College Admissions with Entrance Exams: Centralized versus Decentralized

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1 College Admissions with Entrnce Exms: Centrlized versus Decentrlized Is E. Hflir Rustmdjn Hkimov Dorothe Kübler Morimitsu Kurino My 20, 2015 Abstrct We theoreticlly nd experimentlly study college dmissions problem in which colleges ccept students by rnking students efforts in entrnce exms. Students bility levels ffect the cost of their efforts. We solve nd compre equilibri of centrlized college dmissions (CCA where students pply to ll colleges nd decentrlized college dmissions (DCA where students only pply to one college. We show tht lower bility students prefer DCA wheres higher bility students prefer CCA. Mny predictions of the theory re supported by the experiments, yet we find number of differences tht render DCA less ttrctive thn CCA compred to the equilibrium benchmrk. JEL Clssifiction: C78; D47; D78; I21 Keywords: College dmissions, incomplete informtion, student welfre, contests, ll-py uctions, experiment. We would like to thnk Ken Binmore, Frncis Bloch, Aytek Erdil, Dvid Dnz, Youngwoo Koh, Fuhito Kojim, Ki A. Konrd, Vijy Krishn, Benny Moldvnu, Ariel Rubinstein, Aner Sel, Ron Siegel, Noki Wtnbe, Alistir Wilson nd prticipnts t seminrs in Boston College, Fukuok, Hitotsubshi, HKUST, Lisbon, McMster, Michign, New York University, Otru, Rice, Wsed, the 14th SAET conference t Wsed, the Designing Mtching Mrkets workshop t WZB Berlin, the Economic Theory workshop t Penn Stte University, the Mtch-up conference t the University of Glsgow, for helpful discussions s well s Nin Bonge for progrmming nd helping us to run the experiments nd Jennifer Rontgnger for copy editing. Hflir cknowledges finncil support from Ntionl Science Foundtion grnt SES Kübler cknowledges finncil support from the Deutsche Forschungsgemeinschft (DFG through CRC 649 Economic Risk. Kurino cknowledges finncil support from JSPS KAKENHI Grnt Number 15K All remining errors re our own. Tepper School of Business, Crnegie Mellon University, Pittsburgh, PA, 15127, USA; e-mil: isemin@cmu.edu. WZB, Reichpietschufer 50, 10785, Berlin, Germny; e-mil: rustmdjn.hkimov@wzb.eu WZB, Reichpietschufer 50, 10785, Berlin, Germny; nd Technicl University Berlin; e-mil: kuebler@wzb.eu Fculty of Engineering, Informtion nd Systems, University of Tsukub, Tennodi, Tsukub, Ibrki , Jpn; e-mil: kurino@sk.tsukub.c.jp 1

2 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 then 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 the 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. 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. Currently, the students cnnot pply to both the University of Cmbridge nd the University of Oxford. 3 Moreover, till 1994 the college dmission exms in South Kore ws only offered on two dtes ech yer, nd students were llowed to pply for only one college per exm dte (see Avery, Lee, nd Roth, 2014 for more detils. 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 inbetween the two extreme cses. In this pper, we compre the institutionl effects of different college dmission mechnisms on 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 corresponds to our description nd modeling of the decentrlized mechnism 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. 3 We thnk Aytek Erdil nd Ken Binmore for discussions on college dmission systems in UK. 2

3 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 students hve homogeneous preferences for ttending these colleges. 4 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 the 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 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 with multiple prizes where the plyers hve privte informtion regrding their bilities nd sort themselves into different contests. 5 After solving for the equilibrium of CCA nd DCA, we compre the equilibri in terms of 4 In section 6, we discuss the cse with three or more colleges. 5 There is lrge literture on competing uctions nd competing mechnisms. A notble exception tht nlyzes competing contests with unit prizes nd incomplete informtion is DiPlntino nd Vojnovic (2009. We discuss this literture in the Relted Literture subsection. 3

4 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. 6 The min intuition for this result is tht high-bility students (i cn only get set in the good college in DCA, wheres they cn get sets 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. 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 some mrkets the comprisons of the students ex-nte expected utilities, their effort levels, nd the students preferences regrding the two mechnisms given their bility 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. We lso find significnt differences between the two mechnisms with respect to the sorting of students tht re in prt due to out-of-equilibrium choices of the experimentl subjects. 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 while section 6 provides extensions. Section 7 presents our experimentl results. Finlly, section 8 concludes. Omitted proofs nd dditionl figures 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 6 More specificlly, we obtin single crossing condition: if student who pplies to college 2 in DCA prefers CCA to DCA, then ll higher bility students lso hve the sme preference rnking. 4

5 centrlized college dmission models in the two-sided mtching literture. Recent work regrding centrlized college dmissions with entrnce exms include Abizd nd Chen (2015 nd Tung (2009. Abizd nd Chen (2015 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 the multi-ctegory seril dicttorship (MSD nlyzed by Blinski nd Sönmez (1999 in order to mke students better off. One crucil difference between the modeling in our pper nd the literture should be emphsized: 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. More recently, motivted by the South Koren college dmission system tht went through policy chnge in 1994, Avery, Lee, nd Roth (2014 compre the two (with nd without erly dmissions regimes nd conclude tht lower-rnked colleges my gin in competition with higher-rnked colleges by limiting the number of possible pplictions. 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 centrlized 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 5

6 students pproximtely know their rnkings within the relized smple of privte costs. 7 Similrly, Olszewski nd Siegel (2014 consider contests with mny plyers nd prizes nd show tht the equilibrium outcomes of such contests re pproximted by the outcomes of n ppropritely defined set of mechnisms. In contrst to Hickmn (2009 nd Olszewski nd Siegel (2014, our results re lso pplicble when the number of gents is not 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, 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 others 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 7 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. 6

7 cn be used. Under some conditions, they show tht punishing the bottom is more effective thn rewrding the top. There is lrge literture on competing uctions nd mechnisms; notble exmples re Ellison, Fudenberg, nd Möbius (2004, Biis, Mrtimort, nd Rochet (2000, McAfee (1993, nd more recently, Moldovnu, Sel, nd Shi (2008, Virág (2010, nd Ovdi (2014. Two ppers tht re most relted to our ppers re DiPlntino nd Vojnovic (2009 nd Buyukboyci (2012. DiPlntino nd Vojnovic (2009 consider multiple contests where ech contest gives single prize nd show existence of symmetric monotone equilibrium using the revenue equivlence theorem. They re mostly interested in prticiption rtes mong different contests nd estblish tht in the lrge system limit (i.e., s the popultion gets lrge the number of plyers tht prticipte in given contest clss is Poisson rndom vrible. Buyukboyci (2012, on the other hnd, theoreticlly nd experimentlly compres the performnce of one contest with single prize nd two prllel contests ech with single prize. In her model gents cn be either high bility or low bility type. Her min finding is tht the designer s profit is higher in the prllel tournments when the contestnts low nd high bility levels re sufficiently differentited. This pper lso contributes to the 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 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 do not study 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. 8 For recent exmple of theory nd experiments in school choice literture, see Chen nd Kesten (

8 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 two 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. 9 Ech student s S mkes n effort e s. Ech student is ssigned to one college or no set in ny college by the mechnisms which tke the efforts into ccount while deciding on their dmissions. 10 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 identiclly distributed ccording to T. Tht is, T k,m ( := m j=k ( m T ( j (1 T ( m j. (1 j Moreover, let t k,m ( denote T k,m ( s density: t k,m ( := d d T k,m( = m! (k 1! (m k! T (k 1 (1 T ( m k t(. (2 9 Mny college dmissions, including ones in Turkey nd Jpn, re competitive in the sense tht the totl number of sets in colleges is smller thn the number of students who tke the exms. 10 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. 8

9 0. 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: for ll j, k {0, 1,..., n} nd x [0, 1], ( 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. 11 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], 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 (. 11 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. 9

10 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 (] = 0. Thus, by integrtion nd s the boundry condition is β C (0 = 0, we hve 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. 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}. 12 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 chooses college 2 with probbility 1 γ(, nd mkes effort β D (. (c, 1] chooses college 2 for sure, nd mkes effort β D (. 13 Ech student with type In this equilibrium, lower bility students choose the sme effort level independent of whether they re pplying to college 1 or 2. Note tht this is n equilibrium property, not restriction on effort functions. In other words, 12 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 college to pply to, nd without knowing how mny other students hve pplied, they lso choose their effort level. For ech college 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. 13 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. Therefore, we hve to hve some students using mixed strtegies while choosing which college to pply to. Forml rguments resulting in the nonexistence result re vilble from the uthors upon request. Derivtions show tht in equilibrium, lower bility students would use mixed strtegies, while the higher bility students re certin to pply to the better college. 10

11 students re llowed to choose different effort levels when they re pplying to different colleges, yet they choose the sme effort level in equilibrium. 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. We re now redy to stte the min result of this section, which chrcterizes the unique symmetric Byesin Nsh equilibrium 14 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. 14 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

12 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 where γ( = n 1 A( := v 1 n 1 B( := v 2 ( 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, ( 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 m j=m q ( x( p j,m j. π(c 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 12

13 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 respectively. 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 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, we show tht we cn get the mixed strtegy function γ( by differentiting Eqution (9. (9 13

14 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. 15 βd (. 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 for ll [0, c] 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 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 ( = 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 14

15 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 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, it follows 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 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, it follows 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 15

16 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 = 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 (. 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. 16

17 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 (. 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 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 d EU C ( < d (, such tht EU C ( = EU D ( nd d EU D (, contrdiction. Hence (i is stisfied. Moreover, (ii is obviously d stisfied since whenever EU C ( < EU D (, we hve to hve d d EU C ( > d d EU D (. 17

18 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 whether there is generl ex nte utility rnking between DCA nd CCA. It turns out tht exmples where either DCA or CCA result in higher ex nte utility (or socil welfre cn be found Extensions In this section, we consider two extensions of the model. In the first, we llow for more thn two colleges, gin rnked in terms of qulity. The second extension looks t lrger mrket in the following sense: s before, setup is studied with two types of colleges resulting in utilities v 1 nd v 2 nd with cpcities q 1 nd q 2, but there re k colleges of ech type nd there re k n students. 6.1 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 17 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 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: 16 Specific exmples re vilble from the uthors upon request. 17 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. 18

19 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 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 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] only pply to college l. 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. 6.2 The cse of k-repliction Consider n environment in which we hve, (i k type-1 colleges: C1, 1..., C1 k such tht ech of them hs q 1 sets nd gives utility of v 1 to students, (ii k type-2 colleges: C2, 1..., C2 k such tht ech of them hs q 2 sets nd gives utility of v 1 to students, nd (iii k n students. In other words, in this extension we consider k-repliction of our model. With this extension, in CCA it is esy to see tht there is monotone equilibrium very similr to the originl equilibrium. The students will list ll type-2 colleges bove ll type-1 colleges (in n rbitrry fshion, students with the top k q 2 effort levels will get one of the type-2 colleges sets, nd students with the next top k q 1 effort levels will get one of the type-1 colleges sets. In this equilibrium student with type will choose the effort β C(k ( = ˆ 0 x{f kn kq2,kn 1 (x v 2 (f kn kq2 kq 1,kn 1 (x f kn kq2,kn 1 (x v 1 }dx. Moreover, we hve tht β C(k ( will be very close to β C ( for ll k = 2,...,. In fct, when F 19

20 is uniform we hve ( ( n β C(k q2 n ( = n v q1 q 2 2 n ( = v 2 + q 1v 1 q 2 v 2 n n q 2 v 1 n for ll k = 1, 2,...,. Hence, for uniform distributions, ny k-replic economy bidding function is the sme s in the no-replic economy. In DCA, on the other hnd, one cn observe tht the equilibrium of the k-replic economy essentilly remins the sme s in the no-replic economy: the cutoff c nd equilibrium effort functions will be the sme. The only differences would be tht (i ech student of bility lower thn c will pply to ech type-1 college with probbility γ( nd ech type-2 college with probbility 1 γ( k, nd (ii ech student of bility higher thn c will pply to ech type-2 college with probbility 1 k. Hence, if there re mny students nd mny colleges (belonging to one of the two types, our predictions remin vlid. k 7 The Experiment In this section, we present n experiment 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. We compre the two mechnisms nd study which of them leds to higher (interim nd ex-nte student welfre, higher efforts of the students, nd how they ffect the sorting of students by bility. 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 from 1 to 100. Students hve to choose n effort level e s tht determines their success in the ppliction process. The cost of effort is determined by 100 es s. (Note tht we use the rnge of bilities from 1 to 100 insted of 0 to 1 in order to simplify the clcultions for subjects. Accordingly, we scled up the cost function by constnt of 100. 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 20

21 Tble 1: Overview of mrket chrcteristics Number of sets t [vlue of] college 2 college 1 Predicted utility higher Predicted effort higher 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 some mrkets, one of the two mechnisms domintes the other for ll students. In other mrkets the rnking of the mechnisms depends on the students bility in which cse we compre the expected vlues. 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, see Tble 1. This llows us to investigte behvior under very different mrket conditions. The prmeters in ech mrket were chosen so s to generte clercut predictions regrding the two min outcome vribles, effort nd the expected utility of ech student. Figure 2 shows the interim expected utility of students for ech mrket. We chose the prmeters for mrket 1 nd mrket 2 to ensure tht CCA domintes DCA with respect to interim expected utility of students (mrket 1 nd the reverse (mrket 2. Figure 3 shows the equilibrium effort levels given bilities for ech mrket. We chose prmeters for mrkets 3 nd 4 to ensure tht in ech of the mrkets one mechnism domintes the other with respect to the effort levels of students. 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. We cn lso compre the 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, which is n importnt distinction for the theoreticl predictions. As illustrted by Figure 3, in mrkets 1 nd 4 with n equl number of sets nd pplicnts positive effort in CCA is only exerted by those who cn expect to get into the good college. In DCA, efforts re overll higher in these two mrkets becuse of the risk of miscoordintion. In mrkets 2 nd 3 in contrst, 21

22 Figure 2: Equilibrium expected utility by bility 22

23 Figure 3: Equilibrium efforts by bility 23

24 high-bility students tend to exert less effort in DCA thn in CCA becuse the expected return is higher in CCA: in CCA one cn obtin v 2, v 1 nd 0, while in DCA only v 2 (or v 1 nd 0 re chievble. Note tht our design ims t compring the two mechnisms. We do not study the comprtive sttics of the equilibri of CCA nd DCA by systemticlly vrying one prmeter. This would require completely different design tht is beyond the scope of this study. 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 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 ech round 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. At the beginning of the experiment, printed instructions were given to the prticipnts (see Appendix E. 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. 24

25 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. The significnce level of ll our results is 5%, unless otherwise stted Tretment comprisons: ggregte results We compre the two mechnisms with respect to three properties, summrized in results 1 to 3. The first comprison concerns the utility of students in the two mechnisms which is equl to the number of points erned, due to the ssumption of risk neutrlity. Second, we investigte whether one of the mechnisms induces higher effort levels thn the other mechnism. And the third spect is whether individuls of different bilities prefer different mechnisms. Result 1 (Averge utility: In mrkets 1 nd 4 (where the number of sets equls the number of students, the verge utility of students in CCA is significntly higher thn in DCA, s predicted by the theory. In mrkets 2 nd 3 (where there re less sets thn pplicnts, the verge utility of students in DCA is not significntly 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 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 the utility of students 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 Mrket Utility higher Averge utility higher Averge utility Averge utility Observed utilities for ll students for relized types in CCA in DCA different in (predicted (predicted (observed (observed CCA nd DCA 1 CCA CCA, DCA DCA, N/A DCA, CCA CCA, N/A N/A, Notes: Columns 3 nd 6 show the p-vlues of the Wilcoxon rnk-sum test for equlity of the distributions, bsed on verges of the six mtching groups per tretment. In mrkets 1 nd 4, the equilibrium predictions for the comprison of utilities of students 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 25

26 the gol is to mximize the utility of the students. This is due to the potentil miscoordintion of pplicnts in DCA in these mrkets. However, 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. Note lso tht the verge utility is negtive in the competitive mrket 2 (with only four sets for 12 students such tht, contrry to the prediction, the subjects ern less thn the 2,200 points they re endowed with. Result 2 (Averge effort: In mrkets 1 nd 4 (where the number of sets equls the number of students, the verge effort level of students in DCA is significntly 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 significnt 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 rnksum test of the equilibrium efforts bsed on the relized drw of bilities. We expect effort to only differ significntly between the two mechnisms in mrkets 3 nd 4 (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 becuse observed verge effort is significntly higher in DCA. In mrket 3 verge efforts re higher in 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 N/A DCA, N/A N/A, CCA CCA, DCA DCA, N/A N/A, Notes: Columns 3 nd 6 show the p-vlues of the Wilcoxon rnk-sum test for equlity of the distributions, bsed on verges of the six mtching groups per tretment. Tking results 1 nd 2 together, 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 re not in line with the theory. Mrkets 2 nd 3 should led to higher verge utility of the students in DCA thn in CCA, which is not observed in the lb. 26

27 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 of whether students of different bilities prefer different mechnisms by providing n experimentl test of propositions 2 nd 3. According to Proposition 2, low-bility 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 (where there re less sets thn pplicnts, 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. However, significntly fewer students thn predicted prefer DCA to CCA. There is no significnt difference between the verge utilities of students in DCA nd CCA in mrket 5. Support: In three of our mrkets, nmely 2, 3, nd 5, low-bility students prefer DCA in equilibrium. We refer to the predicted switching point s the mximum bility t which students prefer DCA in equilibrium. The predicted switching points by mrkets re represented in Figure 4 by the intersection of the broken lines. For mrket 2, the switching point is 100, for mrket 3 it equls 81, nd for mrket 5 it equls 26. Figure 4 lso shows the observed switching points s the intersection of the solid lines in mrkets 2, 3, nd 5. The figure revels tht in mrkets 2 nd 3 the observed switching points re substntilly lower thn the predicted switching points. This suggests tht fewer students thn predicted prefer DCA to CCA in these mrkets. Tble 4: Number of switching points in the 50,000 bootstrpped smples, by mrkets Mrket 2 3 Unique switching point in the predicted direction 77.5% 80.1% Two switching points 17.3% 4.5% Three or more switching points 0.8% 4.6% No switching points 4.2% 6.9% Unique switching point in the opposite direction 0.2% 3.9% To ssess the sttisticl significnce of these differences in switching points, we use bootstrpping. Tht is, we smple from the dtset with replcement to generte new smples nd clculte the bootstrp confidence intervls of the observed switching points in mrkets 2 nd 3. 18,19 Before 18 Bootstrp confidence intervls re clculted by the percentile method (Efron, We perform block resmpling to ccount for the dependence of observtions within mtching groups (see Dvison, For ech of 50,000 bootstrp smples, we drw six rndom mtching groups with replcement nd clculte the bootstrp switching point for ech mrket bsed on the polynomil smoothing of the observed utilities (we use lpoly in STATA with bndwidth 15 both for the bootstrp nd for producing Figure We did not clculte bootstrp confidence intervls for mrket 5, becuse it cn be tken from Figure 4 tht 27

28 Figure 4: Predicted expected utilities nd kernel regression of observed utilities by bilities. 28

29 Figure 5: Distribution of observed switching points turning to the bootstrp confidence intervls, we first use the bootstrp smples to ssess the theoreticl prediction of unique switching point with bility types bove the switching point preferring CCA in ll mrkets. The first row of Tble 4 revels tht the vst mjority of the bootstrp smples indeed produce unique switching point in the predicted direction, i.e., lowerbility students prefer DCA while students with bilities bove the switching point prefer CCA. In mrket 2, 77.5% of the bootstrp smples yield unique bootstrpped switching point in the predicted direction. 20 In mrket 3, 80.1% of the bootstrp smples provide bootstrpped switching point tht is unique nd in the predicted direction. Overll, the bootstrp smples confirm the single-crossing property of Proposition 3. Finlly, we study the bootstrp distribution of the observed switching points where we restrict ttention to bootstrp smples with unique switching point in the predicted direction. Figure 5 presents histogrm of the bootstrpped switching points by mrkets. The verge switching point in mrket 2 (left pnel is 48.4 with 95% confidence intervl of [30.5, 68.4], clerly indicting tht the observed switching point is below the theoreticl prediction of 100. In mrket 3 (right pnel, the verge switching point is 37.6 with 95% confidence intervl of [9.2, 69.3], lso indicting tht the observed switching point is below the theoreticl prediction of 81. Thus we cn conclude tht the observed switching points re significntly lower thn the predicted switching points in mrkets 2 nd 3, nd we observe smller rnge of bilities of students preferring DCA thn the theory suggests. there is no significnt difference in the expected utility for high- nd low-bility students in the two systems, s predicted. 20 In this mrket, number of drws resulted in two switching points. This cn be explined by the fct tht in DCA two students with very low bilities of 2 nd 6, respectively, took dominted effort choices by spending ll their endowment, which resulted in utility of points. In some bootstrp smples, these two observtions shift the smoothed line of the expected utility in DCA below the line for CCA for the lowest bility types. 29

30 7.2.2 Point predictions nd individul behvior Next we investigte the individul behvior of subjects in ech mechnism seprtely. In prticulr, we test the point predictions regrding the utility nd 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 from the comprison of the two mechnisms, in prticulr the reltively poor performnce of DCA with respect to student utility. Result 4 (Averge utility nd effort by mrkets: (i The verge utility is significntly lower thn predicted cross ll mrkets nd mechnisms. (ii Averge effort levels in the experiments re higher thn the equilibrium efforts in ll 10 mrkets. This overexertion of effort is significnt in ll five mrkets in DCA nd in three out of five mrkets in CCA. Tble 5: Averge utility nd effort by mrkets Averge Averge Averge Averge equilibrium observed p-vlue equilibrium observed p-vlue utility utility obs.=pred. efforts efforts obs.=pred. (1 (2 (3 (4 (5 (6 CCA Mrket Mrket Mrket Mrket Mrket DCA Mrket Mrket Mrket Mrket Mrket Notes: Column (3 [(6] shows the p-vlues for the significnce of the constnt when regressing the difference between (1 nd (2 [(4 nd (5] on constnt, with stndrd errors clustered t the level of mtching groups. Support: Tble 5 displys the equilibrium nd observed verges for utility nd effort levels by mrkets. Note tht the verge utility of subjects is significntly lower thn predicted in ll five mrkets nd under both mechnisms. In ddition, Figure 4 shows tht verge utility levels re lwys below the predicted level for ll bilities. This is consistent with the fct tht in ll mrkets nd mechnisms, verge effort levels re higher thn predicted, s cn be tken from comprison of columns (4 nd (5 in Tble In CCA the difference is significnt for three out 21 Figure 7 in the online ppendix depicts the observed 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, s predicted. Moreover, the observed effort levels typiclly lie bove the predicted vlues, except for high-bility students in few mrkets. 30

31 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. 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. 22 Moreover, there is no significnt difference with respect to the predictive power of the equilibrium in the different mechnisms (s the predictions for CCA nd the dummy vrible for CCA re both not significnt. As next 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 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. However, high-bility students pply to the good college significntly less often thn predicted in ll mrkets while low-bility students pply to the good college more often thn predicted (significnt in three mrkets. Support: Tble 6 displys the equilibrium cutoff bility for ech mrket in column (1. In column (2 it provides the verge equilibrium probbility of choosing the good college 2 for students with bilities below the cutoff in the respective mrkets. This verge is clculted given the ctul reliztion of bilities in the experiment. It cn be compred to the observed frequency of choosing the good college by these students in column (3 nd the 95% confidence intervls with stndrd errors clustered t the level of mtching groups in column (4. It emerges tht subjects below the cutoff choose the good college 2 more often thn predicted in ll five mrkets. The difference is significnt for mrkets 1, 3, nd 5. Column (5 displys the proportion of subjects bove the cutoff pplying to college 2, followed by the 95% confidence intervl with stndrd errors clustered t the level of mtching groups in column (6. Note tht in equilibrium these high-bility students should pply to college 2 with certinty, but we cn reject this hypothesis in ll five mrkets. 23 Finlly, the lst column of Tble 6 presents the p-vlues for the Wilcoxon rnk-sum test of equlity of the distributions of the choice of college 2 below nd bove the mrket-specific equilibrium cutoff bsed on verges of six mtching groups. In ll mrkets except mrket 4, the differences re significnt t the 1% significnce level, nd the difference is mrginlly significnt for mrket 4. Further evidence of the predictive power of the model is provided by probit regression of observed choices of college 2. The coefficient for the equilibrium probbility of choosing the good college is 22 See Tble 9 in the online ppendix. 23 In mrkets 1, 2, nd 5 the observed proportions re close to the equilibrium. In mrket 3 fewer high-bility students choose the good college, which my be due to the lrge bd college (eight sets reltive to the good college (two sets. In mrket 4, the reltively low rte of high-bility students pplying to the good college my be driven by the similrity of pyoffs for both colleges (1,800 points versus 2,000 points. 31

32 significnt. 24 Thus we conclude tht the choices of the subjects reflect the predicted equilibrium pttern, but tht the point predictions fil. Tble 6: Proportion of choices of good college 2 Equ. prop. p-vlues for of choices Obs. prop. of choices Obs. prop. of choices equlity of Equilibrium of college of college 2 below of college 2 bove proportions bility 2 below the cutoff the cutoff bove nd cutoff the cutoff men 95% conf. int men 95% conf. int below the cutoff (1 (2 (3 (4 (5 (6 (7 Mrket % 33% [25%-44%] 85% [75%-92%] 0.00 Mrket % 51% [41%-61%] 92% [77%-98%] 0.00 Mrket % 27% [20%-36%] 68% [49%-82%] 0.00 Mrket % 17% [11%-27%] 42% [21%-67%] 0.07 Mrket % 64% [54%-72%] 91% [84%-95%] 0.00 Notes: Column (7 displys the p-vlues of the Wilcoxon rnk-sum test for equlity of the distributions, bsed on verges of the six mtching groups. Confidence intervls re estimted with stndrd errors clustered t the level of mtching groups. Next we investigte the ppliction decision of students by bility. Figure 6 presents the choices of subjects in DCA by mrkets nd bility quntiles, together with the equilibrium predictions. Students bove the equilibrium cutoff in mrkets 1, 2, nd 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 is due to students who re slightly bove or below the cutoff. This finding is prticulrly evident in mrkets 1, 2, nd 4. Remember tht the equilibrium is chrcterized by discontinuity regrding 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 lmost 100% probbility. Not surprisingly, in the experiment the choices of universities re smooth round the cutoff. In line with this, we lso do not observe the predicted kink in the effort choices shown in Figure 2. These findings 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. As next step, we consider the effects of the mechnism on the sorting of students into colleges. In prticulr, we compre the two mechnisms, CCA nd DCA, with respect to the resulting verge bilities of the students in ech college by mrkets. Pnels A, B, nd C of Tble 7 present the equilibrium nd observed verge bilities of students ssigned to the good nd bd college nd of unssigned students, respectively. 25 Pnel D presents the equilibrium nd the observed 24 See Tble 10 in the online ppendix. 25 The equilibrium ssignment in CCA is strightforwrd to clculte given the bility drws. For DCA the choice of the college is rndom for students below the bility cutoff. We generte one reliztion of the choice of the college for ll bilities below the cutoff, given the equilibrium probbilities. Given the resulting choices of colleges, the equilibrium lloction is determined nd used for the clcultion in this tble. 32

33 Figure 6: Choice of college in DCA 33

34 percentge of unfilled sets by mrkets. Result 6 (Composition of colleges: (i (Good collegethere is no significnt difference in the verge bility of students in CCA nd DCA. This is in line with the theory except for mrkets 3 nd 4 where significntly higher bility of students in CCA is predicted. (ii (Bd college Ability levels re not significntly different in mrkets 1 nd 5, s predicted. The verge bility of students is lower thn predicted in DCA reltive to CCA in mrkets 2, 3, nd 4. Support: We consider ech mrket seprtely nd minly refer to rows (3 nd (6 in pnels A nd B of Tble 7. In mrkets 1 nd 5, both the theory nd the dt show no significnt difference between bility levels in the good nd bd college when compring CCA with DCA. In mrket 4 where the two colleges hve lmost the sme vlue for the students, the verge bility of students in the good college is predicted to be significntly lower in DCA, nd conversely, the verge bility is predicted to be higher in the bd college in DCA. We fil to see this significnt difference in the dt for both colleges becuse the verge bility levels t both colleges re more similr thn predicted under both mechnisms. Thus, the difference in sorting between the two mechnisms is smller thn predicted. In mrkets 2 nd 3, the observed bilities of students ssigned to the bd college re significntly higher in CCA thn in DCA in mrkets 2 nd 3 (see row (6 of Pnel B. In equilibrium the difference hs the sme sign but is much smller nd is not significnt. Thus, in DCA low-bility students hve better chnce thn predicted to be dmitted to the bd college in mrkets 2 nd 3, t the cost of some high-bility students who remin unssigned (cf. rows (2 nd (4 for mrkets 2 nd 3 in Pnel C. The reson for bilities being higher t the bd college in CCA thn in DCA in these mrkets is due to the following effect: in both mechnisms, students with bilities lower thn predicted re ble to get set in the bd college, due to imperfect sorting. But CCA llows high-bility students who re unble to get into the good college to obtin set in the bd college. This rises the verge bility in the bd college compred to DCA where the students who re unsuccessful t the good college remin unssigned. Tble 7 lso reports on the point predictions for ech mrket seprtely nd their performnce, with test results in rows (7 nd (8 of Pnels A nd B s well s rows (6 nd (7 in Pnel C. The point predictions re rejected in more thn hlf of the cses, but we refrin from discussing them in detil here, s our min focus is on the comprison of the two mechnisms. 7.3 Discussion In this section we discuss possible explntions of the devitions from equilibrium behvior. Overbidding is common finding in ll-py uction experiments (see Brut et l., 2002, nd Noussir nd Silver, 2006 nd our results confirm this in the well-known context of single contest with multiple prizes (CCA, but we lso show it to hold in prllel contests (DCA. Our experiments llow to compre the two mechnisms with the novel result of the reltive unttrctiveness of DCA 34

35 Pnel A Pnel B Pnel C Pnel D Tble 7: Averge bilities of students nd unfilled sets by colleges Mrket Assigned to good college, equil. CCA ( DCA ( CCA=DCA, equil., p-vlue ( Assigned to good college, observed CCA ( DCA ( CCA=DCA, observed, p-vlue ( CCA observed=cca equil., p-vlue ( DCA observed=dca equil., p-vlue ( Assigned to bd college, equil. CCA ( DCA ( CCA=DCA, equil., p-vlue ( Assigned to bd college, observed CCA ( DCA ( CCA=DCA, observed, p-vlue ( CCA observed=cca equil., p-vlue ( DCA observed=dca equil., p-vlue ( Not ssigned, equil. CCA ( DCA ( Not ssigned, observed CCA ( DCA ( CCA=DCA, observed, p-vlue (5 N/A N/A 0.52 CCA observed=cca equil., p-vlue (6 N/A N/A 0.09 DCA observed=dca equil., p-vlue ( Percentge of unfilled sets, equil. CCA (1 0% 0% 0% 0% 0% DCA (2 12.0% 1.4% 3.3% 5.1% 2.2% Percentge of unfilled sets, observed CCA (3 0% 0% 0% 0% 0% DCA (4 10.2% 1.4% 7.8% 9.3% 2.8% Notes: Rows (3 nd (6 of pnels A nd B nd row (5 of pnel C disply the p-vlues of the Wilcoxon rnk-sum test for equlity of the distributions, bsed on verges of the six mtching groups. Rows (7 nd (8 of pnels A nd B s well s rows (6 nd (7 of pnel C disply the p-vlues of t-test of equlity of the verges of the six mtching groups nd the predicted constnt vlue. 35

36 reltive to CCA, even in mrkets where it should be preferred by ll students. One cndidte to explin the difference between predicted nd observed utility levels in the two mechnisms is the number of unfilled sets in DCA. If students coordinte worse thn predicted in equilibrium, the ttrctiveness of DCA is reduced reltive to CCA. Pnel D, Tble 7 presents the equilibrium nd observed shres of unssigned sets by mrkets. The shre of unfilled sets in DCA is somewht higher thn in equilibrium in mrkets 3 nd 4 only, nd the difference is smll. Thus, unfilled sets cn t best prtilly explin the reltive unttrctiveness of DCA in our experiment. Aprt from welfre losses due to sets remining unfilled, the ggregte welfre of students is ffected by their choice of effort levels. Thus, overexertion of effort in DCA reltive to CCA is potentil explntion. Inspecting the observed nd predicted verge effort levels in the two mrkets where DCA should be preferble for students (mrkets 2 nd 3, see Tble 5, it emerges tht overbidding is more pronounced in DCA. 26 Moreover, it cn be tken from Tble 5 tht efforts fil to be significntly higher in CCA thn in DCA in mrket 3, in contrst to the prediction. Thus, the fct tht DCA leds to more overbidding in mrkets 2 nd 3 destroys its reltive dvntge for the students in these two mrkets. To understnd the possible sources of overbidding in DCA reltive to CCA, we investigte whether the students condition their effort choice on the choice of the college in DCA. Tble 8 presents the verge overbidding in terms of cost of effort by mrkets for students bove nd below the equilibrium bility cutoff. One unit of cost of effort corresponds to 50 cents, thus the mximum effort is 11 units. We use the costs of effort insted of effort in order to control for the scle of the overbidding tht depends on bility. 27 Note tht in equilibrium, students with bilities below the cutoff choose to pply to the good nd the bd college with certin probbility, while they exert the sme effort irrespective of the college choice. However, we observe tht prticipnts tend to exert higher effort when pplying to the good s compred to the bd college. 28 We even observe differences in effort by high-bility students, depending on their choice of college lthough they should pply to the good college in equilibrium. Some of these students underbid, especilly when pplying to the bd college, but these re reltively rre instnces. To sum up, reltive overbidding in DCA goes long with students conditioning their effort choice on the choice of the college. Another cndidte explntion for the overexertion of effort especilly in DCA is risk-version. Although we cnnot provide full nlysis of this cse due to technicl difficulties, we would like to elborte on possible effects of risk version in our setup. Fibich, Gvious, nd Sel ( In mrket 2, verge observed efforts nd equilibrium efforts differ by ( =25 points in CCA while the difference is 101 in DCA; similrly for mrket 3 with verge overbidding of 117 in CCA nd 159 in DCA. 27 Presenting this dt in units of effort would not be informtive s the sme devition of, sy, 100 units of effort mens smll devition for high-bility student but very lrge devition for low-bility student. 28 The difference is significnt for two mrkets for low-bility students. Column 4 of Tble 8 presents the p-vlues for the significnce of the dummy vrible for pplying to the good college when regressing overbidding in money terms on the dummy nd constnt for bilities below the theoreticl cutoff in DCA (with stndrd errors clustered t the level of mtching groups. 36

37 Tble 8: Averge overbidding in money terms, given the choice of the college in DCA. Mrket 1 Mrket 2 Mrket 3 Mrket 4 Mrket 5 Ability below cutoff Ability bove cutoff CCA DCA DCA P-vlue CCA DCA DCA P-vlue Bd Good Good= Bd Good Good= college college Bd college college Bd (1 (2 (3 (4 (5 (6 (7 (8 N Overbidding N Overbidding N Overbidding N Overbidding N Overbidding Notes: Columns (4 nd (8 show the p-vlues for the significnce of the dummy vrible for pplying to the good college when regressing overbidding in money terms on the dummy nd constnt for bilities below nd bove the theoreticl cutoff in DCA, respectively, with stndrd errors clustered t the level of mtching groups. hve shown tht in single contest, plyers with high vlues bid higher thn they would bid in the risk-neutrl cse (s compred to low-vlue bidders who will bid less. The intuitive reson for this is tht bidders who bid more (higher-vlued bidders hve more to lose in cse of not winning the prize, due to concve utility functions. Let us use this intuition to compre the overexertion of effort in CCA versus DCA. In CCA, high-bility student cn get high prize (v 2, low prize (v 1, or no prize (0, wheres in DCA, she would get either high prize (v 2 or no prize (0. Therefore, in CCA just filing to win high prize would still give this bidder low prize, wheres this would result in no prize in DCA. In other words, this bidder hs more to lose in decentrlized mechnism. Hence, we cn expect tht overexertion of effort would be more evident in DCA compred to CCA. 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 outperforms the decentrlized mechnism with respect to this criterion (see Tble 2. Averge utilities by mrkets re such tht CCA yields higher verge utility thn DCA in ll five mrkets. However, 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. Finlly, it should be noted tht the choice of the mechnism hs distributionl consequences, both in theory nd in the experiments. DCA is fvored by low-bility students while CCA is better for high-bility students. 37

38 8 Conclusion In this pper, we study college dmissions exms which concern millions of students every yer throughout the world. Our model bstrcts 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 without cost 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. 29 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 college 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, mny predictions of the theory re supported by the dt, despite few importnt differences. We find tht in our 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 the mrkets with n overdemnd for sets, low-bility students prefer decentrlized dmissions mechnism wheres high-bility students prefer centrlized mechnism. However, in these mrkets the predicted superiority of the decentrlized mechnism for the students is weker thn predicted. Thus, only smller group of (low-bility students thn predicted profits from the decentrlized system. 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. References Abdulkdiroğlu, A., nd T. Sönmez (2003: School Choice: A Mechnism Design Approch, Americn Economic Review, 93, We lso discuss the extension to more colleges in section 6. 38

39 Abizd, A., nd S. Chen (2015: Stbility nd Strtegy-proofness for College Admissions with n Eligibility Criterion, Review of Economic Design, 19(1, Amegshie, J. A., nd X. Wu (2006: Adverse Selection in Competing All-py Auctions, Working pper. Avery, C., S. Lee, nd A. E. Roth (2014: College Admissions s Non-Price Competition: The Cse of South Kore, Working pper. 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, Biis, B., D. Mrtimort, nd J.-C. Rochet (2000: Competing Mechnisms in Common Vlue Environment, Econometric, 68(4, Biro, P. (2012: University Admission Prctices - Hungry, Working pper. Buyukboyci, M. (2012: Prllel Tournments, Working pper. 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. DiPlntino, D., nd M. Vojnovic (2009: Crowdsourcing nd All-py Auctions, in Proceedings of the 10th ACM conference on Electronic commerce, pp ACM. 39

40 Efron, B. (1982: The Jckknife, the Bootstrp, nd Other Resmpling Plns, vol. 38. SIAM. Ellison, G., D. Fudenberg, nd M. Möbius (2004: Competing Auctions, Journl of the Europen Economic Assocition, 2(1, 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. 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. McAfee, R. P. (1993: Mechnism Design by Competing Sellers, Econometric, 61(6, 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 (2008: Competing Auctions with Endogenous Quntities, Journl of Economic Theory, 141(1, (2012: Crrots nd Sticks: Prizes nd Punishments in Contests, Economic Inquiry, 50(2,

41 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, Olszewski, W., nd R. Siegel (2014: Lrge Contests, Working pper. Ovdi, D. A. (2014: Sorting through Informtion Disclosure, Working pper. 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. Virág, G. (2010: Competing Auctions: Finite Mrkets nd Convergence, Theoreticl Economics, 5(2, A Appendix A.1 Preliminries The following lemmt re useful for the results given in the rest of the Appendix. 41

42 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, 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 42

43 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. 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 p m,n m 1 (x. 43

44 Differentite ϕ 1 t ech x (0, 1: using Lemm 1, we hve 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 q 2 1 ϕ 2 (π(c = v 2 m=0 q 2 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 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: 44

45 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 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 m p m q1,q 1 1 ( π( π(c m p m q2,q 2 1 ( F ( π( 1 π(c. (11 45

46 Let us define the following functions: n 1 ( π( A( := v 1 p m,n m 1 (π(c m p m q1,q 1 1 > 0, π(c 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( = 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 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 greter 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 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. 46

47 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. 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. βd (c = P [2, b γ, β D ]v 2 βd (b c 47 + βd (c c = U(0, c c + βd (c c + βd (c c βd (c βd (c βd (c

48 Cse 1-3: b > 1 (or e > β D (1. 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. 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 48

49 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. 49

50 C Additionl tbles nd figures Figure 7: Individul efforts by bility 50