College Admissions with Entrance Exams: Centralized versus Decentralized


 Evelyn Howard
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1 Is E. Hflir Rustmdjn Hkimov Dorothe Kübler Morimitsu Kurino College Admissions with Entrnce Exms: Centrlized versus Decentrlized Discussion Pper SP II October 2014 (WZB Berlin Socil Science Center Reserch Are Mrkets nd Choice Reserch Unit Mrket Behvior
2 Wissenschftszentrum Berlin für Sozilforschung ggmbh Reichpietschufer Berlin Germny Copyright remins with the uthor(s. Discussion ppers of the WZB serve to disseminte the reserch results of work in progress prior to publiction to encourge the exchnge of ides nd cdemic debte. Inclusion of pper in the discussion pper series does not constitute publiction nd should not limit publiction in ny other venue. The discussion ppers published by the WZB represent the views of the respective uthor(s nd not of the institute s whole. Is E. Hflir, Rustmdjn Hkimov, Dorothe Kübler, Morimitsu Kurino College Admissions with Entrnce Exms: Centrlized versus Decentrlized Affilition of the uthors: Is E. Hflir Crnegie Mellon University, Pittsburgh Rustmdjn Hkimov WZB Berlin Socil Science Center Dorothe Kübler WZB Berlin Socil Science Center nd Technicl University Berlin Morimitsu Kurino University of Tsukub, Jpn
3 Wissenschftszentrum Berlin für Sozilforschung ggmbh Reichpietschufer Berlin Germny Abstrct College Admissions with Entrnce Exms: Centrlized versus Decentrlized by Is E. Hflir, Rustmdjn Hkimov, Dorothe Kübler nd Morimitsu Kurino * We theoreticlly nd experimentlly study college dmissions problem in which colleges ccept students by rnking students efforts in entrnce exms. Students hold privte informtion regrding their bility level tht ffects the cost of their efforts. We ssume tht student preferences re homogeneous over colleges. By modeling college dmissions s contests, we solve nd compre the equilibri of centrlized college dmissions (CCA in which students pply to ll colleges, nd decentrlized college dmissions (DCA in which students cn only pply to one college. We show tht lower bility students prefer DCA wheres higher bility students prefer CCA. The min qulittive predictions of the theory re supported by the experimentl dt, yet we find number of behviorl differences between the mechnisms tht render DCA less ttrctive thn CCA compred to the equilibrium benchmrk. Keywords: College dmissions, incomplete informtion, student welfre, contests, llpy uctions, experiment JEL clssifiction: C78; D47; D78; I21 * Emil:
4 1 Introduction Throughout the world nd every yer, millions of prospective university students pply for dmission to colleges or universities during their lst yer of high school. Admission mechnisms vry from country to country, yet in most countries there re government gencies or independent orgniztions tht offer stndrdized dmission exms to id the college dmission process. Students invest lot of time nd effort to do well in these dmission exms, nd they re heterogeneous in terms of their bility to do so. In some countries, the ppliction nd dmission process is centrlized. For instnce, in Turkey university ssignment is solely determined by ntionl exmintion clled YGS/LYS. After lerning their scores, students cn pply to number of colleges. Applictions re lmost costless s ll students need only to submit their rnkorder of colleges to the centrl uthority. 1 On the other hnd, Jpn hs centrlized Ntionl Center test, too, but ll public universities including most prestigious universities require the cndidte to tke nother, institutionspecific secondry exm which tkes plce on the sme dy. This effectively prevents the students from pplying to more thn one public university. 2 The dmissions mechnism in Jpn is decentrlized, in the sense tht colleges decide on their dmissions independent of ech other. In the United Sttes, students tke both centrlized exms like the Scholstic Aptitude Test (SAT, nd lso complete collegespecific requirements such s college dmission essys. Students cn pply to more thn one college, but since the ppliction process is costly, students typiclly send only few pplictions (the mjority being between two to six pplictions, see Chde, Lewis, nd Smith, Hence, the United Sttes college dmissions mechnism flls in between the two extreme cses. In this pper, we compre the institutionl effects of different college dmission mechnisms on the equilibrium efforts of students nd student welfre. To do this, we model college dmissions with dmission exms s contests (or llpy uctions in which the cost of effort represents the pyment mde by the students. We focus on two extreme cses: in the centrlized model (s in the Turkish mechnism students cn freely pply to ll colleges, wheres in the decentrlized model (s in the Jpnese mechnism for public colleges students cn only pply to one college. For simplicity, in our min model we consider two colleges tht differ in qulity nd ssume tht 1 Greece, Chin, South Kore, nd Tiwn hve similr ntionl exms tht re the min criterion for the centrlized mechnism of college dmissions. In Hungry, the centrlized dmission mechnism is bsed on score tht combines grdes from school with n entrnce exm (Biro, There re ctully two stges where the structure of ech stge is s explined in Section 4. The difference between the stges is tht the cpcities in the first stge re much greter thn those in the second stge. Those who do not get dmission to ny college spend one yer prepring for the next yer s exm. Moreover, the Jpnese high school dmissions uthorities hve dopted similr mechnisms in locl districts. Although the mechnism dopted vries cross prefectures nd is chnging yer by yer, its bsic structure is tht ech student chooses one mong specified set of public schools nd then tkes n entrnce exm t his or her chosen school. The exms re held on the sme dy. Finlly, institutionspecific exms tht prevent students from pplying to ll colleges hve lso been used nd debted in the United Kingdom, notbly between the University of Cmbridge nd the University of Oxford. We thnk Ken Binmore for pointing this out. 2
5 students hve homogeneous preferences for ttending these colleges. 3 More specificlly, ech of the n students gets utility of v 1 by ttending college 1 (which cn ccommodte q 1 students nd gets utility of v 2 by ttending college 2 (which cn ccommodte q 2 students. We suppose 0 < v 1 < v 2, nd hence college 2 is the better nd college 1 is worse of the two colleges. Students utility from not being ssigned to ny college is normlized to 0. Following the mjority of the literture on contests with incomplete informtion, we suppose tht n bility level in the intervl [0, 1], is drwn i.i.d. from the common distribution function, nd the cost of exerting n effort e for student with bility level is given by e. Thus, given n effort level, the higher the bility the lower the cost of exerting the effort. In the centrlized college dmissions problem (CCA, ll students rnk college 2 over college 1. Hence, the students with the highest q 2 efforts get into college 2, students with the next highest q 1 efforts get into college 1, nd students with the lowest n q 1 q 2 efforts re not ssigned to ny college. In the decentrlized college dmissions problem (DCA, students need to simultneously choose which college to pply to nd how much effort to exert. Then, for ech college i {1, 2}, students with the highest q i efforts mong the pplicnts to college i get into college i. It turns out tht the equilibrium of CCA cn be solved by stndrd techniques, such s in Moldovnu, Sel, nd Shi (2012. In this monotone equilibrium, higher bility students exert higher efforts, nd therefore the students with the highest q 2 bility levels get dmitted to the good college (college 2, nd students with bility rnkings between q 2 +1 nd q 1 +q 2 get dmitted to the bd college (college 1 (Proposition 1. Finding the equilibrium of DCA is not strightforwrd. It turns out tht in equilibrium, there is cutoff bility level tht we denote by c. All higher bility students (with bilities in (c, 1] pply to the good college, wheres lower bility students (with bility levels in [0, c] use mixed strtegy when choosing between the good nd the bd college. Students effort functions re continuous nd monotone in bility levels (Theorem 1. Our pper therefore contributes to the llpy contests literture. To the best of our knowledge, ours is the first pper to model nd solve competing contests where the plyers hve privte informtion regrding their bilities nd sort themselves into different contests. After solving for the equilibrium of CCA nd DCA, we compre the equilibri in terms of students interim expected utilities. We show tht students with lower bilities prefer DCA to CCA when the number of sets is smller thn the number of students (Proposition 2. The min intuition for this result is tht students with very low bilities hve lmost no chnce of getting set in CCA, wheres their probbility of getting set in DCA is bounded wy from zero due to the fewer number of pplictions thn the cpcity. Moreover, we show tht students with higher bilities prefer CCA to DCA (Proposition 3. 4 The min intuition for this result is tht 3 In Section 6, we discuss the cse with three or more colleges. 4 More specificlly we obtin single crossing condition: if student who pplies to college 2 in the decentrlized mechnism prefers the centrlized mechnism to the decentrlized mechnism, then ll higher bility students lso 3
6 highbility students (i cn only get set in the good school in DCA, wheres they cn get sets in both the good nd the bd school in CCA, nd (ii their equilibrium probbility of getting set in the good school is the sme cross the two mechnisms. We test the theory with the help of lb experiments. We implement five mrkets for the college dmissions gme tht re designed to cpture different levels of competition (in terms of the supply of sets, the demnd rtio, nd the qulity difference between the two colleges. We compre the two college dmission mechnisms nd find tht in most (but not ll mrkets, the comprisons of the students exnte expected utilities, their effort levels, nd the students preferences regrding the two college dmission mechnisms re well orgnized by the theory. However, the experimentl subjects exert higher effort thn predicted. The overexertion of effort is prticulrly pronounced in DCA, which mkes it reltively less ttrctive for the pplicnts compred to CCA. The rest of the pper is orgnized s follows. The introduction (Section 1 ends with discussion of the relted literture. Section 2 introduces the model nd preliminry nottion. In sections 3 nd 4 we solve the model for the Byesin Nsh equilibri of the centrlized nd decentrlized college dmission mechnisms, respectively. Section 5 offers comprisons of the equilibri of the two mechnisms. Section 6 discusses the cse of three or more colleges. Section 7 presents our experimentl results. Finlly, section 8 concludes. Omitted proofs re given in the Appendix. 1.1 Relted literture College dmissions hve been studied extensively in the economics literture. Following the seminl pper by Gle nd Shpley (1962, the theory literture on twosided mtching minly considers centrlized college dmissions nd investigtes stbility, incentives, nd the efficiency properties of vrious mechnisms, notbly the deferredcceptnce nd the top trding cycles lgorithms. The student plcement nd school choice literture is motivted by the centrlized mechnisms of public school dmissions, rther thn by the decentrlized college dmissions mechnism in the US. This literture ws pioneered by Blinski nd Sönmez (1999 nd Abdulkdiroğlu nd Sönmez (2003. We refer the reder to Sönmez nd Ünver (2011 for recent comprehensive survey regrding centrlized college dmission models in the twosided mtching literture. Recent work regrding centrlized college dmissions with entrnce exms include Abizd nd Chen (2011 nd Tung (2009. Abizd nd Chen (2011 model the entrnce (eligibility criterion in college dmissions problems nd extend models of Perch, Polk, nd Rothblum (2007 nd Perch nd Rothblum (2010 by llowing the students to hve the sme scores from the centrl exm. On the other hnd, by llowing students to submit their preferences fter they receive the test results, Tung (2009 djusts multictegory seril dicttorship (MSD nlyzed by Blinski nd Sönmez (1999 in order to mke students better off. One crucil difference between the modelling in our pper nd the literture should be emhve the sme preference rnking. 4
7 phsized: In our pper student preferences ffect college rnkings over students through contests mong students, while student preferences nd college rnkings re typiclly independent in the twosided mtching models nd schoolchoice 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 mrketclering prices. The uthors show tht in equilibrium, collegestudent sorting my fil, nd they lso nlyze the effects of ffirmtive ction policies. In our model, the colleges re not strtegic plyers s in Chde, Lewis, nd Smith (2014. Another importnt difference is tht in our model the students do not only hve to decide which colleges to pply to, but lso how much effort to exert in order to do well in the entrnce exms. Che nd Koh (2013 study model in which two colleges mke dmission decisions subject to ggregte uncertinty bout student preferences nd liner costs for ny enrollment exceeding the cpcity. They find tht colleges dmission decisions become tool for strtegic yield mngement, nd in equilibrium, colleges try to reduce their enrollment uncertinty by strtegiclly trgeting students. In their model, s in Chde, Lewis, nd Smith (2014, students exm scores re costlessly obtined nd given exogenously. Avery nd Levin (2010, on the other hnd, nlyze model of erly dmission t selective colleges where erly dmission progrms give students n opportunity to signl their enthusism to the college they would like to ttend. In nother relted pper, Hickmn (2009 lso models college dmissions s Byesin gme where heterogeneous students compete for sets t colleges. He presents model in which there is n lloction mechnism mpping ech student s score into set t college. Hickmn (2009 is mostly interested in the effects of ffirmtive ction policies, nd the solution concept used is pproximte equilibrium in which the number of students is ssumed to be lrge so tht students pproximtely know their rnkings within the relized smple of privte costs. 5 In our pper, we do not require the number of students to be lrge. In nother recent pper by SlgdoTorres (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. SlgdoTorres (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 llpy uction nd contests literture. Notbly, Bye, 5 In relted pper, Morgn, Sisk, nd Vrdy (2012 study competition for promotion in continuum economy. They show tht more meritocrtic profession lwys succeeds in ttrcting the highest bility types, wheres profession with superior promotion benefits ttrcts high types only under some ssumptions. 5
8 Kovenock, nd de Vries (1996 nd Siegel (2009 solve for llpy 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 llpy contests tht re run simultneously with multiple identicl prizes. They chrcterize set of pure strtegy equilibri, nd symmetric equilibrium tht involves mixed strtegies. In our decentrlized college dmissions model, the corresponding contest model is lso model of competing contests. The min difference in our model is tht we consider incomplete informtion s students do not know ech other s bility levels. A series of ppers by Moldovnu nd Sel (nd Shi studies contests with incomplete informtion, but they do not consider competing contests in which the prticiption in contests is endogenously determined. In Moldovnu nd Sel (2001, the contest designer s objective is to mximize expected effort. They show tht when cost functions re liner or concve in effort, it is optiml to llocte the entire prize sum to single first prize. Moldovnu nd Sel (2006 compre the performnce of dynmic subcontests whose winners compete ginst ech other with sttic contests. They show tht with liner costs of effort, the expected totl effort is mximized with sttic contest, wheres the highest expected effort cn be higher with contests with two divisions. Moldovnu, Sel, nd Shi (2012 study optiml contest design where both wrds nd punishments cn be used. Under some conditions, they show tht punishing the bottom is more effective thn rewrding the top. This pper lso contributes to lrge experimentl literture on contests nd llpy uctions, summrized in recent survey rticle by Dechenux, Kovenock, nd Sheremet (2012. Our setup in the centrlized mechnism with heterogeneous gents, two nonidenticl 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 twosided mtching mechnisms nd 6
9 school choice strting with Kgel nd Roth (2000 nd Chen nd Sönmez ( These studies s well s mny followup ppers in this strnd of the literture focus on the rnkorder lists submitted by students in the preferencereveltion gmes, but not on effort choice. Thus, the rnkings of students by the schools re exogenously given in these studies unlike in our setup where the colleges rnkings re endogenous. 2 The Model The college dmissions problem with entrnce exms, or simply the problem, is denoted by (S, C, (q 1, q 2, (v 1, v 2, F. There re 2 colleges college 1 nd college 2. We denote colleges by C. Ech college C C := {1, 2} hs cpcity q C which represents the mximum number of students tht cn be dmitted to college C, where q C 1. There re n students. We denote the set of ll students by S. Since we suppose homogeneous preferences of students, we ssume tht ech student hs the crdinl utility v C from college C {1, 2}, where v 2 > v 1 > 0. Thus we sometimes cll college 2 the good college nd college 1 the bd college. Ech student s utility from not being ssigned to ny college is normlized to be 0. We ssume tht q 1 + q 2 n. 7 Ech student is ssigned to one college or no set in ny college by the mechnisms nd the mechnisms tke the efforts into ccount while deciding on their dmissions. 8 Ech student s S mkes n effort e s. The students re heterogeneous in terms of their bilities, nd the bilities re their privte informtion. More specificlly, for ech s S, s [0, 1] denotes student s s bility. Abilities re drwn identiclly nd independently from the intervl [0, 1] ccording to continuous distribution function F tht is common knowledge. We ssume tht F hs continuous density f = df > 0. For student s with bility s, putting in n effort of e s results in disutility of e s s. Hence, the totl utility of student with bility from mking effort e is v C e/ if she is ssigned to college C, nd e/ otherwise. Before we move on to the nlysis of the equilibrium of centrlized nd decentrlized college dmission mechnisms, we introduce some necessry nottion. 2.1 Preliminry nottion First, for ny continuous distribution T with density t, for 1 k m, let T k,m denote the distribution of the k th (lowest order sttistics out of m independent rndom vribles tht re 6 For recent exmple for theory nd experiments in school choice literture, see Chen nd Kesten ( Mny college dmissions, including ones in Turkey nd Jpn, re competitive in the sense tht totl number of sets in colleges is smller thn the number of students who tke the exms. 8 In relity the performnce in the entrnce exms is only noisy function of efforts. For simplicity, we ssume tht efforts completely determine the performnce in the tests. 7
10 identiclly distributed ccording to T. Tht is, T k,m ( := m j=k Moreover, let t k,m ( denote T k,m ( s density: 0. t k,m (x := d d T k,m( = ( m T ( j (1 T ( m j. (1 j m! (k 1! (m k! T (k 1 (1 T ( m k t(. (2 For convenience, we let T 0,m be distribution with T 0,m ( = 1 for ll, nd t 0,m dt 0,m /d = Next, define the function p j,k : [0, 1] [0, 1] s follows: given j, k {0, 1,..., n}, for ech x [0, 1], define ( j + k p j,k (x := x j (1 x k. (3 j The function p j,k (x is interpreted s the probbility tht when there re (j+k students, j students re selected for one event with probbility x nd k students re selected for nother event with probbility (1 x. Suppose tht p 0,0 (x = 1 for ll x. Note tht with this definition, we cn write T k,m ( = m p j,m j (T (. (4 j=k 3 The Centrlized College Admissions Mechnism (CCA In the centrlized college dmissions gme, ech student s S simultneously mkes n effort e s. Students with the top q 2 efforts re ssigned to college 2 nd students with the efforts from the top (q to (q 1 + q 2 re ssigned to college 1. The rest of the students re not ssigned to ny colleges. 9 We now solve for the symmetric Byesin Nsh equilibrium of this gme. The following proposition is specil cse of the llpy uction equilibrium which hs been studied by Moldovnu nd Sel (2001 nd Moldovnu, Sel, nd Shi (2012. Proposition 1. In CCA, there is unique symmetric equilibrium β C such tht for ech [0, 1], 9 In setup with homogeneous student preferences, this gme reflects how the Turkish college dmission mechnism works. In the centrlized test tht the students tke, since ll students would put college 2 s their top choice nd college 1 s their second top choice in their submitted preferences, the resulting ssignment would be the sme s the ssignment described bove. In school choice context, this cn be described s the following twostge gme. In the first stge, there is one contest where ech student s simultneously mkes n effort e s. The resulting effort profile (e s s S is used to construct single priority profile such tht student with higher effort hs higher priority. In the second stge, students prticipte in the centrlized deferred cceptnce mechnism where colleges use the common priority. 8
11 ech student with bility chooses n effort β C ( ccording to β C ( = ˆ 0 x {f n q2,n 1(x v 2 + (f n q1 q 2,n 1(x f n q2,n 1(x v 1 } dx. where f k,m ( for k 1 is defined in Eqution (2 nd f 0,m (x is defined to be 0 for ll x. Proof. Suppose tht β C is symmetric equilibrium effort function tht is strictly incresing. Consider student with bility who chooses n effort s if her bility is. Her expected utility is v 2 F n q2,n 1( + v 1 (F n q1 q 2,n 1( F n q2,n 1( βc (. The firstorder condition t = is v 2 f n q2,n 1( + v 1 (f n q1 q 2,n 1( f n q2,n 1( [βc (] Thus, by integrtion nd s the boundry condition is β C (0 = 0, we hve β C ( = ˆ 0 = 0. x {f n q2,n 1(x v 2 + (f n q1 q 2,n 1(x f n q2,n 1(x v 1 } dx. The bove strtegy is the unique symmetric equilibrium cndidte obtined vi the firstorder pproch by requiring no benefit from locl devitions. Stndrd rguments show tht this is indeed n equilibrium by mking sure tht globl devitions re not profitble (for instnce, see Section 2.3 of Krishn ( The Decentrlized College Admissions Mechnism (DCA In the decentrlized college dmissions gme, ech student s chooses one college C s nd n effort e s simultneously. Given the college choices of students (C s s S nd efforts (e s s S, ech college C dmits students with the top q C effort levels mong its set of pplicnts ({s S C s = C}. 10 For this gme, we solve for symmetric Byesin Nsh equilibrium (γ(, β D ( ; c where c (0, 1 is cutoff, γ : [0, c] (0, 1 is the mixed strtegy tht represents the probbility of lower bility students pplying to college 1, nd β D : [0, 1] R is the continuous nd strictly incresing effort function. Ech student with type [0, c] chooses college 1 with probbility γ( (hence 10 In setup with homogeneous student preferences, this gme reflects how the Jpnese college dmissions mechnism works: ll public colleges hold their own tests nd ccept the top performers mong the students who tke their tests. In school choice context, this cn be described s the following twostge gme. In the first stge, students simultneously choose which school to pply to, nd without knowing how mny other students hve pplied, they lso choose their effort level. For ech school C {1, 2}, the resulting effort profile (e s {s S Cs=C} is used to construct one priority profile C such tht student with higher effort hs higher priority. In the second stge, students prticipte in two seprte deferred cceptnce mechnisms where ech college C uses the priority C. 9
12 chooses college 2 with probbility 1 γ(, nd mkes effort β D (. (c, 1] chooses college 2 for sure, nd mkes effort β D (. 11 Ech student with type We now move on to the derivtion of symmetric Byesin Nsh equilibrium. Let symmetric strtegy profile (γ(, β( ; c be given. For this strtegy profile, the exnte 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 exnte 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 exnte probbility tht student pplies to college 1 is π(c, while the probbility tht student pplies to college 2 is 1 π(c. Moreover, p m,k (π(c is the probbility tht m students pply to college 1 nd k students pply to college 2 where p m,k ( is given in Eqution (3 nd π( is given in Eqution (5. Next, we define G( : [0, c] [0, 1], where G( is the probbility tht type is less thn or equl to, conditionl on the event tht she pplies to college 1. Tht is, G( := π( π(c. Moreover let g( denote G( s density. G k,m is the distribution of the k th order sttistics out of m independent rndom vribles tht re identiclly distributed ccording to G s in equtions (1 nd (4. Also, g k,m ( denotes G k,m ( s density. Similrly, let us define H( : [0, 1] [0, 1], where H( is the probbility tht type is less thn or equl to, conditionl on the event tht she pplies to college 2. Tht is, for [0, 1], H( = F ( π( 1 π(c if [0, c], F ( π(c 1 π(c if [c, 1]. Moreover, let h( denote H( s density. Note tht h is continuous but is not differentible t c. Let H k,m be the distribution of the k th order sttistics out of m independent rndom vribles distributed ccording to H s in equtions (1 nd (4. Also, h k,m ( denotes H k,m ( s density. 11 A nturl equilibrium cndidte is to hve cutoff c (0, 1, students with bilities in [0, c to pply to college 1, nd students with bilities in [c, 1] to pply to college 2. It turns out tht we cnnot hve n equilibrium of this kind. In such n equilibrium, (i type c hs to be indifferent between pplying to college 1 or college 2, (ii type c s effort is strictly positive in cse of pplying to college 1, nd 0 while pplying to college 2, hence there is discontinuity in the effort function. These two conditions together imply tht type c + ɛ student would benefit from mimicking type c ɛ student for smll enough ɛ. Forml rguments resulting in the nonexistence result re vilble from the uthors upon request. Therefore, we hve to hve some students using mixed strtegies while choosing which college to pply to. Derivtions show tht in equilibrium, lower bility students would use mixed strtegies, while the higher bility students re certin to pply to the better school. 10
13 We re now redy to stte the min result of this section, which chrcterizes the unique symmetric Byesin Nsh equilibrium 12 of the decentrlized college dmissions mechnism. The sketch of the proof follows the Theorem, wheres the more technicl prt of the proof is relegted to Appendix B. Theorem 1. In DCA, there is unique symmetric equilibrium (γ, β D ; c where student with type [0, c] chooses college 1 with probbility γ( nd mkes effort β D (; nd student with type [c, 1] chooses college 2 for sure nd mkes effort β D (. Specificlly, ˆ β D ( = v 2 x 0 n 1 m=q 2 p n m 1,m (π(ch m q2 +1,m(xdx. The equilibrium cutoff c nd the mixed strtegies γ( re determined by the following four requirements: (i π(c uniquely solves the following eqution for x q 1 1 v 1 m=0 q 2 1 p m,n m 1 (x = v 2 m=0 (ii Given π(c, c uniquely solves the following eqution for x q 2 1 v 1 = v 2 m=0 n 1 p n m 1,m (π(c + v 2 m=q 2 p n m 1,m (π(c p n m 1,m (x. m j=m q 2 +1 ( F (x π(c p j,m j. 1 π(c (iii Given π(c nd c, for ech [0, c, π( uniquely solves the following eqution for x( n 1 v 2 m=q 2 p n m 1,m (π(c m j=m q 2 +1 (iv Finlly, for ech [0, c], γ( is given by γ( = ( F ( x( n 1 p j,m j = v 1 p m,n m 1 (π(c 1 π(c m=q 1 π(cb( (1 π(ca( + π(cb( (0, 1, m j=m q 1 +1 ( x( p j,m j. π(c 12 More specificlly, we chrcterize the unique equilibrium in which (i students use mixed strtegy while deciding which college to pply to, nd (ii effort levels re independent of college choice nd monotone incresing in bilities. 11
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