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An auction-theoretical model of grade inflation

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Abstract and Figures

The grade inflation is a substantial and long-sustained increase in test scores. I incorporate 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 admission. 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.
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An auction-theoretical model of grade inflation
Sergey Alexeev
The University of New South Wales
Sydney, NSW 2052, Australia
November 1, 2021
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.
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
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
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
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.
Figure 1: Exam marks elevation
2012 2013 2014 2015 2016 2017
% of intakes with marks above 70
Exam marks
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
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
Figure 2: Growth of TP industry and CCA exemptions
2012 2013 2014 2015 2016 2017
Number of test-preparation schools
Federal cities
Regional capitals
Towns and villages
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
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).
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
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 w1w. 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.
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 βbi
/viif admitted and bi
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:
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)
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
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.
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)
=βFV(x1) + β(1 FV(x1))FW(b(x1)) b(x1)
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) = β
yfV(y)dy. (5)
(b) For all v˜v,b(v)is the strictly increasing solution of the differential equation
b0(v) =
satisfying the boundary condition bv) = 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) +
dtU1(b(t)|v, w)t=y
dy. (7)
4See Fang and Parreiras (2002) and Kotowski and Li (2014) for a similar approach.
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
dtU1(b(t)|v, w) = d
dtβFV(t) + β(1 FV(t))FW(b(t)) b(t)
=b0(t)β(1 FV(t))fW(b(t)) 1
v+βfV(t)(1 FW(b(t)))
= (vt)β(1 FW(b(t)))fV(t)
v(1 + (FV(t)1)fW(b(t)))
(tt)β(1 FW(b(t)))fV(t)
t(1 + (FV(t)1)fW(b(t))) = 0.
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.
dtU1(b(t)|v, w) = β
tfV(t)dt β
tfV(t)dt β
tfV(t)dt = 0.
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 bv) = 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
Figure 3: Functions ˆ
b(v) and b(v) from Example 1.
0.2 0.4 0.6 0.8 1
b(v), b(v)ˆ
(b) For all v˜v= 0.57,b(v)is the solution of the differential equation
b0(v) =
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 wb(1), function b(v) never emerges
and applicants of all types behave in accordance with ˆ
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.
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
b(v), b(v), w ˆ
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
(b) lim
(c) bβ0(v)> bβ(v)>ˆ
bβ(v)for β > FW(bβ(v))
Proof. A direct calculation gives (a)
v˜v+b0(v) = lim
βvfV(v)(1 FW(b(v)))
1βvfW(b(v))(1 FV(v))
=β˜vfVv)(1 FW(b(˜v)))
1β˜vfW(bv))(1 FV(˜v))
=β˜vfVv)(1 FW(w))
1β˜vfW(w)(1 FVv))
1β˜vfW(w)(1 FVv))
1β˜vfW(w)(1 FVv))
| {z }
> β˜vfV(˜v) = lim
The third line uses a boundary condition bv) = w, whereas the last line uses a differential
equation for case v˜v.
To see (b), note that the above showed
v˜v+b0(v) = ψ(β) lim
v˜vb0(v) (12)
ψ(β) = 1
1β˜vfW(w)(1 FVv)) >1.(13)
Clearly, ψ0(β)>0 as
∂β 1
1β˜vfW(w)(1 FVv)) !=fW(w)(1 FVv))˜v
((FVv)fW(w)fW(w))˜vβ + 1)2>0.(14)
An increase in βreinforces the gap between b(·) and ˆ
A direct calculation gives (c)
βvfV(v)(1 FW(b(v)))
1βvfW(b(v))(1 FV(v)) > βvfV(v)
β > FW(b(v))
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)=1exp(θ(ww))
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.
Figure 5: Comparative statics in θusing Example 1.
0.2 0.4 0.6 0.8 1
b(v), b(v), w ˆ
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))
1βvfW(b)(1 FV(v)) <βvfV(v)(1 FW0(b))
1βvfW0(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 vv, z), where zsufficiently
small. Fix ˜v < v < z then
fW0(bW0(v)) fW(bW0(v)) fW(bW(v)) (18)
FW0(bW0(v)) FW(bW0(v)) FW(bW(v)).(19)
Therefore, b0
Finally, because bW(v) and bW0(v) can be expressed as integral equation,
bWv) = bW(z)Zz
W(x)dx > bW0(z)Zz
W0(x)dx =bW0v) (20)
which contradicts bW0v) = 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).
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
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
educational institutions. If a country is already in a state of high-income inequality, with polarized
labor and educational markets, then the model suggests that the invalidation of UTSK might be
useful. Theoretically, the invalidation can be achieved by performing centralized testing only
on a number of randomly selected subjects instead of all of them, and the selection should be
announced as late as practically possible.
Another implication of the model is that the CCA exams should not be used as the primary
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
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Title: An auction-theoretical model of grade inflation
Mathematical Social Sciences
Dear Dr Sergey Alexeev,
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Report 1102021.pdf
Report on MS# MSS-D-21-00191
“An auction theoretical model of grade inflation”
by Sergey Alexeev
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
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.
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 βbi
viif admitted to Hand bi
- 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
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.
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
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
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
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.
Report on MSS-D-21-00191 “An auction-theoretical model of grade ination 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
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.
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.
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 pay 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
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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