College Admissions with Entrance Exams: Centralized versus Decentralized


 Evelyn Howard
 2 years ago
 Views:
Transcription
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
All pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationUNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE
UNVERSTY OF NOTTNGHAM Discussion Ppers in Economics Discussion Pper No. 04/15 STRATEGC SECOND SOURCNG N A VERTCAL STRUCTURE By Arijit Mukherjee September 004 DP 04/15 SSN 10438 UNVERSTY OF NOTTNGHAM Discussion
More informationOn the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding
Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl
More informationDlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report
DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationTests for One Poisson Mean
Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution
More informationEconomics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999
Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationThis paper considers two independent firms that invest in resources such as capacity or inventory based on
MANAGEMENT SCIENCE Vol. 5, No., December 006, pp. 93 99 issn 005909 eissn 56550 06 5 93 informs doi 0.87/mnsc.060.0574 006 INFORMS Strtegic Investments, Trding, nd Pricing Under Forecst Updting Jiri
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationCcrcs Cognitive  Counselling Research & Conference Services (eissn: 23012358)
Ccrcs Cognitive  Counselling Reserch & Conference Services (eissn: 23012358) Volume I Effects of Music Composition Intervention on Elementry School Children b M. Hogenes, B. Vn Oers, R. F. W. Diekstr,
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More informationN Mean SD Mean SD Shelf # Shelf # Shelf #
NOV xercises smple of 0 different types of cerels ws tken from ech of three grocery store shelves (1,, nd, counting from the floor). summry of the sugr content (grms per serving) nd dietry fiber (grms
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More informationLecture 3 Basic Probability and Statistics
Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The
More informationAn Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process
An Undergrdute Curriculum Evlution with the Anlytic Hierrchy Process Les Frir Jessic O. Mtson Jck E. Mtson Deprtment of Industril Engineering P.O. Box 870288 University of Albm Tuscloos, AL. 35487 Abstrct
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationHomework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.
Text questions, Chpter 5, problems 15: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed
More informationCOMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT
COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, crossclssified
More informationSmall Businesses Decisions to Offer Health Insurance to Employees
Smll Businesses Decisions to Offer Helth Insurnce to Employees Ctherine McLughlin nd Adm Swinurn, June 2014 Employersponsored helth insurnce (ESI) is the dominnt source of coverge for nonelderly dults
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationValue Function Approximation using Multiple Aggregation for Multiattribute Resource Management
Journl of Mchine Lerning Reserch 9 (2008) 20792 Submitted 8/08; Published 0/08 Vlue Function Approximtion using Multiple Aggregtion for Multittribute Resource Mngement Abrhm George Wrren B. Powell Deprtment
More information2. Transaction Cost Economics
3 2. Trnsction Cost Economics Trnsctions Trnsctions Cn Cn Be Be Internl Internl or or Externl Externl n n Orgniztion Orgniztion Trnsctions Trnsctions occur occur whenever whenever good good or or service
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More informationContextualizing NSSE Effect Sizes: Empirical Analysis and Interpretation of Benchmark Comparisons
Contextulizing NSSE Effect Sizes: Empiricl Anlysis nd Interprettion of Benchmrk Comprisons NSSE stff re frequently sked to help interpret effect sizes. Is.3 smll effect size? Is.5 relly lrge effect size?
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationMATLAB: Mfiles; Numerical Integration Last revised : March, 2003
MATLAB: Mfiles; Numericl Integrtion Lst revised : Mrch, 00 Introduction to Mfiles In this tutoril we lern the bsics of working with Mfiles in MATLAB, so clled becuse they must use.m for their filenme
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationCEO Pay and the Lake Wobegon Effect
CEO Py nd the Lke Wobegon Effect Rchel M. Hyes nd Scott Schefer December 11, 2008 Forthcoming in Journl of Finncil Economics Abstrct The Lke Wobegon Effect, which is widely cited s potentil cuse for rising
More informationThe Relative Advantages of Flexible versus Designated Manufacturing Technologies
The Reltive Advntges of Flexible versus Designted Mnufcturing Technologies George Normn Cummings Professor of Entrepreneurship nd Business Economics Tufts University Medford MA 055 USA emil: george.normn@tufts.edu
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationLower Bound for EnvyFree and Truthful Makespan Approximation on Related Machines
Lower Bound for EnvyFree nd Truthful Mespn Approximtion on Relted Mchines Lis Fleischer Zhenghui Wng July 14, 211 Abstrct We study problems of scheduling jobs on relted mchines so s to minimize the mespn
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationQuality Evaluation of Entrepreneur Education on Graduate Students Based on AHPfuzzy Comprehensive Evaluation Approach ZhongXiaojun 1, WangYunfeng 2
Interntionl Journl of Engineering Reserch & Science (IJOER) ISSN [23956992] [Vol2, Issue1, Jnury 2016] Qulity Evlution of Entrepreneur Eduction on Grdute Students Bsed on AHPfuzzy Comprehensive Evlution
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationParticipation and investment decisions in a retirement plan: the influence of colleagues choices
Journl of Public Economics 85 (2002) 121 148 www.elsevier.com/ locte/ econbse Prticiption nd investment decisions in retirement pln: the influence of collegues choices Esther Duflo,b, *, Emmnuel Sez MIT,
More informationHealth insurance exchanges What to expect in 2014
Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount
More informationUnit 29: Inference for TwoWay Tables
Unit 29: Inference for TwoWy Tbles Prerequisites Unit 13, TwoWy Tbles is prerequisite for this unit. In ddition, students need some bckground in significnce tests, which ws introduced in Unit 25. Additionl
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More informationWeek 7  Perfect Competition and Monopoly
Week 7  Perfect Competition nd Monopoly Our im here is to compre the industrywide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between
More informationNOTES AND CORRESPONDENCE. Uncertainties of Derived Dewpoint Temperature and Relative Humidity
MAY 4 NOTES AND CORRESPONDENCE 81 NOTES AND CORRESPONDENCE Uncertinties of Derived Dewpoint Temperture nd Reltive Humidity X. LIN AND K. G. HUBBARD High Plins Regionl Climte Center, School of Nturl Resource
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationPerfect competition model (PCM)
18/9/21 Consumers: Benefits, WT, nd Demnd roducers: Costs nd Supply Aggregting individul curves erfect competition model (CM) Key ehviourl ssumption Economic gents, whether they e consumers or producers,
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationSmall Business Networking
Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationSection 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables
The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationHealth insurance marketplace What to expect in 2014
Helth insurnce mrketplce Wht to expect in 2014 33096VAEENBVA 06/13 The bsics of the mrketplce As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationOptimal Execution of OpenMarket Stock Repurchase Programs
Optiml Eecution of OpenMrket Stock Repurchse Progrms Jcob Oded This Drft: December 15, 005 Abstrct We provide theoreticl investigtion of the eecution of openmrket stock repurchse progrms. Our model suggests
More informationClearPeaks Customer Care Guide. Business as Usual (BaU) Services Peace of mind for your BI Investment
ClerPeks Customer Cre Guide Business s Usul (BU) Services Pece of mind for your BI Investment ClerPeks Customer Cre Business s Usul Services Tble of Contents 1. Overview...3 Benefits of Choosing ClerPeks
More informationOptimal Redistributive Taxation with both Labor Supply and Labor Demand Responses
Optiml Redistributive Txtion with both Lbor Supply nd Lbor Demnd Responses Lurence JACQUET NHH Preliminry version Etienne LEHMANN y CREST Bruno VAN DER LINDEN z IRES  UCLouvin nd FNRS Mrch 28, 2011 Abstrct
More informationTRUST and reputation are crucial requirements for most
IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, VOL. 9, NO. 3, MAY/JUNE 2012 375 Itertive Trust nd Reputtion Mngement Using Belief Propgtion Ermn Aydy, Student Member, IEEE, nd Frmrz Feri, Senior
More informationEnterprise Risk Management Software Buyer s Guide
Enterprise Risk Mngement Softwre Buyer s Guide 1. Wht is Enterprise Risk Mngement? 2. Gols of n ERM Progrm 3. Why Implement ERM 4. Steps to Implementing Successful ERM Progrm 5. Key Performnce Indictors
More informationSmall Business Cloud Services
Smll Business Cloud Services Summry. We re thick in the midst of historic sechnge in computing. Like the emergence of personl computers, grphicl user interfces, nd mobile devices, the cloud is lredy profoundly
More informationENHANCING CUSTOMER EXPERIENCE THROUGH BUSINESS PROCESS IMPROVEMENT: AN APPLICATION OF THE ENHANCED CUSTOMER EXPERIENCE FRAMEWORK (ECEF)
ENHNCING CUSTOMER EXPERIENCE THROUGH BUSINESS PROCESS IMPROVEMENT: N PPLICTION OF THE ENHNCED CUSTOMER EXPERIENCE FRMEWORK (ECEF) G.J. Both 1, P.S. Kruger 2 & M. de Vries 3 Deprtment of Industril nd Systems
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationUnleashing the Power of Cloud
Unleshing the Power of Cloud A Joint White Pper by FusionLyer nd NetIQ Copyright 2015 FusionLyer, Inc. All rights reserved. No prt of this publiction my be reproduced, stored in retrievl system, or trnsmitted,
More informationResearch Article Competition with Online and Offline Demands considering Logistics Costs Based on the Hotelling Model
Mthemticl Problems in Engineering Volume 4, Article ID 678, pges http://dx.doi.org/.55/4/678 Reserch Article Competition with Online nd Offline Demnds considering Logistics Costs Bsed on the Hotelling
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationPerformance analysis model for big data applications in cloud computing
Butist Villlpndo et l. Journl of Cloud Computing: Advnces, Systems nd Applictions 2014, 3:19 RESEARCH Performnce nlysis model for big dt pplictions in cloud computing Luis Edurdo Butist Villlpndo 1,2,
More informationChapter 8  Practice Problems 1
Chpter 8  Prctice Problems 1 MULTIPLE CHOICE. Choose the one lterntive tht best completes the sttement or nswers the question. A hypothesis test is to be performed. Determine the null nd lterntive hypotheses.
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationExponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.
Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationBiostatistics 102: Quantitative Data Parametric & Nonparametric Tests
Singpore Med J 2003 Vol 44(8) : 391396 B s i c S t t i s t i c s F o r D o c t o r s Biosttistics 102: Quntittive Dt Prmetric & Nonprmetric Tests Y H Chn In this rticle, we re going to discuss on the
More informationDecision Rule Extraction from Trained Neural Networks Using Rough Sets
Decision Rule Extrction from Trined Neurl Networks Using Rough Sets Alin Lzr nd Ishwr K. Sethi Vision nd Neurl Networks Lbortory Deprtment of Computer Science Wyne Stte University Detroit, MI 48 ABSTRACT
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationLumpSum Distributions at Job Change, p. 2
Jnury 2009 Vol. 30, No. 1 LumpSum Distributions t Job Chnge, p. 2 E X E C U T I V E S U M M A R Y LumpSum Distributions t Job Chnge GROWING NUMBER OF WORKERS FACED WITH ASSET DECISIONS AT JOB CHANGE:
More informationAssessing authentically in the Graduate Diploma of Education
Assessing uthenticlly in the Grdute Diplom of Eduction Dr Mree DinnThompson Dr Ruth Hickey Dr Michelle Lsen WIL Seminr JCU Nov 12 2009 Key ides plnning process tht embeds uthentic ssessment, workintegrted
More informationArc Length. P i 1 P i (1) L = lim. i=1
Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x
More informationFDIC Study of Bank Overdraft Programs
FDIC Study of Bnk Overdrft Progrms Federl Deposit Insurnce Corportion November 2008 Executive Summry In 2006, the Federl Deposit Insurnce Corportion (FDIC) initited twoprt study to gther empiricl dt on
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationRedistributing the Gains from Trade through Nonlinear. Lumpsum Transfers
Redistributing the Gins from Trde through Nonliner Lumpsum Trnsfers Ysukzu Ichino Fculty of Economics, Konn University April 21, 214 Abstrct I exmine lumpsum trnsfer rules to redistribute the gins from
More informationPortfolio approach to information technology security resource allocation decisions
Portfolio pproch to informtion technology security resource lloction decisions Shivrj Knungo Deprtment of Decision Sciences The George Wshington University Wshington DC 20052 knungo@gwu.edu Abstrct This
More informationSTA 2023 Test #3 Practice Multiple Choice
STA 223 Test #3 Prctice Multiple Choice 1. A newspper conducted sttewide survey concerning the 1998 rce for stte sentor. The newspper took rndom smple (ssume it is n SRS) of 12 registered voters nd found
More informationpersons withdrawing from addiction is given by summarizing over individuals with different ages and numbers of years of addiction remaining:
COST BENEFIT ANALYSIS OF NARCOTIC ADDICTION TREATMENT PROGRAMS with Specil Reference to Age Irving Leveson,l New York City Plnning Commission Introduction Efforts to del with consequences of poverty,
More informationA new algorithm for generating Pythagorean triples
A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationAlgorithms Chapter 4 Recurrences
Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationQuantity Oriented Resource Allocation Strategy on Multiple Resources Projects under Stochastic Conditions
Interntionl Conference on Industril Engineering nd Systems Mngement IESM 2009 My 1315 MONTRÉAL  CANADA Quntity Oriented Resource Alloction Strtegy on Multiple Resources Projects under Stochstic Conditions
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationBiostatistics 103: Qualitative Data Tests of Independence
Singpore Med J 2003 Vol 44(10) : 498503 B s i c S t t i s t i c s F o r D o c t o r s Biosttistics 103: Qulittive Dt Tests of Independence Y H Chn Prmetric & nonprmetric tests (1) re used when the outcome
More informationResearch Notes. RatSWD. Research Note No. 11. Population Aging and Trends in the Provision of Continued Education
RtSWD Reserch Notes Reserch Note No. 11 Previously relesed s RtSWD Working Pper No. 15 Popultion Aging nd Trends in the Provision of Continued Eduction Regin T. Riphhn, Prvti Trübswetter 2007 Reserch Notes
More informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
More informationData replication in mobile computing
Technicl Report, My 2010 Dt repliction in mobile computing Bchelor s Thesis in Electricl Engineering Rodrigo Christovm Pmplon HALMSTAD UNIVERSITY, IDE SCHOOL OF INFORMATION SCIENCE, COMPUTER AND ELECTRICAL
More informationMultiple Testing in a TwoStage Adaptive Design With Combination Tests Controlling FDR
This rticle ws downloded by: [New Jersey Institute of Technology] On: 28 Februry 204, At: 08:46 Publisher: Tylor & Frncis Inform Ltd Registered in nglnd nd Wles Registered Number: 072954 Registered office:
More information