College Assignment as a Large Contest

Aaron L. Bodoh-Creed and Brent R. Hickman

September 3, 2015

Abstract

We develop a model of college assignment as a large contest wherein studentswith heterogeneous abilities compete for seats at vertically differentiated collegesthrough the acquisition of productive human capital. We use a continuum modelto approximate the outcomes of a game with large, but finite, sets of colleges andstudents. By incorporating two common forms of affirmative action in our model,admissions preferences and quotas, we can show that (legal) admissions preferenceschemes and (illegal) quotas are outcome equivalent. We assess the welfare costs ofusing human capital accumulation to compete for college admissions. While compe-tition is necessary for (welfare enhancing) assortative match, the welfare losses fromthe accumulation of human capital solely to compete for a better college seat are alsosignificant.

Keywords: Affirmative action; welfare costs of competition; contests; approximate equi-librium.

JEL subject classification: D44, C72, I20, I28, L53.

1 Introduction

There are many crucially important economic features of the competition between stu-dents for admission to colleges. An ideal model would include heterogeneity amongstthe colleges in terms of quality, allow for differences amongst the students in terms of exante ability, and endogenize the decisions students make to compete for admission. Formany policy questions, it is also necessary to allow for asymmetric admissions policiesthat allow for easier admissions for the children of alumni or students from underrep-resented demographic backgrounds. While many models include one or two of theseeffects, providing a tractable model of the market that includes all three features hasproven difficult. A primary reason for the difficulty is the dual role played by human

capital (HC) in college admissions: first, a student’s HC is a productive asset, which yieldsa productive channel of investment incentives, and second, students who acquire more HCgain access to higher quality colleges, which yields a competitive channel of incentives.

We model the college admissions market as a contest wherein colleges are rank or-dered and students compete for admission by endogenously choosing the level of HCto accrue prior to the admissions contest. While this model is difficult to solve when itincludes a finite number of students and colleges, we show that a more tractable limitmodel with a continuum of colleges and students closely approximates a model with alarge, but finite, number of students and colleges. To understand why a large marketsetting might simplify things, consider a student (or a student’s parents) that is decidinghow much effort to exert in school with an eye toward her senior year when she willapply to various colleges. If the student wants to ascertain whether she is likely to beadmitted to a school with a given GPA and SAT score, she does not need to considerthe other students who might also be applying (her opponents) or the strategies they areemploying. Instead, she simply consults a college guide that describes the qualificationsof previously admitted students at her favorite schools. Since the aggregated choices ofmany market players produces a high degree of predictability in these thresholds, she canhave confidence that if she meets them then admission will be reasonably likely.1

The combination of model richness and tractability allow us to explore two key aspectsof college matching markets that depend on the interplay between the distinct channels ofincentives and college/student heterogeneity when admissions thresholds may be demo-graphically conditioned. Our first policy analysis goal is to assess the difference, if any,between affirmative action (AA) schemes implemented through an admissions preferencesystem and those implemented through a quota. We use the term “affirmative action”to refer to any policy that discriminates between individuals based on demographic fea-tures. Although this paper focuses on affirmative action in the context of U.S. collegeadmissions, the US government has mandated AA practices in various areas of the econ-omy where it has influence, including education, employment, and procurement. AA hasbeen widely implemented outside the United States as well, in places such as Malaysia,Northern Ireland, India, and South Africa.2

An affirmative action system implemented through a quota reserves separate poolsof seats for different groups of students, and each student can only compete for theseats allocated to his or her group. In an admissions preference scheme, all applicantscompete for the same pool of seats, but the fashion in which the applications are ranked is

1 Although this example and our main analysis are focused on college admissions, the basic insight re-garding the tractability and applicability of limit games as approximations to real-world contests generalizesfar outside the context of college admissions and affirmative action.

2For an in-depth discussion of AA implementations around the world, see Sowell [58].

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dependent on the demographics of the applicant. We show that the quota and admissionspreference schemes are outcome equivalent. In other words, given any equilibrium of anyquota (admissions preference system), one can design an admissions preference system(quota) with an equilibrium that results in both the same school assignments and HCchoices.

On a more practical level, the equivalence of quota and admission preference systemsraises difficult questions regarding the legality of these schemes in the context of U.S.higher education.3 Since the 1978 Supreme Court Case Regents of the University of Cali-fornia v. Bakke, constitutional jurisprudence has held that racial quotas violate the equalprotection clause of the 14th amendment to the U.S. constitution since some studentsare prevented from competing for some of the seats. In contrast, admissions preferenceschemes may be acceptable under constitutional law since any student can compete forany seat, albeit the competition is not on a level playing field.4

Our result calls into question the bright-line delineation of quotas and admissionspreference systems developed by the 1978 opinion of Justice Powell. The later 2003judgements of Grutter v. Bollinger et al. and Gratz et al. v. Bollinger et al. turned onwhether the University of Michigan had implemented an admissions preference schemethat resulted in a de facto racial quota, which both foreshadows and highlights the legalimportance of the perceived differences between quotas and admission preferences. Tothe extent the constitutionality of an affirmative action scheme hinges on the outcomesof these systems and/or the policy-induced competitive conditions (e.g., incentives) thatproduced them, then the existing precedents are self-contradictory.

The second policy question we address is the welfare cost of forcing the students tocompete for college seats through costly acquisition of HC. Once again, there are twoincentives for a student to invest in HC: first, the productive channel–since HC is a pro-ductive asset bearing a direct return–and second, the competitive channel–since HC isalso used to rank order students at the college matching stage. In our model it would bewelfare maximizing to assortatively match students (by innate productivity) and colleges(by institutional quality level) and allow each student to then choose their preferred hu-man capital investment level given his or her college assignment. Unfortunately, since astudent’s type is private information, the contest naturally relies on observable HC outputto differentiate between higher and lower productivity students. The reliance on HC inthe assignment process in turn induces them to choose a wastefully high level of human

3Both of the authors are in favor of affirmative action schemes. Our argument is purely regarding theconsistency (or lack thereof) of prior legal precedent, not an attempt to argue for (or against) the efficiencyor ethics of affirmative action.

4Later cases, which we discuss in depth in section 6, have refined the standards that acceptable admissionspreference schemes must meet.

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capital (i.e., above that resulting from a first-best assignment) to compete for a seat at agood college, and this wasteful over-investment is the cost that society pays to achievematching assortativity in the presence of private information.

A natural question then is, at the societal level, are the benefits of assortativity in thecollege match enough to outweigh the costs of wasteful over-investment? There are somesigns that the waste caused by the competitive channel may be getting worse with time.Anecdotal evidence of the increasing pressure to compete for college seats includes the“tiger mom” phenomenon brought to popular attention by the 2011 book Battle Hymn ofthe Tiger Mother by Amy Chua as well as popular book The Overachievers: The Secret Livesof Driven Kids that documents the competitive pressures placed on high school students.

Combining our model with estimates from Hickman [36] we are able to provide com-parisons between a number of alternative systems to provide perspective on the relativemerits of different college assignment systems. First we compare the welfare generatedby the first-best system and an assortative color-blind contest, which allows us to com-pute the welfare losses caused by competition. Next we compute the welfare generatedby a match that randomly assigns students to schools before the students choose their HClevels. Comparing the outcome of random assignment with the color-blind contest yieldsa measure of the benefits of assortative assignment. We find that the cost of competition($1,225 loss per student per year) entirely wipes out the benefits of an assortative match($822 gain per student per year). In other words student welfare would be improved byrandomly assigning students to colleges instead of allowing the students to compete.5

The final welfare analysis we conduct is a computation of the second-best contestdesign. We use our limit model to formalize an optimal control problem that computesthe welfare maximizing college assignment contest subject to the incentive compatibilityconstraints of the contest structure. Since demography is a sign of underlying ability,demographics are taken into account by the contest. We are able to show that only 26%of the losses between the status quo and the first best can be recovered in the second-bestcontest. We view this as a pessimistic conclusion, since it implies that the losses fromcompetition can not be significantly ameliorated through clever policy intervention.6

The remainder of this paper has the following structure: in section 2, we briefly dis-cuss the relation between this work and the previous literature on college admissions andAA. In section 3, we give an overview of the full model of competitive human capital in-vestment and describe the college assignment contest we study. In section 4 we introduce

5While our model does not include spillovers from the human capital of the college students to broadersociety, at a minimum our model describes the cost component required for a principled cost-benefit analysis.

6The caveat is that policy intervention might help, but only if more information about the students’ exante types is available. However, to the extent that this information is also subject to strategic competition,the benefits of any such policy are ambigous at best.

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the limit model with a continuum of students and prove that equilibria of the limit modelare approximate equilibria of the finite model of section 3. In section 6 we prove thatquota and admissions preference schemes admit the same set of equilibrium outcomesand discuss the practical ramifications of this result. In section 7 we turn to the welfareanalysis of the cost of signaling. Finally, in section 8 we consider a number of extensionsof our model and describe what, if anything, changes about our results.

2 Previous Literature

One clear antecedent of our work is the literature on the college admissions problem (Galeand Shapley [31]), which considers the many-to-one assignment of students to colleges.A significant benefit of this literature relative to our approach is that the students areallowed to have heterogeneous preferences over the colleges and vice versa, but thesepreferences are assumed to be fixed and exogenous. The focus of the literature followingGale and Shapley [31] has largely been on finding efficient mechanisms that give studentsthe incentive to declare their preferences truthfully in a variety of real-world markets. It isunclear how to be include an endogenous choice of human capital that influences collegepreferences and still (relatively) easily compute equilibria in a model of the form used inthis literature, which is a crucial feature for studying the welfare cost of the competitivechannel.

Our work is also related to the literature on assortative marriage markets, Becker [4]providing an early example. To the best of our knowledge, all of these models assumetransferable utility between agents paired in a match as well as complete informationregarding the types of the agents. Since the complete information eliminates the welfarelosses caused from competition in the college admissions contest, these models cannotaddress our welfare concerns.

Since our paper is largely theoretical in nature, this literature review focuses on thetheory part of the affirmative action literature. There is also rich empirical literature onthe effects of affirmative action programs, and we encourage the interested reader to seethe summary provided in Hickman [36].

Previous economic theory has studied AA and effort incentives, but existing modelseither exhibit important limitations or do not facilitate empirical analysis and the devel-opment of data-driven counterfactuals. Fain [21] and Fu [30] study models in the spiritof all-pay contests with complete information (i.e., academic ability types are ex ante ob-servable) where two students compete for a college seat.7 Extrapolating the two-player

7Schotter and Weigelt [56] study a similar setting in the laboratory. Their results suggest that no equity-achievement tradeoff exists.

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insights to real-world settings is difficult because the framework implicitly assumes thatall minority students, even the most gifted ones, are at a disadvantage to even the leasttalented non-minorities. Talented and challenged students exist among all groups, mak-ing considerations of performance incentives more complex. Franke [23] extends thecontest idea to include more than 2 agents, but at the cost of focusing on a specific formof affirmative action program.

The bilateral matching literature has also touched on incentives under AA, an earlyexample being Coate and Loury [17], which studies both human capital investment andachievement gaps. This paper considers two strategic groups of agents, firms and jobapplicants. Job applicants make a binary choice to either become qualified at somecost or remain unqualified given a privately known cost of becoming qualified. Firmsthen observe a noisy signal of the potential employee’s choice and decides to assignthe applicant to one of two positions. Coate and Loury [16] provides mild conditionsunder which discriminatory equilibria exist, analyzes the complex effects of affirmativeaction on equilibrium outcomes, and demonstrates that the equilibrium beliefs aboutthe typical qualifications of minority applications can be worsened under an affirmativeaction program.

There are a number of fundamental distinctions between bilateral matching modelsand a contest framework. Most obviously, the contest framework takes the “prizes” (e.g.,jobs, school placements) and how these prizes are assigned as exogenously given. Ineffect, the colleges in our model are nonstrategic actors. The benefit of the contest ap-proach is that it is easier to incorporate heterogeneity on the part of the applicants (e.g.,job seekers, students), allow the applicants to employ a richer action space, and allowsus to incorporate heterogeneity in the prizes awarded to the agents (e.g., allow somefirms/schools to be more desirable than others). We believe that these features bring uscloser to empirical data and real-world counterfactuals, but we acknowledge that firmand college decision-making is an interesting and important component of these markets.

Chan and Eyster [13] focuses on the effect of affirmative action bans on a single schoolwhen admission can be conditioned on student traits correlated with race. Epple et al.[20] analyzes a similar question, but considers a set of vertically differentiated colleges.Both papers describe how colleges bias their admissions policy to encourage diversity.These papers assume student quality is fixed and exogenous, and so necessarily can’t saymuch about the general equilibrium effect of the admission policy changes on studentincentives. Fryer et al. [27] partially addresses student incentives by including a binaryeffort choice in the spirit of Coate and Loury [16], but Fryer et al. [27] simplifies thesetting by assuming colleges are homogenous.

Chade, Lewis, and Smith [12] studies a matching model of college admissions with

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heterogeneous colleges. However, academic achievement is exogenous, and the analysisfocuses on the role of information frictions within the market (e.g., such as when SATscores are a noisy signal of student ability), as well as the strategic behavior of collegesin setting admissions standards. Our framework is a frictionless matching market, butacademic achievement is endogenous. In that sense, our work and Chade, Lewis, andSmith [12] may be considered complementary for understanding the role of broad marketforces in college admissions.

Although many of these papers make similar points, our conclusions regarding stu-dent welfare would be impossible without incorporating (1) heterogenous colleges, (2)heterogenous and innate student quality, and (3) endogenous human capital accumula-tion choices by the students. Most of the papers above include flexible specifications for1 or 2 of these components, but no prior paper includes rich models of all 3.

Fryer [25] touches on our analysis of the equivalence of quota and admissions prefer-ence schemes. Fryer [25] studies a model of workplace affirmative action wherein firmsthat wish to maximize profit are subject to an affirmative action mandate imposed bythe government and enforced by an auditor. Fryer [25] finds that firms facing a pool ofapplicants with few minorities will act as if they are subject to a quota on the numberof minorities they must hire, which Fryer [25] argues implies an equilibrium equivalencebetween quota and hiring preference systems. In our setting the students are the strategicactors, and our equivalence result is in some ways stronger - not only is the racial balanceat colleges in the two systems the same, but the endogenous quality of the students isidentical. However, the fundamental public policy points are very similar.

Finally, our methodology analyzes approximate equilibria played by a large numberof agents, which has been a prominent theme in the industrial organization and microe-conomic theory literature. Due to the broad scope of this literature, we provide onlya brief survey and a sample of the important papers related to the topic. Early papersfocused on conditions under which underlying game-theoretic models could be used asstrategic microfoundations for general equilibrium models (e.g., Hildenbrand [37] and[38], Roberts and Postlewaite [52], Otani and Sicilian [50], and Jackson and Manelli [41]).Other early papers focused on conditions under which generic games played by a finitenumber of agent approach some limit game played by a continuum of agents (e.g., Green[33] and [34], Sabourian [55]). A more recent branch of this literature applies these ideasto simplify the analysis of large markets and mechanisms with an eye to real-world ap-plications (e.g., Swinkels [59]; Cripps and Swinkels [18]; McLean and Postlewaite [47];Budish [8]; Kojima and Pathak [45]; Fudenberg, Levine and Pesendorfer [28]; Weintraub,Benkard and Van Roy [60]; Krishnamurthy, Johari, and Sundarajan [43]; and Azevedo and

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Leshno [7]). Many of the approximation results we use are based on Bodoh-Creed [2].8

Of these papers on approximate equilibria and large games, we would like to singleout the contemporaneously developed Olszewski and Siegel [49] as particularly relevant.Both this paper and Olszewski and Siegel [49] use the equilibria of contests played by acontinuum of agents as an approximation of a contest played by a large, but finite, setof contenders. Although there are a number of theoretical differences in the projects(e.g., the definitions of an approximate equilibrium differ slightly), the largest differencesbetween our works is the applications of our large contest theories. Olszewski and Siegel[49] does not make any claims regarding the costs of the competition for college seats orthe equivalence of quota and admissions preference systems.

3 The Finite Model

We model the market as a Bayesian game where high school students are characterizedby an observable demographic classification—minority or non-minority—and a privately-known cost type that governs HC production. Students compete for the right to occupyseats at colleges of differing quality and the seats are allocated to students according toa pre-specified rank-order mechanism which is a function of measured HC and demo-graphic classification. Agents can observe the set of potential match partners (colleges)and the form of the sorting mechanism before making decisions, but they must make theirinvest in HC before entering the market. A student’s ex-post payoff is the utility derivedfrom placing at a given college with her acquired human capital minus her investmentcost.

In this section we lay out the model when the number of students and colleges arefinite. Although we spend most of the paper working with a limit approximation of thismodel, the finite model and the limit approximation share the same underlying primitivesdescribed in this section. A secondary goal is to highlight the difficulty of working withthe finite model directly, which highlights the power of our limit approximation.

3.1 Agents, Actions, and Payoffs

The set of all students is denoted K = 1, 2, . . . , K, but there are two demographic sub-groups,M = 1, 2, . . . , KM (minorities) and N = 1, 2, . . . , KN (non-minorities), where

8There also exists a literature on the relationship between large finite and nonatomic games that focuseson games of complete information and does not emphasize the use of nonatomic games as a framework foranalyzing equilibria (examples include Housman [39], Khan and Sun [44], Carmona [9], and Carmona andPodczeck [10], Yang [61] and [11] amongst others). Kalai [42] studies large games show the equilibria of finitegames are robust to modifications of the game form.

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KM + KN = K and demographic class is observable.9 Each agent has a privately-knowncost type θ ∈ [θ, θ] ⊂ R++, and each student views his or her opponents’ unobservabletypes as independent random variables Θ whose realizations follow group-specific dis-tributions, or Θ ∼ Fi(θ), i =M,N . For convenience, we denote the unconditional typedistribution for the overall population by FK(θ). The reader should assume throughoutthat high values of θ are associated with students that have a high cost of accruing humancapital.10

We denote the set of colleges PK = p1, p2, . . . , pK, where pk ∈ [p, p] ⊆ R+ denotesthe quality level of the kth college. We denote the kth order statistic by p(k : K) so thatminkPK = p(1 : K) and maxkPK = p(K : K). The colleges are passive in our model,and each college accepts the students assigned at that college through the admissionscontest.

Each agent’s strategy space, S = [s, ∞) ⊂ R++, is the set of feasible HC levels thatcan be attained. Each students’ choice of HC is observable to the colleges (e.g., through astandardized examination). Human capital s is the minimum level required to participatein the market; in the current context, this would be a minimum literacy threshold re-quired to attend college. In skilled labor markets s might represent a college degree, andinvestment above s could be interpreted as points above the minimum passing collegeGPA.

Agents value both college quality and HC. The gross match utility derived from beingplaced at college k for a student with type θ and HC s is given by U(pk, s, θ). However, inorder to acquire HC s ∈ S , an agent must incur a cost given by C(s; θ), which depends onboth her unobservable type and her HC investment level. The total utility for a studentof type θ that choose human capital level s and is assigned to college k is

U(pk, s, θ)− C(s, θ)

3.2 Allocation Mechanisms

We now describe the contest that allocates students to colleges. Letting si denote studenti’s human capital level and s−i the vector of all other players’ actions, we let Pj(·, s−i) :S → PK, j = M, N , be an assignment mapping that describes the college to whichstudent i is assigned given each possible HC choice. Note that we have deliberately

9The assumption that there are two groups is not restrictive. Results for the two-group case extendstraightforwardly to the case with L > 2 groups.

10Costs can arise in various ways: they could come from a labor-leisure tradeoff, where θ indexes one’spreference for leisure; they could represent psychic costs, indexed by θ, involved in exerting effort to learnnew concepts, where the more able students learn with the least effort; or costs could be interpreted as mon-etary investment required for study aides (e.g., computers, tutors, and private education), where θ representsthe severity of the budget constraint that determines one’s consumption–investment tradeoff.

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allowed the assignment mapping to depend on student i’s demographic classification toreflect potential discrimination between members of the two groups. With this functionin mind, the ex post payoff for student i is then

Πrj (si, s−i; θ) = U

(Pr

j (si, s−i), si, θi

)− C(si, θi), r ∈ cb, q, ap and j =M,N

The baseline color-blind admission rule (involving no diversity preference) is repre-sented by the following assignment mapping which yields positive assortative matching:

PcbM(si, s−i) = Pcb

N (si, s−i) = Pcb(si, s−i) =K

∑k=1

p(k : K)1 [si = s(k : K)] . (1)

In the above expression, 1 is an indicator function equaling 1 if its argument is true and 0otherwise.

We concentrate on two canonical forms of AA that have received attention due towide implementation: quotas and admission preferences. A quota is the practice ofearmarking seats for each demographic group. Within the current modeling environment,this is equivalent to a set of KM seats being reserved for minorities and then allowing anassortative match within each group.

Let PM = pM1, pM2, . . . , pMKM and PN = pN 1, pN 2, . . . , pNKN denote the sets ofseats earmarked for minorities and non-minorities, respectively, and let sM = sM1, sM2, . . . , sMKMand sN = sN 1, sN 2, . . . , sNKN denote the group-specific human capital profiles. Then aquota assignment mapping is represented by the functions

PqM(si, s−i) =

KM

∑m=1

pM(m : KM)1 [si = sM(m : KM)] , and

PqN (si, s−i) =

KN

∑n=1

pN (m : KM)1 [si = sN (n : KN )] .

(2)

The defining characteristic of a quota is a split of the competition into two separate con-tests where students compete only within their own race group.

An admission preference allows the students to compete against all of the other students,but the human capital choices of the members of each group are treated differently. Inpractice, many American admissions committees are thought to treat observed minorityapplicants’ SAT scores (a commonly used measure of HC) as if they were actually higherwhen evaluating them against their non-minority competitors.11 More formally, an ad-

11There has been a fair amount of empirical research estimating a substantial average admission preferencefor minorities at elite American colleges; e.g., Chung, Espenshade and Walling [15] and Chung and Espen-shade [14]. Hickman [36] employs a similar empirical measure for the aggregate US college market and finds

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mission preference is a markup function S : S → R+ through which minority output ispassed to produce a set of transformed HC levels, s = sN 1, . . . , sNKN , S(sM1), . . . , S(sMKM),and allocations are given by the following group-specific functions:

PapM(si, s−i) =

K

∑k=1

p(k : K)1[S(si) = s(k : K)

], and

PapN (si, s−i) =

K

∑k=1

p(k : K)1 [si = s(k : K)] .

(3)

Regardless of whether admissions are color-blind or follow some form of AA, ties be-tween competitors are assumed to be broken randomly.

3.3 Model Assumptions

We now outline a series of assumptions on our model primitives. The assumptions servethree goals:

1. Establish the existence of a monotone equilibrium.

2. Justify the use of a centralized, assortative matching structure.

3. Insure the model is sufficiently well-behaved that our limit approximation is valid.

Although we highlight how the assumptions tie into goals (1) and (2), we defer dis-cussion of (3) until the next section.

Assumption 1. For each (θ, s) ∈ [θ, θ]×R+, we have Cs(s, θ) > 0, Cθ(s, θ) > 0, and Css(s, θ) >

0

Assumption 2. Up(p, s, θ) > 0, Us(p, s, θ) ≥ 0, Uθ(p, s, θ) ≤ 0, and Uss(p, s, θ) ≤ 0

Assumption 1 states that costs are strictly increasing in human capital level s and costtype θ. Moreover, the cost function is (weakly) convex in s for any given θ. Assumption 2states that utility is differentiable, strictly increasing in college quality p, weakly increas-ing and concave in human capital level s and weakly decreasing in cost type θ. Theseassumptions imply that the individual decision problems have global maximizers

Assumption 3. U(p, s, θ)− C(s, θ) = 0 and arg maxs

U(p, s, θ)− C(s, θ) ≤ s

evidence of a substantial admission preference even at lower-ranked colleges.

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Assumption 3 is a boundary condition for our model. First, the assumption requiresthat the lowest type of student is indifferent between participating in the college admis-sions contest and not attending college, which normalizes the utility of not attendingcollege to 0. This has the benefit of allowing us to interpret U(p, s, θ) − C(s; θ) as theequilibrium college premium of the students. Second, the assumption demands that theminimally qualified student who chooses to attend college does not have an interest inacquiring more HC. While this is not usually assumed, we find that the assumption issupported in the data we use to calibrate our model in section 7.

Assumption 4. There exists s such that U(p, s, θ)− C(s, θ) < U(p, s, θ)− C(s, θ)

Assumption 4 requires that there exist a human capital level so large that even thelowest type, the type that gets the largest value from human capital and has the lowestcost for acquiring human capital, would rather not invest in human capital at all and beassigned to the worst school. This implies that we can limit our analysis to human capitallevels within s ∈ [s, s]. Unless otherwise stated, we simply let S = [s, s].

Assumption 5. Ups(p, s, θ) ≥ 0 and Upθ(p, s, θ) ≤ 0

Assumption 5 requires (weak) positive complementarity between student HC and col-lege quality and between student and college qualities. These assumption imply thatthe efficient, decentralized matching college assignment is characterized by positive as-sortative matching, which helps justify our use of the centralized rank-order mechanismsto model the market.

Assumption 6. Usθ(p, s, θ) ≤ 0 and Csθ(s, θ) > 0

Assumption 7. FM(θ) and FN (θ) have continuous and strictly positive densities fM(θ) andfN (θ), on a common support [θ, θ].

Assumptions 6 is key for existence of a monotone pure-strategy equilibrium. Assump-tion 6 states that marginal benefits of HC are decreasing and marginal costs of HC areincreasing in a student’s type. Assumption 7 is a standard regularity conditions on thetype distributions and are supported by the calibration data.

Assumption 8. S(s) is strictly increasing and differentiable almost everywhere.

Assumption 8 assumes that, holding race fixed, admissions officers always prefer toenroll a student with more HC.

In our model the agents’ types are their private information, and the agents choosetheir human capital level given knowledge about he number of competitors from eachgroup KM and KN , the distribution of student types in the economy, the set of seats

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PK, and the the admission rule Prj , r ∈ cb, q, ap and j ∈ M,N. Under the payoff

mapping Πrj (θ, si, s−i) induced by a particular admission rule, students optimally choose

their achievement level based on their type and the types of potential match partners,taking into account opponents’ optimal behavior. The model defined above fits the moldof a contest, which can be thought of as an asymmetric, multi-object, all-pay auction withsingle-unit bidder demands and bid preferences.

We study the Bayes-Nash equilibria of the finite game. An equilibrium of the gameΓ(KM, KN , FM, FN ,PK) is a set of achievement functions σr

j : [θ, θ] → R+, j = M,Nwhich generate optimal choices of human capital s = σr

j (θ) given that all other agentsfollow the equilibrium strategy. It will be convenient at times to denote the inverse equi-

librium achievement functions by ψrj (s) ≡

(σr

j

)−1(s) = θ.

Theorem 1. In the college admissions game Γ(KM, FM, KN , FN ,PK) with r ∈ cb, q, ap, underassumptions 1–8 there exists a monotone pure-strategy (but potentially not group-wise symmetric)equilibrium (σr

M(θ), σrN (θ)). Moreover, any equilibrium of the game must be strictly monotone

with almost every type using pure strategies.

4 The Limit Game

For large K, the equilibrium of our finite model is analytically and computationally diffi-cult because an agent’s decision problem is a complicated function of the order statisticsof opponents’ cost types. However, intuitions based on the law of large numbers wouldsuggest that as the market becomes large (i.e., as K → ∞), the distribution of realizedtypes and human capital choices ought to approach some limit measure. If this intuitionis true, then the mapping between human capital choices and school assignment in thislimit game, which we call the limit assignment mapping, ought to provide a good descrip-tion of the outcomes in games with a sufficiently large, but finite, set of players. Thiswould suggest that an individual agent could come very close to maximizing his or herutility in a large finite game by optimizing against the limit assignment mapping.

To see how this simplifies the student’s problem (and as a result, our analysis), con-sider the plight of a college applicant in the United States. Do would-be college studentsgo to elaborate lengths to determine who else is applying, where those students are ap-plying, and what the other students’ qualifications are? Of course not - to determinewhether an application is likely to be accepted, the would-be college student can simplylook at data on the grades and SAT scores of currently enrolled students. In our model,the SAT score of currently enrolled students is akin to knowing the equilibrium mappingbetween human capital levels and school assignments. Conveniently, metrics such asSAT scores of incoming freshman are stable from year to year.

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We repeatedly suggest that the equilibria of the limit game are easy to compute, whichcan be interpreted in two ways. First (and we think most importantly), the equilibria ofthe limit game are computed using a decision problem that captures the decision processdescribed in the paragraph above. In other words, a student can discover their optimalaction in the limit game through reasoning that we believe is typical of real-life behavior,which we think supports the palusibility of the limit model. Second, even using modernalgorithms, it is difficult to numerically compute the equilibrium of the finite model whenK is large. Our limit model can also be used as a powerful analytical tool by researchers.In either case, our challenge is to formally prove that the equilibrium of the limit gameapproximates the equilibrium of the more realistic finite agent game when K is large.

Much of the underlying mechanics of the convergence of the finite games to the rel-evant limit model are the same across the color-blind, admissions preference, or quotagames. For each type of college assignment game we consider a sequence of finite gamesdenoted Γ(KM, FM, KN , FN ,PK)∞

KN+KM=2, and we use the same utility function in thefinite games and the limit game. We assume that KM

K → µ ∈ (0, 1) as K → ∞, whichimplies that µ is the asymptotic mass of the minority group.12 We let FK

P (p) denotethe distribution of college qualities in the K agent game, and we assume that there is acontinuous CDF FP (p) such that

LimK→∞

supp∈[p,p]

∥∥∥FKP (p)− FP (p)

∥∥∥ = 0 (4)

Alternately, we could assume that college types are drawn in an independent fashionfrom FP in each of the finite games, in which case equation 4 holds almost surely.

In the limit admissions preference game there is a measure µ continuum of minor-ity students with types distributed exactly as FM and a measure 1 − µ continuum ofnonminority students with types distributed exactly as FN . Finally, there is a measure1 continuum of college seats distributed exactly as FP . In order to describe the limitequilibria using an ODE, we must make the following refularity assumption on FP .

Assumption 9. FP (p) has a continuous and strictly positive density fP(p) on support [p, p].

In the following subsections, we address the color-blind, admission preference, andquota games separately. In each subsection we describe the limit assignment mappingand use the mapping to describe the equilibrium strategy in each game. In section 4.4we provide conditions under which there exists an essentially unique PSNE of the limitgame. In the case of color-blind and quota systems, such a unique equilibrium exists

12Although we assume throughout that the convergence is deterministic, one can think of the agents inthe finite games as being created by nature by assigning each agent to groupM with probability µ ∈ (0, 1),after which nature draws a cost type for the student from the corresponding distribution.

14

for any feasible game. Finally, we close with a description of comparative statics resultsin section 4.5, which also serves to highlight how our model enriches some commonpre-existing models in the contest literature.

4.1 Color-blind Admissions Game

Recall that a color-blind outcome yields positive assortative matching of seats and humancapital levels. We denote the endogenous distribution of human capital levels chosen bythe students as Gcb

K (s),j =M, N and r ∈ cb, q, ap. Using this notation, we can describethe match of students to colleges as:

PcbM(s) = Pcb

N (s) = Pcb(s) =F−1P

(GcbK (s)

)=F−1P

(µGcbM(s) + (1− µ)Gcb

N (s))

=F−1P

(1− µFM

(ψcb(s)

)− (1− µ)FN

(ψcb(s)

)).

(5)

The intuition is simple: quantiles of the population HC distribution GcbK (s) are mapped

into the corresponding quantiles of the distribution FP . Since limiting payoffs do notdepend on race, it follows that σcb

M(θ) = σcbN (θ) = σcb(θ); hence, the lack of subscripts on

the inverse strategies in the third line.13

We now turn to describing the equilibrium of the color-blind limit game. Given thatthe equilibrium will, in the end, be assortative, we can write the endogenous assignmentfunction as a function of agent type as follows

PcbM(θ) = Pcb

N (θ) = F−1P (1− µFM (θ)− (1− µ)FN (θ))

If we describe the agent’s problem in revelation mechanism form, then the decision prob-lem reduces to choosing a type θ to declare

maxθ

U(

Pcb(

θ)

, σcb(

θ)

, θ)− C

(σcb(

θ)

, θ)

Manipulating the first order condition for this problem and using the fact that in equilib-

13Theorists with experience in asymmetric auctions may find this statement puzzling, but one must keepin mind that it merely applies to limiting payoffs. In a two-player finite game, differing investment behav-ior arises from the fact that a minority competitor faces a profile of opponents of each type numbering(KN , KM − 1), whereas a non-minority competitor faces profile (KN − 1, KM), and since costs are asym-metrically distributed across groups, a minority and a non-minority with the same private cost will havediffering expectations of their standing in the distribution of realized competition. However, the differencebetween their expected ranks quickly vanishes as the number of players gets large.

15

rium we have θ = θ yields the following differential equation for investment:

dσcb(θ)

dθ= −

Up(

Pcb[σcb(θ)], σcb(θ), θ)· fK(θ)

fP(

F−1P (1− FK(θ))

)· (Cs (σcb(θ), θ)−Us (Pcb (σcb(θ)) , σcb(θ), θ))

,

σcb(θ) = s (boundary condition).

(6)

The boundary condition comes from the fact that a player of type θ will always bematched with the lowest seat in a monotone equilibrium, so she cannot do better thanto simply choose HC level s to complement p. Note that, given the assumptions on themodel primitives, the initial value problem defined by (6) results in a strictly decreasingfunction.

4.2 Admissions Preference Game

As in the finite-agent model, an admission preference rule is modeled as a markup func-tion S : R+ → R+ and seats are matched assortatively with transformed HC. In otherwords, minorities are repositioned ahead of non-minority counterparts with investmentof S(s) or less. The limiting assignment mapping for groupM is

PapM(s) = F−1

P

((1− µ)GN

(S(s)

)+ µGM(s)

)= F−1

P

(1−

((1− µ)FN

(ψ

apN

(S(s)

))+ µFM

(ψ

apM(s)

))) (7)

and limiting allocations for group N are given by

PapN (s) = F−1

P

((1− µ)GN (s) + µGM

(S−1(s)

))= F−1

P

(1−

((1− µ)FN

(ψ

apN (s)

)+ µFM

(ψ

apM

(S−1(s)

)))).

(8)

The intuition for the above expressions is similar as before: limiting mechanisms mapthe quantiles of a distribution into the corresponding college seat quantiles. For non-minorities, it is a mixture of the distributions of non-minority HC and subsidized minorityHC. For minorities, it is a mixture of the distributions of minority HC and de-subsidizednon-minority HC.

Unlike the color-blind or quota case, it is not possible to describe which kind of stu-dent obtains each seat in an admissions preference scheme without first solving for theequilibrium. This feature of the admissions preference mechanism makes it significantlyharder to work with than the other two schemes. However, we prove in section 6 that theadmissions preference and quota schemes are outcome equivalent - for any admissionspreference markup function one can describe a quota scheme that generates the same

16

school assignment and equilibrium human capital choices. Because of this equivalence,for the majority of the paper we work with color-blind and quota systems. However,for completeness, we now provide the differential equations describing the admissionspreference equilibrium when S is differentiable and S(s) = s.

(ψ

apM)′(s) =−

Cs(s, ψ

apM(s)

)−Us

(PapM(s), s, ψ

apM(s)

)Up(

PapM(s), s, ψ

apM(s)

) ·fP(

PapM(s)

)µ fM

(ψ

apM(s)

)−

(1− µ) fN(

ψapN

(S(s)

))µ fM

(ψ

apM(s)

) (ψ

apN)′ (S(s)

) dS(s)ds

and(ψ

apN)′(s) =−

Cs(s, ψ

apN (s)

)−Us

(PapN (s), s, ψ

apN (s)

)Up(

PapN (s), s, ψ

apN (s)

) ·fP(

PapN (s)

)(1− µ) fN

(ψ

apN (s)

)−

µ fM(

ψapM

(S−1(s)

))(1− µ) fN

(ψ

apN (s)

) (ψ

apM)′ (S−1(s)

) dS−1(s)ds

(9)

These equations are described in terms of the inverse strategy functions for notationalease. If S is not differentiable or S(s) 6= s, then the equilibrium may involve jumps. Theinterested reader is (again) referred to appendix D for a description of how to identify thesize and location of these jumps.

4.3 Quota Game

We denote the empirical measure of seats allocated to group j ∈ M,N in the K-agentgame be denoted QK

j . We assume that there exists a distribution of seats Qj(p) such that

LimK→∞

supp∈[p,p]

∥∥∥QKj (p)−Qj(p)

∥∥∥ = 0

In the limit quota game a measure µ of minority students with types exactly distributedas FM compete for a pool of seats exactly distributed as QM. Similarly, in the limitquota game a measure 1− µ of nonminority students with types exactly distributed asFN compete for a pool of seats exactly distributed as QN . The measures QM and QN aresubject to the following feasibility constraint

For all p we have µQM(p) + (1− µ)QN (p) = FP (p)

Finally, we assume that the quota measures admit a density, which will prove useful indescribing the equilibrium strategies.

17

Assumption 10. Qj(p), j ∈ M,N, admits a density on [p, p].

In the finite game, the students are allocated into two contests in which the studentscan compete for the seats allocated to their group. The result is an assortative matchwithin each distinct contest. In the limit game the students from group j ∈ M,Ncompete for the measure of seats Qj allocated to their group. These distinct contestsyield group-specific assignment mappings of the form

Pqj (s) = Q−1

j

(Gq

j (s))= Q−1

j

(1− Fj

(ψ

qj (s)

)), j ∈ M,N. (10)

As in the color-blind case, the quantiles of the group-specific human capital distributionsare mapped into the corresponding quantiles of Qj. We find the following differentialequation describing the equilibrium when Qj have full support.

dσqj (θ)

dθ= −

Up

(Pq

j

(σ

qj (θ)

), σq(θ), θ

)· f j(θ)

fP(

F−1P(1− Fj(θ)

))·(

Cs

(σ

qj (θ), θ

)−Us

(Pq

j

(σ

qj (θ)

), σ

qj (θ), θ

)) ,

σqj (θ) = s (boundary condition).

(11)

When Qj does not have full support, then there will be jumps in the equilibrium strategies

when Pqj

(σ

qj (θ)

)encounters the left edge of a gap in the support. The interested reader

is referred to appendix D for a description of how to identify the location and size of thejumps. Between these jumps, equation 11 describes the equilibrium strategy.

4.4 Equilibrium Existence

Since the limit game is one of complete information, the relevant equilibrium concept is aNash equilibrium. In a setting with a continuum of agents, this equilibrium concept canbe defined as follows:

Definition 1. σ is a Nash equilibrium of the limit game if for j ∈ M,N and all θ ∈ [θ, θ]

σ(θ) ∈ arg maxs

U(

Prj (s), s, θ

)− C(s, θ)

We would like to use the differential equations describing the equilibria under eachcollege assignment model to prove the existence of a unique, pure-strategy Nash equi-librium. While we cannot prove these exact properties, we can prove that they holdalmost everywhere. Consider a pair of behavioral strategies14 (σM, σN). We refer to

14Recall that in a Bayesian game a behavioral startegy is one that maps types into actions and that agentsadopt these strategies prior to learning their types.

18

an equilibrium as essentially group symmetric if for any agent t in group i = M,N , theequilibrium strategy adopted by agent t is equal to σi almost everywhere. Using thetheory of strict monotone comparative statics developed by Edlin and Shannon [19], wecan show both that any equilibrium must be essentially group-symmetric and that almostall of the agents must adopt pure actions in any equilibrium.

The challenge to proving uniqueness is handling points where the strategy must bediscontinuous. For example, in the quota case if Qi lacks support over an interval ofschools [p1, p2], then the strategy of the students must exhibit a discontinuity. For quotasystems we can identify the size each of these jumps must take in equilibrium, and itis easy to prove that the strategy is continuous almost everywhere. In other words,we can prove the equilibrium is unique for any quota scheme except for the measure 0set of types where the strategy exhibits a discontinuity, which we refer to as essentiallyunique.15 Unfortunately, we do not yet have a direct result for the admissions preferencecase where S is nondifferentiable or S(s) = s. Fortunately, since we work exclusively withquota systems in our later analysis, this result suffices.

Theorem 2. In any equilibrium almost all of the agents take pure actions and the strategies areessentially group-symmetric.

Finally, an essentially unique Nash equilibrium exists in the limit model in the following cases

1. The limit game of the admissions preference model with a differential markup function S thatsatisfies S(s) = s and FP has full support.

2. The limit game of the quota system with any feasible choice of QM and QM.

4.5 Comparative Statics of the Limit Model

Our goal in this section is to identify conditions under which encouragement and discour-agement effects occur in our model. Discouragement effects describe situations whereincompetition within a contest intensifies, which causes low ability competitors to exertless effort while at the same time high ability competitors work harder. Since affirmativeaction schemes (most notably admissions preference schemes) in effect make minoritystudents better competitors, one might expect discouragement effects within the nonmi-nority students. In addition, there will be a symmetric encouragement effect within theminority student population.

15At these discontinuities the type could choose the action on either side of the discontinuity or mixarbitrarily over these choices. Since only a measure 0 set of agents face such a discontinuity, the choices ofthese agents has no effect on the incentives of the other agents.

19

Our notion of changing competition is captured by examining changes in the distri-bution of the types of competitors that can be ordered using a single crossing conditionapplied to the density functions of the relevant distributions.

Definition 2. Two functions f (θ) and g(θ) satisfy the strict single crossing condition (SCC)when for any θ > θ we have f (θ) ≥ g(θ) implies f (θ) > g(θ). We denote this relation g <SCC f

Keeping in mind that students with low values of θ choose higher levels of human capital,if f1 and f2 represent distributions of student types, then f1 <SCC f2 implies that f1

represents a stronger student population.Now we turn to our analysis of the discouragement effect. We first consider a color-

blind admission policy and analyze how students react to changes in the distributions oftypes of the other students. Our first result implies a form of the discouragement effect.When there are fewer high ability students (and hence more low ability competitors), thelow ability students make greater investments in human capital.

Theorem 3. Consider two population cost distributions, F1(θ) and F2(θ), that have have fullsupport over [θ, θ]. Assume f1(θ) <SCC f2(θ). Under a color-blind admissions policy, thereexists a point θ∗ such that σcb

1 (θ) < σcb2 (θ) for each θ ∈ (θ∗, θ).

Proof. In the color-blind admissions scheme, we need not differentiate between the mi-nority and nonminority students. Furthermore, I will drop the “cb” superscript fornotational ease. Denote the equilibrium strategies under F1 and F2 by σ1 and σ2, respec-tively. Note that since the distributions of types are different, the school allocated to eachtype in equilibrium will also depend on the distributions F1 and F2, and we denote theassociated assignment functions as P1 and P2.

Using the notation of a revelation mechanism, the FOC is now

Up(P(θ), σi(θ), θ)P′i (θ) + Us(Pi(θ), σi(θ), θ)σ′i(θ) = Cs(σi(θ), θ)σ′i(θ)

We can rewrite the first order condition as

σ′i(θ) = −Up(Pi(θ), σi(θ), θ) fi(θ)

fP (1− F−1i (θ)) [Cs(σi(θ), θ)−Us(P(θ), σi(θ), θ)]

For the lowest ability student, θ = θ, we can write

σ′i(θ) = −Up(p, s, θ) fi(θ)

fP (p)[Cs(s, θ)−Us(p, s, θ)

] (12)

where we use the fact that P(θ) =p, and we have the boundary condition σi(θ) = s fori = 1, 2. The SCC implies f1(θ) < f2(θ), so equation 12 implies σ′2(θ) < σ′1(θ) < 0. This

20

in turn means that σ1(θ) < σ2(θ) within a neighborhood of θ by differentiability (recallalso that σj is strictly decreasing and rises in the leftward direction from the boundarypoint).

Our second result provides a condition under which there is at most one crossing-point between σcb

1 and σcb2 . If there is no crossing point, then all of the students increase

their human capital choices under F1. If there is a crossing point, then we see both anencouragement and a discouragement effect. In other words, bad students choose higherlevels of human capital, good students acquire less human capital, and the gap betweenthe best and worst students contracts. Our proof proceeds by ordering the slope of theequilibrium strategy as function of the underlying primitives.

Theorem 4. Assume all of the conditions of theorem 3 hold. In addition we will assume that thedistribution of seats is potentially different, and we delineate between the distributions using thenotation FP ,1 and FP ,2. In addition, assume:

Up(F−1P ,1 (1− F1(θ)) , s, θ) f1(θ)

fP ,1(F−1P ,1 (1− F1(θ)))

[Cs(s, θ)−Us(F−1

P ,2 (1− F1(θ)) , s, θ)]

>SCCUp(F−1

P ,2 (1− F2(θ)) , s, θ) f2(θ)

fP ,2(F−1P ,2 (1− F2(θ)))

[Cs(s, θ)−Us(F−1

P ,2 (1− F1(θ)) , s, θ)] in θ

(13)

Under a color-blind admissions policy, there exists at most a single point θ∗ ∈ (θ, θ) such thatσcb

1 (θ) < σcb2 (θ) for each θ ∈ (θ∗, θ) and σcb

1 (θ) < σcb2 (θ) for each θ ∈ [θ, θ∗).

Proof. The first part of our result follows the proof of theorem 3 and is omitted. Whatremains is to show that the point θ∗, if it exists, must be unique.

Suppose there are more than two points where σ1(θ) = σ2(θ) and let θ∗ denote thelargest such point and θ∗∗ the second largest. Let σ1(θ

∗) = σ2(θ∗) = s∗ and p∗i = Pi(θ

∗) =

F−1P ,i (1− Fi(θ

∗)). Since σ1 must cross σ2 from above (recall, σ1(θ) < σ2(θ) in (θ∗, θ)), theformer must have a more negative slope. Combining these facts yields

σ′1(θ∗) = −

Up(p∗1 , s∗, θ∗) f1(θ∗)

fP ,1(p∗1)[Cs(s∗, θ∗)−Us(p∗1 , s∗, θ∗)

]< −

Up(p∗2 , s∗, θ∗) f2(θ∗)

fP ,2(p∗2) [Cs(s∗, θ∗)−Us(p∗2 , s∗, θ∗)]= σ′2(θ

∗)

Note that since Ups = 0 we have Cs(s∗, θ∗) − Us(p∗1 , s∗, θ∗) = Cs(s∗, θ∗) − Us(p∗2 , s∗, θ∗).

21

Therefore, it must be the case that

Up(p∗1 , s∗, θ∗) f1(θ∗)

fP ,1(p∗1)>

Up(p∗2 , s∗, θ∗) f2(θ∗)

fP ,2(p∗2)(14)

Now we will argue from contradiction that σ1 and σ2 can cross at most once. Supposethere were a second point θ∗∗ < θ∗ such that σ1(θ

∗∗) = σ2(θ∗∗). In this case, we would

have σ1 crossing σ2 from below, which would imply σ1 is less steep than σ2, which wouldrequire

σ′1(θ∗∗) = −

Up(p∗∗1 , s∗∗, θ∗∗) f1(θ∗∗)

fP ,1(p∗∗1 )[Cs(s∗∗, θ∗∗)−Us(p∗∗1 , s∗∗, θ∗∗)

]> −

Up(p∗∗2 , s∗∗, θ∗∗) f2(θ∗∗)

fP ,2(p∗∗2 ) [Cs(s∗∗, θ∗∗)−Us(p∗∗2 , s∗∗, θ∗∗)]= σ′2(θ

∗∗)

where p∗∗i and s∗∗ are defined analogously. Again, assumption 1 implies Cs(s∗∗, θ∗∗)−Us(p∗∗1 , s∗∗, θ∗∗) = Cs(s∗∗, θ∗∗)−Us(p∗∗2 , s∗∗, θ∗∗). For σ′1(θ

∗∗) > σ′2(θ∗∗) to be true, it must

then be the case that

Up(p∗∗1 , s∗∗, θ∗∗) f1(θ∗∗)

fP ,1(p∗∗1 )<

Up(p∗∗2 , s∗∗, θ∗∗) f2(θ∗∗)

fP ,2(p∗∗2 )

But this is contradicted by assumption 2. Therefore, σ1 and σ2 can cross at most once.

The condition required by our theorem is very complex, but much of this is dueto the fact that we are allowing more complex interactions between HC choice, schoolassignment, and type than the typical contest model. For example, suppose we (as inmost contest models) assume HC is accrued solely to compete and is not a productiveasset. This could be captured by assuming that U(p, s, θ) = p

(θ − θ

), which implies that

good students (i.e., low θ values) have a stronger preference for better schools. In thiscase equation 13 reduces to

f1(θ)

fP ,1(F−1P ,1 (1− F1(θ)))

>SCCf2(θ)

fP ,2(F−1P ,2 (1− F2(θ)))

in θ (15)

Equation 15 requires that the population of competitors relative to the population of seatsfor which they are competing increase with type. For example, this condition wouldbe satisfied if economy 1 (relative to economy 2) had more high cost students and arelatively small proportion of low quality colleges. Such a situation incentives high HCaccumulation from high cost students since (1) small increases in HC allows the studentsto leap in front of a large number of peers and (2) the small number of low quality colleges

22

means such a leap results in placement in a much better college.If the condition of theorem 4 holds,then low ability students (i.e., students with high

values of θ) will reduce their human capital investment. This is an example of the dis-couragement effect which is a common feature of the contests literature: when a givencost type θ falls far enough behind in the sense that the mass of lower-cost competitorsF2(θ) is large, his incentives for investing in costly effort become weak. If there is a cross-ing point θ∗, then we also find an encouragement effect that causes low-cost individuals(θ sufficiently low) to invest more aggressively to keep the larger mass of their like oppo-nents at bay. This means that an increase in the degree of competition will increase thedisparity in achievement between top students and the ones at the bottom.

While theorem 4 is useful for identifying encouragement and discouragement effectsin generic contest models, it will be difficult to determine whether equation 13 holds inthe calibrated model we study below. Fortunately, we can use the underlying data todetermine whether equation 13 is satisfied.

5 Approximating the Finite Game with Limit Equilibria

Having defined the equilibria of the limit model using ODEs and proven that the equilib-rium thus described exists and is essentially unique, we now argue that the equilibrium ofthe relevant limit game is a useful approximation of the equilibria in games with a large,but finite, set of players. We use two notions of approximate equilibrium in this paper.Our first definition provides for an approximation in terms of incentives - agents that fol-low an ε−approximate equilibrium can gain at most ε by deviating. Intuitively, studentslose little utility if they base their actions on the easy-to-solve limit game equilibriumdecision.

Definition 3. Given ε > 0, an ε-approximate equilibrium of the K-agent game is a K-tuple ofstrategies σε = (σε

1, . . . , σεK) such that for all agents i, j ∈ M,N, almost all types θ, and all

human capital choices s′ we have

U(

Prj (σ

εi (θ)), σε

i (θ), θi

)− C(σε

i (θ), θi) + ε ≥ U(

P(s′), s′, θi)− C(s′, θi)

A δ−approximate equilibrium provides a close approximation of the actual HC choicesof each of the agents.

Definition 4. Given δ > 0, a δ-approximate equilibrium of the K-agent game is a K-tupleof strategies σδ = (σδ

1, . . . , σδK) such that for any exact equilibrium of the K-agent game σ =

23

(σ1, . . . , σK) we have for all agents i, j ∈ M,N, and almost all types θ∥∥∥σδi − σi

∥∥∥ < δ in the sup-norm

Our goal in this section is to prove that the equilibria of the limit game are δ-approximateequilibria of admissions games with sufficiently many students. Proving this resultamounts to proving that the limit game is continuous in the appropriate sense.

We require the following assumption on the markup functions to prove our approxi-mation result for the admissions preference system. The assumption bounds the marginalmarkup applied to minority student human capital choices, which helps insure that mi-nority student utility functions are continuous in the limit game.

Assumption 11. There exists λ1, λ2 ∈ (0, ∞) such that for all s, s′ ∈ S , s > s′, we have

λ2(s− s′

)> S(s)− S(s′) > λ1

(s− s′

)We can now state our primary equilibrium approximation result.

Theorem 5. Let σji , i ∈ M,N and j ∈ cb, q, ap denote an equilibrium of the limit game.

Assume one of the following cases holds:

1. The limit game of the admissions preference model with a differential markup function S thatsatisfies S(s) = s and FP has full support.

2. The limit game of the quota system with any feasible choice of QM and QN that admitstrictly positive PDFs over a connected support.

Under assumptions 1 -9, assumption 11, and given ε, δ > 0, there exists K∗ ∈ N such thatfor any K ≥ K∗ we have that σ

ji is a ε-approximate equilibrium of the K-agent game and σ

ji is a

δ-approximate equilibrium for the K-agent game.

When many students follow the equilibrium strategy of the limit game, then the re-alized distribution of human capital and college seats will (with high probability) be ap-proximately the same as the distributions realized in the limit game. If the student utilityfunctions are continuous, then these small differences have a negligible effect on the agentutility for each possible action (and so the maximum utility changes only slightly). Thisimplies that σ

ji is a ε-approximate equilibrium of the K-agent game.

Theorem 5 also implies that every equilibrium of the finite game must be close to theessentially unique equilibrium of the limit game. First, we show that the continuity ofthe student utility functions implies that the equilibrium correspondence is upper hemi-continuous. In other words, an exact equilibrium of the admissions game with many

24

players must be close to some equilibrium of the limit game. However, it could be thecase that there are equilibria of the limit game that are unlike any equilibrium of the finitegame (i.e., lower hemicontinuity of the equilibrium correspondence might fail). To rulethis out, we use the fact that the limit game has a unique equilibrium to prove that theequilibrium correspondence is in fact continuous, which is equivalent to the claim madeby (ii).

In order to prove our theorem, we need to rule our discontinuities in our model, whichcan arise either through the contest structure or the endogenous equilibrium strategy.In appendix C we modify our model by assuming that the exact distribution of schoolseats uncertain, which reflects the fact that students accrue HC over a long period oftime and they cannot be certain of the exact school qualities that will be available whenthey eventually compete for a college seat. With this assumption, our approximationresults hold for any feasible quota. We can also extend the mechanism equivalence result(theorem 6 below) as long as we allow the quotas and admissions preference schemes todepend on the realized distribution of colleges.16

6 Mechanism Equivalence

In this section we make our argument that the quota and admissions preferences systemsare, in fact, equivalent to each other. In any equilibrium of any affirmative action scheme,the agents in the limit game respond optimally to the assignment mapping, P(s), thatdescribes how HC choices lead to college assignments. We prove our equivalence resultby showing that if an assignment mapping P(s) is generated by some equilibrium ofan admission preference (quota) system, then there exists an equilibrium of some quota(admissions preference) system that yields the same P(s). Since the P(s) are the sameunder each system, the optimal agent responses must also be the same. Note that thenotion of equivalence we use implies that not only are the same measures of minorityand nonminority students assigned to each school, but the students at each school choosethe same level of human capital under both systems.17,18

16Olszewski and Siegel [49] does not require continuity to prove their approximation results, but the notionof approximation they use allows for a small measure of students to take actions that differ significantly fromthe predictions of the limit model.

17Note that we have not proven that any choice of Pi(s) can be implemented by either a quota or anadmissions preference scheme. For example, if Pi(s) is strictly decreasing, then it cannot be implemented byany incentive compatible mechanism.

18We have only conjectured that admissions preference schemes with discontinuous equilibria admit aunique equilibrium. If this conjecture is false for some admission preference markup function S, theorem 6will be unaffected. However, it will be the case that each of the equilibria induced by S will be equivalent todifferent quota systems.

25

Theorem 6. Consider some Pi(s) : S → P , i ∈ M,N . Pi(s) : S → P , i ∈ M,N is the resultof an equilibrium of some quota system if and only if there is an equilibrium of an admissionspreference system that also yields these assignment functions and admits the same equilibriumstrategies.

Recall that under a quota system the equilibrium assignment of student types tocolleges within each group is assortative, which means the only unknown endogenousquantity is the HC accumulation strategy. Given an admission preferences system, boththe equilibrium human capital choices and the school assignment need to be computed,which makes the admissions preference schemes more difficult to study. Theorem 6 isuseful from a methodological perspective because it shows that there is no loss in gener-ality from focusing solely on outcomes that can be realized using quota schemes.19

Our result bears a superficial resemblance to the equivalence of quotas and tariffs inan international trade context (e.g., Bhagwati [5]), although our setting is complicated bythe continuum of heterogeneous “goods” (i.e., college seats) being assigned and the con-tinuum of endogenous “prices” (i.e., human capital levels) required to obtain the goods.Even without the insights from the international trade literature, we do not view it assurprising that the same assignment of student types to colleges can be generated usingeither type of mechanism. We find it much more surprising that the endogenous humancapital decisions can also be replicated, and (to the best of our knowledge) there is noanalog of this result in the international economics literature. Of course some insights,such as the breakdown of the equivalence in the presence of aggregate uncertainty, aretrue both in the quota-tariff equivalence and in our college admissions model.

As just hinted, the equivalence of admissions preference and quota schemes relies onthe distribution of student types being fixed and known. If there are aggregate shocks tothe distribution of student types, the equivalence will no longer hold unless the quota andadmissions preference schemes are allowed to be functions of the realized distribution ofapplicant types. For example, under a quota scheme that is not responsive to the distri-bution of college students, a fixed number of minority student are enrolled even in theevent that the minority student population is significantly better than expected. In con-trast, an admissions preference scheme may allow, the additional high quality minorityapplicants to be enrolled. Large aggregate shocks in the distribution of undergraduateapplicants seems unlikely, but it is easy to imagine aggregate shocks such as economiccycles that would affect the pool of applicants to other kinds of educational programssuch as MBA programs.

19Our claim can be easily extended, but the statement is awkward and the value limited. For example,consider an economy where some schools use (potentially different) admissions preference systems and otherschools use quotas. We can design either an admissions preference system or a quota system that generatesthe same equilibrium strategies and assignment of students to colleges.

26

Another practical issue that may play out is that some aspects of a quota or admissionspreference system may be difficult to formalize in terms of the other. For example, ifthere are quotas for “music students of exceptional promise,” can one precisely specifywhen a student’s potential is “exceptional,” so he or she is eligible for the reserved seats,rather than merely “good”? As a practical matter, such comparisons might be easier toformalize on a sliding scale that can be accommodated using an admissions preferencescheme.

From a legal perspective, theorem 6 throws light on why it has proven so difficultfor the court to draw a line between constitutionally permissible and impermissible af-firmative action systems. As we discuss in more depth below, the Supreme Court ju-risprudence has clearly ruled that quotas violate the U.S. Constitution’s 14th amendment’sguarantee of equal protection because nonminority students cannot compete for the seatsreserved for minority applicants. However, the Court has given guidelines under whichadmissions preference schemes are acceptable, although the members of the court havedisagreed as to the extent that admissions preference schemes are functionally differentfrom quotas.

The cornerstone of supreme court jurisprudence regarding affirmative action is the1978 case University of California Regents v. Bakke [53]. Justice Powell’s opinion establishedthat the government has a compelling interest in encouraging diversity in university ad-missions founded on principles of academic freedom and a university’s right to take whatactions it feels necessary to provide a high quality education to its students. Given thiscompelling interest, universities are free to implement affirmative action programs, al-though these programs must be narrowly tailored and are subject to a rigorous “strictscrutiny” standard of review.

Legal challenges to affirmative action schemes (e.g., Gratz et al. v Bollinger [32], Grutterv. Bollinger [35]) turn on whether the affirmative action schemes are narrowly tailored,a term that refers to whether the affirmative action scheme places the smallest possibleburden on disadvantaged groups. In an amicus curiae brief to the case of University ofCalifornia Regents v. Bakke, Harvard College describes their admissions in the followingterms:

"In Harvard College admissions, the Committee has not set target quo-tas for the number of blacks, or of musicians, football players, physicists orCalifornians to be admitted in a given year. . . . But that awareness [ofthe necessity of including more than a token number of black students] doesnot mean that the Committee sets a minimum number of blacks or of peoplefrom west of the Mississippi who are to be admitted. It means only that, inchoosing among thousands of applicants who are not only ’admissible’ aca-

27

demically but have other strong qualities, the Committee, with a number ofcriteria in mind, pays some attention to distribution among many types andcategories of students." (U. of California Regents v. Bakke [53], 438 U.S. 317)

Justice Powell held up the Harvard admissions program as a canonical example ofa narrowly tailored, and hence constitutional, affirmative action system. The medicalschool at the University of California - Davis, the respondent in University of CaliforniaRegents v. Bakke, used an admissions quota scheme, which Justice Powell ruled was un-constitutional.

Justice Powell’s decision in University of California Regents v. Bakke said little directlyabout acceptable or unacceptable outcomes of an affirmative scheme, but the opinionprovides suggestions about constitutionally acceptable procedures for implementing anaffirmative action system. The keys to a constitutionally approved affirmative schemeare:

1. All applicants are in competition for all seats.20

2. To the extent that “odious” distinctions between applications, such as race, are afactor in admissions, the consideration of the applicants must be individualized.21

The first consideration rules out explicit quotas, and most universities in the UnitedStates adapted their admissions policies accordingly in the post-Bakke era. However,Justice Powell’s opinion acknowledges that even admissions preference schemes that treatapplicants as individuals can serve as “... a cover for a functionally equivalent quotasystem” (U. of California Regents v. Bakke [53], 438 U.S. 219).

The 2003 cases Gratz et al. v. Bollinger et al. and Grutter v. Bollinger et al. were thefirst affirmative action cases addressed by the Supreme Court following the ruling in

20As Justice Powell wrote:

“... race or ethnic background may be deemed a "plus" in a particular applicant’s file, yetit does not insulate the individual from comparison with all other candidates for the availableseats.” (Page 438 U.S. 318)

21Returning to Justice Powell’s opinion:

“This kind of program [referring to the Harvard College admissions system] treats each ap-plicant as an individual in the admissions process. The applicant who loses out on the lastavailable seat to another candidate receiving a "plus" on the basis of ethnic background will nothave been foreclosed from all consideration for that seat simply because he was not the rightcolor or had the wrong surname. It would mean only that his combined qualifications, whichmay have included similar nonobjective factors, did not outweigh those of the other applicant.His qualifications would have been weighed fairly and competitively, and he would have no ba-sis to complain of unequal treatment under the Fourteenth Amendment.” (Page 438 U.S. 319 ofBakke)

28

University of California Regents v. Bakke. These cases turned on whether the University ofMichigan admissions preference schemes are narrowly tailored. Interestingly, althoughPowell’s 1978 opinion appeared to say little about acceptable or unacceptable outcomesof affirmative action, the justices in both of these cases looked to the outcomes to judgethe extent to which the systems function as de facto quotas.

Gratz et al. v. Bollinger et al. addressed whether the admissions preference schemeused by the University of Michigan College of Literature, Science, and the Arts (LSA)met the narrow-tailoring criteria. The admissions preference scheme used by the LSAattributed points to applicants based on (for example) academic performance, athleticability, Michigan residency, and race. Applicants that received more than 100 of themaximum possible 150 points received immediate admissions, while those with fewerpoints were accepted later in the admissions process or not at all. Minorities receivedan additional 20 points, while applicants with significant leadership or public serviceachievements received at most 5 points.

The court ruled the LSA admissions preference scheme unconstitutional for two rea-sons. First, the across-the-board attribution of 20 points based solely on minority statuswas not individualized enough to qualify as narrowly tailored.22 Second, “...virtuallyall [minority freshman applicants] who are minimally qualified are admitted...” (Gratz v.Bollinger et al. [32], 539 U.S. 278), which means that these students are de facto not com-peting with nonminority applicants for admission. To summarize, the concurring justicesargue in their opinion that although LSA did not formally use a quota, the results werefunctionally the same.

Grutter v. Bollinger et al. revolved around the admissions process of the Universityof Michigan Law School (Law School), which the Supreme Court ruled constitutional.The key difference between the LSA’s policy and the Law School’s is that each applicantto the law school is given individualized review without points attributed to particulartraits of the applicant. However, if one reads the dissenting opinions, Justices Scaliaand Rehnquist made separate arguments that the Law School admissions process wasfunctionally equivalent to a quota:

“I join the opinion of The Chief Justice. As he demonstrates, the Univer-sity of Michigan Law School’s mystical “critical mass” justification for its dis-crimination by race challenges even the most gullible mind. The admissionsstatistics show it to be a sham to cover a scheme of racially proportionateadmissions.” (Scalia, p. 346 - 347 of Grutter)

22In her dissent, Justice Ginsburg argued that LSA was simply articulating its policy clearly and unam-biguously.

“If honesty is the best policy, surely Michigan’s accurately described, fully disclosed College affirmativeaction program is preferable to achieving similar numbers through winks, nods, and disguises.” (p. 305)

29

“... the ostensibly flexible nature of the Law School’s admissions programthat the Court finds appealing... appears to be, in practice, a carefully man-aged program designed to ensure proportionate representation of applicantsfrom selected minority groups.” (Justice Rehnquist in Grutter v. Bollinger [35],539 U.S. 385)

In the end theorem 6 implies that attempts to differentiate between unconstitutionalquotas and constitutional admissions preferences on the grounds of the outcomes pro-duced will likely prove futile. For supporters of affirmative action, the equivalence ofsophisticated quotas and affirmative action schemes might suggest that all of these sys-tems ought to be constitutional, and universities should be free to use which ever systemhelps them better achieve their diversity goals. For opponents of affirmative action, thismay provide an argument that no admissions scheme that takes race into account ought tobe able to pass constitutional muster. At a minimum these ruling seem self-contradictoryin light of our equivalence result.23

7 The Welfare Cost of Competition

We now use our model to address the welfare cost of the competition for college seats.We compare four economies to tease apart the relative importance of the assortativitythat competition allows relative to the welfare losses caused by the wastefully high levelof human capital the students accrue in the process of competing. The total welfarelosses from competition amount to roughly $1,153 per student per year, yielding a netpresent value aggregated over 1996’s newly enrolled college class of over $15.7 billionfor the newly entered class of students.24 One might have hoped that the benefits of anassortative match, which can only be accomplished in our incomplete information modelvia competition, might counterbalance the losses of competition. However, our metricsalso suggest that the wasteful competition eliminates all of the benefits of assortativematching. Finally, we argue that the second-best college assignment contest is able torestore only 26% of the welfare losses, which suggests the scope for policy interventionsto enhance average welfare are quite limited.

23We mention two recent cases for completeness. Fisher v. University of Texas at Austin [22] addressedissues regarding the application of the strict scrutiny standard without addressing the legality of particularaffirmative action schemes. Schuette, Attorney General of Michigan v. Coalition to Defend Affirmative Action,Integration and Immigration Rights and Fight For Equality By Any Means Necessary et al. [57] addressed the legalstandard under which an amendment to the Michigan state constitution banning affirmative action by stateinstitutions ought to be assessed.

24In the fall of 1996, the Integrated Postsecondary Education System Dataset dataset studied by Hickman[36] recorded 1,056,580 students newly enrolled in 4 year colleges. The net present value was computedusing a 5% discount rate.

30

First, we would like to formally define the source of the welfare loss due to competi-tion. The limiting payoff for agent type θ with HC s is Πcb(s; θ) = U

(Pcb(s), s, θ

)−C(s; θ).

Differentiating, we get the following first-order condition (henceforth, FOC):

∂U∂s︸︷︷︸

Productive Channel

+∂U

∂Pcb(s)· dPcb(s)

ds︸ ︷︷ ︸Competitive Channel

=dCds

The above expression concisely organizes the different aspects of the investment trade-offbeing made by students. It states that the marginal cost of human capital investment(the right-hand side) must be exactly offset by the marginal benefits (the left-hand side),which can in turn be decomposed into two parts. First, there is the direct value accruedto the student of having human capital level s, represented by the term ∂U/∂s; this is theproductive channel of investment incentives. There is also the indirect benefit of improvingthe college to which the student is assigned. An increase in one’s human capital level willimprove the quality of one’s match partner by dPcb(s)/ds, which increments utility by themargin ∂U/∂Pcb(s); the product of these two terms represents the competitive channel ofinvestment incentives.

For the moment, suppose types were observable to a benevolent planner that couldmatch students to colleges assortatively and allow them to invest ex post, in which casethe FOCs would be Us

(Pr∗

j (θ), s, θ)= Cs(s, θ). We refer to the outcome generated by the

social planner’s omniscient assignment and the ex post investments of the students asthe First-Best Outcome. For concreteness, the following theorem compares the color-blindpolicy and the first-best outcome, but similar results could be generated for any of ouradmission schemes. The basic intuition for the result is clear: by shutting down thecompetitive channel, the first-best outcome reduces the incentive for students to acquirehuman capital. The net effect is to reduce the human capital obtained for every type ofstudent.

Theorem 7. HC investment for all types in the color-blind admissions scheme exceeds that in thefirst-best outcome.

Given the incomplete information nature of the contest, one must consider two forceswhen evaluating the welfare effects of the competitive channel. First, color-blind com-petition between the students for better college seats through the use of human capitalaccumulation results in an assortative match, which is the necessary assignment in thefirst-best outcome. Second, the accumulation of human capital merely for the purposeof competing for a better college imposes negative externalities on the other students.

It is not obvious how to maximize surplus. Should the students compete in a color-

31

blind system that insures an assortative match? Or should society seek to minimize thenegative externalities by dampening the competitive channel even though we may endup in a nonassortative match? This is a question that we answer using a model cali-brated from real world data - it is unclear whether any parsimonious theoretical answeris possible.

The remaing results we present in this section are numerical solutions of our modelconducted using the estimated utility functions and type distributions from Hickman[36] to calibrate our model.25 Hickman [36] uses the US News and World Report collegequality index as his metric for college quality, p. A student’s human capital level, s, isrepresented by the student’s SAT score. For the utility functions we used

U(p, s, θ) = ρ(p, s) ∗ u(p)

ρ(p, s) = −0.176 + 0.000774s− 0.00000049s2 + 0.00076ps

u(p) = 40134p0.536

where ρ represents the probability of graduating from college and u is the college pre-mium conditional on graduating. The cost function for the model is

C(s, θ) = θe0.013(s−s), s = 520

where a student’s θ is inferred from his or her behavior under the status quo collegeadmissions contest. Note that under this calibration we can interpret the average utilityof the students as the ex ante expected college premium per student per year.

The analysis of Hickman [36] is conducted using data from 1995-1996 applicationyear. To the extent that increasing concerns about overinvestment in HC are valid, thenone ought to consider the estimates of the welfare cost of competition presented belowto be a lower bound on the present cost. On the other hand, unless college quality haschanged radically in the past two decades, it is likely that our assessment of the benefitsof assortativity are relatively stable.

Our comparative statics are based on the assumption that college quality is exogenousto the HC of the students that are enrolled, which ignores spillovers between students.Our estimate of the welfare costs of competition is unlikely to be affected by allowingspillovers to influence college quality since our computations do not require reallocatingmany students. While the computation of the welfare of the second-best assignment does

25As pointed out by Fryer and Loury [26], many of the theoretical results on the consequences of affirmativeactions systems are contingent on assumptions about the underlying economic primitives. Like the previousliterature, our results are specific to the functional forms and values we have assumed. Our hope is that thecalibration exercise has put the results in close contact with the real world.

32

require significant reallocation of students, we argue that including these spillovers wouldif anything reduce the welfare of the second best, which in turn aggravates our conclu-sion that policy interventions to mitigate the welfare losses of competition are largelyineffective. However, our estimate of the benefits of assortativity remain sensitive tothese spillovers. We address these issues in more depth in section 7.4 and appendix B.

7.1 The Cost of Competition

To estimate the welfare cost of competition we compare the first-best assignment outcomewith the welfare generated by a color-blind contest. Note that both of these assignmentsare perfectly assortative. In the first-best assignment, the student types are known andthey are assigned to colleges prior to choosing their HC level. In the color-blind match,the students’ HC level is the tool used to rank and assign the students. The only dif-ference in the outcome is that the students in the color-blind contest are pushed via thecompetitive channel to accrue more human capital.

The first-best assignment generates $15,415 of social surplus for the students, whereasthe color-blind mechanism generates only $14,262. Taking the difference in these valuesas the cost of competition, we find that the competitive channel results in the loss of$1,153 of welfare per student per year. For a somewhat more granular perspective, wecompare the minority and nonminority student outcomes in the table below.

Social SurplusNonminority

Social SurplusMinority

Social SurplusAverage

First Best $16,584 $10,115 $15,415

Color-blind $15,329 $9,284 $14,262

Cost ofCompetition

-$1,225 -$831 -$1,153

Below we have plotted the welfare costs of competition by student type.Figure 1 reveals that the costs are highest for students in the top half of the distribution

(i.e., those students with θ ≤ 7.56), which makes sense given that a student admitted to acollege of quality p only imposes externalities on students that are eventually admitted toschools with qualities higher than p. Interestingly, the welfare loss begins to decrease forthe very lowest cost students since these students are predisposed to accrue high amountsof HC and the number of high quality college seats is relatively abundant relative to thepopulation of low cost students. The students in the middle, who are in relatively highsupply relative to the supply of middling quality colleges, have the strongest incentiveto compete since small increases in HC would allow them to leapfrog a large mass of

33

Theta10-3 10-2 10-1 100 101 102 103

Co

st o

f C

om

pet

itio

n(T

ho

usa

nd

s o

f $)

0

0.5

1

1.5

2

2.5

Figure 1

competitors and attain a much better seat.Aggregating across the population of 1,056,580 students newly enrolled in 4 year col-

leges in the fall of 1996, the aggregate value of the losses per year to the enrolled studentsis $1.22 billion. If we use a conservative (for a safe asset) discount rate 5% and assume a30 year career, we get a total net present value of over $15.7 billion for the newly enteredclass of students. To the extent that anecdotes about the increasing competition amongststudents for college seats, such as Battle Hymn of the Tiger Mother, are accurate, then thislarge welfare loss ought to be considered a lower bound of the welfare costs facing currentcollege applicants.

7.2 The Benefits of Assortativity

Our second task is to compute the welfare gains from assortativity, which we do by com-paring the first-best assignment with a random-assignment. By a random-assignment,we mean that the students are first randomly assigned to colleges and then are allowedto choose their human capital level. Specifically, the random assignment provides eachstudent, regardless of his or her type, a seat at college p with a probability equal tofP (p). The competitive channel has been shutdown in both the first-best and randomassignments - the difference in outcomes is generated by the fact that the first-best is anassortative assignment, whereas the random assignment yields none of the benefits of

34

assortativity.As noted above, the first-best assignment generates $15,415 of social surplus for the

students. The random assignment generates a social surplus of $14,593. Taking thedifference of these quantities to represent the gains from assortativity, we find a welfaregain of an assortative match equal to $822 per year. Comparing this with the $1,153welfare loss from competition, we see that the competitive channel more than wipesout the gains from assortativity. In other words, social surplus would be improvedif we could eliminate competition by randomly assigning students to colleges before HCdecisions were made! Total yearly value of the benefit of assortativity are $868 million peryear for a total net present value of $13.6 billion for the newly entered class of students.

If we break down the outcome by demographic group we find

Social SurplusNonminority

Social SurplusMinority

Social SurplusAverage

First Best $16,584 $10,115 $15,415

Random $14,886 $13,226 $14,593

Benefits ofAssortativity

$1,698 -$3,331 $822

We see that minority students suffer a significant loss when moving from a random toan assortative match, but this loss is overwhelmed by the gains made by the more nu-merous nonminority students. The fact that minority students do better under a randomassignment may be initially surprising, but this outcome is driven by the fact that minor-ity students have significantly higher costs to accrue human capital and are assigned adisproportionate fraction of seats at lower quality schools under an assortative outcome.

7.3 Computing the Second Best

Our third and final comparative static is to compute the second-best college assignmentcontest to discover what fraction of the welfare gains of the first-best can be recoveredby either admissions policy changes or a clever intervention on the part of the govern-ment. We take as given that colleges all wish to enroll the highest human capital studentspossible. Any move outside of a contest model would require that colleges either aban-don these goals or coordinate on some alternative mechanism, both of which we view asimplausible in the near-term. Similarly, contests with stochastic outcomes (i.e., whereincolleges deliberately randomize their admissions decisions) would mean the randomizingcollege is choosing not to admit the best applicants, which strikes us as implausible given

35

our underlying beliefs about the motivations of colleges.26

Let us now describe the optimal control problem we solve to derive the second-bestcontest. The control, u(p) ∈ [0, 1], represents the fraction of seats at school p that areallocated to minority students. Because the index variable is the college, p, all of thevariables in our problem must be written as functions of p. The state variables of ourcontrol problem are the equilibrium strategies of the students, σM(p) and σN (p), andthe type of the student from each group that is assigned a seat at college p, θM(p) andθN (p). Our optimal control problem can be written

maxu

∫ p

pu ∗ [U(p, σM, θM)− C(σM, θM)] + (1− u) ∗ [U(p, σN , θN )− C(σN , θN )] fP (p)dp

such that·θM(p) = − u ∗ fP (p)

µ fN (θM(p))(16)

·θN (p) = − (1− u) ∗ fP (p)

(1− µ) fN (θN (p))(17)

·σM(p) =

Up(p, σM, θM)

Cs(σM, θM)−Us(p, σM, θM)(18)

·σN(p) =

Up(p, σN , θN )

Cs(σN , θN )−Us(p, σN , θN )(19)∫ p

pu(p) fP (p)dp = µ (20)

σM(p) = σN (p) = 520, θM(p) = θN (p) = 1517 (21)

The objective of the problem is simply a rewriting of the average social surplus using theindex variable of the control problem, p. Equations 16 - 19 denote the laws of motion forthe state variables, and equation 20 insures that enough seats are allocated to minoritiesthat the entire measure µ of minorities obtains a college seat. Equation 21 providesboundary conditions for our state variables that are derived from Hickman [36]. Weimpose the boundary condition σj(p) = s = 520 regardless of whether or not both groupsare assigned seats at the worst school, p. We are, in effect, assuming that both groups areassigned at least a vanishingly small fraction of a seat at every college.27

Since the objective function and the equations of motion are linear in u, we know im-

26In section 8 we consider situations that generalize our contest model. These extensions include anextension wherein colleges get noisy observations of the students’ human capital choice, but the randomnessthis noise generates has a very different interpretation than deliberate randomization by colleges.

27We do this primarily so that the optimal control problems are tractable. An alternative to our approachwould be to allow the boundary conditions for each group to be defined by indifference conditions.

36

mediately that the solution will have a bang-bang structure. In other words, the socialsurplus maximizing affirmative action scheme will generically involve complete segrega-tion - all of the seats in each school will be allocated to one of the two groups. However,it could be that the two groups are allocated essentially identical schools. For example, itcould be that each time a school p is allocated to nonminority students, minority studentsare allocated a school with a quality very close to p.

College Quality0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

% M

ino

rity

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Second BestColor-Blind

Figure 2

However, the second-best outcome assigns all of the low quality college seats to mi-nority students and reserves the best colleges for nonminority students, a result depictedin figure 2. Figure 2 includes the allocation under the color-blind scheme for compari-son. There are two stages to the logic underlying the second-best. First, the incentive tocompete is driven by the difference in the quality of the prizes. To reduce this incentive,we must make the difference between the best and worst prize available to each groupas small as possible, which can be done by breaking the prize space into two intervals.Second, we ought to assign the better prizes to the group that will reap larger complemen-tarities from being matched to a high quality school. Since membership in a nonminoritygroup is a signal that human capital accumulation costs are low, the second-best gives themajority of the seats at high quality colleges to nonminority students.

We compare the welfare from the first best, second best, color-blind, status quo andrandom assignment schemes in the following table. By status quo, we simply mean thesocial surplus estimated by Hickman [36] given the affirmative action policies in place in

37

the 1990s.

Social SurplusSocial Surplus Relative

to Status Quo

First Best $15,415 $1,266

Second Best $14,472 $323

Color-Blind $14,262 $113

RandomAssignment

$14,593 $444

Status-Quo $14,149 $0

The primary takeaway from these results is that most of the losses caused by thecompetition in the status quo contest, represented by the difference between the statusquo and the first best, cannot be recovered by a policy intervention. Of the $1,266 ofwelfare losses per student per year due to competition, only $323 (26%) can be recoveredin the second-best. Given the extreme unfairness of the second-best assignment, it cannotbe taken seriously as a policy recommendation - surely social planners care about manyconcerns other than average welfare (e.g., diversity). However, our result suggests thatsince market interventions cannot seriously address the significant losses resulting fromcompetition, policy-makers ought to focus their attention on other issues.

7.4 Caveats

At this point we would like to discuss our model’s assumption that school quality is fixedand exogenous. Although it is outside of the scope of this paper to provide a model thatendogenizes school quality, it is well known that there are spillovers between studentsin many contexts that make school quality a function of which students enroll. Thespillovers may be a function of student characteristics (e.g., Hoxby [40]) or student effortchoices (e.g., Fruewirth [24]). In addition, if we take at face value the briefs filed by theuniversities in Regent of the University of California v. Bakke [53], Gratz v. Bollinger [32],and Grutter v. Bollinger [35], universities believe that student welfare is directly enhancedby diversity amongst the student body.

Since the first best outcome as well as the color-blind and status quo contests all resultin nearly assortative matches without radical differences in the human capital choicesof the agents, these concerns are less pressing for these cases. However, the secondbest match and the random assignment counterfactuals both imply significantly nonas-sortative outcomes, and these welfare computations should be reviewed in light of theconcerns presented above. Without a well calibrated model of these spillovers, it is hard

38

to say much about the magnitude of these effects. At one extreme, if one believed thatassigning a few great students to a school would create a high quality school, then thebenefits of assortativity would be even lower. At the other extreme, if a few bad studentsdestroys a school’s quality, then assortativity is crucial for student welfare.

To address these concerns more concretely, appendix B repeats the analysis of thissection using an ad hoc model of endogenous school quality that assumes school qualityis generated by the median student quality. We believe this exercise helps to identifykey features of the problem. However, since the model of school quality is one of manypossible choices, we do not view the numbers provided in appendix B as necessarilyrealistic estimates of the cost of competition or the benefits of assortativity when schoolquality is endogenous.

8 Extensions

Our goal in this section is to discuss some of our modeling choices, potential alternativeassumptions, and to what extent the alternatives make a substantive difference in ourmodel. Our first extension is to allow the matching to be imperfectly assortative by in-cluding a noise term in the colleges’ observations of a student’s human capital choice,which slightly alters the ODEs that characterize the equilibria, but does not substantivelychange our results. Second we consider the effect of allowing the colleges to have multi-dimensional types or the students to make multidimensional investment decisions, whichwe find do not make a significant difference in our results.

Finally we consider whether we can incorporate heterogeneous preferences on eitherside of the market. The key to incorporating a diversity of preferences is to constructa well-defined notion of an assortative outcome that pertains to the new model. If thenotion of assortativity either is ill-defined or fails to hold, then modeling the market as acontest becomes implausible. When student preferences are allowed to be heterogeneous,we find that the notion of an assortative outcome fails to be well defined since the collegescannot be completely ordered. However, we can include these feature in our modelbe generalizing our contest mechanism to a serial dictatorship. When we allow forheterogeneity amongst the preferences of the colleges, assortativity continues to be well-defined and hold in equilibrium since students with lower θ endogenously choose higherhuman capital levels along all dimensions.

8.1 Noisy observations of HC

In the real world, the matching between colleges and students is not perfectly assortativeon SAT scores. Presumably this is because the colleges view other attributes of a student’s

39

application as informative about the applicant. One way to model this is to assume thatthe colleges observe a multidimensional investment decision on the part of the students,which we discuss in section 8.2. An alternative route is to assume that the investmentdecision of the student is randomly shocked, which yields a noisy signal of the student’sinvestment decision t = s + ε, where ε refers to the error in the observation. We assumethat ε is common across colleges and that students are matched to schools assortativelybased on t and p.

Given the random shock, the decision problem of the students in the limit gamebecomes

maxs

E [U (P(s + ε), s, θi) |s]− C(s; θi)

where the expectation is over the random shock. All of the assumptions (e.g., super-modularity, monotonicity) placed on the function U (P(s), s, θ) continue to hold for theexpectation E [U (P(s + ε), s, θ) |s]. It is straightforward to show that all of our resultsregarding the existence and uniqueness of equilibria and the approximation of the finiteequilibria with the limit game equilibrium continue to hold.

8.2 Multidimensional investment decisions and college types

In our main model we assume that student human capital is represented by a singlenumber and that colleges are characterized by a unidimensional quality index. Obvi-ously students choose to build many different kinds of human capital (e.g., performancein class, SAT scores, extracurricular activities), and colleges evaluate them along manydifferent dimensions. Similarly, colleges differ along many dimensions (e.g., quality ofdifferent degree programs, appeal of the location) that affect how students evaluate them.As it turns out, allowing for complex action and type spaces does not change our resultsso long as we retain the assumption of a common student preference over the collegetypes and a common college preference over the student investments.

Suppose the colleges have a multidimensional type p ∈ RN+ . If we assume the

(mild) conditions required for the students’ preferences to be represented by a utilityfunction, then we can use the student utility associated with a college as the college’sunidimensional “type” in our model and all of our results carry through immediately.

Now consider the colleges faced with evaluating a pool of applicants that have eachmade a multidimensional investment choice s ∈ RN

+ . In order for our results to hold, weneed to assume some structure on the human capital decision. Namely, we assume thatthe colleges’ preference ranking is strictly monotone increasing in s and that the decisionproblem of the agent

s(θ) ∈ arg maxs∈RN

+

U(P(s), s, θ)− C(s, θ)

40

is pairwise supermodular in (si, sj), i 6= j, and (si, θ) and differentiable in each componentof s. This insures that s(θ) is strictly increasing in θ, which in turn means that P(s(θ)) isstrictly increasing in θ.

With this structure in place, it is possible to characterize an equilibrium of the limitgame using a system of partial differential equations. Establishing the continuity ofthe utility functions in the limit game is straightforward, which allows us to prove thatany equilibrium of the limit game is an ε-approximate equilibrium of a sufficiently largefinite game. Under stricter conditions than assumed in theorem 2 (e.g., under a quotascheme wherein QM and QM have full support), we can prove that the limit equilibriumis unique28 and is a δ-approximate equilibrium of a sufficiently large finite game.

8.3 Diverse preferences amongst students

Once we allow colleges to be characterized by a multidimensional type p ∈ RN+ , it is

natural to consider a setting wherein the students have diverse preferences over p. Forexample, it may be that colleges are characterized by the quality of the education pro-vided and their tuition fees, and students have diverse preferences over these attributeseither because of differences in family wealth or because some students pay a reducedtuition (e.g., in-state v. out-of-state tuition rates at many public universities). Because thecontestants differ in their rankings of the prizes that can be won in the college admissionscontest, this setting does not neatly fit into the contest model we have studied above.

We can incorporate these features into our analysis by recasting our model as a serialdictatorship. In this recasting, the students sequentially choose their preferred school inorder from highest to lowest human capital choice. The outcome remains assortative inthe sense that students with higher human capital investments are allowed to choose theirpreferred school earlier in the queue. However, without some regularity assumptions onthe preferences of the students, it is difficult to apply any of our limit game approximationresults since significant discontinuities may be present.

Let us assume that a student’s preference are described by a multidimensional variablet ∈ Rn with the utility of the student as follows

U(p, s, θ, t)− C(s, θ)

Assume U is continuous in t and increasing in each dimension of p. We can still use afunction P(s) to describe the colleges available to a student with human capital level s,

28When the investment decision is multidimensional, our method for proving the uniqueness of the dis-continuities in the equilibrium strategy described by the differential equations no longer works. However,standard results on the uniquness of solutions of partial differential equations apply if there is no suchdiscontinuity.

41

but this will be a set valued function. First note that in this setting, the colleges availableto a student with human capital level s depends not only on how many students havehigher levels of human capital but on the preferences of those individuals. Second, inany equilibrium we can describe the possible colleges available to a student with humancapital level s as an n− 1 dimension surface of p ∈ RN

+ . This second insight also impliesthat we can describe the assortative match of a quota system without explicitly computingthe strategies, so this model retains some of the tractability of the unidimensional modelstudied above.

Using our redefined function P(s), we try to write-down an ODE describing the equi-librium human capital investments. However, without some structure on how t varieswith θ, we cannot insure that our ODE has a solution (much less a unique one). Forexample, if t is a deterministic and almost everywhere continuous function of θ, then theODE will have a solution. Since the utility each type of student receives varies contin-uously with s in the limit game, one can show that any equilibrium of the limit gameis an ε-approximate equilibrium of sufficiently large finite games. If we have a uniquelimit equilibrium, then the equilibrium of the limit game is a δ-approximate equilibriumof sufficiently large finite games. In other words, our approximation results continue tohold.

8.4 Diverse preferences amongst colleges

Finally, consider the case where students make multidimensional investment decisions∈ Rn and colleges have heterogeneous preferences over these decisions. We assumethat all colleges have monotone increasing preferences over s. We maintain the benchmarkassumption that student preferences over colleges are characterized by a unidimensionalcollege quality type. If we make regularity assumptions as per those of section 8.2, thenwe obtain an assortative match in the sense that s(θ) is increasing in each dimension,which means that all of the colleges agree on the endogenous ranking of the studentssince s(θ) : θ ∈ [θ, θ] is totally ordered in equilibrium. In other words, better students(low θ) make larger human capital investments and are matched to better colleges. Oncewe have this insight regarding the complete ordering of the equilibrium investments andthe assortativity of the match, the discussion of section 8.2 pertains and we see that mostof our results carry though to this setting.

This extension is significantly easier to handle than the case where colleges have mul-tidimensional types and students have diverse preferences (section 8.3). Section 8.3 al-lowed for models wherein the college types were only partially ordered, which allowedfor student preferences to resolve this partial order in different ways. In this section thedifferent possible investment decisions are only partially ordered, but the endogenous

42

decisions are completely ordered. If the equilibrium investment decision were only par-tially ordered, we would have to deviate from the contest structure to a serial dictatorshipor some other description. The failure of the equilibrium investment decisions to be fullyordered might result if (for example) the utility of the agents was not supermodular insi, sj for some pair i 6= j, which would imply s(θ) might not be monotone in all dimen-sions. Any such violation of monotonicity could cause the set of investment decisionsobserved in equilibrium to be partially ordered, which would require an analysis in thespirit of section 8.3.

9 Conclusion

The purpose of this paper has been to introduce a new model of college admissionsand use it to enrich the debate around the differences between quota and admissionspreference systems as well as throwing light on the welfare costs of the competition foradmission, which anecdotal evidence suggests has ratcheted up in intensity in recentyears. Modeling college admissions and affirmative action programs is challenging sinceone must consider the human capital investment decisions of students, heterogeneity inunderlying quality on both sides of the market, and the strategic decisions of universitiesgiven the information they are presented. Prior papers in the literature gain tractabilityby simplifying various components of the problem (e.g., assuming all college are ho-mogenous or student quality is innate), we instead consider a market with a continuumof agents, and the continuum approximation greatly simplifies the analysis and allows usto produce a large number of novel results.

Our first application of the model is to study the difference between quota-based andadmissions preference-based affirmative action systems, and we find that in fact there isno difference in the equilibrium outcomes produced. It is not particularly surprisingthat one can achieve the same diversity levels under each scheme - for example, onemight imagine duplicating the effect of a quota in an admissions preference system bygiving minority applicants a small boost when evaluating their applications. However,we believe it is surprising that the equivalence holds over both the diversity at individualcolleges and the human capital accumulation decisions of those admitted.

Moreover, our equivalence formalizes comments made in the jurisprudence regardingthe difference between quotas and affirmative action schemes. The legality of affirmativeaction turns on what was perceived by Justice Powell as a sharp difference between quotasand well designed admissions preference systems. Later opinions about the legality ofdifferent admissions preference schemes hinged on how closely the respective justicethought the admissions preference scheme mimicked a quota. Our analysis suggests

43

that drawing a sharp line between quotas and admissions preference schemes based onoutcomes will be futile. Not being legal scholars, we have little to say about the eventuallegal ramifications for universities that employ color-sighted affirmative action programs.

Our last contribution is to analyze the welfare effects of competition for college ad-mission using a calibration of our model drawn from Hickman [36] that used admissionsdata from 1996. In an incomplete information world, competition is necessary to achievean assortative match between students and colleges. Unfortunately, our analysis suggeststhat competition wipes out most (and possibly more than all) of the benefits of assortativ-ity. We are able to compute the welfare in the second-best contest, and we find that wecan only recover roughly 20% of the welfare lost due to competition. This suggests thatpolicy or market interventions will have little effect on blunting the disutility of competi-tion unless a radical departure from the current contest-like admissions process becomespossible.

Three other interesting directions for further research exist. First, if we could in-corporate a model of how student assignment endogenously influences college quality,we could discuss with more confidence the general equilibrium implications of massivechanges in the assignment of students to schools. Unfortunately, we are unaware of anystructural models we could use as the basis for this research agenda.

Second, the current model focuses on student behavior, conditional on participationin the college market, but there is another interesting group of individuals to consider aswell: those whose college/work-force decisions may be affected by a given policy. Thisquestion could be addressed by formalizing the “supply-side” comprised of potential col-leges and firms who may enter the market and supply post-secondary education servicesor unskilled jobs. Such a model might illustrate how the marginal agent (i.e., the indi-vidual indifferent between attending college and entering the workforce) is affected by agiven college admission policy. This would help to characterize the effect of AA on thetotal mass of minorities enrolling in college.

Finally, the eventual goal for this line of research should be to answer the question ofhow AA helps or hinders the objective of erasing the residual effects of institutionalizedracism. This will require a dynamic model in which the policy-maker is not only con-cerned with short-term outcomes for students whose types are fixed, but also with thelong-run evolution of the type distributions. Empirical evidence suggests that academiccompetitiveness is determined by factors such as affluence, as well as parental educa-tion. If AA affects performance and outcomes for current minority students, then thenext question is what effect it might that have on their children’s competitiveness whenthe next generation enters high school? If a given policy produces the effect of betterminority enrollment and higher achievement in the short-run, then one might conjecture

44

that a positive long-run effect will be produced. However, given the mixed picture onthe various policies considered in this paper, it seems evident that a long-run model isneeded in order to give meaningful direction to forward-looking policy-makers. It is ourhope that the theory developed here will serve as a basis for answering these importantquestions in the future.

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A Proofs

A.1 Proving Theorem 5

Since the equilibrium strategies are strictly decreasing (proposition 1), we know immedi-ately that the equilibrium strategy must be almost everywhere differentiable. We nowprove that there is a lower bound on the derivative of the equilibrium strategy, whichimplies that the distribution of human capital in any equilibrium must be nonatomic.Moreover, it implies if we look at sequences of equilibrium strategies, the resulting limitstrategy generates a nonatomic distribution of human capital.

Lemma 1. There exists ω < 0 such that for any equilibrium strategy, σ(θ), of either a finite gameor the limit game, we have d

dθ σ(θ) < ω at points where the strategy is differentiable.

49

Proof. We prove our lemma for the color-blind game, but the proof extends directly to theadmissions games with quotas by treating each group separately. Finally, the proof tech-nique easily extends to the score function game, although the notation becomes cumber-some since one must accommodate both groups of students and account for the variationin S.

Suppose there is no such upper bound on the derivative, which means that for anyω < 0 there exists θ such that σ′(θ) > ω. From the a.e. differentiability of σ, there existsan interval [θL, θU ] such that σ′(θ) > ω for all θ ∈ [θL, θU ] where σ′(θ) exists. Withoutloss of generality, we assume σ’(θ) exists at θL. Let sL = σ(θL) and sU = σ(θU), and notethat 0 < sL − sU < ω(θL − θU). Since σ must be decreasing, we have

Prσ(θ) ∈ [sU , sL] = F(θU)− F(θL)

Rearranging this we find

Prσ(θ) ∈ [sU , sL]sL − sU

=F(θU)− F(θL)

σ(θU)− σ(θL)>−1ω

F(θU)− F(θL)

θU − θL

Let ηθ = infθ

f (θ) > 0. Taking limits we find

LimθU→θL

Prσ(θ) ∈ [σ(θU), σ(θL)]σ(θU)− σ(θL)

>−1ω

LimθU→θL

F(θU)− F(θL)

θU − θL=−1ω

f (θL) >−1ω

ηθ > 0

This means that in intervals where σ’(θ) is close to 0, the “density” of individualsmaking the associated human capital choices is arbitrarily large. We call such a point ofhigh density a pseudo-atom.

Let δp = supp

fP (p) < ∞. Increasing the human capital choice from sL to sU yields a

minimal benefit of increasing the rank of one’s school in the limit game by

− 1ω

ηθ

δp(sL − sU)

If we let ηU = mins,p,θ

Up(p, s, θ) > 0, then the utility benefit must be at least

− 1ω

ηθ

δpηU(sL − sU)

Let the maximum marginal cost of human capital that we can observe in any equilibriumbe denoted

δC = maxs∈S ,θ∈Θ

Cs(s, θ) < ∞

50

This means the cost of deviating from sU to sL is bounded from above by

(sL − sU)δC

For such a deviation to be suboptimal, we must have

(sL − sU)δC ≥ −ηUω

ηθ

δp(sL − sU)

which requiresσ′(θ) ≤ ω ≤ −

ηθηUδpδC

In the game with K students, the formation of pseudoatoms is probabilistic. Consideran agent with type θU who in equilibrium chooses sU = σK(θ). Suppose such a studentconsiders increasing her human capital choice to sL. For each student she leap frogs,her school placement improves by at least (δp)−1 1

K , which generates a utility benefit of atleast

ηUδp

1K

For each student, there is a probability of at least − ηθω (sL− sU) of observing a human cap-

ital choice in [sU , sL], which yields a lower bound on the expected benefit of the deviationequal to

ηUδp

1K

E [i]

where i is distributed binomially with K draws using a parameter equal to − ηθω (sL − sU).

Given the distribution of i, we can write

ηUδp

1K

E [i] =ηUδp

1K

[−ηθ

ω(sL − sU)K

]= −

ηUδp

ηθ

ω(sL − sU)

The remainder of the argument proceeds as above.

Before proving our approximation results, we provide a few background results fromthe theory of the weak convergence of empirical processes.

Theorem 8. Consider a random variable X : Ω → Rd, d < ∞, with measure π0 and associatedCDF F(y) =

∫Ω 1x ≤ y ∗ π0(dx). For N i.i.d. realizations, X1, ..., XN, drawn from π0,

denote the N realization empirical CDF as FN(y). Then we have

supy∈Rd‖FN(y)− F(y)‖ → 0 almost surely as N → ∞

51

Proof. Follows from van der Vaart et al. [?], p. 135 and noting that sets of the formx : x ≤ y for y ∈ Rd are lower contours and form a VC Class.

The topology over the space of measures generated by the sup-norm over the space ofCDFs is referred to as the Kolmogorov topology. It is straightforward to show that the Kol-mogorov topology and the weak-* topology29 are identical over any space of nonatomicmeasures.

Corollary 9. Define the empirical measure generated by the counting measure over X1, ..., XNas πN . Then πN → π0 almost surely in the weak-* topology over ∆(Rd)

Proof. From Billingsley (p. 18, [?]) we have that FN(y)→ F(y) at continuity points of F im-plies πN → π0 almost surely in the weak-* topology. Since we have uniform convergenceFN(y)→ F(y) for all y almost surely, we have πN → π0 in the weak-* topology.

Corollary 10. Consider a random variable X : Ω → Rd, d < ∞, and associated CDF F(y).Denote the N realization empirical CDF as FN(y). Then

Pr√

N supy∈Rd‖FN(y)− F(y)‖ > t ≤ C ∗ e−2t2

where the constant C > 0 depends only on the dimension d.

Proof. This result follows directly from Theorems 2.6.7 and 2.14.9 of van der Vaart andWellner [?].

With this background in hand, we can now proceed with the proof of our approxi-mation results. We prove our results in two steps. First we provide a weaker versionof theorem 5 that assumes that the measures defining the quota schemes have strictlypositive PDFs. This assumption allows us to prove continuity results that imply ourclaim. We then prove our original result by arguing that even when we allow QM andQN to have disconnected supports, the equilibrium conditions of our model insure thatequilibria of the limit game remain equilibria of the finite game with sufficiently manyplayers.

Theorem 11. Let σji , i ∈ M,N and j ∈ cb, q, ap denote an equilibrium of the limit game.

Assume one of the following cases holds:

1. The limit game of the admissions preference model with a differential markup function S thatsatisfies S(s) = s and FP has full support.

29The sup-norm and the Prokhorov-Levy metric that metricizes the weak-* topology are identical whenone of the measures is nonatomic.

52

2. The limit game of the quota system with any feasible choice of QM and QN that admitstrictly positive PDFs over a connected support.

Under assumptions 1 -9, assumption 11, and given ε, δ > 0, there exists K∗ ∈ N such thatfor any K ≥ K∗ we have that σ

ji is a ε-approximate equilibrium of the K-agent game and σ

ji is a

δ-approximate equilibrium for the K-agent game.

Proof. We prove our theorem through a series of lemmas steps. These lemmas are pri-marily necessary for the application of the theorems in Bodoh-Creed [6].

Since our proofs rely on results on the convergence of empirical processes to their truedistributions, we will need to define the spaces in which these measures live. The truedistributions of the student types are FM(θ) or FN (θ). We have assume these distributionshave full support, which means that their respective PDFs are bounded from below. Thismeans that we can define a compact set ∆Θ of measures over Θ such that FM(θ), FN (θ) ∈∆Θ and all of the measures in ∆Θ have PDFs that are uniformly and strictly boundedabove 0. We endow the space ∆Θ with the Kolmogorov topology.

First we establish some initial properties of the objects we are working with. Lemma1 establishes that the equilibrium strategies have an upper bound on their derivative,which implies that pseudo-atoms cannot exist. Let ΣR be the set of strategies that adhereto the bound prescribed by lemma 1, and we endow this space with the sup-norm. Let∆R(S) denote the space of pushforward measures generated by a strategy σ ∈ ΣR and adistribution of student types, Fi(θ) ∈ ∆Θ. The set ∆R(S) is a tight family of measures andis compact as a result (Theorem 15.22 of Aliprantis and Border [3]). We endow the space∆R(S) with the weak-* topology. Since the measures in ∆R(S) are all nonatomic, theweak-* topology is equivalent to the Kolmogorov topology generated by the sup-normapplies to the space of CDFs.

Let ∆K(X) denote the set of empirical measures generated K draws from the set X.We will let GKN

N denote the CDF of the empirical measure of human capital choices forthe nonminority students when there are KN such students in the economy, and we usethe notation ∆KN (S) to refer to the set of such CDFs. GKM

M denotes the empirical measureof human capital choices for the minority students when there are KM such students inthe economy, and we use ∆KM(S) to refer to the set of such CDFs. We endow ∆KN (S)and ∆KM(S) with the weak-* topology. Note that since σ ∈ ΣR are strictly monotone, thenall of the elements of ∆KN (S) and ∆KM(S) must be nonatomic, which implies that theweak-* topology and the Kolmogorov topology are equivalent.

We now prove two useful lemmas about inverse functions that, for some reason, wecannot find in the existing literature.

53

Lemma 2. Let Gi, i =M,N be the CDF of the pushfoward measure generated by σi ∈ ΣR andthe type distribution Fi ∈ ∆Θ. Then Gi is uniformly continuous in (σi when the space of CDFsis endowed with the sup-norm.

Proof. Consider σi with the associated pushforward measure Gi. Then

Gi = Fi

(σ−1

i (s))

Since σi ∈ ΣR has a slope that is bounded from below, we know that σ−1i is uniformly

continuous in σi under the sup-norm.30 Since Fi is nonatomic, this implies that Gi isuniformly continuous in σi.

Lemma 3. Let K be any increasing function where for some γ > 0 and any t > t′ we haveK(t)− K(t′) ≥ γ(t− t′). Then K−1 (·) is uniformly continuous.

Proof. Consider k, k′ where K(t) = k and K(t′) = k′. Then we have

∥∥k− k′∥∥ > γ

∥∥t− t′∥∥∥∥t− t′

∥∥ <1γ

∥∥k− k′∥∥

which implies that K−1 (·) is uniformly continuous.

Our next proves that the limit assignment mapping must be continuous in both sand the distributions of agent actions, Gi, i = M,N . To this end, let Pr

i (·; GN , GM) ,i = M,Nand r ∈ cb, q, ap, denote the assignment mapping generated if the agentactions are distributed as per (GN , GM) .

Lemma 4. Pri (s; GN , GM), i = M,Nand r ∈ cb, q, ap, is uniformly continuous in (s, GN , GM).

Proof. We provide a proof for the admissions preference game since the other systems arespecial cases of an admissions preference scheme. In the admissions preference game,we have Pap

M(s) = F−1P

((1− µ)GN

(S(s)

)+ µGM (s)

). Let

q(s) = (1− µ)GN(

S(s))+ µGM (s)

q(s′) = (1− µ)GN(

S(s′))+ µGM

(s′)

30A quick proof of this claim starts by noting that for any θ

σi(θ − ε) < σi(θ)−ωε

σi(θ + ε) < σi(θ) + ωε

For any ε choose δ < ωε and choose σi such that ‖σi(θ)− σi(θ)‖ < δ, which implies σi(θ) ∈[σi(θ)− δ, σi(θ) + δ] ⊂ (σi(θ)−ωε, σi(θ) + ωε)which implies σ−1

i (θ) ∈ (θ − ε, θ + ε) as required.

54

GN and GM must be uniformly continuous with respect to s since the slopes of σN andσM are strictly bounded away from 0, and S is uniformly continuous from assumption 11.Therefore, for any γ > 0 we can choose δ > 0 sufficiently small that for any (s, GN , GM)

and(

s′; GN , GM)

that satisfy

∥∥∥GN − GN∥∥∥+ ∥∥∥GM − GM

∥∥∥+ ∥∥s− s′∥∥ < δ

we have

∥∥q(s′)− q(s)∥∥ <

∥∥∥q(s)−[(1− µ)GN

(S(s′)

)+ µGM

(s′)]∥∥∥+ δ

<∥∥∥q(s)−

[(1− µ)GN

(S(s)

)+ µGM (s)

]∥∥∥+ δ +γ

2= δ +

γ

2< γ

Since fP has a lower bound, then FP has a lower bound on it’s slope, so from lemma 3F−1P is uniformly continuous. Therefore for any ε > 0 we can choose δ > 0 sufficiently

small that for any (s, GN , GM) and(

s′; GN , GM)

that satisfy

∥∥∥GN − GN∥∥∥+ ∥∥∥GM − GM

∥∥∥+ ∥∥s− s′∥∥ < δ

we have ∥∥∥Pri (s; GN , GM)− Pr

i

(s′; GN , GM

)∥∥∥ < ε

which establishes our claim.

We use the notation Πrj (s, θ; GKN

N , GKMM , K) to refer to the expected utility in the K-agent

game of an agent of type θ in demographic group j =M,N that chooses human capitallevel s given GKN

N and GKMM and admissions system r = cb, q, ap. Let Πr

j (s, θ; GKNN , GKM

M )

refer to the utility received by an agent of type θ that chooses human capital level s givenGKNN and GKM

M in the limit game. Finally we use our lemmas to prove that the agentutility is continuous, which is the linchpin of our ε-approximate equilibrium result.

Lemma 5. Πrj (s, θ, GN , GM) is uniformly continuous in (s, θ, GN , GM) ∈ S × Θ × ∆(S) ×

∆(S).

Proof. The continuity result follows from lemma 4 and the continuity of U and C withrespect (p, s, θ). The uniform equicontinuity comes from the compactness of S × Θ ×∆(S)× ∆(S).

We now employ a bit of sleight of hand to prove the final result. In our model we

55

assume that all ties are broken randomly. For the next result we define an assignmentmapping Pr

i (s; GN , GM) in the limit game and Pri (si, s−i; K) that instead assign each of the

students in the tie the best school available in the random tie-break. We let the associatedpayoff functions be

Πrj (s, θ; GKN

N , GKMM , K) = E

[U(

Pri (si, s−i; K), s, θ

)− C(s, θ)

]Πr

j (s, θ; GKNN , GKM

M ) = E[U(

Pri (s; GN , GM, s, θ

)− C(s, θ)

]Note that in cases when GKN

N , GKMM are nearly nonatomic, these modified payoff functions

will be almost the same as Πrj (s, θ; GKN

N , GKMM , K) and Πr

j (s, θ; GKNN , GKM

M ). We refer to anygame that uses Pr

i (s; GN , GM) or Pri (si, s−i; K) as a modified game.

Lemma 6. For all ε > 0 there exists K∗ such that for all K > K∗ and all (s, θ; GKNN , GKM

M ) ∈S ×Θ× ∆KN (S)× ∆KM(S) we have∥∥∥Πr

j (s, θ; GKNN , GKM

M , K)− Πrj (s, θ; GKN

N , GKMM )

∥∥∥ < ε

Proof. The minority student assignment mapping in the limit game is

PapM(s; GN , GM) = F−1

P

((1− µ)GN

(S(s)

)+ µGM (s)

)It will be useful to define GN and GM to be the empirical CDFs of the other agents’actions in the K agent game, which allows us to write the assignment mapping of thefinite game as

PapM(s; GN , GM, K) = F−1

P

((1− µ)GN

(S(s)

)+ µ

KM − 1KM

GM (s) +1K

)As K → ∞ we have

(1− µ)GN(

S(s))+ µ

KM − 1KM

GM (s) +1K→ (1− µ)GN

(S(s)

)+ µGM (s)

uniformly over (s, θ; GKNN , GKM

M ). From lemma 3 and the lower bound on the slope of FP ,we know that F−1

P is uniformly continuous. Together these imply that for any γ > 0 wecan choose K sufficiently large that for all s∥∥∥Pap

M(s; GN , GM)− PapM(s; GN , GM, K)

∥∥∥ < γ

Since U is uniformly continuous in p, this immediately implies for for any ε > 0 we can

56

choose K sufficiently large that∥∥∥Πrj (s, θ; GKN

N , GKMM , K)− Πr

j (s, θ; GKNN , GKM

M )∥∥∥ < ε

Finally, we establish the link between the original and modified game.

Lemma 7.(σ

apM, σ

apN)

is an equilibrium of the modified K−agent (limit) game if and only if it isan equilibrium of the original K-agent (limit) game.

Proof. Since Πrj (s, θ; GKN

N , GKMM ) = Πr

j (s, θ; GKNN , GKM

M ), the result is immediate for the limitgame. The payoffs in the original and modified game only differ in the event that twoagents choose the same HC level. Since this is a measure 0 event, the payoffs are almostsurely the same for any choice in the modified and original K-agent game. Therefore,(σ

apM, σ

apN)

is an equilibrium of the modified K−agent game if and only if it is an equilib-rium of the original game.

We now establish our two approximation results in separate lemmas.

Lemma 8. If(σ

apM, σ

apN)

is an exact equilibrium of the limit game, then for any ε > 0 we canchoose K∗ such that for any K > K∗

(σ

apM, σ

apN)

is an ε−approximate equilibrium of the K-agentgame

Proof. Given our previous results on uniform convergence and uniform continuity ofthe modified game, theorem 8 of Bodoh-Creed [6] implies that if

(σ

apM, σ

apN)

is an exactequilibrium of the modified limit game, then for any ε > 0 we can choose K∗ such that forany K > K∗

(σ

apM, σ

apN)

is an ε−approximate equilibrium of the modified K-agent game.Combined with lemma 7 we have our result.

Proof. Our next result deals with sequences of equilibrium strategies. We use the notation(σ

apM(K), σ

apN (K)

)to denote an exact equilibrium of the limit game.

Lemma 9. Consider any sequence(

σapM(K), σ

apN (K)

)∞K=2 such that

(σ

apM(K), σ

apN (K)

)→(

σapM, σ

apN). Then

(σ

apM, σ

apN)

is an exact equilibrium of the limit game.31

31One might have thought theorem 6 of Bodoh-Creed [6] would provide this result. Unfortunately, theproof of that theorem required the strategy space to be compact, which is not be the case in our application.

57

Proof. Let (GKN , GK

M) refer to the pushforward measure generated by(σ

apM(K), σ

apN (K)

)and the student type distributions. From lemma 2 we have (GK

N , GKM) → (GN , GM) in

the sup-norm. Let(

GKN , GK

M

)denote an empirical measure defined by random draws

from (GKN , GK

M) in the K-agent game. From theorem 8 for any ρ, δ > 0 we can choose Ksufficiently large that with probability at least 1− ρ∥∥∥GK

N − GKN

∥∥∥+ ∥∥∥GKM − GK

M

∥∥∥ < δ

Suppose(σ

apM, σ

apN)

is not an equilibrium of the modified limit game. For concreteness,let us assume that minority students of some type θ where σ

apM(θ) = s∗ have a profitable

deviation. Formally that means there exists s′ and ε > 0 such that

ΠrM(s∗, θ; GN , GM) + ε < Πr

M(s′, θ; GN , GM)

We now translate this into a statement regarding payoffs in the K-agent game. Forsufficiently large K we have with probability at least 1− ρ

ΠrM

(s∗, θ; GK

N , GKM

)+ ε < Πr

M

(s′, θ; GK

N , GKM

)ΠrM

(σ

apM(K), θ; GK

N , GKM

)+

ε

2< Πr

M

(s′, θ; GK

N , GKM

)ΠrM

(σ

apM(K), θ; GK

N , GKM, K

)< Πr

M

(s′, θ; GK

N , GKM

)The first line follow from our continuity result (lemma 5). The second line followsfrom our continuity result (with respect to action) and the fact that σ

apM(K) → s∗ as

K → ∞. The third line follows from our uniform convergence result (lemma 6). Thethird line implies that σ

apM(K) is in the K-agent game with probability at least 1− ρ. Since

the utility function is bounded, this implies that for ρ sufficiently small s′ represents aprofitable deviation for an agent of type θ in the K−agent game. From this contradictionwe conclude that

(σ

apM, σ

apN)

is an exact equilibrium of the limit game

Since lemma 9 holds for all such sequences of strategies, we have from Theorem 17.16of Aliprantis and Border [3] that the equilibrium correspondence is upper hemicontinu-ous.

Our claim regarding δ-approximate equilibria could still fail if there were equilibria ofthe limit game that were not close to any equilibrium of arbitrarily large finite game. Inother words, the argument might fail if the equilibrium correspondence were not lowerhemicontinuous in K. Since there is a unique equilibrium of the limit game, it must bethe case that the sequence σK : Θ→ ∆(S)∞

K=1 converges in the sup-norm to the uniqueequilibrium of the limit game.

58

A.2 Remaining Proofs

The remaining results in the paper are produced roughly in the order they appear in themain body.

Theorem 1. In the college admissions game Γ(KM, FM, KN , FN ,PK, r, U, C,S) with r ∈ cb, q, ap,under assumptions 1–8 there exists a monotone pure-strategy (but potentially not group-wise sym-metric) equilibrium (σr

M(θ), σrN (θ)). Moreover, any equilibrium of the game must be strictly

monotone with almost every type using pure strategies.

Proof. Existence and weak monotonicity is a straightforward application of Athey [1, The-orem 3] who establishes these conditions in a general class of auction-related games towhich our model belongs. Strict monotonicity follows from lemma 1.

Now we prove that all of the equilibria must be increasing and that almost all typesmust use pure strategies. First recall that E [P(si, s−i)], although endogenous, must beincreasing in si. Let the maximizers of the problem faced by an agent of type θi be

C(θi) = arg maxs

E[U(Pj(s, s−i), s, θi)]− C(s, θi), j ∈ M,N

The super modularity of the objective function in (s,−θi) and theorem 1 of Edlin andShannon [19] implies that for θ > θ′, s ∈ C(θ), and s′ ∈ C(θ′) we have s > s′. Sincein equilibrium σj(θi) ∈ C(θi) we must have σj(θ) > σj(θ

′). Since the strategy is strictlyincreasing and constrained to lie within the compact set S , then for each σj , j ∈ M,N,

there can only be a countable set of points ΘD =

θ : Lim

t→θ−σj(t) < Lim

t→θ+σj(t)

. Since such

a discontinuity is required for a type to mix in equilibrium and the set ΘD is countable(and hence measure 0 with respect to FK(θ)), we have that almost all types must use apure strategy in any equilibrium.

Theorem 2. HC investment for all types in the color-blind admissions scheme exceeds that in thefirst-best outcome.

Proof. Let σcb be the equilibrium strategy under color-blind admissions and σFB be the expost investment strategy in the first-best benchmark. The boundary condition for bothproblems is the same:

σcb(θ) = σFB (θ) = s

Since the first order condition must hold at θ = θ in the color-blind scheme, we have

∂U(

P(s), s, θ)

∂p· dP(s)

ds+

∂U(

P(s), s, θ)

∂s=

dC(s; θ)

ds≥∂U(P(s), s, θ)

∂s

59

This can only hold if dP(s)ds = 0, which requires ∂σcb(θ)

∂θ → −∞ as θ → θ, or if ∂σFB(θ)∂θ = 0.

In either case, we know for an interval of θ in the neighborhood of θ that σcb(θ) ≥ σFB(θ)

where the inequality is strict for θ 6= θ in that neighborhood.Now assume that σcb(θ∗) = σFB (θ∗) for some θ < θ. In this case, we have, as per the

argument above, that either dP(s)ds = 0 or ∂σFB(θ)

∂θ = 0, which implies σcb(θ) > σFI(θ) for θ

sufficiently close to, but less than, θ∗. Therefore, it cannot be the case that σcb and σFB

ever cross - σcb is always weakly greater than σFB.

Theorem 3. In any equilibrium almost all of the agents take pure actions and the strategies areessentially group-symmetric.

Finally, an essentially unique Nash equilibrium exists in the limit model in the following cases:

1. The limit game of the admissions preference model with a differential markup function S thatsatisfied S(s) = s and FP has full support.

2. The limit game of the quota system with any feasible choice of QM and QM that admitstrictly positive PDFs over a connected support.

Proof. We begin our analysis with two lemmas that imply we can focus on the groupsymmetric equilibria in pure strategies defined by the ODEs.

Lemma 10. All but a measure 0 set of agents have a unique optimal action in any equilibrium.

Proof. First define the following equilibrium value function

V(s, θ) = U(Pji (s), s, θ)− C(s, θ), j = q, ap and i =M,N

From assumptions 5 and 6 we have that Vsθ(s, θ) > 0. From theorem 1 of Edlin and Shan-non [19] we have that if θ > θ, s∗ ∈ arg max

sV(s, θ), and s∗ ∈ arg max

sV(s, θ) then it must

be the case that s∗ > s. Since all agents within the same group have the same decisionproblem, this implies that the actions of the different agent types within a demographicmust be ordered in this way.

Note that if an agent is indifferent between two actions s, s′ where s > s′, then it mustbe that no other type of agent within that demographic group chooses an action in theinterval [s′, s]. Since at most a countable number of such “jumps” can occur within thebounded space S , it must be that only a countable number of types of agent in eachgroup has such an indifference. Finally, since the type distributions are nonatomic, thiscountable set must have measure 0.

Lemma 10 has two crucial implications. First, for a mixed action to be optimal, itmust be that the agent is indifferent between the pure actions over which he or she mixes.

60

Lemma 10 implies that only a measure 0 set of agents can mix in this fashion. Second,for two agents to use different strategies σ, σ′ in equilibrium, it must be that there exists atype θ where the agent is indifferent between σ (θ) and σ′(θ). Since this can happen onlyfor a measure 0 set of agents, then the strategies used by the agents in each demographicgroup must be the same almost everywhere.

First consider the admissions preference case and let σapi be defined by equation 9

and is therefore continuous. Standard results on differential equations imply that σapi is

uniquely defined. First note that equation 9 can by treated a direct mechanism whereinthe agent’s problem is to choose a declared type θ where i =M,N

maxθ

U(Papi (θ), σ

api (θ), θ)− C(σap

i (θ), θ)

If one computes the first order conditions with respect to θ and recognizes that in equi-librium Pap

i (θ) = Papi (σ

api (θ)), we find that the local incentive compatibility conditions

are already “built in” to the differential equations. It remains to prove that local in-centive compatibility implies global incentive compatibility. Suppose θ < θ and letPap

i (θ) = p > p = Papi (θ) and σ

api (θ) = s > s = σ

api (θ). We then have from local incentive

compatibility that

Up(p, s, θ)dPap

i (θ)

dθ

∣∣∣∣∣θ=θ

+ Us(p, s, θ)dσ

api (θ)

dθ

∣∣∣∣∣θ=θ

= Cs(s, θ)dσ

api (θ)

dθ

∣∣∣∣∣θ=θ

Since θ < θ, we have from assumption 5 that Up(p, s, θ) > Up(p, s, θ), and from assump-tion 6 that Us(p, s, θ) > Us(p, s, θ) and Cs(s, θ) < Cs(s, θ). The value of the first ordercondition that results if type θ deviated from truthfulness upwards by declaring θ = θ is

Up(p, s, θ)dPap

i (θ)

dθ

∣∣∣∣∣θ=θ

+ Us(p, s, θ)dσ

api (θ)

dθ

∣∣∣∣∣θ=θ

> Cs(s, θ)dσ

api (θ)

dθ

∣∣∣∣∣θ=θ

which, since there is an inequality, implies declaring θ = θ cannot be optimal if the truetype is θ. Similar arguments imply that deviating downward from truthfulness alsocannot be optimal.

As a final step we must rule out cases where it might be optimal for the agent tochoose a human capital level outside of the range of σ

api . Since σ

api is continuous, this

implies that the deviation satisfies s > s = σapi (θ). Let p = Pap

i (θ). We know from our

61

first order conditions that32

Up(p, s, θ)dPap

i (θ)

dθ

∣∣∣∣∣θ=θ

+ Us(p, s, θ)dσ

api (θ)

dθ

∣∣∣∣∣θ=θ

= Cs(s, θ)dσ

api (θ)

dθ

∣∣∣∣∣θ=θ

Increasing from s to s does not change the school p, and so it can only be optimal if

Us(p, s, θ) = Cs(s, θ) (22)

Since Uss < 0, Css > 0, and Us(p, s, θ) ≤ Cs(s, θ), equation 22 cannot hold, which meansdeviating to s cannot be optimal.

Now consider case 2 and assume that σqi is defined by equation 11 over the intervals

where Qi has support. Again, by standard results in differential equations, if [p, p′] isan interval where Qi has support and type θ is assigned to college p with human capitallevel σ

qi (θ) = s, then the strategy is uniquely defined by equation 11 for all types that are

(in equilibrium) assigned a seat at a college in [p, p′]. The only way there could exist twoequilibria σ

qi and σ

qi (θ) is if there is some discontinuity at θ shared by both strategies such

thatLimε→0+

σqi (θ + ε) 6= Lim

ε→0+σ

qi (θ + ε)

In other words, the jump over an interval in which Qi lacks support is not uniquelydefined. We prove that the first such jump in the strategy must be uniquely defined. An(omitted) induction step using essentially the same argument can be used to prove thatall of the jumps must be uniquely defined.

Suppose that Qi lacks support over the interval (p, p′) and Qi has full support overboth [p, p] and [p′, p′ + δ] for some δ > 0. Let θ satisfy Pq

i (σqi (θ)) = p - in other words, θ

is where the first jump must occur. In equilibrium it must be the case that θ is indifferentabout whether to make this jump, so

U(p, s, θ)− C(σqj (θ), θ) = U(p′, s′, θ)− C(s′, θ)

where s = σqi (θ) and s′ = Lim

ε→0+σ

qi (θ + ε). Suppose there was a second equilibrium σ

qi

starting at type θ such that s′′ = Limε→0+

σqi (θ + ε) > s′. Then it must be the case that

U(p, s, θ)− C(σqj (θ), θ) = U(p′, s′, θ)− C(s′, θ) = U(p′, s′′, θ)− C(s′′, θ)

But since Uss is strictly concave and Css is strictly convex, this cannot be true. Therefore

32We have written the first order condition with the derivatives defined using limits from the right (i.e.,using sequences contained in Θ).

62

s′ is uniquely defined.

Theorem 5. Consider some Pi(s) : S → P , i ∈ M,N . Pi(s) : S → P , i ∈ M,N is the resultof an equilibrium of some quota system if and only if there is an equilibrium of an admissionspreference system that also yields these assignment functions and admits the same equilibriumstrategies.

Proof. Suppose Pqi : S → P , i ∈ M,N , is the result of an equilibrium of some quota

system and denote the equilibrium strategies σqi : Θ → S . Since Pq

i and σqi are strictly

monotone, the functions are invertible. Let Pap(s), the assignment function under admis-sions preferences, be Pap(s) = Pq

N . Since the assignment functions are the same for thenonminority students, Pap and Pq

N generate identical decision problems for the nonmi-norities. Therefore, if σ

qN was an equilibrium for nonminority students under Pq

N , thenσ

apN (θ) = σ

qN (θ) will be an equilibrium for the nonminority students under Pap(s).

To construct the outcome equivalent score function, let

S(s) =(

PqN)−1

(PM(s))

A minority student who chooses human capital level s will then be assigned to college

Pap(S(s)) = PqN (S(s)) = Pq

N ((

PqN)−1

(PqM(s))) = Pq

M(s)

Since the assignment functions are the same for the minority students, Pap and PqM gen-

erate identical decision problems for the minorities. Therefore, if σqM was an equilibrium

for minority students under PqM, then σ

apM(θ) = σ

qM(θ) will be an equilibrium for the

nonminority students under Pap(s).Now suppose Pap : S → P with score function S is the result of an equilibrium of

some admissions preference system and denote the equilibrium strategies σapi : Θ → S ,

i ∈ M,N . To define the equivalent quota system, we need to define allocations of seatsto each group. Let these distributions be denoted Qi, i ∈ M,N , and define them as

For all p let QM(p) = 1− Fi

[ψ

api

(S−1

((Pap)−1 (p)

))]For all p let QN (p) = 1− Fi

[ψ

api

((Pap)−1 (p)

)]Note that the total measure of nonminority students choosing s and minority students

choosing S−1(s) under Pap (in equilibrium is)

1− µFM[ψ

apM

(S−1

((Pap)−1 (p)

))]− (1− µ)FN

[ψ

api

((Pap)−1 (p)

)]=

µQM(p) + (1− µ)QN (p) = FP (p)

63

which implies QM and QN are feasible quotas. Written more formally, PqN (s) = Pap(s)

and PqM(s) = Pap(S(s)). As argued above, since the decision problems for the agents are

the same, the equilibrium strategies in the original admissions preference scheme and theconstructed quota are the same.

B Endogenous School Quality

In this section we provide an analysis of a version of our model that includes a stylizedmodel of endogenous school quality, and our goal is to repeat the calculations of section7. The precise numbers we provide in this section are probably not comparable to real-world outcomes since the model of school quality we have chosen is ad hoc. However,the differences between the results presented in this section and those provided in section7 help illustrate important features of the analysis that may change when endogenousschool quality is taken into account.

The model we employ is identical to the calibrated model used in section 7, except thatinstead of assuming that the distribution of college qualities is exogenous and equal tothe distribution estimated by Hickman [36], we assume that the “effective” school qualityis a function of the types of agents assigned to the school.33 Suppose in equilibriumthat school p has a fraction u(p) of it’s student body drawn from the minority studentpopulation and that the type of minority and nonminority student assigned to school pare θM(p) and θN (p). The effective school quality, pE, is then equal to

pE = F−1P [µ (1− FK(θM(p)) + (1− µ) (1− FK(θN (p))] (23)

where FP refers to the estimate of the distribution of school qualities provided by Hick-man [36]. To decompose this, first note that

µ (1− FK(θM(p)) + (1− µ) (1− FK(θN (p))

represents the average cost-type distribution quantile of the students assigned to schoolp, which we take to represent the average ability of the students assigned to school p.We then assign school p the effective quality, pE, of the school of the same quantile indistribution FP .

Let’s take a few moments to discuss whether this is a reasonable model of school33Since agent HC level is a strictly monotone function of the agent’s type in equilibrium, we could also

treat this as a model of spillovers in agent HC. One might wonder whether the externalities imposed onother agents through the effect of HC choice on school quality introduce new strategic factors to consider.In the limit game these effects are not present. We conjecture that as long as the school quality is continuousfunction of agent HC, the effects will vanish as the economy grows.

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quality. If the assignment of students to schools is perfectly assortative, as is the case forthe first-best and color-blind assignments, then equation 23 predicts an effective schoolquality distribution equal to FP . Since the assignments of students to schools in thereal-world (as estimated by Hickman [36]) had only small deviations from assortativity,our model would predict almost the same distribution of school qualities as we observein reality. This suggests that our model is consistent with the real-world. Unfortunately,without observing how significant variations in the assignment influence school quality,we cannot say much about how well our model captures the effect of large deviationsfrom the status-quo assignment observed in the real-world.

Using our model of spill-overs we recompute the assignments used in section 7 tocompute the benefits of assortativity, the cost of competition, and the second-best out-come. Note that, as expected, the perfectly assortative first-best and color-blind assign-ments yield the same welfare since the distribution of effective school qualities is identicalto FP .

Social Surplus

First Best $15,415

Second Best $14,556

Color-Blind $14,262

Random Assignment $13,959

When we use our comparative statics to compute the benefits of assortativity and thecost of competition, we find

Social Surplus

Benefits of Assortativity $1,456

Cost of Competition -$1,153

The cost of competition is unchanged since the assignment and school quality in the first-best and color-blind assignment are the same in both the endogenous and exogenousschool quality models. However, the benefits to assortativity are 77% higher when schoolquality is endogenous. In summary, the benefits of assortativity are about 26% higherthan the costs of competition when school quality is endogenous, whereas in the caseof exogenous school quality we found that the cost of competition eliminated all of thebenefits of assortativity.

To see why the benefits of assortativity are higher when school quality is endoge-nous, let us decompose the source of the assortativity benefits into two pieces. First, forassortativity to matter there must be increasing returns to assigning high HC students(i.e., low cost students) to high quality schools. Second, there must be some variation

65

in school quality so that there are higher (lower) quality schools to which high (low) HCstudents can be assigned. When school quality is exogenous (as in section 7), we onlylose the first component of the benefit - some low cost students still get assigned to highquality schools. However, when school quality is endogenous, we lose all of the benefitsof assortativity - since the colleges all have the same median quality (pMED = 0.5630), wereap no complementarities between student type/HC level and school quality.

C Smoothing the Model

In this section we present a modification of the model that smoothes the discontinuities inthe equilibrium strategies and insures that the equilibria of the limit quota model alwaysapproximate the equilibria of the finite game (assuming the quota is feasible). The basicintuition is that our approximation can fail when small changes in the students’s HC cancause large changes in his or her assignment. This can occur because of gaps in thesupport of the college seats the students are competing for. When there is a gap in thesupport, the limit game equilibrium will involve discontinuities. As the HC starts toexceed the HC choice of the type assigned to the school on the left of the gap, marginalincreases in HC do not change the assignment. Once the HC level finally surpasses thelevel chosen by the type at the right of the gap, the assignmed discretely jumps in qualityto the school at the right of the gap.

The random distribution of colleges is denoted with the CDF FP (; ω) where ω is anelement of the probability space (Ω,B (Ω)) with underlying measure ν. This distributionrepresents the final quality of the colleges to which a student can be assigned, which weassume is unknown at the time that the HC is being accumulated. This lack of knowledgecould reflect actual changes in the quality of the schools or the innate uncertainty of thestudents that is only resolved at the time of college application, the point at which moststudents begin relatively detailed research into potential colleges.

We let Qj(; ω), j ∈ M,N, denote the quota given state ω, while Qj() denotes therandom CDF of the quota. If we let the empirical distribution of seats in the K−agentgame allocated to group j in stated of the world ω be denoted QK

j (; ω), then we requirethat the measure of seats assigned to each group converge uniformly over p and ω.

Assumption 12. LimK→∞

supp,ω

∥∥∥Qj(p; ω)−QKj (p; ω)

∥∥∥→ 0

In addition to the convergence assumption, we require that (random) assignment ofstudents to schools is continuous.

Assumption 13. Q−1j (r), j ∈ M,N, is continuous in r under the weak-* topology.

66

This assumption is (for example) fully compatible with the measures Qj(; ω) ad-mitting disjoint supports, but the edges of the support cannot vary too much and thedistributions over each interval of the support cannot radically change at many points.As we show below, the continuity of the assignment mapping implies that there are nodiscontinuities in the equilibrium strategy.

The ODE describing the equilibrium strategies becomes

dσqj (θ)

dθ=

∂∂θ

E[U(

Q−1j

(1− Fj

(θ))

, σqj (θ) , θ

)]∣∣∣θ=θ

Cs

(σ

qj (θ), θ

)− E

[Us

(Q−1

j (1− Fi(θ), σqj (θ), θ

)]σ

qj (θ) = s (boundary condition).

(24)

Although we have stated the ODE governing the equilibrium relatively abstractly, thereare a number of conditions under which a solution exists. For example, if Qj(p; ) isalmost everywhere continuous for all p, then assumption 13 holds. If we further assumethat Qj(; ω) admits a PDF fQJ (; ω) and there exists c > 0 such that fQJ (; ω) > c andfQJ (; ω) is continuous over the support of fQJ (; ω) for all ω, then

∂

∂θE[U(

Q−1j

(1− Fj

(θ))

, σqj (θ) , θ

)]∣∣∣∣θ=θ

= −E

Up

(Q−1

j

(1− Fj (θ)

), σ

qj (θ) , θ

)fQJ (1− Fj (θ))

f j(θ)

will be continuous, which implies that the solution to equation 24 is well-defined andunique.

The following proposition is proven by arguing that the agent utility is continuous asa function of s in the limit game, which then allows us to use the bulk of the argument inthe proof of theorem 5 to prove our claims.

Proposition 1. Assume that a solution to equation 24 exists and assumption 13 holds. Then forany ε, δ > 0 there exists K∗ such that σ

qj is an ε−approximate equilibrium of the K-agent game

and σqj is a δ−approximate equilibrium of the K-agent game.

Proof. First we argue that there are no jumps in σqj . By contradiction, suppose there exists

θ such that Limγ→0+

σqj (θ−γ) = s < Lim

γ→0+σ

qj (θ +γ) = s′. From the convexity of C with respect

to s and concavity of U with respect to s, we have U(Q−1j (1 − Fj (θ)), s, θ) − C(s, θ) >

U(Q−1j (1− Fj (θ)), s′, θ)− C(s′, θ). But for the discontinuity in σ

qj to be an equilibrium,

the type θ must be indifferent between s and s′, which provides our contradiction.Given that the strategy is strictly increasing and continuous, we immediately have that

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the assignment mapping Pqj (s) is uniformly continuous in the weak-* topology, which

implies that E[U(Pq

j (s), s, θ)]− C(s, θ) is uniformly continuous s. It is a matter of slight

changes in notation (e.g., using expected utility as opposed to utility) to modify lemmas6, 7, 8, and 9 to the setting with uncertainty, which yields the ε−approximate equilibriumclaim. The uniqueness of the equilibria of the limit game then implies our δ−approximateequilibrium.34

D Identifying Discontinuities in the Equilibrium

In this appendix we briefly describe how to identify discontinuities in the equilibrium ofthe limit game. This section will be of practical interest primarily to practitioners whowish to use equations 11 and 9 to compute equilibria numerically.

First let us consider quota schemes, where jumps are caused by gaps in the supportof Qj. The size of the jump must make the types on the edge of the gap indifferentabout making the jump. Formally written, suppose (pL, pU) is an interval such thatQj((pL, pU)) = 0 and for all ε > 0 we have Qj([pL − ε, pU ]) > 0 and Qj([pL, pU + ε]) > 0.Let θ be such that P(σq

j (θ)) = pL. Then it must be that for s = Limε→0+

σqj (θ + ε) (i.e., the

human capital choice on the other side of the jump) we have

U(pL, σqj (θ), θ)− C(σq

j (θ), θ) = U(pU , s, θ)− C(s, θ)

which identifies s.Now we discuss how to identify gaps in an admissions preference scheme, which are

caused by kinks or discontinuities in S. When these issues arise, the marginal incentivesfor both groups will change. We handle each possible issue in turn.

First consider a kink in S at s such that dds S jumps at s. Without loss of generality,

assume that the strategies are lower semicontinuous35 and let θN = ψapN (s) and θM =

ψapM(S−1(s)), and consider the first order conditions that would have to hold at s if the

strategies are continuous

For i = N , Up(Pap (s′) , s′, θN )dPap (s)

ds

∣∣∣∣s=S(s′)

+ Us(Pap (s′) , s′, θ) = Cs(s′, θ)

For i = M, Up(Pap(

S(s′))

, s′, θ)dPap (s)

ds

∣∣∣∣s=S(s′)

dS(s)ds

∣∣∣∣∣s=s′

+ Us(Pap(

S(s′))

, s, θ) = Cs(s′, θ)

34See the proof of theorem 5 for the details of the argument.35Since the edges of these jumps are described by an indifference condition, we could just as easily con-

struct an upper semicontinuous equilibrium.

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For both group’s strategies to be continuous, we would need the first order conditionsacross the discontinuity in d

ds S to be continuous. In other words, we would require

dPap (s)ds

=dPap (s)

dsdS(s)

ds

which is clearly impossible at the discontinuity in dS(s)ds .

To resolve this problem, one of the groups must jump. We will construct an equilib-rium where the minority students jump, but the construction and the test for the validityof the construction is symmetric in the case where the nonminority students jump. Ifthe minority student strategy exhibits a jump, then it must be the case that the first ordercondition for the nonminority students is smooth across the discontinuity, so dPap(s)

ds mustbe continuous. Although convoluted, we write the equation for dPap(s)

ds below for clarity:

dPap (s)ds

= − ΦfP (1−Φ)

where Φ =(1− µ) fN (ψ

apN (s))(

σapN)′(s)

+µ fM(ψ

apM(S−1(s)))(

σapM)′(s)S′(s)

and we let(σ

api

)′be infinity if group i has jumped across that level of human capital. In

other words, if the minority students jump, it must be that(σ

apN)′(s) drops discontinu-

ously to keep dPap(s)ds constant at s. For the duration of the minority student jump, we

can use the differential equations 9 to describe the nonminority student strategy. Thisconstruction is successful if in the gap we have

Up(Pap(

S(s))

, s, θ)dPap (s)

dsdS(s)

ds+ Us(Pap

(S(s)

), s, θ) ≥ Cs(s, θ)

If the inequality is reversed, then it must be the case that nonminority student jump andminority students do not. Finally, we need to define the size of the jump in the minoritystudent strategy. Suppose the minority strategy is lower semicontinuous and we haveσ

apM(θ) = s (i.e., θ is the type of minority student that jumps). To define the jump we

need to compute s′ such that

U(Pap(

S(s))

, s, θ)− C(s, θ) = U(Pap(

S(s′))

, s′, θ)− C(s′, θ)

and let the minority strategy jump to s′ at θ. Again, it will often be the case that the firstorder conditions for the two groups will not align and

dPap (s′)ds

6= dPap (s′)ds

dS(s′)ds

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If this occurs, it is treated as noted above. It is easy to construct examples wherethe groups repeatedly jump and never compete at the same college. For example, ifS(s) = s + ∆, ∆ > 0, the equilibrium has this structure.

Second, assume that there is a discontinuity in S at human capital level s. Sincewe have assumed S is increasing, S must jump upwards. Let σ

qj (θ) = s and assume σ

qj

is lower semicontinuous at θ.36 In this case, there must be a jump in the equilibriumstrategy of the minority students that is defined by

U(Pap(

S(s))

, s, θ)− C(s, θ) = U(Pap(

S(s′))

, s′, θ)− C(s′, θ)

and we let the value of the minority student strategy jump to s′ at θ. Again, if thefirst order conditions cannot both line up, then we will have to allow one of the groups’strategies to jump again, which requires the construction techniques outlined above.

36The construction would be essentially the same if we chose to let σqj be upper semicontinuous at θ.

70