tigers and flies: corruption, discretion and expertise in ... · tigers and flies: corruption,...
TRANSCRIPT
Tigers and Flies: Corruption, Discretion and
Expertise in a Hierarchy∗
Philip H. Dybvig†
Washington University in Saint Louis
Yishu Fu‡
Southwestern University of Finance and Economics
(Preliminary Draft)August 28, 2018
∗We are grateful for comments from Michael Brennan, Pingyang Gao, Pierre Liang,Tiejun Wang, Jun Yang, Gaoqing Zhang and participants of workshops at WashingtonUniversity in Saint Louis, SWUFE. All errors are our own. Yishu Fu gratefully acknowl-edge financial support from Southwestern University of Finance and Economics Instituteof Chinese Financial Studies.†John M. Olin School of Business, Washington University in Saint Louis, Campus Box
1133, One Bookings Drive, Saint Louis, MO 63130-4899. Email: [email protected]‡Institute of Chinese Financial Studies, Southwestern University of Finance and E-
conomics, No.55 Guanghuacun Street, Qingyang District, Chengdu, P.R.China, 610074.Email: [email protected]
Abstract
We build a stylized theoretical model of decision-making in a hierar-
chy. The model is motivated by the ongoing anti-corruption campaign
in China, but the analysis should be applicable to the policing of con-
flicts of interest in many hierarchies in governments, firms and other
organizations. Borrowing terminologies from China, we can choose to
reduce corruption incentives of tigers (high-level potentially corrupt
officials) and flies (low-level potentially corrupt officials). We have
three main results. First, the anti-corruption campaign should go af-
ter both tigers and flies, the stated goal in China. Second, fighting
corruption and imposing rigid constraints are substitutes: doing ei-
ther one reduce the value of the other. Third, fighting corruption and
training the flies are complements: doing one increases the value of
the other. Fighting corruption and training can be used together.
JEL classification:
Key Words: Anti-corruption Campaign, China, Tigers and Flies,
Information, Discretion, Expertise
1 Introduction
Managing conflicts of interest is a big challenge for hierarchies in governments
and organizations. We can think of the conflict of interest as agency prob-
lems as modeled by Ross (1973) and Holmstrom (1979), or as rent-seeking
behaviour described by Krueger (1974) as the primary source of inefficiency
in developing countries. Inspired by the ongoing anti-corruption program in
China, we present a stylized model of decision-making in a hierarchy with
high-level agents (tigers) and low-level agents (flies), all of whom are conflict-
ed. The agency problems can be ameliorated by fighting corruption (which
we will take to mean the same as policing conflict of interest) or by imposing
rigid constraints. We find that (1) if corruption is big, we want to reduce cor-
ruption of both tigers and flies (a stated goal of the anti-corruption campaign
in China), (2) fighting corruption and imposing rigid rules are substitutes in
the sense that if we do one, the other is not so useful, and (3) training the
flies and fighting corruption are complements in the sense that doing one is
more useful if we do the other. Fighting corruption plus training the flies
can move from a strict bureaucracy to a technocracy.
Our analysis is motivated by the large-scale anti-corruption campaign s-
tarted by Xi Jingping after he became the President of China in 2012. The
ongoing campaign has many features, including an austerity campaign, a
public-relations campaign, and an actual anti-corruption campaign.1 One
feature of the anti-corruption campaign is to reduce the corruption oppor-
tunities at different levels, so-called going after both the “tigers” and the
“flies.” 2 Our model adopts the colorful terminology from the Chinese anti-
1The austerity campaign restricts the use of the public funds for food, drink, gifts andentertainment. The public-relation campaign restricts self-promoting announcements andostentatious behaviors. For instance, there should not be a welcoming banner, red carpet,floral arrangement, or a grand reception celebrating official visits.
2SC²µ�j±/Pm0!/ñG0�å�"(Xi, Jinping: We must crack down on both“tigers” and “flies”.)
1
corruption campaign. In the model, a tiger decides how much discretion
to give the fly. The fly and the tiger have different objectives, but the fly
has information the tiger does not have, and the objectives are somewhat
aligned, so the tiger will give the fly some discretion. The choice of how
much discretion reflects the tiger’s rational anticipation of the fly’s use of
the discretion to help the tiger against the fly’s use of the discretion in ways
that hurt the tiger. When the fly’s incentives are well-aligned with the tiger,
as when the fly’s corruption opportunities are few, the tiger gives the fly a
lot of discretion. Alternatively, when the incentives are not well-aligned, as
when the fly’s corruption is high, the tiger will constrain the fly to have little
discretion. Given the level of abstraction in the model, it can be useful for
understanding many countries with different political systems, only with a
modest change in the interpretations of the social goals.
Our theoretical model looks at corruption from the perspective of three
different players: society, a tiger, and a fly. Preferences for society are hard
to agree on in practice but simple in our model as the expected quadratic
deviation from an exogenous random ideal point. In China, this ideal point
might be determined by the Communist Party’s assessment of what is best
for society. In a western democracy, it might be some economically efficient
benchmark. We are looking for results that are not dependent on the in-
terpretation of the social ideal point or the government’s political structure.
Discretion in our model is not freedom or democracy; rather, discretion is
only granted to the extent that incentives are there to serve the centralized
goals of people higher in the government hierarchy.
Both the tiger and the fly have their own preferences for deviation from
what is socially optimal, so we can talk separately about fighting corrup-
tion of the tiger and corruption of the fly. We abstract from the detailed
mechanics of the anti-corruption campaign, including the exact process for
identifying and punishing corruption, and the nature of the costs of the var-
ious elements of the campaign. Instead, we use a reduced form in which the
2
policy variable for fighting corruption of the tiger is the standard deviation
of the difference between tiger’s ideal point and the social optimum. For
each degree of corruption (for the tiger and for the fly) there is some cost
(in units of social welfare) to society of reducing corruption. For example, in
China, a clerk whose job it is to provide a stamp on verifying the necessary
steps have been taken to start a company might require dinner and a cash
gift (ù�—red envelops) before granting the stamp. Other examples include
granting monopoly power, cartel enforcement protecting financial interest of
big players. The hold-up problem includes taking bribes (such as equity in
the project, cash, real estate, etc.) to grant approval. This is a waste of
time and an additional cost of doing business, and some beneficial business
might not be started because of this. The cost of fighting corruption includes:
commitment of resources to investigators intervening, time spent providing
documentation to investigators, the anxiety of people being investigated, di-
rect and indirect damage from these positive people accused wrongly, and
paralysis from people afraid of being accused wrongly. Only the fly has any
information, in the form of a noisy signal of what the tiger wants. Everyone
has the same priors on the distribution of the random variables in the model.
Everyone knows the value of the policy variables, which are the extent of
fighting the tiger’s corruption, the extent of fighting the fly’s corruption, and
the expertise of the fly. To keep the algebra simple, we assume that society’s
ideal point, the two deviations, and the noise in the fly’s information are
independent normal variates.
Although this is an agency problem in that the tiger and the fly are
making decisions but their incentives may not be aligned with society, we do
not model this using an agency problem with optimal incentive contracting
as in Ross (1973) or Holmstrom (1979). Instead, we think of the tiger and
the fly as being separated by at least several layers of hierarchy and therefore
the tiger does not have direct control over compensation. Instead, the tiger
has limited control over the fly through rule-setting that imposes constraints
on the fly’s actions, and the policy variables are the two levels of corruption
3
(the result of the anti-corruption campaign targeted at the tiger and at the
fly) and the amount of expertise for the fly (the result of training). Because
the fly has superior information not available to the tiger and some common
interests, the tiger wants to give the fly at least some discretion. When
corruption opportunities for the fly are scarce (due to the anti-corruption
campaign), the fly’s interests are closely aligned to the tiger’s interests, and
the tigers will choose to give the fly a lot of discretion because the fly will
make choices similar to what the tiger would choose.
Our first theoretical result shows it is essential to fight corruption for
both tigers and flies, not just for the flies. If corruption is large, it might be
tempting to fight corruption for the flies and leave the tigers alone, since the
tigers are powerful and can fight back. However, reducing flies’ corruption
opportunities helps tigers’ corruption more than any social goals and makes
society worse off. We also show that tigers would like to impose a more
severe anti-corruption campaign on flies than is socially optimal. This is
because tigers get more than proportional benefits from the anti-corruption
campaign and bear less than proportional costs. Our second result finds
that part of the benefit of anti-corruption campaign for flies is the direct
impact from aligning incentives and part is the indirect benefit from taking
advantage of the better incentives by giving flies more discretion. In other
words, the anti-corruption campaign and strict rules are substitutes, and the
full benefits of an anti-corruption campaign come if we relax rules. Probably
it is tempting to impose stricter rules at the same time when we introduce an
anti-corruption campaign, thinking we are clamping down on everything, but
our model illustrates the strict rules reduce the benefit of the anti-corruption
campaign. Our third theoretical result shows that fighting corruption and
enhancing expertise are complements. If corruption is high, it makes sense
to constrain flies a lot, which neutralizes expertise. With low corruption and
high expertise, the economy can flourish.
The paper is organized as follows. Section 2 presents and solves a the-
4
oretical model of society, tigers and flies. Section 3 shows that when cor-
ruption opportunities are great, the anti-corruption campaign should crack
down on both tigers and flies to benefit society. Section 4 shows strict rules
and fighting corruption are substitutes. Section 5 shows fighting corruption
and enhancing expertise are complements. Section 6 closes the paper and
summarizes our results.
2 Society and Equilibrium with Tigers and
Flies
In this section, we present a theoretical model of an anti-corruption cam-
paign. We abstract from the hierarchy and consider two levels: the tiger and
the fly. Probably it is useful to think of these two levels as being separated
by several levels in a hierarchy, so the connection between the two is not so
tight. We assume that the tiger has different objectives from the society’s
because of the corruption. The fly is the tiger’s subordinate and consequent-
ly his preferences are similar to the tiger’s, but he also has his own objective
due to corruption. The tiger has imperfect control over the fly, executed by
issuing rules that determine how much discretion the fly has. This limited
control is why we think it makes sense to think of the tiger as being several
levels above the fly in the hierarchy. The fly can collect extra rents (for both
the fly and for the tiger) by deviating from the social optimum.
We assume the tiger and the fly jointly make a choice represented by
a real number X, subject to constraints imposed by the tiger. The tiger
does not have control over contracting as in the traditional agency literature
following Ross (1973) and Holmstrom (1979). Instead, the tiger can only
impose a constraint that the fly must choose X in some interval [¯X, X̄]. In
5
other words, the endpoints of [¯X, X̄] are chosen by the tiger.3 The tiger
knows that the fly is bribed but does not have any information to condition
on, so¯X and X̄ are constants. This limited degree of control is consistent
with what is reasonable when the tiger and the fly are separated by at least
several levels in the hierarchy.
The tension in the model comes from the fact that the tigers have the
authority but the flies have the information. To make this simple, we will
assume the flies know everything in the model, and the tigers know nothing.
We can think of this as conditioning on what the tigers know and computing
payoffs given what both know, without the algebraic burden of modeling this
explicitly. We assume that the society’s ideal choice is S. The social welfare
function is the expectation of
W S(X;S) = −(X − S)2 − CT (γT )− CF (γF )− Cn(σn),(1)
where S ∼ N(0, σ2S). The tiger likes to be close to society’s ideal point S, but
can seek some private rents from deviating from society’s ideal point. The
level of the anti-corruption campaign depends on two policy choices, γT and
γF . They represent the level of corruption for tigers and flies, respectively,
which can be reduced by the anti-corruption campaign. The fly’s expertise
level is inverse related with noise σn. CT (γT ), CF (γF ) and Cn(σn) give the
costs of fighting the tiger’s corruption, fighting the fly’s corruption, and train-
ing the fly, respectively. With smaller γT and γF indicating fewer corruption
opportunities and smaller σn indicating better skill for the fly. The three cost
functions CT (·), CF (·), Cn(·) have similar properties, and limγT→∞
CT (γT ) = 0
similar for others. For i = T, F and n, we assume that for x > 0, Ci(x) > 0,
C ′i(x) < 0, and C ′′i (x) > 0, etc. For simplicity, we assume the tiger and the
fly do not bear any of the cost of the anti-corruption campaign, although
3As seems natural given the quadratic loss function and multivariate normal setting, itcan be proven that if the tiger can choose any closed subset of < as the restriction on X,the optimal choice would be a interval.
6
what is important is the costs they bear are less than proportional to the
benefit. Thus, tiger’s utility is
UT (X;S, ζT ) = kTE[−(X − S)2 + 2ζT (X − S)](2)
= kT [γ2T + E(−(X − T )2)]
where ζT ∼ N(0, γ2T ), drawn independently of S, kT > 0 and
T ≡ S + ζT ,(3)
where T ∼ N(0, σ2T ) for
σ2T = σ2
S + γ2T(4)
because S and ζT are drawn independently. We take kT = 1, which is without
loss of generality.4 We interpret T as the tiger’s ideal choice if the tiger knew
S and ζT . In fact, neither S nor ζT is known by the tiger, or else the tiger’s
choice would depend on the information. For example, if the tiger knew both
S and ζT , the tiger would choose¯X = X̄ = S + ζT to force the fly to choose
the tiger’s ideal point. The rents extracted by the tiger from the deviation
from the social ideal point S are given by 2ζT (X − S).
The fly does not know the tiger’s ideal choice T. However, the fly knows
some information I, which consists of T and noise term εn ∼ N(0, σ2n). We
can write
I = T + εn(5)
for σ2I = σ2
T + σ2n. smaller σn indicates better information, or equivalently
4Society and the tiger do not receive the same scale of benefits from the anti-corruptioncampaign. However, taking kT = 1 simplifies the algebra without affecting our resultsbecause multiplying the objective function by a constant does not change the optimalchoice or ordering of alternatives.
7
more expertise, for the fly. If σn = 0, the fly knows I, while in the limit
σn ↑ ∞, the fly knows nothing about I. As mentioned earlier, we assume for
simplicity that εn is independent of S and T.
Given information I, the tiger’s ideal choice T is
T = βII + ηI(6)
where βI = cov(T, I)/var(I) = σ2T/σ
2I = σ2
T/(σ2T +σ2
n), var(ηI) = σ2T (1−βI),
and βII = E[T |I]. We can think of βI as a measure of the fly’s expertise.
The fly likes the outcome to be near to the tiger’s ideal point T, but can
get some private rents from deviating from the tiger’s ideal point. The fly’s
utility is expressed as
UF (X;T, ζF ) = kFE[−(X − T )2 + 2ζF (X − T )](7)
= kF [γ2F + E(−(X − F )2)− var(ηI)]
where ζF ∼ N(0, γ2F ), drawn independently of S and ζT , is known by the
fly. A case can be made that ζF is correlated with S-T, since bribes could
be for activities that produce benefits for society as well as for rent-seeking
activities. We included these feature in our original analysis but assuming
independence simplifies the algebra without changing the results significantly.
As for kT , kF > 0 and we set kF = 1 for simplicity. Based on (7), the fly’s
ideal choice F of X absent constraints given I can be expressed as
F ≡ βII + ζF(8)
where F ∼ N(0, σ2F ) for
σ2F = (βI)2σ2
I + γ2F = βI(σ2
T/σ2I )σ
2I + γ2
F = βIσ2T + γ2
F(9)
because information I and ζF are drawn independently. The fly knows I and
8
ζF , but the tiger only knows the joint distribution of these variables. The
privacy of the fly’s superior information prevents the tiger from forcing the
fly to choose X=T. The rents extracted by the fly from the deviation are
given by 2ζF (X − T ). Since ζF has mean zero, the tiger cannot anticipate
the directions of the fly’s preferred deviation.
Given joint normality, the conditional expatiations of T given F is given
by a linear regression
T = βTF + ηT ,(10)
where
βT =cov(F, T )
var(F )=βIσ2
T
σ2F
=βI(σ2
S + γ2T )
βI(σ2S + γ2
T ) + γ2F
=1
1 + γ2F/(β
Iσ2T )
and ηT ∼ N(0, σ2T (1 − βIβT )). There is no constant term in the regression
because F and T both have mean zero. The regression coefficient βT , a
number between 0 and 1, can be interpreted as the degree of alignment of the
fly’s preferences with the tiger’s. We can see that the alignment is increasing
in the fly’s expertise γF and decreasing in the level of the fly’s corruption
opportunities γF . The alignment of preferences is best when βT = 1 (and
var(ηT ) = 0), and worst when βT approaches 0 (and var(ηT ) approaches
σ2T ).
Given the choice of¯X and X̄ by the tiger, the fly’s optimal response X
is the projection of F on [¯X, X̄], given by
X = π(F,¯X, X̄) =
¯X, if F <
¯X;
F, if¯X ≤ F ≤ X̄;
X̄, if X̄ < F .
(11)
9
From (2) and (11) the tiger chooses¯X and X̄ to maximize expected utility
γ2T + E[−(π(F,
¯X, X̄)− T )2].(12)
Also, given that¯X and X̄ are chosen by the tiger, social welfare is
E[−(π(F,¯X, X̄)− S)2 − CT (γT )− CF (γF )− Cn(σn)].(13)
The first order condition for an optimum from Equation (12), derived in
Appendix B, implies that{¯X = −σF
¯x(βT )
X̄ = −¯X = σF x̄(βT )
(14)
where x̄(βT ) is the solution of βT = x̄N(−x̄)/n(x̄).
We do not know how to solve this explicitly for x̄ given βT , but almost
just as useful, this is a parametric solution for βT in terms of x̄ and σF . The
optimal x̄ is increasing in βT , with x̄ moving from 0 to ∞ as βT varies from
0 to 1, as shown in Appendix B.
3 Both Tigers and Flies
In this section, we fix noise σn and discuss the first implications of our full
model: consequences of reducing tiger’s corruption γT and fly’s corruption
γF for society. We show that when corruption opportunities are abundant,
reducing fly’s corruption (γF ) without reducing the tiger’s corruption (γT )
makes society worse off. In other words, when both the tiger and the fly are
very corrupt, the anti-corruption campaign should get the flies and the tiger-
s together to be effective. A second result in this section shows that tigers
always prefer a relatively higher level of anti-corruption campaign than is so-
10
cially optimally. This is because reducing flies’ corruption opportunities γF
always makes tigers better off. Besides, tigers receive more than proportion-
ally the benefits but bear less than proportionally the costs from reducing
flies’ corruption.
We illustrate this section by using figures first and then state them for-
mally in theorems. Figures 1A and 1B show how tiger’s utility (blue curves)
and social welfare (black curves) vary with different levels of the corruption
opportunities (γT and γF ), when noise σn is fixed (σn = 0 and σn = 1.5).
The contours of tiger’s utility shows that the tiger is always better off with
the increases in γT and decreases in γF . The contours of the society’s utili-
ty shows that various degrees of anti-corruption campaign leads to different
impacts on the social welfare. The maximum utility that the society can
achieve is denoted by the red dot. When both the tiger and the fly are very
corrupt (large γT and γF ), reducing fly’s corruption (γF ) only makes the so-
ciety worse off because it aids the tiger’s corruption. When the tiger is very
corrupt but the fly’s corruption opportunity is scarce (γF → 0), reducing
fly’s corruption (γF ) makes the society worse off because it is more costly to
the society. Formally,
Theorem 3.1. The anti-corruption campaign should crack down on both
tigers and flies together to be effective. It is because when both the fly and the
tiger are very corrupt, reducing the fly’s corruption opportunities γF reduces
the social welfare. We can write,
dW S
dγF=∂W S
∂γF|¯X,X̄ +
∂W S
∂¯X|γF ∗
d¯X
dγF+∂W S
∂X̄|γF ∗
dX̄
dγF
When γF and γT are large enough, we have ∂W S/∂γF |¯X,X̄ < 0, ∂W S/∂
¯X|γF ∗
d¯X/dγF = ∂W S/∂X̄|γF ∗ dX̄/dγF > 0, and dW S/dγF > 0.
Proof: See Appendix C.
11
1A. S
ocia
l Wel
fare
and
Tig
er’s
Util
ity w
ith la
rge
γ T w
hen
σ n =
0
Tig
er’s
cor
rupt
ion
oppo
rtun
ities
γT
Fly’s corruption opportunities γF
−1.
8 −
1.7
−1.
6 −
1.5
−1.
4 −
1.3
−1.
2 −
1.1
−1
−0.
9
−0.
8
−0.
7
−0.
6
0.0
0.5
1.0
1.5
0.00.51.01.5
−0.
4
−0.
2
0
0.2
0.4
0.6
0.8
1
1.2 1.4 1.6
1.8 2 2.2
1B. S
ocia
l Wel
fare
and
Tig
er’s
Util
ity w
ith la
rge
γ T w
hen
σ n =
1.5
Tig
er’s
cor
rupt
ion
oppo
rtun
ities
γT
Fly’s corruption opportunities γF
−1.
9 −
1.8
−1.
7 −
1.6
−1.
5 −
1.4
−1.
3 −
1.2 −
1.1
0.0
0.5
1.0
1.5
0.00.51.01.5
−0.6
−0.4
−0.2
0
0.2 0.4 0.6 0.8
Fig
ure
1Ash
ows
that
the
anti
-cor
rupti
onca
mpai
gnsh
ould
goaf
ter
bot
hth
eti
gers
and
the
flie
sto
be
effec
tive
.It
isb
ecau
sew
hen
bot
hth
eti
ger
and
the
fly
are
very
corr
upt
(lar
geγT
andγF
),figh
ting
fly’s
corr
upti
onγF
only
mak
esso
ciet
yw
orse
off(b
lack
curv
es).
Incr
easi
ng
tige
r’s
corr
upti
onop
por
tunit
iesγT
and
reduci
ng
fly’s
corr
upti
onop
por
tunit
iesγF
alw
ays
mak
esth
eti
ger
bet
ter
off(b
lue
curv
es).
Fig
ure
1Bsh
ows,
when
com
par
ing
opti
mal
leve
lsof
anti
-cor
rupti
onca
mpai
gnfo
rth
eti
gers
(gre
enso
lid
line)
and
for
the
soci
ety
(red
solid
line)
,ti
gers
alw
ays
hav
eth
ete
nden
cyto
choos
em
ore
stri
ctle
vel
ofan
ti-
corr
upti
onca
mpai
gnth
anis
soci
ally
opti
mal
.It
isb
ecau
seti
gers
rece
ive
hig
her
pro
por
tion
ofth
eb
enefi
tsbut
bea
rle
ssth
anpro
por
tion
ofth
eco
sts.
12
We also show optimal levels of anti-corruption campaign to reduce fly’s
corruption γF for the tiger (green solid line) and for the society (red solid
line). For any level of γT , the tiger always wants the fly’s corruption opportu-
nity is zero (γF = 0), indicated by the horizontal line. It is because reducing
γF always makes the tiger better off, and the tiger gets more than propor-
tionally the benefits but bear less than proportionally the costs. However,
for each level of γT , the society always wants a lower level of anti-corruption
campaign against the fly (γF > 0) because the society bears all the cost of
the anti-corruption campaign. Formally,
Theorem 3.2. The tiger always prefers a higher level of anti-corruption
campaign for the flies than is socially optimal. In particular, given σS, γT
and σn, the tigers want the flies to have no corruption (γF = 0), but society
is better off with some positive level of corruption (γF > 0).5
Proof: See Appendix C and D.
4 Information and Discretion
In this section, we fix noise σn, tiger’s corruption γT , and discuss the second
implications of our model: consequences of reducing fly’s corruption γF for
the tiger and for the fly. We show that the anti-corruption campaign directed
at the flies generate both direct and indirect benefits for the tiger. The direct
benefit is that anti-corruption reform aligns the fly’s incentives closer to the
tiger’s, inducing a choice closer to what the tiger wants given the same level
of discretion, i.e., the same choice of¯X and X̄. The indirect effect is that
5Note: The tiger wants to reduce the fly’s corruption opportunity γF to zero becausethe tiger does not bear any of the cost. If the tiger have a fraction of the cost, the tigerwould still want less corruption for the fly than is socially optimal.
13
the tiger optimally takes advantage of the improved alignment of incentives
by decentralizing more power and giving the fly more discretion by choosing
a larger range [¯X, X̄], putting more of the decision in the hands of the agent
with superior information by increasing the range of choices to the fly.
A second result in this section shows that increasing the fly’s corruption
opportunities γF makes the fly better off initially when γF is small, but then
worse off when γF is big. Increasing γF has two effects. It gives the fly more
profits given¯X and X̄, but it also induces the tiger to shrink the interval
[¯X, X̄] in anticipation of the fly deviating more from what the tiger wants,
which reduces profits for the fly. When γF is small, discretion is large, the first
effect dominates, and the fly profits from taking advantage of the increased
corruption opportunities. When γF is large, the second effect dominates, and
the fly is worse of when γF increases, because the main effect is that the tiger
reduces discretion.
We illustrate this section by using figures first, and then state them for-
mally in theorems. Figures 2A displays how the alignment of incentives
βT changes with the availability of relative corruption opportunities γF/σT ,
when relative noise σn/σT is fixed at different levels. We show that the align-
ment of incentives βT decreases in the relative corruption availability γF/σT .
Given the same level of relative corruption opportunities γF/σT , when the
fly has lower level of expertise (indicating by larger σn/σT ), the incentives
between the tiger and the fly are worse aligned (smaller βT ). Given the same
level of alignment βT between the tiger and the fly, expertise could help fly
to seek more rents from corruption opportunities (higher γF/σT ).
14
02
46
8
0.00.20.40.60.81.0
2A. A
lignm
ent o
f Inc
entiv
es β
T a
nd
Rel
ativ
e C
orru
ptio
n O
ppor
tuni
ties
γ Fσ T
Rel
ativ
e C
orru
ptio
n O
ppor
tuni
ties
γ Fσ T
Alignment of Incentives βT
σ nσ T
= 0
σ nσ T
= 1
σ nσ T
= 2
σ nσ T
= 3
σ nσ T
= 4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0102030
2B. R
elat
ive
Dis
cret
ion
Xσ T
and
Rel
ativ
e C
orru
ptio
n O
ppor
tuni
ties
γ Fσ T
Rel
ativ
e C
orru
ptio
n O
ppor
tuni
ties
γ Fσ T
Relative Discretion XσT
σ nσ T
= 0
σ nσ T
= 1
σ nσ T
= 2
σ nσ T
= 3
σ nσ T
= 4
Fig
ure
2Ash
ows,
give
nσ
2 T,
hav
ing
mor
ere
lati
veco
rrupti
onop
por
tunit
ies
(hig
herγF/σ
T)
reduce
sth
eal
ign-
men
t(β
T)
ofth
efly’s
and
tige
r’s
pre
fere
nce
s(a
bse
nt
any
const
rain
t).γF/σ
Tis
the
scal
eof
pro
fita
bilit
yfo
rth
ere
lati
veco
rrupti
onop
por
tunit
ies.σn/σ
Tis
the
leve
lof
rela
tive
noi
se.βT
isth
ere
gres
sion
coeffi
cien
tof
tige
r’s
idea
lp
oint
Ton
the
fly’s
idea
lp
oint
F.
The
tige
rch
oos
esth
era
nge
of[ ¯X,X̄
]to
const
rain
the
fly’s
choi
ces
ofX
.F
igure
2Bsh
ows,
hav
ing
mor
ere
lati
veco
rrupti
onop
por
tunit
ies
(hig
herγF/σ
T)
reduce
sth
edis
cret
ion
give
nto
the
fly.
The
opti
mal
inte
rval
issy
mm
etri
car
ound
0,i.e.
¯X=−X̄
ineq
uilib
rium
andX̄>
0.L
arge
rX̄/σ
Tco
rres
pon
ds
tola
rger
dis
cret
ion
give
nto
the
fly.
Ther
efor
e,th
ela
rgerγF/σ
Tis
,th
ew
orse
the
alig
n-
men
tb
etw
een
the
tige
r’s
and
fly’s
ince
nti
ves,
and
the
smal
ler
the
dis
cret
ion
gran
ted,
i.e.
,th
em
ore
the
tige
rco
nst
rain
sth
efly.
15
0.0
0.5
1.0
1.5
2.0
2.5
−3.0−2.5−2.0−1.5−1.0−0.5
3A. T
iger
’s U
tility
UT a
nd F
ly’s
Cor
rupt
ion
γ F
Fly
’s C
orru
ptio
n O
ppor
tuni
ties
γ F
Tiger’s Utility UT
Opt
imal
XX
= 0
.3X
= 0
.5X
= 1
X =
2
0.0
0.5
1.0
1.5
2.0
2.5
−1.0−0.50.00.51.01.52.02.5
3B. F
ly’s
Util
ity U
F a
nd F
ly’s
Cor
rupt
ion
γ F
Fly
’s C
orru
ptio
n O
ppor
tuni
ties
γ F
Fly’s Utility UF
Opt
imal
XX
= 0
.3X
= 0
.5X
= 1
X =
2
Fig
ure
3Ash
ows
that
,gi
venX̄
,th
eti
ger
isal
way
sb
ette
roff
ifth
eco
rrupti
onop
por
tunit
iesγF
for
the
fly
are
reduce
db
ecau
seth
isal
igns
thei
rin
centi
ves
bet
ter.
The
tige
rca
nfu
rther
explo
itth
eim
pro
ved
alig
nm
ent
ofin
centi
ves
by
givin
gth
efly
mor
edis
cret
ion.
When
γF
issm
all,
larg
edis
cret
ion
toth
efly
mak
esth
eti
ger
bet
ter
offb
ecau
sem
ore
jobs
are
del
egat
edto
the
agen
tw
ith
bet
ter
info
rmat
ion.
When
γF
isla
rge,
larg
edis
cret
ion
toth
efly
mak
esth
eti
ger
wor
seoff
bec
ause
the
fly
care
sm
ore
abou
tth
ere
nts
from
bri
ber
yth
anth
eti
ger’
sob
ject
ive.
Fig
ure
3Bsh
ows
that
the
anti
-cor
rupti
onca
mpai
gnm
akes
the
fly
bet
ter
offw
hen
γF
issm
all
and
wor
seoff
when
γF
isbig
.W
hen
γF
issm
all,
the
anti
-cor
rupti
onca
mpai
gnm
akes
the
fly
wor
seoff
bec
ause
ben
efits
from
corr
upti
onop
por
tunit
ies
are
larg
erth
anth
ein
crea
ses
inth
edis
cret
ion.
When
γF
isla
rge,
the
anti
-co
rrupti
onca
mpai
gnm
akes
the
fly
bet
ter
offb
ecau
seth
ein
crea
sein
the
dis
cret
ion
ism
ore
imp
orta
nt
than
the
rents
collec
ted
from
the
corr
upti
onop
por
tunit
ies.
16
Figure 2B displays how the amount of relative discretion X̄/σT changes
with the availability of relative corruption opportunities γF/σT , when relative
noise σn/σT is fixed at different levels. We show that the fly’s relative discre-
tion X̄/σT decreases with the relative corruption availability γF/σT . When
the relative corruption availability γF/σT is small, better aligning the inter-
ests for the fly and the tiger (increasing βT in Figure 2A), so the tiger gives
more discretion to the fly (increasing X̄/σT ). When the relative corruption
opportunity γF/σT is large, anticipating that the fly will collect more rents
and care less about the tiger’s ideal choice (decreasing βT in Figure 2A), the
tiger gives the fly very small range of X to choose (decreasing X̄/σT ). Given
the same level of relative corruption opportunities (γF/σT ), the tiger would
like to give more relative discretion X̄/σT to the fly when the fly is more
skillful (smaller σn/σT ). Given the same level of relative discretion X̄/σT ,
expertise could help fly to collect more rents from corruption opportunities
(larger γF/σT ).
Figure 3A shows the change in the tiger’s utility UT with the change
in the corruption opportunities γF at different levels of discretion X̄, when
noise σn is fixed at 1. Reducing fly’s corruption γF always makes the tiger
better off and the tiger can achieve the highest utility at the optimal level
of X̄. When fly is very corrupt (large γF ), large discretion (X̄) to the fly
makes the tiger worse off. It is because within this range, the fly cares more
about the rents collected from bribery than the tiger’s preference. When fly’s
corruption opportunities are scarce (small γF ), large discretion (X̄) to the
fly makes the tiger better off due to the benefits from delegating the jobs to
the agent with better decision making. Formally,
Theorem 4.1. Reducing the fly’s corruption opportunities γF always makes
the tiger better off because the reduction aligns the fly’s incentives better with
the tiger’s. Reducing the fly’s corruption opportunities γF also generates
indirect benefits to the tiger because the tiger optimally gives the fly more
discretion so that more decisions are made by the agent with superior infor-
17
mation.
dUT
dγF=∂UT
∂γF|¯X,X̄ +
∂UT
∂¯X|γF ∗
d¯X
dγF+∂UT
∂X̄|γF ∗
dX̄
dγF
We have
(1) ∂UT/∂γF |¯X,X̄ < 0 (direct benefit)
which says that given the level of discretion, reducing γF makes the tiger bet-
ter off.
(2) d¯X/dγF > 0 and dX̄/dγF < 0 (indirect benefit)
which says that the tiger prefers to give the fly more discretion as γF decreas-
es.
Proof: See Appendix D.
Figure 3B shows how the fly’s utility UF varies with the availability of
corruption γF at different levels of discretion X̄, when noise σn is fixed at
1. When X̄ is fixed, given the same availably of the corruption γF , the fly
is always better off with more discretion. Besides, the fly’s utility with the
optimal X̄ indicates that anti-corruption campaign makes the fly sometimes
worse off and sometimes better off. For instance, when γF is small, anti-
corruption makes the fly worse off. Because in this range, the reduction in
corruption opportunities is more important than the increase in discretion,
which is already large. When γF is large, anti-corruption makes the fly better
off. Because in this range, the additional discretion is more important than
the reduction in corruption opportunities. Formally, we have
Theorem 4.2. Reducing the fly’s corruption opportunities γF makes the fly
worse off initially when γF is small and then better off when γF is big. We
can write
dUF
dγF=∂UF
∂γF|¯X,X̄ +
∂UF
∂¯X|γF ∗
d¯X
dγF+∂UF
∂X̄|γF ∗
dX̄
dγF
We have
18
(1) ∂UF/∂γF |¯X,X̄ > 0, but ∂UF/∂
¯X|γF ∗ d¯
X/dγF = ∂UF/∂X̄|γF ∗ dX̄/dγF <0, which does not allow us to sign dUF/dγF .
(2) dUF/dγF > 0 when γF is close enough to 0, and
(3) dUF/dγF < 0 when γF is large enough.
Proof: See Appendix D.
5 Expertise
In this section, we discuss the last implications from our model: consequences
of reducing noise σn for the tiger and the society. We show that fighting
corruption and enhancing expertise are complements. From the tiger’s per-
spective, if corruption is high,¯X and X̄ will be close to zero, and expertise
will not matter much. Strict rules tend to neutralize expertise. If expertise
is low, the benefit of discretion is small no matter how low corruption is, and
corruption will not matter much. Fighting flies’ corruption probably will not
be worth the cost. In addition, both the direct and indirect benefits of re-
ducing corruption are small. Conversely, if corruption is small and expertise
is high, the range of [¯X, X̄], will be chosen to be large and the expertise will
have a big impact on the choice. The results for society are similar, but if
the tiger’s corruption is large enough both expertise and anti-corruption for
flies may benefit the tiger but not society.
19
02
46
810
−6−4−20244A
. Soc
ial W
elfa
re, T
iger
’s u
tiliti
y an
d no
ise
σ n w
ith s
mal
l cor
rupt
ion
(γT=
γ F=
0.1)
nois
e σ n
Social Welfare and Tiger’s utility
Soc
ial W
elfa
re w
ith c
ost
Soc
ial W
elfa
re w
ithou
t cos
t T
iger
’s u
tility
02
46
810
−6−4−2024
4B. S
ocia
l Wel
fare
, Tig
er’s
util
itiy
and
nois
e σ n
with
larg
e co
rrup
tion
(γT=
γF=
2)
nois
e σ n
Social Welfare and Tiger’s utility
Soc
ial W
elfa
re w
ith c
ost
Soc
ial W
elfa
re w
ithou
t cos
t T
iger
’s u
tility
InF
igure
4A,
when
corr
upti
onis
smal
l,tr
ainin
gth
eflie
s(t
ore
duceσn)
hel
ps
soci
ety
ifth
eco
sts
are
not
too
hig
h.
Inth
isca
se,
the
mai
ndiff
eren
ceb
etw
een
soci
ety’s
pre
fere
nce
and
that
ofti
gers
isth
eco
stof
trai
nin
g.In
Fig
ure
4B,
when
corr
upti
onis
larg
e,tr
ainin
gth
atb
enefi
tsth
eti
gers
hurt
sso
ciet
yb
ecau
seit
hel
ps
the
tige
rsto
imple
men
tth
eir
corr
upti
on.
Inb
oth
case
s,ti
gers
pre
fer
trai
nin
gth
eflie
sb
ecau
seth
eydo
not
bea
rth
eco
stan
dth
eflie
sar
ew
orkin
gfo
rth
em.
20
We illustrate this section by using figures first, and then state them for-
mally in theorems. Figures 4A and 4B show how the tiger’s utility and social
welfare (with and without cost) change with noise σn when corruption op-
portunities are either very small or very large, respectively. We show that
training flies (through reducing σn) always makes tigers better off. Tigers
prefer training because they do not bear any cost and flies are working for
them. In Figure 4A, when the corruption opportunities are scarce (where
γF = γT = 0.1), training flies makes the society better off if the costs are not
too high. In this case, the main difference between society’s preference and
that of tigers is the cost of training. In Figure 4B, when both tigers and flies
are very corrupt, training that benefits tigers hurts society because it helps
tigers to implement their corruption. Formally, we have
Theorem 5.1. Increasing the fly’s expertise (through training that reduces
σn), always makes the tiger better off because the expertise aligns the fly’s
incentives better with the tiger’s. Increasing the fly’s expertise also generates
indirect benefits to the tiger because the tiger optimally gives the fly more
discretion so that more decisions are made by the agent with superior infor-
mation. We can write
dUT
dσn=∂UT
∂σn|¯X,X̄ +
∂UT
∂¯X|γF ,βI ∗
d¯X
dσn+∂UT
∂X̄|γF ,βI ∗
dX̄
dσn
We have
(1) ∂UT/∂σn|¯X,X̄ < 0 (direct benefit)
which says that given the level of discretion, reducing σn makes the tiger bet-
ter off.
(2) d¯X/dσn > 0 and dX̄/dσn < 0 (indirect benefit)
which says that the tiger prefers to give the fly more discretion as σn decreas-
es.
Proof: See Appendix E.
Theorem 5.2. The anti-corruption campaign should be accompanied by train-
21
ing flies to have more expertise. With low corruption and high expertise, the
economy can flourish. We can write
dW S
dσn=∂W S
∂σn|¯X,X̄ +
∂W S
∂¯X|γF ,βI ∗
d¯X
dσn+∂W S
∂X̄|γF ,βI ∗
dX̄
dσn
When γF and γT are small, we have ∂W S/∂σn|¯X,X̄ < 0, ∂W S/∂
¯X|γF ,βI ∗
d¯X/dσn = ∂W S/∂X̄|γF ,βI ∗ dX̄/dσn > 0, and dW S/dσn < 0. When γF and
γT are very large, we have ∂W S/∂σn|¯X,X̄ < 0, ∂W S/∂
¯X|γF ,βI ∗ d¯
X/dσn =
∂W S/∂X̄|γF ,βI ∗ dX̄/dσn > 0, and dW S/dσn > 0.
Proof: See Appendix E.
In another case (not shown), when the flies are much more corrupt than
the tigers, training the flies does not have a significant impact on the society,
because when the flies are very corrupt, tigers are not going to give the flies
much discretion. The benefit to the society, if any, will be less than the
cost. Strict rules neutralize expertise. However, when the tigers are much
more corrupt than the flies, training the flies will cause a loss to the society,
because when the flies’ corruption opportunities are scarce, tigers are going
to give the flies more discretion. This induces the flies to do more what the
corrupt tigers want, and makes the society worse off.
6 Conclusion
This paper builds a theoretical model of an anti-corruption campaign like the
ongoing campaign in China. The issues of balancing control with provision
of incentives is more universal and arises in all governments. We focus on the
true anti-corruption campaign and discuss effects of reductions in corruption
opportunities to society, tigers and flies. We have three main conclusions.
22
First, fighting the flies without fighting the tigers reduces welfare. If
the tigers are very corrupt, reducing flies’ corruption helps the tigers to im-
plement their corruption. Second, strict rules and fighting corruption are
substitutes. To take full advantage of the aligned incentives, a reduction
of corruption should be accompanied by more discretion. Third, fighting
corruption and enhancing expertise are complements. If corruption is high,
discretion will be close to zero, and expertise will not matter much. Strict
rules tend to neutralize expertise. If expertise is low, the benefit of discre-
tion is small no matter how low corruption is, and corruption will not matter
much. If expertise is low, both the direct and indirect benefits of reducing
corruption are small. Conversely, if corruption is small and expertise is high,
discretion will be chosen to be large and the expertise will have a big impact
on the choice. An effective fight against corruption at all levels can make
society more efficient by aligning incentives and allowing the allocation of
control rights to the people with the information needed to make decisions.
The greatest improvement will occur if fighting of corruption is accompa-
nied by efficient delegation of decision-making and the development of the
appropriate level of expertise.
Appendix A Facts about Normal Distribution
Lemma A1. Let n(x) = 1√2πe−
x2
2 and N(x) =∫ xy=−∞ n(y) dy be the unit
normal density and cumulative distribution functions, respectively. Thus,(1) N ′(x) = n(x), n′(x) = −xn(x), n′′(x) = −n(x) + x2n(x);(2) N(x) is an increasing 1-1 function mapping (−∞,+∞) onto (0,1);(3) For x < 0, g(x) ≡ N(x)+n(x)/x+N(x)/x2 is an increasing 1-1 functionmapping (−∞, 0) to (0,+∞);(4) For x < 0, p(x) ≡ g(x)x2/N(x) is an increasing 1-1 function mapping(−∞, 0) to (0,1);(5) For x > 0, f(x) ≡ xN(−x)/n(x) is a monotonically increasing 1-1function mapping (0,+∞) onto (0,1).
23
Proof of Lemma A1(3).
dg(x)
dx=
[n(x)− n(x)− n(x)
x2+n(x)
x2− 2N(x)
x3
]= −2N(x)
x3
Therefore, g(x) is increasing with x when x < 0. As x → −∞,g(x)→ 0 because each term goes to 0; As x→ 0, g(x)→ +∞. Thus,for x ∈ (−∞, 0), g(x) ∈ (0,+∞). �
Proof of Lemma A1(4).
g(x) =N(y)
y2
∣∣∣∣xy=−∞
−∫ x
y=−∞
n(y)
y2dy =
N(x)
x2−∫ x
y=−∞
n(y)
y2dy
p(x) =
(N(x)
x2−∫ x
y=−∞
n(y)
y2dy
)x2
N(x)= 1− 1
N(x)
∫ x
y=−∞
x2n(y)
y2dy
Then
0 <1
N(x)
∫ x
y=−∞
x2n(y)
y2dy = 1− p(x) <
1
N(x)
∫ x
y=−∞
y2n(y)
y2dy = 1
Thus, for x ∈ (−∞, 0), p(x) ∈ (0, 1). �
Proof of Lemma A1 (5).
df(x)
dx=
d
dx
(xN(−x)
n(x)
)=N(−x)
n(x)− x+
x2N(−x)
n(x)
=x2
n(x)︸ ︷︷ ︸>0
(N(−x) + n(−x)/(−x) +N(−x)/(−x)2︸ ︷︷ ︸
g(−x)>0
)> 0
Thus, f(x) is strictly increasing with x.
When x → 0, f(x) → 0. When x → +∞, N(−x) → 0, n(x)/x → 0,
24
thus
limx→+∞
N(−x)
n(x)/x= lim
x→+∞
dN(−x)/dx
d(n(x)/x)/dx= lim
x→+∞
−n(x)
−n(x)− n(x)/x2= lim
x→+∞
1
1 + 1/x2= 1
Thus, for x > 0, f(x) ≡ xN(−x)/n(x) is a monotonically increasing1-1 function mapping (0,+∞) onto (0,1). �
Appendix B Characterizing optimal¯x and x̄
Since T and F are jointly normal with 0 mean, we can write T = βTF + ηT ,where ηT is independent of F, βT = cov(T, F )/var(F ) = βIσ2
T/σ2F , and
σ2T = βIβTσ2
T + var(ηT ). Therefore, var(ηT ) = σ2T (1− βIβT ).
Tiger’s utility is
E[UT ] = γ2T + E[−(X − T )2](15)
or alternatively
E[(X − T )2] = E[(X − (βTF + ηT ))2](16)
= E[(X − βTF )2 − 2ηT (X − βTF ) + ηT2]
= E[(X − βTF )2] + var(ηT )
E[ηTF ]=0 by the property of regression (10), and E[ηTX] = 0 becauseE[ηT ] = 0 and the fly’s choice of X depends on realized X but not realized
25
ηT (or equivalently depends on F but not T). From (11), (15) and (16),
E[UT ] = γ2T − E[(π(F,
¯X, X̄)− βTF )2]− var(ηT )(17)
= γ2T −
1√2πσF
∫¯X
F=−∞(¯X − βTF )2e
− F2
2σ2F dF
− 1√2πσF
∫ X̄
F=¯X
(F − βTF )2e− F2
2σ2F dF
− 1√2πσF
∫ +∞
F=X̄
(X̄ − βTF )2e− F2
2σ2F dF − var(ηT )
Let ϕ ≡ F/σF ∼ N(0, 1) and x̄ = X̄/σF . Thus, the tiger’s first-ordercondition for X̄ of the above is
0 =∂
∂X̄E[(−π(F,
¯X, X̄)− βTF )2](18)
= − 1√2πσF
∫ +∞
F=X̄
2(X̄ − βTF )e− F2
2σ2F dF
= −2σF
∫ +∞
ϕ=x̄
(x̄− βTϕ)1√2πe−
ϕ2
2 dϕ
= −2σF
(x̄
∫ +∞
ϕ=x̄
n(ϕ) dϕ− βT∫ +∞
ϕ=x̄
ϕn(ϕ) dϕ
)= −2σF x̄
(N(ϕ)|+∞x̄ − βTn(ϕ)|+∞x̄
)= −2σF [x̄N(−x̄)− βTn(x̄)]
= −2σFn(x̄)
[x̄N(−x̄)
n(x̄)− βT
]Similarly, the tiger’s first-order condition for
¯X is
0 =∂
∂¯XE[(−π(F,
¯X, X̄)− βTF )2](19)
= −2σFn(¯x)
[¯xN(
¯x)
n(¯x)
+ βT]
Since the positive coefficient σF does not affect Equation (18), the optimal x̄is independent of σF (given βT ), i.e. X̄ = σF x̄ where x̄ is a constant solving
26
Equation (18) given βT . Thus,
βT =x̄N(−x̄)
n(x̄)=−
¯xN(
¯x)
n(¯x)
(20)
Thus, Lemma A1(5) can be used to show that, 1) the optimal choice of¯x
and x̄ is determined uniquely by the first order condition (18), and 2) theoptimal x̄ (respectively
¯x = −x̄) is increasing (respectively decreasing) in
βT . From Lemma A1(5), Equation (18) has a unique solution given βT , callit x̄(βT ). Furthermore, by Equation (18) and Lemma A1(5), dUT/dX̄ > 0when x̄ < x̄(βT ) and dUT/dX̄ < 0 when x̄ > x̄(βT ). It implies that x̄(βT ) (ormore precisely X̄ = σF x̄(βT )) is the tiger’s unique optimal choice. This proofis similar that
¯X = −σF x̄(βT ) is also optimal. Furthermore, Equation (18)
and Lemma A1(5) imply that x̄(βT ) is an increasing function.
Appendix C Both Tigers and Flies
C.1 Social Welfare and γF
Proof of Theorem 3.1. Since S and T are jointly normal with 0 mean, wecan write
S = βSF + ηS,(21)
where ηS is independent of F, βS = cov(S, F )/var(F ) = βIσ2S/σ
2F , and σ2
S =βIβSσ2
S + var(ηS). Therefore, var(ηS) = σ2S(1− βIβS).
Social welfare is
E[W S] = E[−(X − S)2 − CT (γT )− CF (γF )− Cn(σn)](22)
27
or alternatively
E[(X − S)2] = E[X − (βSF + ηS)2](23)
= E[(X − βSF )2 − 2ηS(X − βSF ) + η2S]
= E[(X − βSF )2] + var(ηS)
E[ηSX] = 0 because E[ηS] = 0 and the fly’s choice of X depends on realizedX but not realized ηS (or equivalently depends on F but not S). E[ηSF ] = 0by the property of regression (21). From (11), (22) and (23),
E[W S] = E[−(π(σFϕ,¯X, X̄)− βSF )2]− var(ηS)− CT (γT )− CF (γF )− Cn(σn)(24)
= E[−(π(σFϕ,¯X, X̄)− βSσFϕ)2]− var(ηS)− C(γT )− C(γF )− Cn(σn)
=
∫¯X/σF
ϕ=−∞−(
¯X − βSσFϕ)2n(ϕ) dϕ+
∫ X̄/σF
ϕ=¯X/σF
−(σFϕ− βSσFϕ)2n(ϕ) dϕ
+
∫ +∞
ϕ=X̄/σF
−(X̄ − βSσFϕ)2n(ϕ) dϕ− var(ηS)− CT (γT )− CF (γF )− Cn(σn)
=
∫¯X/σF
ϕ=−∞(−
¯X2 + 2
¯XβIσ2
S
σFϕ− (βI)2σ4
S
σ2F
ϕ2)n(ϕ) dϕ
+
∫ X̄/σF
ϕ=¯X/σF
(−σ2Fϕ
2 + 2βIσ2Sϕ
2 − (βI)2σ4S
σ2F
ϕ2)n(ϕ) dϕ
+
∫ +∞
ϕ=X̄/σF
(−X̄2 + 2X̄βIσ2
S
σFϕ− (βI)2σ4
S
σ2F
ϕ2)n(ϕ) dϕ
− σ2S − CT (γT )− CF (γF )− Cn(σn)
=
∫¯X/σF
ϕ=−∞(−
¯X2 + 2
¯XβIσ2
S
σFϕ)n(ϕ) dϕ+
∫ X̄/σF
ϕ=¯X/σF
(−σ2Fϕ
2 + 2βIσ2Sϕ
2)n(ϕ) dϕ
+
∫ +∞
ϕ=X̄/σF
(−X̄2 + 2X̄βIσ2
S
σFϕ)n(ϕ) dϕ− σ2
S − CT (γT )− CF (γF )− Cn(σn)
We have E[W S] depends on¯X, X̄, σF , [βI ],6 [σS], where
(1) σF depends on [σT ], [σn], γF ;
6We use [ ] to present this parameter is fixed as a constant in the proof through thissection.
28
(2)¯X, X̄ depend on σF , βT ;
(3) βT depends on [σn], [σT ] and γF .
Then,
dE[W S]
dγF=∂E[W S]
∂σF∗ dσFdγF
+∂E[W S]
∂CF (γF )∗ dCF (γF )
dγF(25)
+∂E[W S]
∂¯X
∗(∂
¯X
∂σF∗ dσFdγF
+∂
¯X
∂βT∗ dβ
T
dσF∗ dσFdγF
)+∂E[W S]
∂X̄∗(∂X̄
∂σF∗ dσFdγF
+∂X̄
∂βT∗ dβ
T
dσF∗ dσFdγF
)
I. Fixed¯X and X̄
For now, we are fixing σ2T , σ2
n, σ2S, γT ,
¯X and X̄ throughout this section.
Since σ2F = (σ2
T )2/σ2I + γ2
F = (σ2T )2/(σ2
T + σ2n) + γ2
F , then
dσFdγF
=d
dγF
([(σ2
T )2/(σ2T + σ2
n) + γ2F ]
12
)=γFσF
> 0(26)
29
From equations (24) and (25),
∂E[W S]
∂σF=
∫¯X/σF
ϕ=−∞−2
¯XβIσ2
S
σ2F
ϕn(ϕ) dϕ+
∫ X̄/σF
ϕ=¯X/σF
−2σFϕ2n(ϕ) dϕ(27)
+
∫ +∞
ϕ=X̄/σF
−2X̄βIσ2S
σ2F
ϕn(ϕ) dϕ
= 2¯XβSn(ϕ)|¯X/σF−∞ − 2σF (−ϕn(ϕ))|X̄/σF
¯X/σF
− 2σFN(ϕ)|X̄/σF¯X/σF
+ 2X̄βSn(ϕ)|+∞X̄/σF
= 2¯XβSn(
¯X/σF ) + 2σF
X̄
σFn(X̄/σF )− 2σF ¯
X
σFn(
¯X/σF )
− 2σF [N(X̄/σF )−N(¯X/σF )]− 2X̄βSn(X̄/σF )
= 2σF (βS − 1)[¯X/σFn(
¯X/σF )− X̄/σFn(X̄/σF )]− 2σF [N(X̄/σF )−N(
¯X/σF )]
= 2σF (βS − 1)[¯xn(
¯x)− x̄n(x̄)]− 2σF [N(x̄)−N(
¯x)]
= 4σF (1− βS)x̄n(x̄)− 2σF [2N(x̄)− 1]︸ ︷︷ ︸kS(x̄)
Let kS(x̄) ≡ 4σF (1− βS)x̄n(x̄)− 2σF [2N(x̄)− 1], when x̄ > 0
k′S(x̄) = 2σF [2(1− βS)(n(x̄)− x̄2n(x̄))− 2n(x̄)]
= 2σF [2n(x̄)− 2x̄2n(x̄)− 2βSn(x̄) + 2βSx̄2n(x̄)− 2n(x̄)]
= 2σF [−2βSn(x̄)︸ ︷︷ ︸<0
+ 2(βS − 1)x̄2n(x̄)︸ ︷︷ ︸<0
] < 0
Thus, kS(x̄) is decreasing with x̄ when x̄ > 0. When x̄ → 0, kS(x̄) → 0.Thus, for 0 < x̄ < +∞, k′S(x̄) < 0 implies that kS(x̄) < 0.
When γF is very large, X̄ → 0, CF (γF )→ 0, C ′F (γF )→ 0, ∂E[W S]/∂σF <0. Therefore,
∂E[W S]
∂γF|¯X,X̄ =
∂E[WS]
∂σF∗ dσFdγF
+∂E[W S]
∂CF (γF )∗ dCF (γF )
dγF(28)
= 2γF (βS − 1)[¯xn(
¯x)− x̄n(x̄)]− 2γF [N(x̄)−N(
¯x)]− C ′F (γF ) < 0
Thus, when both¯X and X̄ are fixed, reducing the fly’s corruption op-
portunities (γF ) makes the society better off when γF is very large.
30
II. Tiger Chooses¯X and X̄ Optimally in Response to γF
Now, we consider what happens to social welfare with changes in γF whenthe tiger chooses
¯X and X̄ optimally in response to γF .
∂E[W S]
∂X̄=
∫ +∞
X̄/σF
(−2X̄ + 2βIσ2
S
σFϕ)n(ϕ) dϕ(29)
= −2X̄N(−X̄/σF ) + 2βIσ2
S
σFn(X̄/σF )
= −2σFn(X̄/σF )
[(X̄/σF )N(−X̄/σF )
n(X̄/σF )− βIσ2
S
σ2F
]= −2βIγ2
Tn(X̄/σF )
σF= −2βIγ2
Tn(x̄)
σF
From equation (20)
∂βT
∂X̄
∣∣∣∣σF
=∂
∂X̄
{(X̄/σF )N(−X̄/σF )
n(X̄/σF )
}=N(−X̄/σF )
σFn(X̄/σF )+−X̄n(X̄/σF )
σ2Fn(X̄/σF )
+X̄2N(−X̄/σF )n(X̄/σF )
σ3Fn
2(X̄/σF )
=1
σFn(¯X/σF )
(N(−X̄/σF )− X̄
σFn(X̄/σF ) +
X̄2
σ2F
N(−X̄/σF )
)=
1
σFn(¯x)
[N(−x̄)− x̄n(x̄) + x̄2N(−x̄)]
Thus,
∂X̄
∂βT
∣∣∣∣σF
=σFn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x)(30)
When βT is fixed, X̄/σF is also fixed. Thus, ∂(X̄/σF ) = 0, ∂log(X̄/σF ) =∂log(X̄)− ∂log(σF ) = 0, ∂X̄/X̄ = ∂σF/σF , and
∂X̄
∂σF
∣∣∣∣βT
=X̄
σF= x̄(31)
31
dβT
dγF=
d
dγF
(σ4T
σ4T + σ2
Tγ2F + σ2
nγ2F
)= − 2γFσ
4T (σ2
T + σ2n)
(σ4T + σ2
Tγ2F + σ2
nγ2F )2
(32)
= − 2γFσ4T (σ2
T + σ2n)
[(σ2T + σ2
n)(σ4T
σ2T+σ2
n+ γ2
F )]2= −2γFσ
4T (σ2
T + σ2n)
(σ2T + σ2
n)2σ4F
= − 2γFσ4T
(σ2T + σ2
n)σ4F
= −2γFβIσ2
T
σ4F
= −2γFβT
σ2F
∂E[W S]
∂X̄∗(∂X̄
∂σF∗ dσFdγF
+∂X̄
∂βT∗ dβ
T
dγF
)(33)
= −2βIγ2Tn(x̄)
σF∗[x̄ ∗ γF
σF+
σFn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x)∗(− 2γFβ
T
σ2F
)]= −2βIγ2
TγFn(x̄)
σ2F
∗[x̄− 2βTn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x)
]=
2βIγFγ2Tn(x̄)
σ2F
[2βTn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x̄)− x̄]
Similarly,
∂E[W S]
∂¯X
∗(∂
¯X
∂σF∗ dσFdγF
+∂
¯X
∂βT∗ dβ
T
dγF
)(34)
=∂E[W S]
∂X̄∗(∂X̄
∂σF∗ dσFdγF
+∂X̄
∂βT∗ dβ
T
dγF
)=
2βIγFγ2Tn(x̄)
σ2F
[2βTn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x̄)− x̄]
32
From Equations (28), (33) and (34)
dE[W S]
dγF= 2γF (βS − 1)[
¯xn(
¯x)− x̄n(x̄)]− 2γF [N(x̄)−N(
¯x)]− C ′F (γF )(35)
+ 2 ∗(
2βIγFγ2Tn(x̄)
σ2F
)∗[
2βTn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x)− x̄]
= 4γF (1− βS) ∗ x̄n(x̄)− 2γF [N(x̄)−N(¯x)]− C ′(γF )
4 ∗(βIγFγ
2Tn(x̄)
σ2F
)∗[
2βTn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x)− x̄]
= 4γF x̄
[(1− βS)n(x̄)− N(x̄)−N(
¯x)
2x̄
+βIγ2
Tn(x̄)
σ2F x̄
∗(
2x̄N(−x̄)
N(−x̄)− x̄n(x) + x̄2N(−x̄)− x̄)]− C ′F (γF )
= 4γF x̄
[(1− βS)n(x̄)− N(x̄)−N(
¯x)
2x̄
+βIγ2
Tn(x̄)
σ2F
∗(
2
1− x̄n(x)/N(−x̄) + x̄2− 1
)]− C ′F (γF )
When both γT and γF are very large, x̄→ 0, βS → 0, CF (γF )→ 0,C ′F (γF )→0. Thus, Equation (35) can be written as
dE[WS]
dγF≈ 4γF x̄
[(1− βS)n(0)− n(0) +
βIγ2Tn(0)
σ2F
∗ (2 + 4n(0)x̄− 1)
]− C ′F (γF )
≈ 4γF x̄
[− βSn(0) +
βIγ2Tn(0)
σ2F
∗ (1 + 4n(0)x̄)
]− C ′F (γF )
≈ 4γF x̄
[− βIσ2
S
σ2F
n(0) +βIγ2
Tn(0)
σ2F
+4βIγ2
Tn(0)2
σ2F
x̄)︸ ︷︷ ︸→0
]− C ′F (γF )
≈ 4βIγF x̄n(0)
σ2F
(γ2T − σ2
S)︸ ︷︷ ︸>0 when γT is large enough
−C ′F (γF )︸ ︷︷ ︸>0
> 0
When both γF and γF are large enough, social welfare is increasingwith γF . Thus, when both the fly and the tiger are very corrupt,reducing fly’s corruption opportunities γF without reducing tiger’s
33
corruption opportunities γT makes the society worse off.
dE[WS]
dγF≈ 4γF x̄
[(1− βS)n(0)− n(0) +
βIγ2Tn(0)
σ2F
∗ (2 + 4n(0)x̄− 1)
]− C ′F (γF )
≈ 4γF x̄
[− βSn(0) +
βIγ2Tn(0)
σ2F
∗ (1 + 4n(0)x̄)
]− C ′F (γF )
≈ 4γF x̄
[− βIσ2
S
σ2F
n(0) +βIγ2
Tn(0)
σ2F
+4βIγ2
Tn2(0)x̄
σ2F︸ ︷︷ ︸→0
]− C ′F (γF )
≈ 4βIγF x̄n(0)
σ2F
(γ2T − σ2
S)︸ ︷︷ ︸>0 when γT is large enough
−C ′F (γF )︸ ︷︷ ︸>0
> 0
�
34
Appendix D Information and Discretion
D.1 Tiger and γF
Proof of Theorem 4.1. Based on equation (12), the tiger’s utility is
E[UT ] = γ2T + E[−(π(σFϕ,
¯X, X̄)− βTF )2]− var(ηT )(36)
= γ2T + E[−(π(σFϕ,
¯X, X̄)− βTσFϕ)2]− var(ηT )
= γ2T +
∫¯X/σF
ϕ=−∞−(
¯X − βTσFϕ)2n(ϕ) dϕ+
∫ X̄/σF
ϕ=¯X/σF
−(σFϕ+ βTσFϕ)2n(ϕ) dϕ
+
∫ +∞
ϕ=X̄/σF
−(X̄ + βTσFϕ)2n(ϕ) dϕ− var(ηT )
= γ2T +
∫¯X/σF
ϕ=−∞(−
¯X2 + 2
¯XβIσ2
T
σFϕ− (βI)2σ4
T
σ2F
ϕ2)n(ϕ) dϕ
+
∫ X̄/σF
ϕ=¯X/σF
(−σ2Fϕ
2 + 2βIσ2Tϕ
2 − (βI)2σ4T
σ2F
ϕ2)n(ϕ) dϕ
+
∫ +∞
ϕ=X̄/σF
(−X̄2 + 2X̄βIσ2
T
σFϕ− (βI)2σ4
T
σ2F
ϕ2)n(ϕ) dϕ− σ2T +
(βI)2σ4T
σ2F
= γ2T +
∫¯X/σF
ϕ=−∞(−
¯X2 + 2
¯XβIσ2
T
σFϕ)n(ϕ) dϕ+
∫ X̄/σF
ϕ=¯X/σF
(−σ2Fϕ
2 + 2βIσ2Tϕ
2)n(ϕ) dϕ
+
∫ +∞
ϕ=X̄/σF
(−X̄2 + 2X̄βIσ2
T
σFϕ)n(ϕ) dϕ− σ2
T
E[UT ] depends on σF , [σT ], [βI ], where σF depends on [σT ], γF , [σn].
dσFdγF
=γFσF
(37)
35
∂E[UT ]
∂σF=
∫¯X/σF
ϕ=−∞−2
¯XβIσ2
T
σ2F
ϕn(ϕ) dϕ+
∫ X̄/σF
ϕ=¯X/σF
−2σFϕ2n(ϕ) dϕ
+
∫ +∞
ϕ=X̄/σF
−2X̄βIσ2T
σ2F
ϕn(ϕ) dϕ
= 2¯XβTn(ϕ)|¯X/σF−∞ − 2σF (−ϕn(ϕ))|X̄/σF
¯X/σF
− 2σFN(ϕ)|X̄/σF¯X/σF
+ 2X̄βTn(ϕ)|+∞X̄/σF
= 2¯XβTn(
¯X/σF ) + 2σF
X̄
σFn(X̄/σF )− 2σF ¯
X
σFn(
¯X/σF )
− 2σF [N(X̄/σF )−N(¯X/σF )]− 2X̄βTn(X̄/σF )
= 2σF (βT − 1)[¯X/σFn(
¯X/σF )− X̄/σFn(X̄/σF )]− 2σF [N(X̄/σF )−N(
¯X/σF )]
= 2σF (βT − 1)[¯xn(
¯x)− x̄n(x̄)]− 2σF [N(x̄)−N(
¯x)]
= 4σF (1− βT )x̄n(x̄)− 2σF [2N(x̄)− 1]︸ ︷︷ ︸kT (x̄)
Let kT (x̄) ≡ 4σF (1− βT )x̄n(x̄)− 2σF [2N(x̄)− 1], when x̄ > 0
k′T (x̄) = 2σF [2(1− βT )(n(x̄)− x̄2n(x̄))− 2n(x̄)](38)
= 2σF [2n(x̄)− 2x̄2n(x̄)− 2βTn(x̄) + 2βT x̄2n(x̄)− 2n(x̄)]
= 2σF [−2βTn(x̄)︸ ︷︷ ︸<0
+ 2(βT − 1)x̄2n(x̄)︸ ︷︷ ︸<0
] < 0
Thus, kT (x̄) is decreasing with x̄ when x̄ > 0. When x̄ → 0, kT (x̄) → 0.Thus, for 0 < x̄ < +∞, k′T (x̄) < 0 implies that kT (x̄) < 0.
∂E[UT ]
∂γF|¯X,X̄ =
∂E[UT ]
∂σF∗ dσFdγF
< 0(39)
For fixed¯X and X̄, tiger’s utility is decreasing with γF . Al-
so, this is true if the tiger chooses¯X and X̄, since the tiger will
choose¯X and X̄ to maximize utility and the pointwise maximum
of decreasing functions is decreasing.
36
From Equations (30), (31), (32),
dX̄
dγF=∂X̄
∂σF∗ dσFdγF
+∂X̄
∂βT∗ dβ
T
dγF(40)
= x̄ ∗ γFσF
+σFn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x)∗(− 2γFβ
T
σ2F
)=γFσF∗(x̄− 2βTn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x̄)
)=γF x̄
σF
(1− 2N(−x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x̄)
)=γF x̄
σF
[1− 2N(−x̄)
(−x̄)2 ∗ (N(−x̄)/(−x̄)2 + n(−x̄)/(−x̄) +N(−x̄))
]=γF x̄
σF
(1− 2
p(−x̄)
)< 0
The final inequality follows from Lemma A1(4), which defines p(x) and im-plies p(x) ∈ (0, 1), so 2/p(x) ∈ (2,+∞). Similarly,
d¯X
dγF= − dX̄
dγF> 0(41)
Thus, discretion is decreasing with fly’s corruption opportunitiesγF . �
37
D.2 Fly and γF
Proof of Theorem 4.2. Based on (7), the fly’s utility is
E[UF ] = γ2F + E[−(π(σFϕ,
¯X, X̄)− σFϕ)2]− var(ηI)(42)
= γ2F +
∫¯X/σF
ϕ=−∞−(
¯X − σFϕ)2n(ϕ) dϕ+
∫ +∞
ϕ=X̄/σF
−(X̄ − σFϕ)2n(ϕ) dϕ− σ2T (1− βI)
= γ2F +
∫¯X/σF
ϕ=−∞(−
¯X2 + 2
¯XσFϕ− σ2
Fϕ2)n(ϕ) dϕ
+
∫ +∞
ϕ=X̄/σF
(−X̄2 + 2X̄σFϕ− σ2Fϕ
2)n(ϕ) dϕ− σ2T (1− βI)
E[UF ] depends on σF , [βI ],¯X, X̄, where
(1) σF depends on [σT ], [σn], γF ;(2)
¯X, X̄ depend on σF , βT ;
(3) βT depends on γF , [σT ] and [σn].
Thus,
dE[UF ]
dγF=∂E[UF ]
∂σF∗ dσFdγF
+∂E[UF ]
∂¯X
∗(∂
¯X
∂σF∗ dσFdγF
+∂
¯X
∂βT∗ dβ
T
dγF
)(43)
+∂E[UF ]
∂X̄∗(∂X̄
∂σF∗ dσFdγF
+∂
¯X
∂βT∗ dβ
T
dγF
)
I. Fixed¯X and X̄
For now, we are fixing σT ,¯X and X̄ throughout this section. From Equations
38
(42),
∂E[UF ]
∂σF|¯X,X̄ = 2γF ∗
σFγF
+
(− ¯X
σ2F
)(−
¯X2 + 2
¯XσF ∗ ¯
X
σF− σ2
F ∗ ¯X2
σ2F
)n(¯X/σF )(44)
+
∫¯X/σF
ϕ=−∞
(2
¯Xϕ− 2σFϕ
2
)n(ϕ) dϕ
+
(X̄
σ2F
)(−X̄2 + 2X̄σF ∗
X̄
σF− σ2
F
X̄2
σ2F
)n(X̄/σF )
+
∫ +∞
ϕ=X̄/σF
(2X̄ϕ− 2σFϕ
2
)n(ϕ) dϕ
= 2σF +
∫¯X
σF
ϕ=−∞
(2
¯Xϕ− 2σFϕ
2
)n(ϕ) dϕ+
∫ +∞
ϕ=X̄/σF
(2X̄ϕ− 2σFϕ
2
)n(ϕ) dϕ
= 2σF + 2¯X[−n(ϕ)]|¯X/σF−∞ − 2σF
∫¯X/σF
ϕ=−∞(n′′(ϕ) + n(ϕ)) dϕ
+ 2X̄[−n(ϕ)]|+∞ϕ=X̄/σF
− 2σF
∫ +∞
ϕ=X̄/σF
(n′′(ϕ) + n(ϕ)) dϕ
= 2σF − 2¯Xn(
¯X/σF ) + 2σF (
¯X/σF )n(
¯X/σF )− 2σFN(
¯X/σF )
+ 2X̄n(X̄/σF )− 2σF (X̄/σF )n(X̄/σF )− 2σF (1−N(X̄/σF ))
= 2σFN(X̄/σF )− 2σFN(¯X/σF ) = 2σF [N(x̄)−N(
¯x)] > 0
From Equations (26), (43), and (44),
∂E[UF ]
∂γF|¯X,X̄ =
∂E[UF ]
∂σF|¯X,X̄ ∗
dσFdγF
(45)
= 2σF [N(x̄)−N(¯x)] ∗ γF
σF= 2σF [N(x̄)−N(
¯x)] ∗ γF
σF> 0
Therefore, when¯X and X̄ are fixed, the fly’s utility is increasing
with γF .
II. Tiger Chooses¯X and X̄ Optimally
Now consider what happens when the tiger chooses¯X and X̄ optimally in
39
response to γF ,
∂E[UF ]
∂X̄
∣∣∣∣σF
= − 1
σF(−X̄2 + 2X̄2 − X̄2)n(X̄/σF ) +
∫ +∞
ϕ=X̄/σF
(−2X̄ + 2ϕσF )n(ϕ) dϕ(46)
= −2X̄N(ϕ)|+∞X̄/σF
− 2σFn(ϕ)|+∞X̄/σF
= −2X̄N(−X̄/σF ) + 2σFn(X̄/σF )
= −2σFn(X̄/σF )
(X̄/σFN(−X̄/σF )
n(X̄/σF )− 1
)= −2σFn(X̄/σF )(βT − 1) =
2γ2Fn(x̄)
σF
∂E[UF ]
∂X̄∗(∂X̄
∂σF∗ dσFdγF
+∂
¯X
∂βT∗ dβ
T
dγF
)(47)
=2γ2
Fn(x̄)
σF∗[x̄ ∗ γF
σF+
σFn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x)∗(− 2γFβ
T
σ2F
)]=
2γ3Fn(x̄)
σ2F
[x̄− 2βTn(x̄)
N(−x)− x̄n(x̄) + x̄2N(−x̄)
]= −2γ3
Fn(x̄)
σ2F
[2βTn(x̄)
N(−x)− x̄n(x̄) + x̄2N(−x̄)− x̄]
Similarly,
∂E[UF ]
∂¯X
∗(∂
¯X
∂σF∗ dσFdγF
+∂
¯X
∂βT∗ dβ
T
dγF
)(48)
=∂E[UF ]
∂X̄∗(∂X̄
∂σF∗ dσFdγF
+∂
¯X
∂βT∗ dβ
T
dγF
)= −2γ3
Fn(x̄)
σ2F
[2βTn(x̄)
N(−x)− x̄n(x̄) + x̄2N(−x̄)− x̄]
From Equations (45), (47), and (48), Equation (43) can be expressed as
dE[UF ]
dγF= 2γF [N(x̄)−N(
¯x)]− 2 ∗ 2γ2
Fn(x̄)
σF
[2βTn(x̄)
N(−x)− x̄n(x̄) + x̄2N(−x̄)− x̄]
(49)
40
When γF is very large,¯X → 0, X̄ → 0 and βT → 0. Since f ′(x) ≡
lim∆→0
f(x+∆)−f(x)∆
, and f ′(x) ≡ lim∆→0
f(x+∆)−f(x−∆)2∆
, we have
N(x̄)−N(¯x)
2x̄→ n(0)(50)
γ2F
σ2F
= 1− βT = 1− (X̄/σF )N(−X̄/σF )
n(X̄/σF )≈ 1− (X̄/σF )(1/2− n(0)X̄/σF )
n(0)(51)
= 1− 1
2n(0)
(X̄
σF
)+
(X̄
σF
)2
≈ 1− 1
2n(0)
(X̄
σF
)= 1− x̄
2n(0)
41
Thus, equation (49) can be written as
dE[UF ]
dγF≈ 4x̄2γF
[N(x̄)−N(−x̄)
2x̄2− γ2
F
σ2F
n(x̄)
x̄2
(2βTn(x̄)
N(−x̄)− x̄n(x̄)− x̄)]
≈ 4x̄2γF
[n(0)
x̄− γ2
F
σ2F
n(0)
x̄2
(2x̄N(−x̄)
N(−x̄)− x̄n(x̄)− x̄)]
= 4x̄2γF
[n(0)
x̄− γ2
F
σ2F
n(0)
x̄
(2N(−x̄)
N(−x̄)− x̄n(x̄)− 1
)]= 4x̄2γF
[n(0)
x̄− γ2
F
σ2F
n(0)
x̄
(2
1− x̄ n(x̄)N(x̄)
− 1
)]≈ 4x̄2γF
[n(0)
x̄− γ2
F
σ2F
n(0)
x̄(2 + 2
n(x̄)
N(x̄)∗ x̄− 1)
]≈ 4x̄2γF
[n(0)
x̄− γ2
F
σ2F
n(0)
x̄(2 + 4n(0)(x̄)− 1)
]≈ 4x̄2γF
[n(0)
x̄−(
1− 1
2n(0)(x̄)
)∗ n(0)
x̄∗ (1 + 4n(0)(x̄))
]= 4x̄2γF
[n(0)
x̄−(n(0)
x̄− 1
2
)∗ (1 + 4n(0)x̄)
]= 4x̄2γF
[n(0)
x̄−(n(0)
x̄+ 4n(0)2 − 1
2− 2n(0)x̄
)]= 4x̄2γF
[n(0)
x̄− n(0)
x̄− 4n(0)2 +
1
2+ 2n(0)x̄
]= 4x̄2γF [−4n(0)2 +
1
2+ 2n(0)x̄]
= 4x̄2γF
[−4n(0)2 +
1
2︸ ︷︷ ︸<0
]< 0
Therefore, the fly’s utility is decreasing with γF when γF is large.
When γF is small,¯X → −∞, X̄ → +∞ and βT → 1.
γ2F
σ2F
=γ2F
βIσ2T + γ2
F
≈ 0
42
Thus, Equation (49) can be written as
dE[UF ]
dγF≈ 4x̄2γF
[N(x̄)−N(−x̄)
2x̄2− γ2
F
σ2F
n(x̄)
x̄2
(2βTn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x̄)− x̄)]
= 4x̄2γF
[N(x̄)−N(−x̄)
2x̄2− γ2
F
σ2F
n(x̄)
x̄2
(2x̄N(−x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x̄)− x̄)]
= 4x̄2γF
[N(x̄)−N(−x̄)
2x̄2− γ2
F
σ2F
n(x̄)
x̄
(N(−x̄) + x̄n(x̄)− x̄2N(−x̄)
N(−x̄)− x̄n(x̄) + x̄2/N(−x̄)
)]= 4x̄2γF
[N(x̄)−N(−x̄)
2x̄2− γ2
F
σ2F
n(x̄)
x̄
(N(−x̄)/x̄2 + n(x̄)/x̄−N(−x̄)
N(−x̄)/x̄2 − n(x̄)/x̄+N(−x̄)
)]= 4x̄2γF
[N(x̄)−N(−x̄)
2x̄2− γ2
F
σ2F
n(x̄)
x̄
(2N(−x̄)/x̄2 − g(−x̄)
g(−x̄)
)]= 4x̄2γF
[N(x̄)−N(−x̄)
2x̄2− γ2
F
σ2F
n(x̄)
x̄
(2N(−x̄)
g(−x̄)x̄2− 1
)]= 4x̄2γF
[N(x̄)−N(−x̄)
2x̄2− γ2
F
σ2F︸︷︷︸→0
n(x̄)
x̄︸ ︷︷ ︸→0
(2
p(−x̄)︸ ︷︷ ︸→∞
−1
)]︸ ︷︷ ︸
?
The fly’s utility is increasing with γF when γF is close to 0. �
Appendix E Expertise
E.1 Tiger’s utility and σn
Proof of Theorem 5.1. Now, we consider what happens to the tiger in re-sponse to fly’s expertise σn. From (36), we have E[UT ] depends on σF , βI ,[σT ], where(1) σF depends on [σT ], [γF ], σn;(2) βI depends on [σT ], σn.
43
dE[UT ]
dσn=∂E[UT ]
∂σF∗ dσFdσn
+∂E[UT ]
∂βI∗ dβ
I
dσn(52)
+∂E[UT ]
∂¯X
∗(∂
¯X
∂σF∗ dσFdσn
+∂
¯X
∂βT∗ dβ
T
dσn
)+∂E[UT ]
∂X̄∗(∂X̄
∂σF∗ dσFdσn
+∂X̄
∂βT∗ dβ
T
dσn
)
∂E[UT ]
∂βI=
∫¯X/σF
ϕ=−∞2
¯Xσ2T
σFϕn(ϕ) dϕ+
∫ X̄/σF
ϕ=¯X/σF
2σ2Tϕ
2n(ϕ) dϕ+
∫ +∞
ϕ=X̄/σF
2X̄σ2T
σFϕn(ϕ) dϕ(53)
= −2¯Xσ2T
σFn(ϕ)
∣∣¯X/σF−∞ + 2σ2
T
∫¯X/σF
ϕ=¯X/σF
[n′′(ϕ) + n(ϕ)] dϕ− 2X̄σ2T
σFn(ϕ)
∣∣+∞X̄/σF
= −2¯Xσ2T
σFn(
¯X/σF ) + 2σ2
T
[− ϕn(ϕ)
∣∣X̄/σF¯X/σF
+N(ϕ)∣∣X̄/σF
¯X/σF
]+ 2X̄
σ2T
σFn(X̄/σF )
= −2¯xσ2
Tn(¯x)− 2σ2
T x̄n(x̄) + 2σ2T¯xn(
¯x) + 2σ2
T [N(x̄)−N(¯x)] + 2x̄σ2
Tn(x̄)
= 2σ2T [N(x̄)−N(
¯x)]
∂E[UT ]
∂σn|¯X,X̄ =
∂E[UT ]
∂σF∗ dσFdσn
+∂E[UT ]
∂βI∗ dβ
I
dσn(54)
=
(2σF (βT − 1)[
¯xn(
¯x)− x̄n(x̄)]− 2σF [N(x̄)−N(
¯x)]
)∗(− σnσ
4T
σFσ4I
)+
(2σ2
T [N(x̄)−N(¯x)]
)∗(− 2σ2
Tσnσ4I
)=
4σnσ4T (βT − 1)x̄n(x̄)
σ4I
+2σnσ
4T [N(x̄)−N(
¯x)]
σ4I
− 4σnσ4T [N(x̄)−N(
¯x)]
σ4I
=4σnσ
4T (βT − 1)x̄n(x̄)
σ4I︸ ︷︷ ︸
<0
− 2σnσ4T [N(x̄)−N(
¯x)]
σ4I︸ ︷︷ ︸
>0
< 0
For fixed¯X and X̄, tiger’s utility is decreasing with σn. Also, this is
true if the tiger chooses¯X and X̄, since the tiger will choose
¯X and
44
X̄ to maximize utility and the pointwise maximum of decreasingfunctions is decreasing.
Since σ2F = (σ2
T )2/σ2I + γ2
F = σ4T/(σ
2T + σ2
n) + γ2F , then
dσFdσn
=d
dσn
([σ4T/(σ
2T + σ2
n) + γ2F ]
12
)=
1
2σF∗(− σ4
T ∗ 2σn(σ2
T + σ2n)2
)(55)
= −σ4Tσnσ4IσF
= −(βI)2σnσF
< 0
dβT
dσn=
d
dσn
(σ4T
σ4T + σ2
Tγ2F + σ2
nγ2F
)= − 2σ4
Tσnγ2
(σ4T + σ2
Tγ2F + σ2
nγ2F )2
(56)
= − 2σ4Tσnγ
2F[
(σ2T + σ2
n) ∗(
σ2T
σ2T+σ2
n+ γ2
F
)]2 = −2σ4Tσnγ
2F
σ4I ∗ σ4
F
= −2(βI)2σnγ2F
σ4F
From Equations (30), (31), (55), (56),
dX̄
dσn=∂X̄
∂σF∗ dσFdσn
+∂X̄
∂βT∗ dβ
T
dσn(57)
= x̄ ∗(− (βI)2σn
σF
)+
σFn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x̄)∗(− 2(βI)2σnγ
2F
σ4F
)=
(−(βI)2σn
σF︸ ︷︷ ︸<0
)∗(x̄+
2γ2F ∗ n(x̄)/x̄2
σ2F ∗ [N(−x̄) + n(−x̄)/(−x̄) +N(−x̄)/(−x̄)2]︸ ︷︷ ︸
>0
)< 0
The final inequality follows from Lemma A1(3), which defines g(x) and im-plies g(x) ∈ (0,+∞). Similarly,
d¯X
dσn= − dX̄
dσn> 0(58)
Thus, discretion is decreasing with noise σn.
45
�
E.2 Social welfare and σn
Proof of Theorem 5.2. Next, we consider what happens to the society inresponse to fly’s expertise σn. From (24), we have E[W S] depends on
¯X, X̄,
σF , βI , [σS], where(1) σF depends on [σT ], [γF ], σn,(2) βI depends on [σT ], σn,(3)
¯X, X̄ depend on σF , βT ,
(4) βT depends on [γF ], [σT ] and σn.
dE[W S]
dσn=∂E[W S]
∂σF∗ dσFdσn
+∂E[W S]
∂βI∗ dβ
I
dσn+∂E[W S]
∂Cn(σn)∗ dCn(σn)
dσn(59)
+∂E[W S]
∂¯X
∗(∂
¯X
∂σF∗ dσFdσn
+∂
¯X
∂βT∗ dβ
T
dσn
)+∂E[W S]
∂X̄∗(∂X̄
∂σF∗ dσFdσn
+∂X̄
∂βT∗ dβ
T
dσn
)
I. Fixed¯X and X̄
Now, let’s fixed σ2T , σ2
S, γF , γT ,¯X and X̄.
dβI
dσn=
d
dσn
(σ2T
σ2T + σ2
n
)= − 2σ2
Tσn(σ2
T + σ2n)2
= −2σ2Tσnσ4I
(60)
46
∂E[W S]
∂βI=
∫¯X/σF
ϕ=−∞2
¯Xσ2S
σFϕn(ϕ) dϕ+
∫ X̄/σF
ϕ=¯X/σF
2σ2Sϕ
2n(ϕ) dϕ+
∫ +∞
ϕ=X̄/σF
2X̄σ2S
σFϕn(ϕ) dϕ(61)
= −2¯Xσ2S
σFn(ϕ)
∣∣¯X/σF−∞ + 2σ2
S
∫¯X/σF
ϕ=¯X/σF
[n′′(ϕ) + n(ϕ)] dϕ− 2X̄σ2S
σFn(ϕ)
∣∣+∞X̄/σF
= −2¯Xσ2S
σFn(
¯X/σF ) + 2σ2
S
[− ϕn(ϕ)
∣∣X̄/σF¯X/σF
+N(ϕ)∣∣X̄/σF
¯X/σF
]+ 2X̄
σ2S
σFn(X̄/σF )
= −2¯xσ2
Sn(¯x)− 2σ2
Sx̄n(x̄) + 2σ2S¯xn(
¯x) + 2σ2
S[N(x̄)−N(¯x)] + 2x̄σ2
Sn(x̄)
= 2σ2S[N(x̄)−N(
¯x)]
Therefore, from Equations (26), (59), (60) and (61)
∂E[W S]
∂σn
∣∣∣∣¯X,X̄
(62)
=∂E[W S]
∂σF∗ dσFdσn
+∂E[W S]
∂βI∗ dβ
I
dσn+∂E[W S]
∂Cn(σn)∗ dCn(σn)
dσn
=
(2σF (βS − 1)[
¯xn(
¯x)− x̄n(x̄)]− 2σF [N(x̄)−N(
¯x)]
)∗(− σnσ
4T
σFσ4I
)+ 2σ2
S[N(x̄)−N(¯x)] ∗
(− 2σ2
Tσnσ4I
)− C ′n(σn)
=4σnσ
4T
σ4I
(βS − 1)x̄n(x̄) +2σnσ
4T
σ4I
[N(x̄)−N(¯x)]− 4σnσ
2Sσ
2T
σ4I
[N(x̄)−N(¯x)]− C ′n(σn)
=4σnσ
4T
σ4I
(βS − 1)x̄n(x̄) +2σnσ
2T (σ2
T − 2σ2S)
σ4I
[N(x̄)−N(¯x)]− C ′n(σn)
Case one:When both γF and γT are small,
¯X → −∞, X̄ → +∞.
∂E[W S]
∂σn
∣∣∣∣¯X,X̄
≈ 4σnσ4T
σ4I
(βS − 1)x̄n(x̄) +2σnσ
2T (σ2
T − 2σ2S)
σ4I
[N(+∞)−N(−∞)]− C ′n(σn)
≈ 4σnσ4T
σ4I
(βS − 1)x̄n(x̄)︸ ︷︷ ︸<0
+2σnσ
2T (γ2
T − σ2S)
σ4I︸ ︷︷ ︸
<0
−C ′n(σn)
47
Thus, for fixed¯X and X̄, when both tiger and fly are not very corrupt,
society is worse off with the increase in the noise σn before cost.
Case two:When both γF and γT are very large,
¯X → 0, X̄ → 0.
∂E[W S]
∂σn
∣∣∣∣¯X,X̄
≈ 4σnσ2T x̄
σ4I
[σ2T (βS − 1)n(0) + (σ2
T − 2σ2S)n(0)]− C ′n(σn)(63)
=4σnσ
2T x̄
σ4I
[(σ2Tβ
S − σ2T + σ2
T − 2σ2S)n(0)]− C ′n(σn)
=4σnσ
2T x̄
σ4I
[(σ2Tβ
S − σ2T + σ2
T − 2σ2S)n(0)]− C ′n(σn)
=4σnσ
2T x̄
σ4I
[(σ2S
βIσ2T
σ2F
− 2σ2S)n(0)]− C ′n(σn)
=4σnσ
2T x̄
σ4I
[σ2S(βT − 2)n(0)]︸ ︷︷ ︸<0
−C ′n(σn) < 0
Thus, for fixed¯X and X̄, when both tiger and fly are very corrupt, society
is worse off with the increase in the noise σn before cost.
II. Tiger chooses¯X and X̄ optimally in response with γF
Next, we consider what happens to the social welfare with changes in σnwhen the tiger chooses
¯X and X̄ in response to γF .
∂E[W S]
∂X̄∗(∂X̄
∂σF∗ dσFdσn
+∂X̄
∂βT∗ dβ
T
dσn
)(64)
= −2βIγ2Tn(x̄)
σF∗[x̄ ∗(− (βI)2σn
σF
)+
σFn(x̄)
N(−x̄)− x̄n(x̄) + x̄2N(−x̄)∗(− 2(βI)2σnγ
2F
σ4F
)]=
2(βI)3γ2Tσnn(x̄)
σ2F
[x̄+
2γ2Fn(x̄)
σ2F ∗ (N(−x̄)− x̄n(x̄) + x̄2N(−x̄))
]
48
Similarly,
∂E[W S]
∂¯X
∗(∂
¯X
∂σF∗ dσFdσn
+∂
¯X
∂βT∗ dβ
T
dσn
)(65)
=2(βI)3γ2
Tσnn(x̄)
σ2F
[x̄+
2γ2Fn(x̄)
σ2F ∗ (N(−x̄)− x̄n(x̄) + x̄2N(−x̄))
]
Case one: When both γF and γT are small,¯X → −∞, X̄ → +∞.
49
Equation (59) can be expressed as:
dE[W S]
dσn≈ 4σnσ
4T
σ4I
(βS − 1)x̄n(x̄) +2σnσ
2T (γ2
T − σ2S)
σ4I
∗ [N(x̄)−N(¯x)]− C ′n(σn)(66)
+4(βI)3γ2
Tσnn(x̄)
σ2F
[x̄+
2γ2Fn(x̄)
σ2F ∗ (N(−x̄)− x̄n(x̄) + x̄2N(−x̄))
]≈ 4σnσ
2T x̄
σ4I
[(βS − 1)σ2
Tn(x̄) + (σ2T − 2σ2
S) ∗ N(x̄)−N(¯x)
2x̄
+σ4Tγ
2Tn(x̄)
σ2Iσ
2F
(1 +
2γ2Fn(x̄)
σ2F ∗ [N(−x̄)− x̄n(x̄) + x̄2N(−x̄)]
)]− C ′n(σn)
≈ 4σnσ2T x̄
σ4I
[(βS − 1)σ2
Tn(x̄) + (σ2T − 2σ2
S) ∗ N(x̄)−N(¯x)
2x̄
+ βTγ2Tn(x̄) +
2βTγ2Fγ
2Tn
2(x̄)
σ2F ∗ x̄[N(−x̄)− x̄n(x̄) + x̄2N(−x̄)]
)]− C ′n(σn)
≈ 4σnσ2T x̄n(x̄)
σ4I
[(βS − 1)σ2
T + βTγ2T +
2γ2T (1− βT )
p(−x̄)
]+
4σnσ2T x̄
σ4I
∗ (σ2T − 2σ2
S)[N(x̄)−N(¯x)]
2x̄− C ′n(σn)
≈ 4σnσ2T x̄n(x̄)
σ4I
[(βS − 1)σ2
T + βTγ2T +
2γ2T (1− βT )
p(−x̄)
]+
4σnσ2T
σ4I
∗ (γ2T − σ2
S)
2− C ′n(σn)
≈ 4σnσ2T x̄n(x̄)
σ4I
[βSσ2
T − σ2T + βTγ2
T +2γ2
T (1− βT )
p(−x̄)
]+
4σnσ2T (γ2
T − σ2S)
2σ4I
− C ′n(σn)
≈ 4σnσ2T x̄n(x̄)
σ4I
[βTσ2
S − σ2S − γ2
T + βTγ2T +
2γ2T (1− βT )
p(−x̄)
]+
4σnσ2T (γ2
T − σ2S)
2σ4I
− C ′n(σn)
≈ 4σnσ2T x̄n(x̄)
σ4I
[(βT − 1)(σ2
S + γ2T ) +
2γ2T (1− βT )
p(−x̄)
]+
4σnσ2T (γ2
T − σ2S)
2σ4I
− C ′n(σn)
≈ 4σnσ2T x̄n(x̄)
σ4I
[(βT − 1)︸ ︷︷ ︸
<0
(σ2S + γ2
T −2γ2
T
p(−x̄)︸ ︷︷ ︸−∞
)]+
4σnσ2T (γ2
T − σ2S)
2σ4I︸ ︷︷ ︸
<0
−C ′n(σn)
︸ ︷︷ ︸?
50
Thus, for fixed¯X and X̄, when both tiger and fly are not very corrupt,
society is worse off with the increase in the noise σn before cost.
Case two: When both γF and γT are large,¯X → 0, X̄ → 0. From
equation (66), Equation (59) can be expressed as:
dE[W S]
dσn≈ 4σnσ
2T x̄
σ4I
[(βS − 1)σ2
Tn(x̄) + (σ2T − 2σ2
S) ∗ N(x̄)−N(¯x)
2x̄(67)
+ βTγ2Tn(x̄) +
2βTγ2Fγ
2Tn
2(x̄)
σ2F ∗ x̄[N(−x̄)− x̄n(x̄) + x̄2N(−x̄)]
)]− C ′n(σn)
≈ 4σnσ2T x̄
σ4I
[(βS − 1)σ2
Tn(0) + (σ2T − 2σ2
S)n(0)
+ βTγ2Tn(0) +
2γ2T (1− βT )n(0)
p(−x̄)︸ ︷︷ ︸→1
]− C ′n(σn)
≈ 4σnσ2T x̄n(0)
σ4I
[βSσ2
T − σ2T + σ2
T − 2σ2S + βTγ2
T + 2γ2T − 2βTγ2
T
]− C ′n(σn)
=4σnσ
2T x̄n(0)
σ4I
[βSσ2
T − βTγ2T + 2γ2
T − 2σ2S
]− C ′n(σn)
=4σnσ
2T x̄n(0)
σ4I
[βT
σ2S
σ2T
σ2T − βTγ2
T + 2(γ2T − σ2
S)
]− C ′n(σn)
=4σnσ
2T x̄n(0)
σ4I
[βT (σ2
S − γ2T ) + 2(γ2
T − σ2S)
]− C ′n(σn)
=4σnσ
2T x̄n(0)
σ4I
[(2− βT )(γ2
T − σ2S)︸ ︷︷ ︸
>0
]− C ′n(σn)
Thus, for fixed¯X and X̄, when both tiger and fly are very corrupt, society
is better off with the increase in the noise σn before cost.
�
51
02
46
810
−10−50510
Soc
ial W
elfa
re, T
iger
’s u
tiliti
y an
d no
ise
σ n w
ith la
rge
γ T s
mal
l γF (
γ T=
3, γ
F=
0.1)
nois
e σ n
Social Welfare and Tiger’s utility
Soc
ial W
elfa
re w
ith c
ost
Soc
ial W
elfa
re w
ithou
t cos
tT
iger
’s u
tility
02
46
810
−6−4−2024
Soc
ial W
elfa
re, T
iger
’s u
tiliti
y an
d no
ise
σ n w
ith la
rge
corr
uptio
n (γ
T=
γF=
2)
nois
e σ n
Social Welfare and Tiger’s utility
Soc
ial W
elfa
re w
ith c
ost
Soc
ial W
elfa
re w
ithou
t cos
tT
iger
’s u
tility
52
References
Holmstrom, B. (1979). Moral hazard and observability. The Bell Journal of
Economics, 10(1):74–91.
Krueger, A. O. (1974). The political economy of the rent-seeking society. The
American Economic Review, 64(3):291–303.
Ross, S. A. (1973). The economic theory of agency: The principal’s problem.
The American Economic Review, 63(2):134–139.
53