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An auction-theoretical model of grade inﬂation

Sergey Alexeev

The University of New South Wales

Sydney, NSW 2052, Australia

s.alexeev@unsw.edu.au

November 1, 2021

Abstract

The grade inﬂation is a substantial and long-sustained increase in test scores. I incorpo-

rate unproductive test-speciﬁc knowledge (UTSK) into a standard auction-theoretical model

of centralized college admission (CCA) and show that the model in this form explains the

grade inﬂation and predicts colleges’ reluctance to participate in centralized college admis-

sion. This result argues for more moderate use of small-dimension measures of performance

(e.g., Scholastic Aptitude Test) in educational selection mechanisms (e.g., college admission),

particularly in the presence of income inequality or competitive educational opportunities.

Keywords: Auctions; Higher Education; Market Design; Matching Theory.

JEL codes: D44; I23; D47; C78.

Declarations of interest: none

This research did not receive any speciﬁc grant from funding agencies in the public, commercial,

or not-for-proﬁt sectors.

The funding sources had no involvement in the conduction of the research and/or preparation of

the article; in the collection, analysis and interpretation of data; in the writing of the report; and

in the decision to submit the article for publication.

Highlights:

•The adverse consequences of UTSK on the equality of opportunity in education are well

established by educational scientists

•However, game-theoretical modeling of college admission assumes away the inﬂuence of

UTSK

•My model considers a simple college admission model with UTSK

•The model is able to explains the grade inﬂation and colleges’ reluctance to participate in

centralized college admission system

1 INTRODUCTION 2

1 Introduction

Following the seminal work of Roth and Xing (1997), which delineates and solves the problem

of congestion (a failure by participants to make suﬃcient oﬀers and acceptances to clear the

market), centralized matching has assumed a prominent place in economic theory and practice.

For example, centralized matching has been successfully applied to medical residencies and public

school choices. College admission represents a similar market and is known to suﬀer from a lack

of coordination. It is believed that improved coordination by centralized matching would result in

welfare beneﬁts; however, the admission process in many countries still operates in a decentralized

fashion similar to labor markets (Abdulkadiro˘glu and S¨onmez 2003; Gale and Shapley 1962).

A better understanding of the reluctance to adopt a centralized college admission system is

an important aspect of the market design. This paper explores one possible explanation for this

reluctance. Unlike in the medical residency context, in which centralized matching begins after

applicants and hospitals interview one another in a decentralized fashion, CCA uses standardized

examination marks taken to assess second school students as a proxy for applicants’ skills while

colleges are assumed to prefer applicants with high marks to applicants with low marks. CCA

has been studied extensively in game-theoretical literature; however, to date, no papers have

acknowledged that standardized examination mark may not reﬂect true students’ ability due to

the availability of commercial test-preparation (TP) cramming schools (e.g., Haladyna, Nolen,

and Haas 1991; Mehrens and Kaminski 1989).

Since the 1980s, educational scientists started documenting the growth of the college admission

TP industry and aggressive TP practices by parents and their children (Aurini, Dierkes, and

Davies 2013; Bray and Lykins 2012). CCA examinations, which vary little in their general designs

year to year, enable the TP industry to accumulate UTSK (Anastasi 1981; Reeve, Heggestad, and

Lievens 2009). Private clients use UTSK to achieve higher examination marks by complementing

their skills (Dang and Rogers 2008; Haddon and Post 2006).

This perpetuates social inequalities, negatively aﬀects the sorting properties of the public

system of education and led to resources being consumed that could be better used in other

places (Gorgodze 2007; Hurwitz and Lee 2018; Nordin, Heckley, and Gerdtham 2019). Aggressive

TP practices by parents and their children are also one of the reasons for grade inﬂation: the

substantial and long-sustained increase in test scores that started in the 1980s (Bar, Kadiyali, and

Zussman 2009,2012; Jewell, McPherson, and Tieslau 2013; Rojstaczer and Healy 2012; Shepherd

2011). The connection between test familiarity and grade inﬂation was ﬁrst speculated by Flynn

(1984) and is now often referred to as the Flynn eﬀect.

This paper uses a college admission model framed as a contest (cf. Bodoh-Creed and Hickman

2018; Hafalir et al. 2018), but instead of the usual assumption that the variation in the student

skills drives the variation in the exam marks, I assume that the exam marks is a complementing

mixture of the skills and UTSK. On a technical level, I merge Moldovanu and Sela (2001,2006)’s

contest model and Che and Gale (1998)’s constrained auction model.

I show that if access to UTSK is explicitly considered, examination marks are less informative

about applicants’ underlying skills, and applicants who do not necessarily possess high levels of

skills could be admitted. I perform comparative static analysis and show that colleges’ sorting

becomes more problematic in the presence of highly competitive colleges or income inequality.

The model of my paper provides a uniﬁed explanation for several observations. It shows why top

2 A CASE STUDY 3

universities, for example, in Japan or Russia or medical specializations in Ireland or Australia, are

exempted from CCA. It also shows that rising inequality of opportunities may explain the grade

inﬂation that started in the 1980th even if skill level had not changed. Perhaps that is why the

United States never employed CCA and unlikely will, as shown by growing organized eﬀorts to

stop using standardized tests for college admission, known as ‘The Opt-Out Movement’ (Supovitz

et al. 2016).

My work complements existing game-theoretical arguments that exam marks do not accurately

reﬂect skills. One argument is that when employers cannot tell whether a school truly has many

good students or just gives easy grades, a school has incentives to inﬂate grades to help its

mediocre students. The inability to commit to an honest grading policy reduces job assignment

eﬃciency (Chan, Hao, and Suen 2007; Himmler and Schwager 2013; Ostrovsky and Schwarz 2010;

Zubrickas 2015). Another argument is that teachers send messages (possibly lying or failing to

withstand some students’ pestering) to their students after learning about the students’ skills.

Then, students choose the eﬀort they would invest in the subject at university. This introduces

gender and other diﬀerences in education and labor markets (Franz 2010; Mechtenberg 2009).

The remainder of the paper proceeds as follows. Section 3delineates the model’s primitives.

Section 4characterizes equilibrium of the college admission problem in the presence of skill-

UTSK complementary. In particular, the existence of a unique symmetric equilibrium in strictly

increasing strategies is demonstrated. The analysis indicates that under CCA, certain types of

applicants with better access to UTSK make a larger investment in TP to gain an advantage over

those with less access to UTSK. Section 5shows how test-taking equilibrium strategies change

in response to change in college wage premium. I demonstrate that the prestigious colleges are

overwhelmed by applicants with high examination marks but moderate skill levels. A similar

behavior emerges in the context of inequality of opportunities: a better prospect for defeating

an applicant with no access to UTSK rationalizes more aggressive test-taking behaviors. In the

end, I demonstrate a reduction in the quality of sorting that results from this behavior. Section 6

concludes and discusses the model’s practical implications and limitations.

2 A case study

The enormous growth of college admission TP has been evident globally (Aurini, Dierkes, and

Davies 2013; Bray 2011; Bray and Lykins 2012). In some countries, TP industry has risen to

occupy a grotesquely large role in households’ lives. For example, in Turkey, Korea, Azerbaijan,

and Mauritius, students skip public high schools en masse (or use them to sleep) to attend TP

schools later in the day (Bhorkar and Bray 2018; Bray 2017). To understand how the TP industry

emerges to occupy such a prominent role, a recent wave introduction of CCA in ex-Soviet republics

is particularly helpful.1Unlike many countries that have embraced large-scale test standardization

decades ago, the households in the ex-Soviet republics, due to certain shared past, had college

and often department-speciﬁc admission exams – an antipode to CCA. Therefore it serves as a

rare and useful quasi-experiment.

Unique insight on the practical implementation comes from Georgia and is provided by Gor-

godze (2007). The Georgian capital, which encompasses all universities, allocates the college

1Years of implementation: Azerbaijan 1992, Kazakhstan 1999 – 2004, Russia 2001 – 2009, Kyrgyzstan 2002,

Ukraine 2004 – 2005, Georgia 2005.

2 A CASE STUDY 4

Figure 1: Exam marks elevation

2012 2013 2014 2015 2016 2017

25

30

35

40

45

50

Years

% of intakes with marks above 70

Exam marks

450

500

550

600

650

International comparison of skills

Skills measured by PISA

Notes: Orange line show an increase in the fraction of students with exceptionally high centralized exam marks

(scoring more than 70 out of 100). The black line shows average skills in reading, mathematics, and science of

15-year-old students measured by the Programme for International Student Assessment. Marks for 2013 are

ignored because questions for the centralized exam were leaked to the public right before the test day (Gushchin

2013).

Sources: M. S. Dobryakova (2017), OECD (2017)

degrees for the whole country. The introduction of CCA has indeed equalized admission op-

portunities for out-of-capital households, but only for few admission cycles. Afterward, capital

residents have discovered that instead of paying semi-institutional bribes, they can invest money

into TP schools. Due to the systematic increase of their marks, the eventual allocation of degrees

is now close to before the policy change.

Russia went through a remarkably similar experience. CCA has been sequentially introduced

during 2001–2009, which is fortunate to illustrate that CCA with TP industry and without sort

applicants diﬀerently. A report for the government by Efendiyev and Reshetnikov (2010) on the

experimental introduction of CCA in several regions in 2001 – when TP schools have not yet

been massively deployed – have shown an increase in admission of students with a less opportune

background. Contrary, more than a decade later, Prakhov and Yudkevich (2017) have shown

that the engagement of TP schools by applicants with higher income substantially and reliably

increases their centralized exam mark, which naturally crowds out applicants with lesser income.

The above suggests a regularity. Centralization introduces an exam with the publicly known

general design. Preparation becomes a homogeneous service which creates an organized TP

market. Over time learning by doing allows TP to accumulate knowledge of how to pass the

centralized exam, a fraction of that knowledge is unproductive in the sense that it does not

increase applicants’ skills and can not be used in future studies. As a result, UTSK creates a

group of applicants with higher marks, but only because they know the test better. Since marks

become systematically independent of skills, it creates a problem when colleges are forced to the

marks to sort students.

The following two facts summarize the key observations that accompanied the introduction

in Russia. They are used to motivate the model in the next section.

Fact 1. After CCA has been introduced, the right tale of centralized exam marks distribution has

3 THE MODEL 5

Figure 2: Growth of TP industry and CCA exemptions

2012 2013 2014 2015 2016 2017

−5

0

5

10

15

20

25

30

35

40

45

50

55

60

65

Years

Number of test-preparation schools

Federal cities

Regional capitals

Towns and villages

45

50

55

60

65

70

75

80

Number of colleges exempted from CCA

Exempted colleges

Notes: Bars demonstrate that the growth of private preparation schools is concentrated in high income areas.

The black line demonstrates the total number of colleges allowed to have complementary exams to sort

applicants.

Sources: 4EGE (2017), YP (2017), Spark-interfax (2017)

constantly been elevating. Concurrently, an alternative measure of skills indicates that the level

of high school students has been approximately the same (Figure 1).

The fraction of applicants with exceptionally high marks on the centralized exam constantly

goes up. Indicating that the aggregate capacity to do well on the centralized exam is increasing.

Fact 2. The number of TP schools is increasing, and they tend to concentrate in higher-income

areas. Concurrently, the number of colleges that have requested to use others tests in addition to

a centralized exam mark has been strictly increasing (Figure 2).

This paper takes this fact as suggestive evidence that the TP industry drives the elevation

of the exam marks by systematically accumulating UTSK and dispensing it to those applicants

who can aﬀord it. As a result, colleges that participate in CCA suﬀer a reduction in utility. For

the US, a similar concentration of TP schools in higher-income areas is shown recently by Kim,

Goodman, and West (2021).

3 The model

The key challenge of modeling the college admission problem with UTSK is to separate the

productive and unproductive aspects of TP activity. TP schools, as any private education, play an

important role in human capital formation. Middle and high-income families choose to investment

in early childcare (Ramey et al. 2010) and extracurricular activities (Friedman 2013), and view

kindergarten as a time for academic focus rather than play and socializing (Bassok, Latham,

and Rorem 2016). Families also demand more intensive and competitive secondary education,

pushing for dual enrollment programs and Advanced Placement classes (Davies and Hammack

2005). Low-income families may also improve human capital using TP educational services, as

shown by the No Child Left Behind Act (Ascher 2006; Chappell et al. 2011).

3 THE MODEL 6

The existing modeling approaches deﬁne an exam mark as a monotone function b(vi), with vi

being an unobserved skill level of a student i(e.g., Balinski and S¨onmez 1999). The skill level

includes training from both public and private educational institutions, including the productive

aspect of TP activity. The validity of this modeling choice informs a simple empirical test. A

student’s rank in the CCA exam mark should be highly predictive of a student’s rank in the

mark in the ﬁrst year of college. Because this empirical test is rarely passed (e.g., Rothstein

2004), other students’ characteristics (hidden in the residual of an econometric model) inﬂuence

the marks. UTKS is one of those characteristics that have a ﬁrst-order inﬂuence on the marks.

UTKS is distinct from skills that students accumulated because their parents care for their

education. The examples include practicing questions from previous exams or closely resembling

them, paying more attention to typical exam topics and vocabulary. It also includes learning

nonobvious but eﬃcient approaches to certain test-speciﬁc questions as well as time and stress

management techniques speciﬁc to particular test structures. It also extends to test-taking con-

ditions. TP often consult on the best time and location to sit the exam. TP schools are uniquely

positioned to accumulate UTKS knowledge as they can aggregate tacit knowledge across genera-

tions of students. A deﬁning feature of UTSK is that they do not have an independent eﬀect on

exam marks. They only complement existing skills.

That is why, in this study, I model the exam mark of student iwith skill viand UTSK wi

as a complementing mixture. To accentuate that there is no substitutability between these two

characteristics and that they are distinct, the exam mark is deﬁned as the Leontief production

function

bi(vi, wi) := min{b(vi), wi} ∈ [0, wi].(1)

Complementary2captures an observation that an excess of UTSK does not on its own increase the

applicant’s marks; similarly, being exceptionally skillful does not translate into higher marks in the

absence of UTSK. Applicants could be highly skilled, but if they have no training on, for example,

the questions typical to the examination, their marks may not reﬂect their skills. Further, a higher

value of wis required to signal a higher v. For example, highly skilled applicants still have to do

full-ﬂedged practice tests to achieve a result that signals their skill level. Conversely, to achieve

a moderate result, a moderately skilled applicant may only need to complete practice questions

from free textbooks or on-line materials.

The current paper focuses on one adverse consequence of UTSK – reduction in sorting quality

for colleges that are required by law to use CCA to admit students. In the model, there are

two secondary school students (college applicants). Each student i= 1,2 has two private values,

viand wi. The values of each student are jointly distributed according to a continuous density

deﬁned on [0,1] ×[w, w], where 0 ≤w≤1≤w. Variable Vwith CDF FVgoverns distribution

of viand variable Wwith CDF FWgoverns distribution of wi. The applicants simultaneously

submit bids (exam marks).

This setup accentuates the uncertainty faced by the colleges at the moment when they are

allocated an applicant. At this particular moment, the colleges see UTSK and skills as if they are

given exogenously to an applicant. The formation of human capital and UTSK are maintained

in the background. Importantly, because b(v) is a component of b(v, w), the productive activity

2The degree of complementary is regulated by an implied coeﬃcient αin the expression min{b(vi), αwi}. The

degree is assumed to be unity and ﬁxed throughout the paper.

4 EQUILIBRIUM TEST-TAKING BEHAVIOR 7

of TP schools, similarly to previous studies, is implied in the model.

There are two colleges in the model, denoted Hand L, which are not decision-makers. The

higher bid student is admitted to college Hand obtains an extra wage premium β(a diﬀerence

between wage premium granted at Hand L). College Hcould also be interpreted as a cluster of

better colleges, more prestigious specializations, or higher education levels (i.e., secondary school

vs. college degree). Students are then interpreted as two aggregated groups characterized by skill

level and access to UTSK (or lack of access if the group is ‘unprivileged’). Either way, high β

captures the presence of a highly beneﬁcial educational opportunity. The payoﬀ to student iwith

bid biis modeled as β−bi

/viif admitted and −bi

/viotherwise.3

College Hprefers to admit the higher ability student, not necessarily the student with the

highest bid. If college Hadmits the most skilled student, it obtains a payoﬀ equal to v(1), the

ﬁrst order statistic. If not, a payoﬀ is v(2), the second-order statistic. College Hpayoﬀ function

Πdistinguishes three states:

Π=

v(1) if b(v(1))< w

v(1) if b(v(1))> w > b(v(2) )

v(2) if b(v(1))> w(2) and b(v(2))> w(1)

.(2)

In the ﬁrst case, the most skilled applicant is not restricted and can signal his or her type. The

best applicant is restricted in the second case, but the least skilled applicant (whether restricted

or not) still has a lower bid. In the last case, where w(1) and w(2) denote, respectively, the ﬁrst and

the second-order statistics of W, both applicants are restricted, and the most skilled applicant

has tighter restrictions. Taking the expectation over all possibilities produces college Hexpected

utility UH:

UH= 2 w

Z

w

w

Z

0

v(1)dFB1dFW+

w

Z

w

w

Z

1

1

Z

w

v(1)dFB1,2dFW

+

w

Z

w

1

Z

w(2)

dFB1dFW2

w

Z

w

1

Z

w(1)

v(2)dFB2dFW1!,

(3)

where FB1,FB2,FW1and FW2are, respectively, CDFs of order statistic random variables b(V(1)),

b(V(2)), W(1) and W(2).

4 Equilibrium test-taking behavior

I will focus on the symmetric equilibrium in which the bidding function takes the form (1) and

b(·) is a strictly increasing and piecewise diﬀerentiable function. The key to ﬁnding the solution is

to characterize b(v), which one could think of as the examination mark of an applicant who is not

3This work focuses on the equilibrium of the form min{b(v), w}by construction, as it closely resembles the

notions of test ‘sophistication’ (Anastasi 1981) or ‘familiarity’ (Reeve, Heggestad, and Lievens 2009). At the same

time it can be shown, using the argument of Lemma 1 of Che and Gale (1998), that in the presence of constraint

b(v, w) = min{b(v), w}without loss of generality.

4 EQUILIBRIUM TEST-TAKING BEHAVIOR 8

limited by UTSK, but who understands that the competing applicant could be constrained, and

so on.4I will refer to b(v) as the ‘unconstrained’ bidding function (exam mark). For expositional

ease, I will refer to bidder 1 as ‘she’ and bidder 2 as ‘he.’

Suppose that bidder 2 follows the bidding rule b(·,·) as in (1), and consider the optimal

strategy for bidder 1 who has a UTSK wand a skill x1. Bidder 1 will not herself be constrained,

but she understands that her opponent may be constrained and may expect him to be constrained

with positive probability, and so on. By the rules of CCA, if she bids band if bidder 2 is of type

(w2, x2) and is following the bidding strategy (1) bidder 1 solves the following problem:

U1(b(x1)|v2, w2) = βP [v2< x1] + β P [v2> x1]P[w2< b(x1)] −b(x1)

v1

=βFV(x1) + β(1 −FV(x1))FW(b(x1)) −b(x1)

v1

.

(4)

She defeats two categories of opponents. First she defeats all types who have v2< x1. Second,

she defeats all types who have w2< b(x1). Given the maintained assumptions, ties are probability-

zero events. The ﬁrst-order condition for truth-telling to be an equilibrium is that the derivative

of U1(b(x1)|v2, w2) with respect to x1evaluated at vis equal to zero. Applying the condition and

performing some manipulations results in the following proposition.

Proposition 1. There exists a symmetric equilibrium of the form b(v, w) = min{b(v), w}in

strictly increasing piecewise diﬀerentiable strategies

(a) For all v < ˜v,b(v) = ˆ

b(v)

ˆ

b(v) = β

v

Z

0

yfV(y)dy. (5)

(b) For all v≥˜v,b(v)is the strictly increasing solution of the diﬀerential equation

b0(v) =

βvfV(v)1−FWb(v)

1−βvfWb(v)1−FV(v),(6)

satisfying the boundary condition b(˜v) = w;

Proof. Note that if β= 1 multiplying Equation (4) by v1results in the objective function of all-

pay-auction analyzed by Kotowski and Li (2014). Furthermore, for any positive β, but P[w2<

b(x)] = 0, Equation (4) becomes the objective function of contest analyzed in Moldovanu and

Sela (2001). For the complete proof, let U1(b(x)|v, w) be the expected utility of applicant iwith

skill lever vthat places the bid b(x)≤w.U1(b(x)|v, w) is continuous and can be written as

U1(b(x)|v, w) = U1(b(˜v)|v, w) +

x

Z

˜v

d

dtU1(b(t)|v, w)t=y

dy. (7)

4See Fang and Parreiras (2002) and Kotowski and Li (2014) for a similar approach.

4 EQUILIBRIUM TEST-TAKING BEHAVIOR 9

I now verify that no applicant wishes to deviate to an alternative (feasible) bid. Consider an

applicant with skills v < ˜v. When following the strategy b(v, w), this applicant puts an eﬀort that

yields him or her an exam mark b(v) = ˆ

b(v). From Moldovanu and Sela (2001), we know that this

applicant will not have a proﬁtable deviation to any eﬀort level b(x), x∈[0,˜v]. Suppose instead

that this applicant contemplates about b(x) for some x > ˜v. The expected payoﬀ from this eﬀort

level is given by Equation (7). It is suﬃcient to verify that when v < x,d

dt U1(b(x)|v, w)≤0

d

dtU1(b(t)|v, w) = d

dtβFV(t) + β(1 −FV(t))FW(b(t)) −b(t)

v

=b0(t)β(1 −FV(t))fW(b(t)) −1

v+βfV(t)(1 −FW(b(t)))

= (v−t)β(1 −FW(b(t)))fV(t)

v(1 + tβ(FV(t)−1)fW(b(t)))

≤(t−t)β(1 −FW(b(t)))fV(t)

t(1 + tβ(FV(t)−1)fW(b(t))) = 0.

(8)

Consider instead an applicant with a skill level v≥˜v. An argument parallel to the preceding

case conﬁrms that an eﬀort that results in b(x), x>v, will not be a proﬁtable deviation. Finally,

consider a deviation to b(x), x < ˜v. It is suﬃcient to show that d

dt U1(b(t)|v, w)≥0 for all t < ˜v.

d

dtU1(b(t)|v, w) = β

v

Z

0

tfV(t)dt −β

t

Z

0

tfV(t)dt

≥β

t

Z

0

tfV(t)dt −β

t

Z

0

tfV(t)dt = 0.

(9)

The above analysis is exhaustive of all the cases; thus, the considered strategy proﬁle is a

symmetric equilibrium. Finally as b(v) is strictly increasing, then there is a unique value ˜vsuch

that b(˜v) = w.

In the absence of UTSK, the applicants behave according to ˆ

b(v). This strategy has been

characterized in Moldovanu and Sela (2001). With UTSK, the applicant behaves according to

b(v). The diﬀerential Equation (6) is deﬁned over a set {v∈[0,1] : b(v)≥w}and it accounts

for the change in marginal incentives faced by bidder 1. A slight increase in investment in marks

not only allows her to outperform other applicants with slightly higher skill levels, v, but also to

outperform applicants with suﬃciently low access to UTSK, w, regardless of his skill level, v. A

property of the diﬀerential equation (6) is that it permits an explicit solution (5) when b(v)< w.

The technical reason for this is that the value of b(v) of this size is outside of the support of W.

The economic meaning is that an applicant with low skills will not strategize against applicants

who might have less fortunate TP opportunities.

Example 1. Assume Vi.i.d.

∼ U[0,1],Wi.i.d.

∼ U[0.5,2.08] and β= 3 then

(a) For all v < ˜v= 0.57,b(v) = (3

/2)v2

5 IMPLICATIONS FOR COLLEGE ADMISSION 10

Figure 3: Functions ˆ

b(v) and b(v) from Example 1.

0.2 0.4 0.6 0.8 1

0.4

0.8

1.2

1.6

v

ˆ

b(v), b(v)ˆ

b(v)

b(v)

w

(b) For all v≥˜v= 0.57,b(v)is the solution of the diﬀerential equation

b0(v) =

3v1−b(v)−0.3

2.08−0.5

1−3v1−v

2.08−0.5,(10)

satisfying the boundary condition b(0.57) = 0.5.

Function ˆ

b(v) in Figure 3represents a mark that an applicant with skill level, v, would have

had if UTSK were assumed away.5In an explicit formulation of UTSK, applicants with skill

levels of v < 0.57 will still receive the same mark; however, applicants with skill levels of v > 0.57

will choose to invest more in preparation and will be awarded marks in the size of b(v). The

parameter wplays a deﬁning role here; in the case when w≥b(1), function b(v) never emerges

and applicants of all types behave in accordance with ˆ

b(v).

5 Implications for college admission

This section explores a change in the equilibrium behaviors and sorting outcomes in response to

a change in the college wage premium, β, and a change in the distribution of UTSK, FW.

5.1 Change in wage premium

A change in βcaptures the change in equilibrium test-taking behaviors for diﬀerent educational

opportunities. Intuitively, a higher wage premium justiﬁes a higher investment into activities that

will lead to higher marks being achieved.

Denote by bβan equilibrium strategy for β. Figure 4uses Example 1to show how applicants

respond to β= 2 and β0= 6. When the wage premium is equal to 2, the presence of UTSK

changes practically nothing in the applicants’ behavior, as shown by the diﬀerence between blue

5In all examples, plots of numerical solutions are obtained using the Runge-Kutta method.

5 IMPLICATIONS FOR COLLEGE ADMISSION 11

and black solid lines, whereas when the wage premium is equal to 6, the blue and black dashed

lines are radically diﬀerent. All applicants with v > 0.57 choose to exhaust their access to UTSK

of 2.08 and have similar exam marks. It shows that applicants’ behavior in the presence of UTSK

is acutely sensitive to the size of the college wage premium. These results hold generally.

Figure 4: Comparative statics in βusing Example 1.

0.2 0.4 0.6 0.8 1

0.4

0.8

1.2

1.6

2

v

ˆ

b(v), b(v), w ˆ

bβ(v)

ˆ

bβ0(v)

bβ(v)

bβ0(v)

w= 0.5

Proposition 2. Suppose β0> β, i.e., the college wage premium is higher. Then, under the

conditions of Proposition 1:

(a) lim

v→˜v+b0(v)>lim

v→˜v−b0(v);

(b) lim

v→˜v+b0

β0(v)−lim

v→˜v−b0

β0(v)>lim

v→˜v+b0

β(v)−lim

v→˜v−b0

β(v)>0.

(c) bβ0(v)> bβ(v)>ˆ

bβ(v)for β > FW(bβ(v))

fW(bβ(v))

1

1−FV(v)

1

v.

Proof. A direct calculation gives (a)

lim

v→˜v+b0(v) = lim

v→˜v+

βvfV(v)(1 −FW(b(v)))

1−βvfW(b(v))(1 −FV(v))

=β˜vfV(˜v)(1 −FW(b(˜v)))

1−β˜vfW(b(˜v))(1 −FV(˜v))

=β˜vfV(˜v)(1 −FW(w))

1−β˜vfW(w)(1 −FV(˜v))

=β˜vfV(˜v)

1−β˜vfW(w)(1 −FV(˜v))

=1

1−β˜vfW(w)(1 −FV(˜v))

| {z }

<1

β˜vfV(˜v)

> β˜vfV(˜v) = lim

v→˜v−b0(v).

(11)

The third line uses a boundary condition b(˜v) = w, whereas the last line uses a diﬀerential

equation for case v≤˜v.

5 IMPLICATIONS FOR COLLEGE ADMISSION 12

To see (b), note that the above showed

lim

v→˜v+b0(v) = ψ(β) lim

v→˜v−b0(v) (12)

where

ψ(β) = 1

1−β˜vfW(w)(1 −FV(˜v)) >1.(13)

Clearly, ψ0(β)>0 as

∂

∂β 1

1−β˜vfW(w)(1 −FV(˜v)) !=fW(w)(1 −FV(˜v))˜v

((FV(˜v)fW(w)−fW(w))˜vβ + 1)2>0.(14)

An increase in βreinforces the gap between b(·) and ˆ

b(·).

A direct calculation gives (c)

b0(v)>ˆ

b0(v)

βvfV(v)(1 −FW(b(v)))

1−βvfW(b(v))(1 −FV(v)) > βvfV(v)

β > FW(b(v))

fW(b(v))

1

1−FV(v)

1

v.

(15)

Thus, Part (a) of Proposition 2shows that UTSK always encourages more aggressive test-

taking behaviors among some applicants (due to the realization of their relative advantage). Part

(b) shows that an increase in the wage premium always encourages more types to behave more

aggressively. Part (c) shows that for higher values of β, all types of bidders bid more aggressively.

Higher wage premium colleges are more likely to be aﬀected by gaming against applicants with

fewer preparation opportunities.

5.2 Change in access to UTSK

TP schools tend to be located in higher-income areas. The tendency produces areas where

applicants are more deprived of UTSK in comparison to applicants from higher-income areas.

This mechanism can be studied in the current model as an increase in the probability that one’s

opponent will fail to broadcast their skill level due to a lack of UTSK.

Denote by W0a random variable with corresponding CDF denoted by FW0that stochastically

dominates FW0. Further, represent by bWan equilibrium strategy for W. Consider Example 1, and

replace distribution of UTSK with an exponential distribution FW(w;θ)=1−exp(−θ(w−w))

distributed on [w, ∞) and ﬁx w= 0.5. The exponential distribution belongs to a family of

parametric functions that satisfy the increasing reverse hazard rate condition. That is, an increase

in the parameter θmakes one’s opponent more likely to be constrained.

Figure 5shows equilibrium test-taking behavior for θ= 0.5 and θ= 1.2 (analogue of FW

and FW0, respectively). A better prospect for defeating an applicant with no access to UTSK

rationalizes more aggressive test-taking behaviors across applicants of all skill levels. This result

holds generally.

5 IMPLICATIONS FOR COLLEGE ADMISSION 13

Figure 5: Comparative statics in θusing Example 1.

0.2 0.4 0.6 0.8 1

0.4

0.8

1.2

1.6

2

v

ˆ

b(v), b(v), w ˆ

b(v)

bW(v)

bW0(v)

w= 0.5

Proposition 3. Suppose FW0(w)≤FW(w), i.e., the likelihood that one’s opponent would fail to

complement their skills with UTSK is higher. Then, under the conditions of Proposition 1:

bW0(v)> bW(v),(16)

Proof. Let b>bW0(˜v) = bW(˜v). Then FW0(w)< FW(w) and fW0(w)> fW(w). Thus, for all b

where Equation (6) is positive:

0<βvfV(v)(1 −FW(b))

1−βvfW(b)(1 −FV(v)) <βvfV(v)(1 −FW0(b))

1−βvfW0(b)(1 −FV(v)).(17)

If bW0(v) and bW(v) intersect, the former is steeper than the latter, and this implies that they

intersect once at most. Additionally, bW(v) would cross bW0(v) from above.

Thus, if bW(v)> bW0(v) for any v > ˜v, then bW(v)> bW0(v) for v∈(˜v, z), where zsuﬃciently

small. Fix ˜v < v < z then

fW0(bW0(v)) ≥fW(bW0(v)) ≥fW(bW(v)) (18)

and

FW0(bW0(v)) ≤FW(bW0(v)) ≤FW(bW(v)).(19)

Therefore, b0

W0(v)≥b0

W(v).

Finally, because bW(v) and bW0(v) can be expressed as integral equation,

bW(˜v) = bW(z)−Zz

˜v

b0

W(x)dx > bW0(z)−Zz

˜v

b0

W0(x)dx =bW0(˜v) (20)

which contradicts bW0(˜v) = bW(˜v). Therefore, bW0(v)> bW(v).

There is another subtle economic counterpart to the comparative statics of Proposition 3,

which reveals the unearned power that the TP industry possesses due to the presence of informa-

tion asymmetry. A Bayes Nash equilibrium requires that characterization of a bivariate random

variable (V, W ) be commonly known (Morris 1995).

6 CONCLUSION 14

In practice, guessing the extend of the access to UTSK requires an inference, which can be

manipulated. In a typical advertisement campaign, a TP school contends that its paying clients

will gain access to an exclusive bank of questions that mimics the real test. Such TP schools are

eﬀectively declaring that they can help applicants who attend their school to outperform those

who will not have access to such questions. The moment parents are convinced that they acquire

unique UTSK, the mechanics of Proposition 3apply.

Therefore, to explain the elevation of exam marks, it only takes a noisy advertisement cam-

paign by a few TP schools, through which parents would attempt to infer the distribution of the

availability of UTSK and rationally respond by over-engaging the TP industry. This outcome is

in line with an idea of the normalization of aggressive TP practices, which educational scientists

have extensively documented. Once aggressive TP practices enter a society, it is impossible to

remove them (Bray 2009). The model exposes the force that makes aggressive TP practices so

persistent: information.

5.3 Inﬂuence on college’s utility

Thus far, the model has shown an increase in the wage premium or the probability that a com-

peting applicant lacks UTSK intensiﬁes test-taking behaviors and elevates marks. The following

summarizes the consequences of such behaviors on the quality of sorting.

Theorem 1. UHdecreases if β0> β or FW0(w)≤FW(w)

Proof. The results in Proposition 2and Proposition 3show that the conditions of this theorem

guarantee that b(v) attains a higher value for all v. Denote the more aggressive and less bidding,

respectively, by b(v) and b(v) Now deﬁne FBand FBas CDFs that correspond to those biddings.

Uniform elevation of b(v) implies that FB(x)< FB(x), for all x. Therefore FB(x) stochastically

dominates FB(x). It also follows that FB1(x)< FB1(x), FB2(x)< FB2(x) and FB1,2(x)< FB1,2(x).

That is the destribution of highest order, lowest order and joint order statistic of more aggressive

bid stochastically dominates that of less aggressive bid (Krishna 2009, App. C). From the

deﬁnition (3), the college utility is statewise lower when the bidding is more aggressive.

Proposition 2and Proposition 3show what could drive the skills-irrelevant elevation of exam

marks, while Theorem 1shows that the same forces rationalize the college’s reluctance to partic-

ipate in CCA. Facts 1and 2are explained.

6 Conclusion

This paper uses auction-theoretical modeling to demonstrate a potentially harmful tendency

among college applicants. It is shown that rising inequality may rationalize more aggressive TP

practices. This makes the sorting of applicants troublesome for colleges, especially those that

are highly competitive and may make the CCA unstable. The key economic message is that the

functionality of the CCA depends on students’ access to a stock of UTSK, which is created and

managed by the TP industry and cannot be directly controlled by a regulator. The TP industry

is an integral element of CCA, which beneﬁts from an increase in the examination marks but

ignores the role of those marks in resolving uncertainty. The mechanics analyzed in the paper

REFERENCES 15

applicable to everything from CCA exams to Scholastic Aptitude Tests to English-language tests

and selective-high-school tests.

The model provides a limited theoretical case against CCA. Factors outside the model may

still be in favor of CCA. These factors are numerous, as the model in this paper is highly stylistic

and aims to demonstrate a problem rather than oﬀer a solution. The results might diﬀer if more

students or colleges or explicit formation of skills and UTSK are included. The model also does

not model a decentralized admission (or the utility of college L). Implicitly my model assumes

that there is an exogenous utility level below which colleges quit CCA. The choice between

decentralized and CCA may also depend, in practice, on political economy considerations which

would be hard to model rationally.

Keeping all of these limitations in mind, the model suggests that the policymakers may need to

be on the lookout if a country’s education system features a highly competitive college, clusters

of colleges, or specializations, especially in the presence of income inequality. An increase in

the exam marks might be masking a reduction in the quality of matching between students and

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measure in the school accountability systems (Figlio and Loeb 2011). There should be one exam

to access schools’ performance and another exam to select applicants into colleges. If this is not

the case, then UTSK may exacerbate education inequalities throughout the system (Neal and

Schanzenbach 2007).

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“An auction theoretical model of grade inﬂation”

by Sergey Alexeev

Summary:

The current paper makes a theoretical contribution by analyzing the economic ineﬃcien-

cies in the College Admissions due to the existence of “unproductive test speciﬁc knowledge”

(UTSK). In particular, the author provides an auction thoretical model, which (he claims)

explains grade inﬂation and also the reluctance of colleges to participate in a centralized

admissions.

Even though coordination in a centralized matching mechanism seems to be better in

terms of welfare, many colleges seem to be reluctant to adopt such a mechanism. This is

partly possibly due to the fact that standardized test scores may not reﬂect the students’

true types (abilities) —possibly due to existence of commercial test-preparation (TP) mar-

ket. Such a market help their clients (students who are to take the standardized tests) to

accumulate and use UTSK to achieve higher grades. This not only leads to social inequali-

ties but also negatively aﬀects the sorting properties of education.

The paper contains a case study section where the author provides examples of the

experience in centralized college admission (CCA) with Test Preparation (TP) schools in

ex-Soviet countries. In particular, the paper argues that even though in the beginning,

CCA allows an increase in the enrollment of students with less opportune backgrounds in

colleges, eventually, with the help of TPs, high income students crowd out these students

by acquiring UTSK. In particular, the case study presents the following two stylized facts

that are aligned with the results the paper provides:

Fact 1: Even though PISA scores do not show any increase in average skills of high school

students in reading, math, and science, the fraction of exceptionally high cetralized

exam scores increases signiﬁcantly.

Fact 2: The number of TP schools is increasing and concentrates in higher income areas.

As a result, the number of colleges that requests to use other tests in addition to a

centralized standardized test has increased.

The paper then builds an auction theoretical model that predicts these facts.

1

The model / Technical contribution :

The paper provides a simple and highly stylized model that complements existing game-

theoretical models by allowing for standardized test scores to not accurately reﬂect students’

essential skills relevant for education. The model in the paper can be summarized as follows:

- There are two students i= 1,2 and two colleges Hand L.

- The types of each student can be summarized by a tuple (vi, wi) where videnotes the

skill of the student whereas widenotes the UTSK level that the student accummulated.

- These skills are complementary to produce the test score of a student. In particular,

the exam score of student iis assumed to be given by the Leontieﬀ production function

bi(vi, wi) = min{b(vi), wi}.

- The applicants simultaneously submit their exam scores and the student with the

higher score is admitted to Hand as a result obtains an extra wage premium β.

- The payoﬀ to student iwith bid (exam score) biis β−bi

viif admitted to Hand −bi

vi

otherwise.

- College Hwould like to admit the higher skilled student. The payoﬀ to college H

is the highest skill level, v(1), if the college succeeds in admitting the higher skilled

student, otherwise, it is the lowest skill level, v(2).

Summary and interpretations of the Results:

Proposition 1 characterizes the unique symmetric strictly increasing and diﬀerentiable BNE.

Proposition 2 shows that in the equilibrium described in Proposition 1, an increase in the extra wage

premium will lead to an increase in the submitted bid of each skill level. The author

interprets this as more agressive test-taking behavior emerges if college Hguarantees

a higher wage.

Proposition 3 shows that if the probability that your opponent would fail to complement their

skills with UTSK increases, then the submitted bid in the equilibrium described in

Proposition 1 increases for each skill level as well.The author interprets this result as

more agressive test-taking behavior emerges in high income areas where there is more

access to UTSK compared to the low income areas with little access to UTSK.

Theorem 1 shows that the equilibrium payoﬀ to College H(in the equilibrium described in Propo-

sition 1) decreases when either the extra wage premium increases or the probability

that your opponent would fail to complement their skills with UTSK increases. The

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author interprets this result as an increase in the high quality college’s reluctance

when the extra wage premium increases or when one type of student has more access

to UTSK then the other.

Evaluation:

The current paper provides a highly stylistic model where there are only two students

and two colleges. Each student has a two-dimensional type: a skill type and a UTSK type.

The author is pretty honest about the limitations of his model. The model, even though it

is simple and highly stylized, seems to do a reasonable job in explaining stylized facts we

observe in real-life in the college admissions market.

I have the following comments/ questions that I see as problems for the current paper:

•I am confused about what it means for students to make bids in terms of exam scores?

As viand wiare ﬁxed (interim) once and for all, aren’t their exam scores given by

bi(vi, wi) are also ﬁxed? What does an ”unconstrained bidding function” b(v) mean?

I am not feeling comfortable to see the exam scores as bids. The author should clearly

explain this.

•How do we know the equilibrium described in Proposition 1 arises? In particular, how

do we know that there are no other equilibria?

•College admissions are repeated every year. On the other hand, the paper presents

a static model. It might be more realistic but technically challenging to model the

students as short-run players whereas the college as a long-lived player.

•Two students, two (non-decision maker) colleges is perhaps too simple. One wonders

how much of the results carry over to many students and many colleges.

•Can we explain all these phenomena with costly signalling? Here is a simple thought

experiment:

–UTSK is non-essential for college education but costly to acquire.

–High ability students might be investing in UTSK to convince colleges that they

are highly skilled given that everyone believes that high scores means on aver-

age high ability. (I would buy that UTSK level is highly correlated with real

skill/ability.)

–Colleges might be willing to enroll low-skilled students with rich or inﬂuential

parents. These might oﬀset the short-run costs with long-run beneﬁts. Hence,

this is why they are not in favor of a centralized admission. (e.g., There are

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many stories online regarding whether rich parents can buy their sons/daughters

a ticket to Harvard like schools.)

Comments on Typos and Exposition:

The following are my comments on exposition and typos:

•The paper is well-organized and relatively well-written.

•I believe there is something wrong in Equation (2) on page 6. I understand College

H’s payoﬀ from the words but I do not understand it stated as Π in Equation (2).

What is win Π? It is supposed to be a UTSK level but UTSK level of whom? This

causes confusion in the ﬁrst part of Equation (3) as well.

•What is FB1,2in the ﬁrst line of Equation (3)? Is this a typo?

•It is really confusing to use b(·,·) as a production function that gives the exam score

but also b(v) as a bidding function. The author should deﬁnitely ﬁx this.

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Report on MSS-D-21-00191 “An auction-theoretical model of grade in‡ation” by Sergey

Alexeev, for Mathematical Social Sciences

The paper considers a model equivalent to the private-value all-pay auction with

private budget constraints. Overall, it is a special case of the model studied by Kotowski

and Li (2014). The paper investigates the e¤ect of changes in the bidders’marginal value

of the object (represented by the parameter in the paper) and budget constraints

(represented by the distribution of budgets FW) on symmetric equilibrium bidding

functions and the college value function (which is similar but not equivalent to the social

welfare).

The key idea of the paper is to represent this setup as a model of centralized

college admission in which bidders’values represent applicants’abilities, bidding functions

represent exam grades, and the budgets represent unproductive test-speci…c knowledge

(UTSK). Selling an object is equivalent to admittance to a college. The paper establishes

that an increase in the bidders’ marginal value and relaxing their budget constraints

results in the uniformly more aggressive bidding (the latter result is doubtful as explained

below). As a result, the college value function decreases. Based on these results, the paper

claims that it explains the test grade in‡ation, which is equivalent to more aggressive

bidding, and predicts colleges’reluctance to participate in centralized college admission,

because their value function decreases.

Assessment

I don’t think the proposed model and the results are applicable to centralized college

admission. My main critiques are as follows:

1. The budget constrains do not play a role of unproductive test knowledge. First, they

are given exogenously in the paper, whereas the amount of e¤ort put in the test

preparation is endogenous. Second, the proposed exam-mark function (equivalent

to the equilibrium bidding function in an auction) of the form

bi(vi; !i) = min fbi(vi); !ig;

where viis the applicant ability and !iis the test-speci…c unproductive knowledge,

is not a good approximation of the actual function. The paper claims that

“complementary captures an observation that an excess of UTSK does not on

its own increase the applicant’s marks; similarly, being exceptionally skillful does

not translate into higher marks in the absence of UTSK ”, but it does not sound

convincing. The key reason for using test-speci…c preparation is that is a less-costly

substitute (rather than a complement) for knowledge and skill accumulation.

2. The paper assumes that the an extra wage premium (equivalent to the marginal

value of the object in an auction) is independent on applicants’ skills. It does not

seem correct in general.

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3. The college value function is such that “if college admits the most skilled student,

it obtains a payo¤ equal to v(1), the …rst order statistic. If not, a payo¤ is v(2), the

second-order statistic”. On the other hand, the value function is de…ned as

= v(2) if bv(1)> !(2) and bv(2)> !(1) ,

where (i)is the i-th order statistic. However, if !(2) is very low, then the inequalities

above hold and b(1) = min bv(1); !(1) >min bv(2); !(2)=b(2). That is, the

more skilled applicant is admitted. Why is the payo¤ of the college determined by

v(2) in this case?

4. I don’t think Proposition 3 is correct. First, the proof claims “Let b>bW0(~v) =

bW(~v). Then FW0(!)< FW(!)and fW0(!)> fW(!).” In general, the

…rst-order stochastic dominance does not impose any relationship on the densities

of comparable distributions (even with the same support). Thus, I don’t see why

fW0(!)> fW(!). Second, the model is equivalent to that by Kotowski and Li

(2014) who performed a similar comparative statics (pp. 91). They conclude that

“a bidder’s strategy adjustment as budget constraints are relaxed is not monotone

across types ”. According to them, bW0(:)is neither greater nor less than bW(:)even

for more restrictive likelihood-ratio dominance and “the same qualitative ordering

that exists for the second-price auction does not carry over to the case of the all-pay

auction.”

To sum up, the paper creates an impression that the author attempts to represent

the centralized college admission as a rather narrow auction and use speci…c results from

the auction theory to make strong statements about the existing phenomena in college

admission processes. It does not seem to be a correct approach.

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