versus Decentrlized SPII SPII SPII SPII Reserch Unit: Economics of Chnge Jn Grohn, Steffen Huck, Justin Mttis Vlsek SP II A note on empthy in gmes Mj Aden SP II Tx-price elsticity of chritble dontions - Evidence from the Germn txpyer pnel Orzio Attnsio, Britt Augsburg, Rlph De Hs, Eml Fitzsimons, Heike Hrmgrt Group lending or individul lending? Evidence from rndomized field experiment in Mongoli SP II Britt Augsburg, Rlph De Hs, Heike Hrmgrt nd Costs Meghir SP II Microfinnce t the mrgin: Experimentl evidence from Bosni nd Herzegovin All discussion ppers re downlodble:

52 Andrew Schotter, Isbel Trevino SP II Is response time predictive of choice? An experimentl study of threshold strtegies Luks Wenner SP II Expected Prices s Reference Points - Theory nd Experiments WZB Junior Reserch Group: Risk nd Development Ferdinnd M. Vieider, Abebe Beyene, Rndll Bluffstone, Shn Dissnyke, Zenebe Gebreegzibher, Peter Mrtinsson, Alemu Mekonnen Mesuring risk preferences in rurl Ethiopi: Risk tolernce nd exogenous income proxies SP II All discussion ppers re downlodble: