Department of Policy and Planning Sciences
Discussion Paper Series

No.1325

Incentives and information order withapplications

by
Michiko OGAKU

Mar 2015

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPANIncentives and information order with applications∗†
Michiko Ogaku‡
Preliminary
20 March 2015
Revised 29 February 2016
This version: 13 December 2017

Abstract
One might predict students to be less motivated to work when evaluated according to a coarse rating pattern, such as pass or fail, rather than a primary score(1, 2, . . . , 100). However, coarser rating patterns often induce more eﬀort. We ask
what conditions will guarantee that the primary score is the most motivational grading. This paper provides a simple suﬃcient condition for this: the primary scoreis the most motivational grading when a higher rank is associated with greater
ability by the market, and when greater eﬀort increases the chance to achieve ahigher rank at any level of eﬀort. The proposed condition is given via two simple
functions, a market belief and a likelihood ratio, which do not require knowledge
for the distribution of ability. Thus, this may open an avenue for examinations ofa theory on the impact of information on implicit incentives by experimental andempirical research.

Key words: career concerns; Blackwell; incentives; grading; transparency.
JEL: D86; D83.
∗

I thank participants at EBES, EARIE and BEAM Kyoto International Conference for helpful comments; and Ishii Memorial Security Research Promotion Foundation for financial support.
†
Earlier versions of this paper were circulated under the title “Career concerns and negatively correlated signals conditional on garblings thereof.”‡
Faculty of Economics, Nagasaki University, 4-2-1 Katafuchi, Nagasaki, 850-8506 Japan.

1Introduction

One might predict that students would be less motivated to work when a coarse scoresuch as pass or fail is disclosed to parties such as university committees, graduate schoolsor employers (hereafter, the market) rather than the primary score, 1, 2, 3, . . . , 100. In
fact, the answer to this question is mixed.
The purpose of this paper is to explore how this garbling of information, which is atypical example of a transformation of information structure in the sense of Blackwell(1951, 1953), has an impact on incentives for individuals (e.g. students, researchers andCEOs) who care about the evaluation their ability by the market.
A prominent study by Dewatripont, Jewitt and Tirole (1999, hereafter, DJT) provided
conditions for identifying the impact of garbling on eﬀort. However, these conditions are
not satisfactory for rating problems such as grading students. Because their conditions
require a joint probability density function for a garbled report and ability, one cannot
directly apply the conditions to a rating pattern that has a probability distribution with,for example, masses of 1/6, 1/2 and 1/3 at A, B and C, respectively.
The main contribution of this paper is to provide a suﬃcient condition for primaryscore reporting to be the most motivational grading: (i) if the market belief about ability
after observing the primary score, E[θ|Xâ = x], is an increasing function; and (ii) if thelikelihood ratio of the primary score is an increasing function, which is often referred as
the monotone likelihood ratio property (MLRP), at any level of eﬀort.
The conditions presented extend the corresponding conditions given by DJT to include
grading reports such as a letter grade. In addition, our conditions are more accurate withregard to equilibrium analysis.
Importantly, because this paper’s conditions do not require knowledge of the distribution of ability, it is easier to test in experimental and empirical research. In the context ofgrading, the market belief about ability may be increasing with respect to score. Therefore, if the primary score satisfies MLRP at any level of eﬀort, then the primary scorewill be the best grading for motivating students to work.
Interestingly, Dubey and Geanakoplos (2010) investigated a similar question and pro-2

vided an answer that is inconsistent with this paper’s conclusion: the primary score
cannot be better than a properly garbled score. The reason that the diﬀerence arises is
as follows. They assumed that students who compete with each other for relative ordering are ordered by ability ex ante, and no chance is given to change this order by eﬀort.
Under this assumption, each student knows his/her ability and the primary score reveals
their ranking, while a coarser letter rank gives an opportunity to be mixed with higher
ranking students through eﬀort. On the other hand, this paper supposes that the ability
is not known by everyone ex ante, which is perhaps in the spirit of a remark in Lazear
and Rosen (1981): contests are feasible only when chance is an important factor.
Rodina (2016) independently showed a result similar to ours in a study of principal’s
information design.1 Their result shows a suﬃcient condition under which fully revealing
the output gives the best pure strategy equilibrium. The major diﬀerence with this paper
is that Rodina (2016) does not assume diﬀerentiability of the distribution of signals withrespect to eﬀort, but instead assumes supermodularity of the conditional expectation ofability given eﬀort and output with respect to eﬀort and output.
However, it may be diﬃcult to test the supermodularity in Rodina (2016), or whether
an increase in the market belief about ability under an increase in output does not decrease
when greater eﬀort is conjectured by the market. Our suﬃcient condition is much easier
to check. In addition, our proof is simpler and straightforward.
This paper also relates to the literature on the relationship between formal incentive contracts and information order. The important implication from the literature isthat garbled information makes second-best contracts more expensive (Holmström, 1979;
Grossman and Hart, 1983; Kim, 1995). Thus, more information is better for the principal if formal contracts based on the information are feasible. Such a formal contract isinfeasible for the motivational grading studied in this paper.
The rest of this paper is organised as follows. The next section presents a motivating
example in which pass/fail reporting is a better incentive device than full reporting.
Section 3 contains the main result. Section 4 summarises the analysis and provides
1

I would like to thank Professor Barton L. Lipman for kindly pointing out this article.

3

conclusions.

2

Motivating example

The example consists of a market, a professor and a representative student. The professor’s problem is choosing a disclosure rule that encourages the student to work. The
student’s problem is choosing an optimal level of eﬀort to raise the market belief about
his/her ability as much as possible.2 The timeline of events for this example is as follows.
1. The professor decides the disclosure rule; full or pass/fail.
2. The student decides the level of eﬀort.
(The student does not know his/her own ability ex ante)
3. The signal for ability is realised.
4. The signal is reported in accordance with the rule.
5. The market updates beliefs about ability according to the report.

Signals
We define a probability space (Ω, Σ, P ) for signals as follows. Let Ω = {wi , 1 ≤ i ≤ 4},
where the elements wi of R2 are specified by

w1 = (0, 0), w2 = (1, 1), w3 = (5, 5),
w4 = (0, 1).

Let Σ be the power set of Ω, and let P [{w1 }] = P [{w4 }] = 1/6 and P [{w2 }] = P [{w3 }] = 1/3.
Next, consider a process X1 , X2 , where Xi (ω) is the ith component of wi , so that
X1 (w4 ) = 0, X2 (w4 ) = 1. Let X1 be the grade without eﬀort. For simplicity, let X1 be
the perfect signal for ability in the sense that X1 = θ. Let X2 be the grade with eﬀort.
2

A similar model can be found in DJT. We change it so as to satisfy the condition used in Section 3:
any realisation of the signal whose distribution is chosen strategically by the student is measurable using
the distribution of signals that the market conjectures.

4

Reward and cost
The student’s reward is the market’s belief of his/her ability after observing the report.
The report is either full disclosure of the grade in the exam or the garbled disclosure ofwhether the student passed or failed the exam. We assume that those whose grade is 1
or 5 pass the exam and fail otherwise. We assume that no-one knows his/her ability exante, and that the student and the market know the distributions of signals.
The student’s eﬀort incurs a private cost c(index) that satisfies12< c(2) − c(1) < .
65

Expected rewards
Table 1 indicates the student’s rewards and the corresponding probabilities under the full
disclosure rule. The expression E[θ|X1̂ = x] (resp. E[θ|X2̂ = x]) represents the market
belief about ability after observing the grade x when the market conjectures that thestudent shirks (resp. works). If the student does what the market has conjectured, thenhis/her expected reward is E[θ] = 2. If the student works when the market conjecturesthat he/she shirks, then the student’s expected reward is: E[E[θ|X1̂ ]] = 1×3/6+5×2/6 =13/6. If student shirks when the market conjectures that he/she works, then the student’s
expected reward is: E[E[θ|X2̂ ]] = 1/3 × 2/3 + 1/3 × 5 = 17/9.
Table 1: Rewards and probabilities under the full disclosure rule
E[θ|X1̂ = x]

x=00

E[θ|X2̂ = x]

0
1
3
1
6

P [X1 = x|a = 1]
P [X2 = x|a = 2]

x=1
1
2
3
1
3
3
6

x=5
5
5
1
3
2
6

Rewards and probabilities corresponding to the pass/fail disclosure rule are obtainedin a similar manner.
5

Table 2: Rewards and probabilities under the pass/fail disclosure rule
E [θ|X1̂ ∈ {{0}, {1, 5}}]

fail: {0}0E[θ|X2̂ ∈ {{0}, {1, 5}}]0

P [X1 ∈ {{0}, {1, 5}}|a = 1]

1
3
1
6

P [X2 ∈ {{0}, {1, 5}}|a = 2]

pass: {1, 5}
3
12
5
2
3
5
6

Perfect Bayesian equilibrium
In equilibrium, the student shirks under the full disclosure rule. This is because, whenthe market conjectures that the student shirks and the student decides to work, then the
expected reward is 13/6. However, the net benefit is less than the cost: 13/6 − 2 = 1/6 <c(2) − c(1). Second, when market conjectures that the student works and the student

decides to work, the expected reward is 2. However, the net benefit is again less than the
cost: 2 − 17/9 = 1/9 < c(2) − c(1).
On the other hand, the student works under the pass/fail disclosure rule in equilibrium. Because the net benefit of working is more than the cost no matter what themarket conjectures. Thus, the professor would be better oﬀ if he chooses the pass/faildisclosure rule ex ante.
Remark 2.1. The model structure in this example is similar to that in Grossman andHart (1983). However, this result contradicts one of their results which shows that adecrease in informativeness in the sense of Blackwell makes the second-best contract lesseﬃcient. This is because the market belief is not the second-best contract controlled by
the principal: if the market belief were a contract, then the principal could oﬀer 12/5 fora grade of either 1 or 5, which would induce the student to work under the full disclosure
rule also.

6

MLRP is important
The above result is not entirely rigid. For example, suppose that those with an ability of
1 have a chance to obtain the highest score with probability 1/2;

w1 = (0, 0), w2 = (1, 1), w3 = (5, 5),
w4 = (0, 1), w5 = (1, 5).

Then, the full disclosure rule induces more eﬀort. In both examples, E[θ|Xâ = xi ] isincreasing and the first-order stochastic dominance holds;
P [X2 ≤ xi ] ≤ P [X1 ≤ xi ]∀xi .

The key diﬀerence is that MLRP is not satisfied in the former example, while it is satisfied
in the latter example. To see this, let pij be the probability that x = xj when the eﬀort
is i under the full disclosure rule. Then, under the full disclosure rule, the diﬀerence
between the expected reward of working and shirking when the market conjectures thatthe student works is
(

)

(

)

p22 − p12
p21 − p11
E[θ|X2̂ = x1 ] · p21 +E[θ|X2̂ = x2 ] · p22
p21
p22
(
)
p23 − p13+
E[θ|X2̂ = x3 ] · p23
p23

(1)

Similarly, let qij be the probability that z = zj when the eﬀort is i under the pass/faildisclosure rule. Then, under the pass/fail disclosure rule, the diﬀerence between the
expected reward of working and shirking when the market conjectures that the student
works is[]p2i − p1i(q21 − q11 )E[θ|X2̂ = x1 ] + E
X2 ∈ {x2 , x3 } (p22 + p23 )E[θ|X2̂ ∈ {x2 , x3 }]. (2)p2i

7The diﬀerence of (1) and (2) can be written as the following conditional covariance:[(p2i − p1i
Cov E [θ|X2̂ = xi ] ,
p2i

)

]

X2 ∈ {x2 , x3 } (p22 + p23 ).

Observing the above conditional covariance, it seems that the full disclosure rule induces
more eﬀort if the market belief after observing the signal, E[θ|Xâ = xi ], and the likelihood
ratio of the signal, (p2i − p1i )/p2i , are conditionally positively correlated given the resultof pass or fail. This observed relationship holds more generally, which is verified in thenext section.

3

Model

Consider a family of random variables, {θ, Xa , a ∈ (a, ā)}, on a probability space, (Ω, Σ, P ).
Let Xa be a signal about the agent’s unknown ability θ when the unobservable eﬀort isa ∈ (a, ā), and let PXa (·) be the distribution of Xa :{PXa (B) = P ω : Xa (ω) ∈ B a}, B ∈ B(R).

Suppose that for a fixed B ∈ B(R), PXa is a Borel-measurable function of a that is
diﬀerentiable with respect to a ∈ (a, ā) and the partial derivative ∂PXa /∂a is absolutely
continuous with respect to PXa . Then, by the Radon-Nikodým theorem, there is a Borelmeasurable function h such that
∫
∂PXa
(B) =
h dPXa for all B ∈ B(R).
∂a
B

(3)

We call h the likelihood ratio of Xa , written as fa /f (x|a).
In addition, suppose that for any a ∈ (a, ā), Xa is σ(X)-measurable and vice versa,
σ(Xa ) = σ(X), and that for any a ∈ (a, ā), {Xa (B) : B ∈ σ(X)} = {X(C) : C ∈ σ(X)},
where X is a signal about ability when eﬀort is not determined, and σ(X) is the smallest
σ-algebra generated by X.
The agent’s reward is the market’s ex post belief about ability after observing a report8

based on the signal. To alter the report by eﬀort a ∈ (a, ā), the agent privately incurs
the cost of eﬀort, c(·) : (a, ā) → R, which is strictly increasing and convex; c′ > 0, c′′ > 0.
The timeline is the same as in the example in Section 2.

3.1

Marginal reward

Consider the case where the authority chooses full disclosure of Xa . Then, the agent’s
expected reward when the market conjectures eﬀort â is∫E[θ|Xâ = x]dPXa (x).

The agent’s objective function is given by∫R

E[θ|Xâ = x]dPXa (x) − c(a).

(4)

Assuming an interior solution, the marginal reward is given by∫

E[θ|Xâ = x]fa(x|a)dPXa (x).
f

(5)

3.2 A garbled report
To define a garbled report, which includes coarse ratings such as A, B and C, we use
the concept of suﬃciency and measurability. First, we restate the concept of a stochastictransformation and suﬃciency from Blackwell (1953).
Definition 3.1. (Stochastic transformation) Let B, C be Borel fields of subsets of Ω1 , Ω2 ,respectively. A stochastic transformation T is a function Q(x, C) defined for all x ∈ Ω1
and C ∈ C. For fixed C, Q(·, C) is a B-measurable function of x; for fixed x, Q(x, ·) is a
probability measure on C. For any probability measure m on B, the function∫
M (C) =Q(x, C)dm(x)is a probability measure on C, denoted by T m.
Definition 3.2. (Suﬃciency) Consider two experiments, {mi , i = 1, . . . , N } and {Mi , i =1, . . . , N } with mi and Mi defined on Borel fields B and C of Ω1 , Ω2 , respectively. We
9say that {mi , i = 1, . . . , N } is suﬃcient for {Mi , i = 1, . . . N } if there is a stochastictransformation T with T mi = Mi for all i = 1, . . . , N .
In Blackwell (1951), i = 1, . . . , N denotes an unknown state of nature. This is slightly
modified in our model to consider the unobservable eﬀort a ∈ (a, ā). Suppose that{ma , a ∈ (a, ā)} is suﬃcient for {Ma , a ∈ (a, ā)}. Then, we can consider corresponding
random variables X and Z by specifying the following joint distribution. The conditional
distribution of X given a is ma = PXa ; the conditional distribution of Z ∈ C given a, x
is Q(x, C), which is a Borel-measurable function of x only. Then, Ma is the conditional
distribution of Z given a, namely Ma = PZa such that∫

PZa (C) =Q(x, C)dPXa (x)(6)

holds.
In addition to the concept of suﬃciency, we assume that there is a measurable function
h : R → R such that Za = h(Xa ). By the Doob-Dynkin lemma, this holds if and only
if Za is σ(Xa )-measurable, that is, σ(Za ) ⊂ σ(Xa ) (see e.g. Rao and Swift, 2006). This
assumption guarantees that anyone who knows the realisation of Xa also knows the rank
in the ranking report for Za for sure. An equivalent assumption is made in DJT using thestatistic T = T (X) as a garbled report in (see Lemma 5.1 in Dewatripont et al., 1999).
Definition 3.3. (Garbled report) Let {PXa , a ∈ (a, ā)} be suﬃcient for {PZa , a ∈ (a, ā)},and let Z be σ(X)-measurable. Then Za is a garbled report made with Xa .
Remark 3.4. Letter grading can be seen as a partition of scores, and thus is a garbled
report. This is easily checked by setting Q(x, C) to be an index function: For example,
consider a partition P = {Ai , i = 1, . . . , n} on Ω1 = Ω2 , and let Q(x, Ai ) = I{x∈Ai } , Ai ∈P.
The concept of suﬃciency implies the following useful lemma.
Lemma 3.5. (Martingale property) If {PXa , a ∈ (a, ā)} is suﬃcient for {PZa , a ∈ (a, ā)},10then the martingale property:[gafa(z|a) = E(x|a) Za = zgf]a.e. [PZa ] f or a ∈ (a, ā)(7)holds, where ga /g(z|a) is the likelihood ratio of Za .
Proof. Consider (Ω1 , F1 ) and (Ω2 , F2 ), and let Ω = Ω1 × Ω2 , F = F1 × F2 , Xa (x, z) = x
and Za (x, z) = z.
Suppose that {PXa , a ∈ (a, ā)} is suﬃcient for {PZa , a ∈ (a, ā)}. Then, the conditional
distribution of Z ∈ C ∈ F2 given x ∈ Ω1 , a is written as Q(x, C), which is a function ofx only. Let D((z, a), B) be the conditional distribution of x ∈ B ∈ F1 given z ∈ Ω2 , a.
Let µa be the unique measure on F determined by PXa and Q(x, ·). Then, µa is
determined by D((z, a), ·) and PZa as well, since by Fubini’s theorem for B ∈ F1 and forC ∈ F2∫ ∫µa (B, C) =

B

∫ ∫
C

Q(x, dz)dPXa (x) =C

B

D((z, a), dx)dPZa (z).

Applying Fubini’s theorem further, the following equations hold:
∫

∫
fa
fa
(Xa |a)dµa =(Xa |a)I{Za ∈C} dµaΩ f{Za ∈C} f∫ ∫fa=
IC (z) (Xa (x, z)|a)D((z, a), dx)dPZa (z)
f
Ω2 Ω1
]
∫ [∫
fa
=
(x|a)D((z, a), dx) dPZa (z)
C
Ω1 f
[
]
∫
fa
=
E
(x|a) Za dPZa (z)
f
C

(8)
(9)

However, (8) can be written as
∫ ∫
fa
fa
(x|a)D((z, a), dx)dPZa (x) =(x|a)Q(x, dz)dPXa (x)Ω1 C fΩ1 f∫fa=(x|a)Q(x, C)dPXa (x)Ω1 f∂PZa (C)
=
∂a
∫
ga
=
(z|a)dPZa .
C g

∫ ∫
C

11

(10)
(11)

Equations (10) and (11) are obtained by diﬀerentiating (3) and (6) with respect to a,respectively.
We obtain (7) from (9) and (11).

3.3 Comparison of marginal reward
Using Lemma 3.5, the diﬀerence in marginal reward under Xa and under Za can be
written as the conditional covariance of the market belief after observing Xa and thelikelihood ratio of Xa given Za . More precisely, we have the following result.
Lemma 3.6. Suppose that {PXa , a ∈ (a, ā)} is suﬃcient for {PZa , a ∈ (a, ā)}, and that
Za is σ(Xa )-measurable. Let the equilibrium eﬀort under Za be a∗ . Then, if the market
belief after observing Xa , E[θ|Xâ = x], and the likelihood ratio of Xa when the eﬀort isa∗ , fa /f (x|a∗ ), are conditionally strictly positively correlated given Za∗ , then the agent
will make strictly greater eﬀort when the market observes Xa .
Proof. Since {PXa , a ∈ (a, ā)} is suﬃcient for {PZa , a ∈ (a, ā)}, using Lemma 3.5, themarginal reward under the garbled report when the market conjectures eﬀort ǎ is given
by
∫

[

]

fa
E[θ|Zǎ = z]E(x|a) Za = z dPZa (z).
f(12)

Since Za is σ(Xa )-measurable, Za is σ(Xâ )- measurable as well and (12) can be written
as
∫

[

]

[

]

fa
(x|a) Za = z dPZa (z),E E[θ|Xâ = x] Zǎ = z Ef(13)

which follows from the commutativity property of the conditional expectation operator(see e.g. Rao and Swift, 2006).
Let a∗ be the perfect Bayesian equilibrium eﬀort under Za . Then, the equilibriumeﬀort is a = ǎ = a∗ . Setting a = ǎ = a∗ and subtracting (13) from (5) yields the diﬀerence12

in the marginal reward under Xa and that under Za when the eﬀort is a = ǎ = a∗ :
∫

[

]

fa
E E[θ|Xâ = x] (x|a∗ ) Za∗ = z dPZa∗ (z)
f
] [
]
∫ [
fa
− E E[θ|Xâ = x] Za∗ = z E(x|a∗ ) Za∗ = z dPZa∗ (z)
f
[
]
∫
fa
∗
= Cov E[θ|Xâ = x], (x|a ) Za∗ = z dPZa∗ (z).
f

(14)

From the given hypotheses in this lemma, the sign of (14) is positive.
Let a∗∗ be the perfect Bayesian equilibrium eﬀort under Xa . Then, a∗∗ maximises theagent’s objective function under Xa in (4). Thus, the equilibrium marginal reward under
Xa must be higher than the marginal reward under Xa when the eﬀort is a∗ . Since themarginal cost function c′ is strictly increasing with respect to eﬀort, a∗∗ > a∗ holds.
Remark 3.7. In the construction of a garbled report, Xa and Za are supposed to be
univariate. However, Lemma 3.6 holds if Xa and Za are random vectors that satisfy themartingale property.
Remark 3.8. Lemma 3.6 extends the corresponding lemma in DJT (Lemma 5.1) toinclude ranking reports such as letter grades. In addition, Lemma 3.6 follows from amore accurate analysis of the diﬀerence in equilibrium eﬀort under Xa and under Za
than Lemma 5.1 in DJT.
Using Lemma 3.9 below, the suﬃcient condition given in Lemma 3.6 can be simplified
so that the correlation is not required to be conditional.
Lemma 3.9. (Sign of conditional covariance) Let (Ω, Σ, P ) be a probability space, and
let G ⊂ Σ be any σ-algebra. Let X be a real valued random variable, and let g and h be
two bounded increasing real functions. Then∫GGE [g(X)h(X)]dP −

∫

∫

G

E [g(X)]dPGGThe proof is in the Appendix.

13

EG [h(X)]dP ≥ 0

G ∈ G.

(15)

Proposition 3.10. Suppose that {PXa , a ∈ (a, ā)} is suﬃcient for {PZa , a ∈ (a, ā)} andthat Za is σ(Xa )-measurable. Let the equilibrium eﬀort under Za be a∗ . Then, if themarket belief after observing Xa , E[θ|Xâ = x], and the likelihood ratio of Xa when theeﬀort is a∗ , fa /f (x|a∗ ), are concordant functions of x, then the agent will make greater
eﬀort when the market observes Xa . Furthermore, if E[θ|Xâ = x] and fa /f (x|a) areconcordant for any eﬀort a, then Xa will be the most motivational reporting.
Proof. Let G = σ(Za ). Let h(Xa ) = fa /f (Xa |a), and let g(Xa ) = E[θ|Xâ ], which follows
from the Doob-Dynkin lemma; σ(Xâ ) ⊂ σ(Xa ) if and only if there is a measurable
function k : R → R such that Xâ = k(Xa ).
From Lemma 3.9, if g(Xa ) and h(Xa ) are increasing functions, then the sign of (14)is nonnegative. If −g(Xa ) and −h(Xa ) are increasing functions, then the sign of (14) isnonnegative.
Thus, if g(Xa ) and h(Xa ) are concordant for a = a∗ , then from Lemma 3.6, the agent
will make greater eﬀort. The last part of the proposition holds because g(Xa ) and h(Xa )are concordant for any a ∈ (a, ā).
Remark 3.11. The corresponding proposition in DJT (Proposition 5.2) requires thejoint p.d.f. of θ and x which satisfies the concept of aﬃliation, and the joint p.d.f. ofθ and z such that the p.d.f. of z conditional on (θ, x, a) is independent of a and θ. On
the other hand, Proposition 3.10 does not specify the distribution of θ as strictly. With
regard to θ Proposition 3.10 requires only the market belief after observing the signal Xa ,which enable the implications to be tested more easily.

4

Conclusion

This paper studies the impact of the primary score and coarser gradings on a student’s
motivation to work, and provides a suﬃcient condition for the primary score to be themost motivational grading. Specifically, the primary score is the most motivational whena higher rank is associated with higher ability by the market, and when greater eﬀort
increases the chance to increase rank at any level of eﬀort. If data are available, this canbe tested.
14

Appendix
Lemma 3.9 shows that conditional covariance also satisfies Harris’s inequality, which is
the well known inequality for covariance.

Proof of Lemma 3.9
Proof. Let Y be another random variable which is stochastically identical to X, that is,
X =st Y , and independent of X. Then, for any G ∈ G
0≤

∫
∫

G

=

(g(X) − g(Y ))(h(X) − h(Y ))dP∫

g(X)h(X)dP +
G

G

[∫

=2
[∫

G

g(X)h(X)dP −G=2Gg(Y )h(Y )dP −∫∫g(X)dPGE [g(X)h(X)]dP −∫∫G

g(X)h(Y )dP −∫]g(Y )h(X)dPG

h(X)dPGG∫

E [g(X)]dP
GG]

E [h(X)]dP .
GThe first inequality is true since for any ω ∈ G either X(ω) ≤ Y (ω) or X(ω) ≥ Y (ω),
and thus the two factors on the right-hand side have the same sign. From the bracket ofthe right-hand side of the last equation, we obtain the inequality in (15).

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