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of economics
A THEORY OF COLLECTIVE REPUTATIONS
with applications to the persistence of
corruption and to firm quality
Jean Tirole
June 1993
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
A THEORY OF COLLECTIVE REPUTATIONS
with applications to the persistence of
corruption and to firm quality
Jean Tirole
No.
93-13
June 1993
M.I.T.
DEC
LIBRARIES
- 7 1993
RECEIVED
A
Theory of Collective Reputations
with applications to the persistence of corruption
and to firm aualitv
Jean Tirole*
June
7,
1993
"Institut d'Economie Industrielle, Toulouse, MIT and Ceras, Paris. This paper was
prepared under a cooperative agreement between the Institute for Policy Reform (IPR) and
Agency
for International
Development (USAID), Cooperative Agreement No.PDC#0095-
A-00- 1126-00. Views expressed in this paper are those of the author and not necessarily
IPR or USAID. The author is grateful to Jacques Cremer, Drew Fudenberg,
David Martimort, Patrick Rey, Paul Seabright, Lars Stole, and Barry Weingast for helpful
comments.
those of
Abstract
The
topic of this paper, the
phenomenon
neglected in economic theory despite
Because a group's reputation
is
its
of group reputation, has been
importance
for the social sciences.
only as good as that of
its
members, we focus
on the interaction between individual incentives and collective reputation.
Stereotypes are viewed as stemming from history dependence rather than
from
specific cultural or racial traits.
JEL
n: 026,511
Keywords: Stereotypes, history dependence, corruption, firm quality
Collective reputations.
1
Collective reputations play an important role in economics
sciences. Countries, ethnic, racial or religious groups are
working, honest, corrupt, hospitable or belligerent.
from their reputations
stantial rents
Some departments
for
and the
known
Some
to
social
be hard-
firms enjoy sub-
producing high-quality products.
are reported to treat their faculty or students
fairly.
It
seems
futile to ascribe
tions.
Rather, we view collective reputations as the outcome of group history.
By
such stereotypes to intrinsic features of the popula-
definition, the collective reputation of a
behavior of
a)
A
members. This implies
its
group
member
's
is
reputation
is
group
reflects the
average past
that:
only as good as that of
its
members.
Each
characterized by traits such as talent, diligence or honesty.
Past individual behavior conveys information about these traits and
generates individual reputations.
b)
By
contrast with group belonging, individual past behavior
fectly observed.
If
is
past individual behavior were fully unobserved,
imper-
mem-
bers of the group would have no incentive to sustain the reputation of
the group.
if
Conversely, the collective reputation would play no role
individual behaviors were perfectly observed.
ity of individual
Imperfect observabil-
behavior thus underlies the phenomenon of collective
reputation.
c)
Hence, the past behavior of the member's group
individual behavior.
affected
is
used
to predict his
Each member's welfare and incentives are thus
by the group's reputation.
d)
And
new members of a group depends on
therefore, the behavior of
the
past behavior of their elders.
Despite their pervasiveness, collective reputations have not, to the best
of
my
knowledge, been formally investigated. This paper
filling this
gap. Its key feature
lective reputations.
We
offer
is
may
two variants of the same model. In the
or
is
may
exclusion by the trading partner.
the
member
explain
(e.g.
We
why
first,
not belong to the group). The incentive
By
this
member's
we mean that the
fear of direct
individual rep-
induce the trading partner to behave in a way undesirable for
may
affected.
col-
imperfectly observed by his potential
to sustain an individual reputation stems from the
utation
an attempt at
the interplay between individual and
a member's individual reputation
"trading partner" (who
is
by not trading), while the belonging to the group
is
not
apply the direct exclusion variant to the issue of corruption to
corruption
is
a societal
phenomenon and why
it
tends to persist.
(See section 2 for an overview of the application to corruption).
This direct exclusion variant does not seem appropriate when the trading
partner has a low probability of knowing the member's past behavior.
The
buyer of a car does not even know the names of the worker, foreman and
engineer
who
car market.
built the car.
Yet brand image
is
an important factor
The reason why the car manufacturer's employee has an
to maintain quality
is
the fear of delegated or internal exclusion:
the interest of the firm to
fire
It
in the
incentive
may be
in
employees who have demonstrated undesirable
traits.
In this case, the worker
is
not concerned by the possibility that his sup-
plying poor quality will have a significant impact on buyers'
work, but rather by the possibility of being
fired.
So
in
demand
for his
the delegated exclu-
sion variant the trading partner (the buyer) reacts to the collective reputation
and the group (the firm) excludes on the
basis of individual reputation, while
in the direct exclusion variant the trading
partner reacts to both collective
and individual reputations and the group does not necessarily control memYet the two variants are formally very similar because imperfect
bership.
observability of individual behavior plays the
Modeling
study
its
in industrial organization
same
central role
1
has viewed the firm as a black box to
quality choices and has ignored the question of
why workers have
individual incentives to defend the firm's collective reputation
offers
one insight into
is
2
.
Our work
this black box.
Before developing the formal analysis,
reputation
.
neither a convention nor a
it is
worth noting that a
norm3 A convention
.
collective
refers to the
coordination on a particular Nash equilibrium in a situation of multiple Nash
equilibria.
Many models
in
economics have multiple equilibria, for example
coordination games, repeated games, macroeconomic models with aggregate
demand
externalities or
The
will later discuss).
particular equilibrium
(1992),
models of
is
racial
and sexual discrimination (which we
interpretation of a convention as the selection of a
stressed for
example
in
Cole et
al.
(1992), Kandori
and Seabright (1992). Kreps (1990) compares corporate culture
to
'if workers' individual behavior were not observed within the firm, there would be
no incentive to sustain individual reputations, and firms could not build reputations for
high quality. If workers' individual behavior were perfectly observed within the firm,
workers would have no incentive to sabotage the firm's reputation, at least under the
classic conditions under which individual reputation is sustainable.
J
The works of Cremer (1986) and Kreps (1990) are exceptions to this rule. Among other
things, our work departs from theirs in that the behavior of a firm's employees is truly
history dependent. Cremer and Kreps develop repeated-game models of organizations
with overlapping generations of workers in which future generations may punish current
ones if these do not behave well. Kotlikoff et al (1988), in a similar spirit, show that in an
overlapping generations framework, the young may refrain from taxing the olds' capital by
fear that the next generation
at each point of time
3
is
would tax their
capital. In these models, the set of equilibria
history-independent.
We use the sociological definition of convention and norm (see, e.g., Elster (1989),
Sugden (1989) and Ullman-Margalit (1977)). Economists often seem to call a norm what
a sociologist would label a convention.
a convention, in that corporate culture in a firm
is
meant
to
communicate
The
to its employees the (focal) behavior that they are expected to follow.
norm, unlike that of a convention, seems to
sociological notion of a
stress
psychological factors at the expense of methodological individualism. That
individuals are eager to be approved by others generates
consumption norms 4
be
.
Unlike convention behaviors,
in one's self-interest (at least
me").
("I will not litter in the
Thus neither conventions nor norms
between individual and
park even
if
noone
refer to the interplay
collective reputations described above.
An overview
2
norm behavior need not
narrowly defined). Alternatively individuals
want to be approved by themselves
will see
norms of etiquette or
The
of the argument:
case of
corruption.
It
is
commonplace
development
to observe that corruption
policies.
laws, collection of taxes
It
a central issue faced by
affects all aspects of public
and
tariffs,
management
subsidies, police work, credit, building
many
is
countries, corruption has
enforcement of
of public contracts, housing
and business permits, and so
become
scales of charges for public services,
life:
institutionalized.
and markets
forth. In
There are agreed
for public offices are well
developed (for instance, superintending engineers' posts on the coastal deltas
in India cost
up
to 40 times the annual salary for that rank, for an expected
duration on the job of two years
The
epochs
).
large disparity in corruption patterns across countries
is
puzzling.
rupt while another
4
5
An
African country
(e.g.,
Kenya)
will
(e.g.,
Zaire) will
have kept a clean
and across
be completely
civil service.
cor-
Most
Such norms might also be rationalized by the economic theory of wasteful signaling,
but the emphasis is rather on the eagerness to be approved.
6
The Economist, May 4, 1991, India survey, pages 15-18.
LDCs
are affected by the plague to a
countries, the recent
much
growth of corruption
Italian scandals notwithstanding.
larger extent than developed
in
France or the Japanese and
Corruption was pervasive
in
England and
several other
European countries two centuries ago and has much subsided
While
factors related to the social fabric such as a family-centered
since.
ethos or the existence of tightly-knit clans certainly play a role,
tile
it
seems
to ascribe corruption to particular cultures or racial groups. Rather,
fu-
it is
important to understand the historical reasons and institutional factors that
make some
societies
more corrupt than
others.
have a good grip on how to tackle the issue6
We
will
we be
able to
.
argue that history matters. In particular, a society in which corrup-
tion develops unfettered today
is
more
an identical society that takes a better
as a surprise to those
tices
Only then
likely to
start.
who have observed
be corrupt tomorrow than
This conclusion
will
not come
the persistence of corrupt prac-
and witnessed the many unsuccessful attempts to eradicate them.
It is
nevertheless important to identify the causes of hysteresis.
Our purpose
we want
is
not to investigate the
to build an abstract
many
facets of corruption.
Rather
framework that can be used to study the dy-
namics of collective reputation
in a
wide range of circumstances.
We
first
develop a stylized model in which economic activity requires trust between
contracting parties that they will not engage in corrupt practices.
ties
make
The
par-
inferences about the honesty of their potential trading partners
on
the basis of an imperfect observation of their track record, namely whether
"The
many
articles and books [see, e.g., Gould (1980), Hager (1973), Klitgaard
Myrdal (1970), Lui (1986), Noonan (1984), Rose-Ackerman (1978), and
Theobald (1990)], corruption hasn't yet attracted much attention from economic theorists,
and therefore its analysis lacks adequate foundations based on information economics and
game theory (note that Robert Klitgaard's fascinating books on the topic constantly point
at the relevance of information economics). A proper understanding of the phenomenon
seems to require an examination of its microstructure.
topic of
(1986, 1989, 1991),
they engaged in corrupt practices in the past. Because one's real track record
is
partially observed
by potential trading partners, individuals
incentive to develop or maintain a reputation for honesty.
hand, because this track record
is
may
On
have an
the other
not perfectly observed, inferences are also
based on the society's behavior as a whole. This combination of individual
and
collective
equilibria.
viduals
trust
is
what
in
society as a whole
If
whom
them
stigmas
in
our model
is
may
create a scope for multiple
honest, people are willing to trust indi-
they have not heard to be corrupt.
the future
if
And because
society will
they keep a clean record, individuals are willing
to invest in a
good reputation.
eral suspicion
makes honesty a low-yield investment, and
In contrast, in a corrupt society, the gendistrust
indeed
is
justified (section 3).
We then
sitivity of
is
study the issue of persistence of corruption by analyzing the sen-
equilibrium to
initial conditions.
In the benchmark, the
a stationary equilibrium and has a low level of corruption.
in
slightly perturb the
there
is
economy by assuming that
in the corrupt activity at that date.
1, 2,
•
•
•.
We
The
then
date (date
agents alive at date
The economy
0),
otherwise unchanged
is
economy
the low steady state level of corruption. Interestingly,
omy must remain
engage
then ask whether the temporary increase in corruption
necessarily has lasting effects, or whether the
Our
We
a one-shot increase in the gain to being corrupt (or a relaxation in
the enforcement of anticorruption laws).
at date
at the initial
economy
is
able to go back to
we
find that the econ-
corrupt not only in the short run, but also in the long run.
analysis unveils two effects: First, the agents
who were
alive at date
have smeared their reputation. In our model, they have more incentives to
engage in corrupt
activities
than
if
they had always behaved honestly.
They
are thus locked into corruption. This idea explains the short-run persistence
of corruption. Shortly after date 0, there are lots of agents locked into cor-
This
ruption.
first effect
however does not explain why the steady state
affected by this one-shot increase in corruption, since
new ones
are progressively replaced by
ping generations). Namely,
why do
(that
is,
the agents
we assume
our model
who
is
that agents
one of overlap-
arrive with an
unsmeared
Why
(individual) reputation also necessarily engage in corrupt activities?
the young inherit the corrupt practices of their elders?
the early periods after date
records, the large
number
if
of agents
their "age" (or
nities to get corrupt earlier)
who have been
more
is
realistically,
not observed.
are victims of this suspicion for at least a
large
enough (that
to remain honest.
is, if
and
corrupt at date
This suspicion affects
whether they had opportu-
Agents who arrive at date
number
T
of periods and,
if
T
1
is
This implies that the number of agents with a smeared
of this suspicion for at least
We
that in
is
agents are not replaced very fast), have no incentives
record does not decrease. In turn, agents
so forth.
The answer
do
and because of imperfect observation of track
0,
therefore remain corrupt raises a general suspicion.
new agents
is
T
periods,
who
arrive at date 2 are victims
and decide
to
become
corrupt.
therefore obtain a vicious circle of corruption, where the
And
new
generations suffer from the original sin of their elders long after the latter
are gone.
It
should also be noted that in this model, corruption ratchets up and not
down, in the sense that a one-shot reduction
in corruption due, say, to
enforcement of anticorruption laws has no lasting
number
takes a
minimum
of periods without corruption to upset the corrupt equilibrium. At
this stage
level of
effect. It
tough
we have but an example, and no general
corruption in society increases faster than
the example suggestive of
why
result
it
showing that the
decreases, but
we
find
short run crackdowns on corruption often have
limited efficiency.
We
hope that further work
will investigate
the generality
of this conclusion.
While sections
nomic
activity
3
and
4 analyze the possible
breakdown of desirable
eco-
due to lack of trust and widespread corruption, section 5
uses similar modeling and ideas to study the development of other, undesirable activities.
More
precisely,
we analyze the phenomenon
Suppose a foreign company wants to do business
whether
it
of extortion.
a country and wonders
in
should bribe low-or high-level government employees to process
goods through customs, issue work permits
for
company personnel or
building
permits for plants, grant a government contract or provide police protection.
It
has been well documented by Jacoby et
is
unfortunately one of the
ing aside any moral issue,
with different
first
al
(1977) and
we ask whether there can
This
levels of extortion.
are offered no bribe, firms can get
officials
that this
questions business persons confront. Leav-
is
exist multiple equilibria
indeed the case. In a noncorrupt
equilibrium, government officials do what they are
ment
many others
away by
meant
offering
to
do even
if
they
no bribe, and govern-
have no incentive to give them trouble given that they
will
not be offered bribes in the future. In a corrupt equilibrium, firms attach a
low probability of being able to conduct business without giving bribes, and
they do offer bribes.
Government
officials
are reluctant to do their job in
the absence of a bribe because this might reveal their "softness" to future
bribers. Again, the multiplicity of equilibria stems
from the combination of
individual and collective reputations (section 5).
Before turning to the analysis,
to the view that corruption
in
is
some academic
circles in
is
a
I
should point out that
lesser
harm.
It
I
do not subscribe
has become fashionable
the last thirty years to argue that corruption
a market mechanism that frees the economy from the
10
evils of excessive
bureaucracy. While this view has the merit of questioning the organization
of bureaucracy,
it
ignores the substantial efficiency costs of corruption, not to
mention moral and social
and the implications
effects
income distribution
for
(including those due to the diversion of international aid). These efficiency
costs include,
civil
among
servants, the
others, the selection of incompetent contractors
many
and
barriers to entry into business, the shortage of tax
and duty income, and the costs associated with tolerated pollution and job
safety infringements. Accordingly,
I
will
model corruption
as a socially costly
activity.
The
Individual and social stigmas:
3
case of
trust.
This section develops a simple model in which the
economic activity requires a minimum
ties.
More
precisely, a principal (the
level of trust
efficient
between contracting par-
buyer of a service)
agent (the supplier of the service) only
if
she
is
agent will not engage in corrupt activities.
organization of
will contract
with an
sufficiently confident that the
The
principal has some, albeit
imperfect information about the agent's track record, namely about whether
the agent has engaged in corrupt activities in the past.
Matching.
date
t
consider a stationary
remain in the economy up to
A € (0,1).
is
We
offset
With
by the
constant.
The model
is
(at least)
which agents
in
date
t
+
1
with probability
a new agent, so that the population of agents
a matching model. At each date
t,
each
(alive)
is
matched with a new principal 7 The principal decides whether to
1
or task 2 to the agent.
.
7
alive at
death process", we assume that each quit
this "Poisson
arrival of
economy
Task
1 is
the efficient task. Task 2
Principals can be either short lived or long lived.
11
is
a
is
agent
offer task
less efficient
task, but, for the principal,
corrupt.
less sensitive to
it is
the agent's choosing to be
a slightly different version of the model, task 2 corresponds to
[In
the absence of a
We
hire].
will
make an assumption guaranteeing
that
it is
always optimal for the principal to at least offer task 2 to the agent rather
than not hiring him.
Once
the corrupt activity, that
principal's payoff
and
1
is
D
if
is
from task
1
hired, the agent chooses
in
whether "to cheat" (behave dishonestly). The
in
the period
is
H
if
the agent behaves honestly
he cheats. Similarly her payoffs from task 2 are h and
more
d.
That task
sensitive to corruption than task 2 (given that the principal faces
a nontrivial choice)
means that
H >h>d>
We
whether to engage
also
assume that d >
so that
it is
D.
optimal to hire the agent.
Agents' preferences. There are three types of agents: "honest", in proportion a, "dishonest", in proportion
where a
7,
+
and therefore
for
f3
+
7
=
8
l
.
whom
are the
same
for
proportion
each cohort
Honest agents have a strong distaste
and
has a
directly punished, "honest" agents might
being punished
is
very costly). Dishonest agents always
cheat, for instance because they derive a high benefit from
in
in
in corrupt activities (alternatively, if corruption
probability of being exposed
for
and "opportunistic",
The proportions
for the entire population.
and never engage
be ones
/?,
it
a slightly different model, they might be transient agents
(alternatively,
who do
not care
about their reputation). Because honest and dishonest agents behave mechanistically (never
and always cheat,
respectively), the focus of our analysis
on opportunists. These have no aversion to being corrupt, but trade
current benefit from corruption and the loss in reputation.
8
This formulation of preferences
is
standard
(1991).
12
in reputation
off
is
the
Their benefits
models, see,
e.g.,
Diamond
from being hired
in tasks
1
and 2 and not cheating are
B
and
respectively,
6,
where
B>
They enjoy an
That
task.
>
additional short-run gain
is
same
the
in
we do not model
also that
The
G
b
0.
G>
from being corrupt
in either
both tasks simplifies the formal analysis. Note
explicitly the role of anti-corruption campaigns.
simplest, albeit extreme interpretation of the
model
is
that there
is
no
hard evidence that could lead to the indictment of a corrupt agent. Alternatively,
G
could be an expected gain from being corrupt, which would allow
a probability of confronting legal sanctions. Last the agents' discount factor
is
8q
<
1.
We
will let 8
8q\ denote the "relevant discount factor".
Agents know their own preferences (that
Information.
Principals
=
know the proportions
a,fl,"f
There are several ways of
malizing the imperfect observability of the track record.
in order to easily illustrate the
main
their types).
and imperfectly observe the track
record of the agent they are matched with.
one
is,
ideas.
The
We
for-
choose a simple
principal has probability
I* of finding out that the agent has engaged in the past at least once in a
corrupt activity
when the agent has
track record, that
with
is
binary.
is
The
9
.
So the observed
the information of the principal the agent
principal
is
matched
knows that the agent has been corrupt
once, or has no such knowledge.
9
in fact cheated k times
The assumption
at least
that the principal does not
would be interesting to extend the analysis to alternative information technologies.
it would seem reasonable to allow for forgetfulness (witnesses or evidence
disappear over time). Our insights ought to carry over to such specifications, but new
insights (such as the possibility of an individual's resuming an honest behavior after being
corrupt) would arise.
We have performed a different check of robustness by assuming that once an individual
is exposed a public file exposes him for the rest of his life. The expressions of Y and Z
below are slightly altered, but the analysis goes through under the same assumptions 1
through 4. See also section 6 in which, once exposed, the agent is excluded for the rest of
It
In particular
his
life.
13
.
know the
agent's age
and giving
important for the second
is
rise to everlasting effects of
may
Of
a one-time shock in corruption.
course this assumption should not be taken too
the idea that the principal
effect unveiled in section 4
literally. It is
a metaphor for
not be fully informed about the
number
of
times the agent had an opportunity to be corrupt in the past.
Assumption
=
x
1:
<
<
X\
<
£2
<
X3
•
<
•
1
and
zjt+i
Assumption
comes more
-
xk
<x k - Xfc-i
for all k.
says that the leakage of information about corruption be-
1
when the agent has cheated more
likely
in the past;
and that
this
increase occurs at a decreasing rate. This assumption simplifies the analysis
by garanteeing that an individual
corrupt a certain
We now
a)
Low
corruption equilibrium. Suppose that
been corrupt
agent
of times.
demonstrate the possibility of coexistence of two
A
have honestly.
since d
number
locked in corruption after having been
is
may be
agent
In contrast,
when the
is
economy
agents with a clean track record
The
10
and
no such information, the
The proportion
is
is
(a
+
A)
[l
+
A
(1
-
ix )
+
A2
Y
fiY where
(1
-
and op-
of honest
The proportion
7).
that past corruption activities go unnoticed
-
she knows has
honest or opportunistic, or else be a dishonest agent with a
portunistic agents in the
(1
who
necessarily a dishonest agent
principal has
deceivingly clean observed track record.
Y=
opportunists always be-
principal offers task 2 to an agent
in the past, since the
> D.
all
equilibria.
of dishonest
the average probability
is
10
x2 )
:
+
•
•
•
+
A*
(1
-
xk )
+
•
•]
probability that the agent will not cheat given a clean observed record
The proportion
of "newborns" (who therefore have not yet cheated)
proportion of "one-period old" (who have cheated once)
14
is
(1
—
is (1
—
A)A, and so forth.
A), the
is
(a + 7)/(a+7 + /?F).
The
principal offers task
1 if
and only
if
the following
assumption holds:
Assumption
2:
"+^
Do
1.
Their payoff
is
By never being
is
offered
= B/(l— S). Suppose that
B + 8B + S*B-\
therefore
.
and observed) record and are always
they instead cheat today and keep cheating
payoff
m- d)>0
opportunists have an incentive not to become corrupt?
corrupt, they keep a clean (real
task
0Y
. h)+
:{H
in the future.
Their expected
then
(B + G)
+ 6(B +
-6)-Z] + 6(b +
G)[l/(1
G)Z,
where
Z=
is
xi
+
Sx 2
+
2
S x3
-\
the present discounted probability of being found out in the future given
that one has cheated once and will continue cheating. So, a necessary condition for a low corruption equilibrium
Assumption
Appendix
assumptions
1
1
3:
-
G/(\
is:
< S(B -
6)
b)Z.
shows that the low corruption equilibrium indeed
through
3.
The
intuition
is
that from assumption
exists
1
,
under
the agent
has more incentive to be corrupt, the more he has been corrupt in the past.
In this sense, agents are locked into corruption once they start being corrupt.
Note also that a low corruption equilibrium
are not poorly informed
u
.
exists only
if
the principals
Agents must have enough incentives to maintain
their reputation for honesty.
b)
High corruption equilibrium. Suppose now that opportunists are always
corrupt and principals always offer task
11
If the
x
s are close to zero,
Z
is
2.
close to zero
15
Because keeping a clean slate has
and assumption 3
is
violated.
no value,
indeed optimal for opportunists to be always corrupt.
it is
optimal for a principal to
agent
is
or dishonest with probability
Assumption
an agent with a clean slate? Such an
offer task 2 to
honest with probability a/[a
(/?
+
(/?
+
+ i)YI\ct +
"i)Y]
(0
*
+ f)Y}. We
(/?
-h)+
:{H
v
+ 7 )r
'
thus
i£pK- D- d)<
0S + )r
a+
when
that assumption 4 holds
when the
(
T
The high corruption equilibrium exists if and only
We
and either opportunistic
make
4-
a+
agents and
Is it
if
o.
.
assumption 4 holds. Note
there are enough opportunistic and dishonest
principals' information
is
not very precise.
conclude that the low and high corruption equilibria both exist when
assumptions
1
through 4 hold
12
.
The
role of imperfect observability
highlighted by the facts that assumption 3
formation
very bad and that assumption 4
is
information
is
is
violated
is
if
violated
is
the principals' inif
the principals'
very good.
Remark (comparison*
with the economic theory of discrimination):
imperfect observability assumption
is
reminiscent of that
made
in
Arrow's
(1973) statistical theory of discrimination of minorities by employers
Arrow looks a one-shot employment decision and assumes that workers
(secretly) invest in skills
and then the employers run an imperfect
Our
13
.
first
test of the
13
Sah (1991) has developed a theory of crime in which the multiplicity of equilibria has
a different origin. In Sah's model, the probability of being caught and punished for a
crime decreases with the number of other criminals, assuming that the budget for crime
investigation is not very responsive to the level of crime. The individuals' choices of
whether to commit a crime are therefore strategic complements: The more people commit
commit a crime. While the multiplicity of
a static framework, Sah's model is actually an intertemporal
one in order to highlight the idea of osmosis; the focus is not on reputation as in the
present paper, but on local learning about the probability of punishment. Individuals
learn slowly about this probability by observing whether their neighbors get punished
when they commit a crime.
13
See also Akerlof (1976), Coate-Loury (1991), Kremer (1992), Lundberg-Startz (1983),
Milgrom-Oster (1987), and Phelps (1972) for related ideas.
crime, the
more
incentives the individual has to
equilibria can be illustrated in
16
Because the
resulting ability.
beliefs
test is imperfect, the
about whether the worker has invested
ability. If
employer uses the prior
in assessing the worker's true
a higher prior belief that the worker has invested also makes
profitable for the worker to invest, there
is
it
more
scope for multiple equilibria. The
literature has interpreted the multiplicity of equilibria as the possibility of a
differential
treatment of workers based on their race, sex or other observable
characteristics.
There
an analogy between the theory of discrimination
is
and the (more dynamic) theory of corruption developed
here. In the corrupt
equilibrium, agents face a general suspicion of corruption and do not gain
from not becoming corrupt,
same way that a discriminated against
in the
group has (under certain conditions)
little
employer puts more weight on prior
incentive to invest in skills
beliefs
if
the
than on imperfectly measured
ability.
There
is
is
however a sense
not about societal behavior;
crimination model
which the
in
for,
statistical discrimination theory
the multiplicity of equilibria in the dis-
independent of whether there are other employers or
is
workers besides the employer and the worker in question 14
the statistical discrimination theory, which
collective reputations,
an
intrinsically
is
Furthermore,
.
a static theory,
dynamic phenomenon,
is
not about
either.
Persistence of corruption.
4
We now
librium.
investigate the effect of a one-time shock in corruption on the equi-
To keep the
analysis simple,
we
specialize the
model further by
making
Assumption
14
5:
X\
=
ij
=
•
•
•
=
€
x
(0, 1).
The low and high
corruption equilibria in our model could similarly coexist with
single long-lived principal and agent, but only under the implausible assumption that the
principal does not observe her per- period payoff (otherwise the principal perfectly
the agent's track record).
17
knows
That
is,
the probability of exposure of corrupt activities
number
the
Assumption 5 implies
of past corrupt acts.
an opportunist remains corrupt once he has started;
Y = -Ax
and
1
Z=
-
x/(l
we
shock at date
all
will
0.
the gain
G
exists
assume. Suppose
and only
if
now that
The gain from being corrupt
agents alive at date
independent of
in particular that
it
also implies that
6).
The low corruption equilibrium
hold, which
is
get corrupt.
assumptions 2 and 3
the economy faces a temporary
at that date
The parameters
from cheating) are unchanged
if
is
of the
very large, and so
model (including
at dates 1,2, ••.
We
show that
under an additional assumption, the economy cannot go back to the low
corruption equilibrium. Indeed, the unique equilibrium exhibits a high level
of corruption forever.
Let us perform the following thought experiment. Suppose that the opportunistic agents born at date
date
t.
date
t.
1
through
t
behave honestly before and at
This presumption gives the best chance to the existence of trust at
The
probability of honest behavior at date
t
given an observed clean
record and given that opportunists born at or before date
corruption
P(t)
is
=
[a
+ 7 (1 -
["
a + 7 (l-A<)
+ 7(l-A')] + [/?K + 7(l-z)A<r
+
(1
A) (1
+
•
•
•
+
A'" 1 )]
+
\fiY
+ (1 - p(l))(I> - d) <
— p(oo))(Z? —
d)
>
we
let
(this is
T
p(T)(H -
is,
+
h)
increasing function,
That
A
-
Suppose p(l)(H
h)
are locked into
+
(1
*)(1
-
A) (A'
t
is
X t+1
-
an
such that
- p(T))(D - d) <
0.
under the most optimistic assumption, principals
18
+
Recalling that p(oo)(H
assumption 2) and noting that p
denote the largest
h)
0.
+ 7 (1 -
still
do not
trust
+
+
agents with observed clean records at date T; thus (T
now
length for suspicion to phase out. Suppose
6:
Assumption
6 states that
date
1
to cheat at date
be trusted before
is
G (l + 8 +
Assumption
+
it is
task
1
right
side
(T
after date
G<
x6{B -
> xd^(B -
6)/(l
-
6).
[T
left
hand
through date
1
T
not
side of assumption 6
(discounted at date
Note that assumption 6 requires
1).
ijt
+ 1). The
will
1),
an upper bound on the cost of not being offered
is
+
too small, since with
)
that
(and therefore forever) given that the agent
1
(at best) date
hand
7 "" 1
minimum
a dominant strategy for an agent born at
the gain from cheating from date
and the
a
a
1) is
>
constant for k
1
assumption 3
is
T
not to be
equivalent to
b).
Consider now the generation born at date
2.
All its elders have
been
corrupt in the past, and assumption 6 ensures similarly that cheating at date
2
and thereafter
a dominant strategy.
is
By
generations. Corruption has ratcheted up
same
induction, the
is
true for
and does not subside even
all
after the
generation that has committed the original sin has by and large disappeared15
15
There
once
exist other reasons
in place.
other corrupt
official
Corruption
officials to
than those unveiled here why corruption tends to persist
may
also persist because corrupt officials are likely to choose
work with them and to succeed them.
from having a corrupt subordinate
is
that the
official
A
benefit for a corrupt
can extort the subordinate and
obtain some of the bribes he collects. For example, the subordinate
who
.
may be a
tax collector
gives back a fraction of the bribes to his boss. Another benefit for a corrupt official
from being surrounded by other corrupt officials is that these colleagues will be reluctant to
denounce him by fear that they themselves might be exposed in a retaliation [see AndvigMoene (1990) for a model in which a bureaucrat's cost of being corrupt decreases with
the
number of corrupt
colleagues (such colleagues can be bribed not to report corrupt
transactions).] Last, a corrupt official
corrupt successor
will
is
likely to prefer
having a corrupt successor, since a
not perform as well as an honest one and therefore
will
not disparage
performance. For these three reasons, corruption in hierarchies such
as government, courts and political organizations is likely to have a life of its own. This
the departing
official's
explanation for persistence,
likely to
be
if it is
relevant, suggests that
efficient if it fries big fish, since
hierarchy run by corrupt
an anticorruption campaign is
move up a
honest individuals cannot easily
officials.
Another factor of persistence is the possibility of a low-budget trap (see, e.g., Klitgaard
(1988)). A government official, like any economic agent, has an incentive to behave only if
the cost of cheating (the probability of being punished times the extent of the punishment)
exceeds the benefit of misbehaving (the bribe). The monetary punishment when caught
19
This simple model also illustrates the possible failure of a short-run anticorruption campaign. Suppose that at date
1
(or,
equivalently at any later
date) the government runs a tough anticorruption campaign that lasts one
period and makes
it
unprofitable for opportunists to engage in corruption at
that date. Suppose further that the following strengthening of assumption 6
holds:
G (l +
Then
it is
at date 2,
• •
•
+
S
T- 2
6)/(l
-
S).
a dominant strategy for generations born at dates
and corruption
prevails at all dates after date
campaign only implies a decrease
no
> x6 T -\B -
)
effect thereafter.
in corruption
1.
1
and 2 to cheat
The anticorruption
during the campaign and has
Corruption does not ratchet down.
Extortion.
5
We now
apply similar ideas to study extortion
briber (the principal)
is
sufficiently
16
.
Extortion occurs
if
the
convinced that the bribee (the agent)
will
can be the loss of a well-paid job (plus, possibly, the confiscation of personnal assets). In
government officials may act as a potential deterrent to corrupcountry with a low level of tax collection or with high procurement expenditures
pays low wages to its civil servants, who are then encouraged to become corrupt. Corparticular, high wages for
tion.
A
ruption in turn reduces tax collection and raises procurement expenditures, creating
budgetary problems. This yields a poverty
An
objection to the previous reasoning
new
cycle.
is
that the government could borrow internally
or externally in order to give decent wages to the civil servants, get rid of corruption,
escape the poverty trap and then reimburse
its
debt.
Let us note however that
not be easy to borrow internally substantial sums of private
which, furthermore, the rich prefer to put their
money abroad
money
in
for safety
it
may
a poor country
and
(in
confidentiality
reasons). Borrowing abroad is not easy either, if only because foreign creditors are worried
by the possibility of repudiation of the debt. It is interesting to note in this respect that
major international lenders often require tough budgetary discipline as a precondition for
their loans. Future research ought to investigate the feasibility of an escape from the
poverty trap in a situation of imperfect capital markets.
16
See Strand (1990) for a different model of extortion. There, a bureaucrat asks for a
bribe from a firm. The firm may accept the deal or report the attempt to extort to a
government controller, who himself may or may not be corrupt. The firm is blacklisted if
the controller is corrupt and receives a reward otherwise. A corrupt controller demands a
bribe from the bureaucrat instead of punishing him.
20
By analogy with
not provide a service in the absence of a bribe.
the model of
section 3 where the agent wanted to develop a reputation for trustworthiness,
the agent here wants to look tough and convince principals that he will not
provide services for them unless they give a bribe.
of similarities with the previous one,
and
will
The model
shares a
number
purposedly share some of
its
notation.
As
before, the
model
one of matching. In each period, the agent (the
is
government
official,
the briber).
The timing within
whether or not to
the bribee)
is
matched with a new principal (the
the period
is
as follows: First, the firm decides
offer a bribe to the official.
the size of the bribe.
The
firm gains
firm,
V>B
if
For simplicity, we
let
B
denote
the agent provides the service.
Second, the agent decides whether to provide the service. There are three
types of agents: "honest", in proportion a, "corrupt", in proportion
"opportunist", in proportion 7, where
same
for
each cohort. Honest
a + /3 + *f =
officials
1.
The proportions
/?,
and
are the
always provide the service. Corrupt
never provide the service unless they receive bribe B. Opportunists,
officials
when they are
offered
no
bribe, trade off a short-term cost c
>
of not
providing the service and the long-term loss of reputation for being tough.
They provide the
either
service
if
offered a bribe.
One can
think of c as coming
from scruples associated with not doing one's job or from a probability
of being caught
and punished. The probability of survival A and the relevant
discount factor 6 are denned as before.
We
again posit imperfect information about the agent.
probability x^ of finding out that the agent has been
The
weak
principal has
at least once in
In Cadot (1987), a bureaucrat administers a test to grant a permit. There are two kinds
of bureaucrats: "honest" (who grant the permit if and only if the candidate passes the test)
if and only if they receive a bribe). The candidate,
a bribe, can accept the deal or denounce the bureaucrat to a controller.
Denunciation delays the permit (and, if the candidate does not know his ability, may not
and "corrupt" (who grant the permit
when asked
for
succeed).
21
the past,
when the agent has
or "giving in"
No
been weak k times, where "being weak"
means that the agent provides the
does not offer the bribe 17
a)
in fact
.
The x^ sequence
when they
never offer a bribe even
the government
official
gave
in.
satisfies
assumption
who
1.
no extortion equilibrium, the firms
In a
extortion equilibrium.
service to a principal
don't
know
of any occurrence in which
an equilibrium, opportunists always
In such
give in, since they will never be offered a bribe in the future. Is this rational
a firm not to offer a bribe when
for
does not know
it
its
faces
an honest or
opportunistic agent? Let
Y=
(1
-
A)
[l
+
A
(1
-
Xl )
+
A2
(1
-
i2 )
+
•
•
•]
denote the average probability over the population of opportunists and honest
agents that an opportunist or honest agent
in the past.
know
not
if
The
if
official
whose type
it
does
the following assumption holds:
7:
B>
Assumption 7
not observed to have been weak
firm does not offer a bribe to an
and only
Assumption
is
»
+
(a
•'
+ -,)Y
states that the size of the bribe exceeds the conditional prob-
ability that the official is corrupt times the value of the service to the firm.
Note that the no extortion equilibrium
exists only
informed about the agent's track record
close to
and assumption
7
is
who
known
to have given in;
17
It
This
are not
known
the firm
is
not perfectly
A and the is are close to
1,
Y
is
violated).
b) Extortion equilibrium.Suppose
agents
(if
if
now
that the firms offer a bribe to those
to have given in,
and no bribe to those who are
and that opportunists do not give
in (unless they
would be worth investigating alternative assumptions on individual reputations.
but simple assumption allows us to make direct use of the preceding
restrictive,
analysis.
22
have already given in at
>
when
times, in which case they give in)
1
no bribe.
offered
If
least k*
the firm knows that the
no bribe,
this official
has given in at least once when offered
official
must be honest and therefore
not to offer a bribe. In contrast,
if
it is
the firm does not
optimal
know
has given in in the past, the firm optimally offers a bribe
for the firm
that the
if
official
the probability
that the service will not be provided in the absence of a bribe times the value
of the service exceeds the bribe:
Assumption
8:
b<
q
/3
In an extortion equilibrium,
it
Z+z-v.
+ 7 + aK
must
be the case that when offered no
also
bribe an opportunist does not want to give
discounted expected number of bribes that the
and continuing to give
in
Let z denote the present
in.
every time that he
official receives
is
by giving
not offered a bribe
necessary condition for the existence of the extortion equilibrium
Assumption
(rr6
-
>c
z
the principals' information
is
close to
B/(l
—
6)
is
not too imprecise
and assumption 9
equilibria coexist.
-
1).
is
Note that
(if
it
can exist only
the xs are close to
0,
z
violated).
thus conclude that under assumptions
no extortion
that
assumptions 8 and 9 hold (the
if
almost identical to that in Appendix
if
We
is
A
)
Conversely, the extortion equilibrium exists
is
.
9:
6
proof
18
in
The formal
7,
8
and
analysis
is
9,
the extortion and
almost identical to
that of section 3. Yet the economics of trust (section 3) and extortion (this
18
z
is
given by the following recursive equation:
opportunist
who
Let
Vk
denote the valuation of an
has given in k times in the past, and gives in whenever he has not been
offered a bribe:
Let x
=
linu^oo * 4
Vk =z k 6Vk+l + (l-z k )(B + 6Vk ).
and
V*, = (1 - x)B/(l - S). Then z = V
x
23
.
section) differ in a few respects.
In the extortion context, individuals
want
to build a reputation for the behavior that society tries to eradicate. In the
trust context, they
will
want to build a reputation
for honesty.
This distinction
have implications when adapting the design of anticorruption policies to
the targeted form of corruption.
A
careful analysis of this conjecture falls
outside the modest scope of this exploratory paper.
Exclusion from the group: The case of a
firm's reputation for quality.
6
When
the trading partner hardly observes the past individual behavior of
member, the
the
behave well can only come from the
latter's incentive to
threat of retaliation by the group
to the group generates a rent but
itself.
is
We now
no longer a
assume that belonging
fait
While we
acccompli.
develop the analysis in the context of a firm's reputation for quality,
equally well to any organization or group that coopts
its
it
applies
members and can
freely exclude them.
We
consider a stylized model of a firm as a workers' cooperative.
Each
period the workers share and consume the firm's profit. [We abstract from
issues such as
unequal treatment, hierarchies and delayed compensation in
der to better focus on that of collective reputation.
The assumptions
or-
driving
the results are that incentive problems are not perfectly solved by alternative
methods and that the workers enjoy a higher
firm.]
Demand
workers
is
is
inelastic
and constant over time; the
(large)
number
of
constant and, by normalization, equal to one worker per unit of
good produced
in a period.
with Poisson probability
screening
rent in a higher reputation
among workers
(1
Workers who either quit (which occurs,
—
as earlier,
A)) or are fired are immediately replaced.
at the hiring stage
24
is
feasible in our
model.
No
We also
assume
for the
moment
that there
no cost
is
for the firm of firing a
worker
and hiring a new one.
The consumers
in
each period observe the firm's track record, namely the
average quality of items produced
equal to
H
(high) or
L
(low), with
reservation price at date
t
is
each past period.
in
H
> L >
individual track record of the worker
if
u
is
t
item's quality
is
The consumers' (common)
equal to the expected quality produced by the
firm conditional on the firm's track record.
they buy. So,
0.
An
Consumers do not observe the
who has produced
the particular item
the consumers' posterior probability of buying a high
quality item given the track record, the firm charges price p t
=
i/ t
Producing a low-quality item costs nothing to the worker
H+(\ — u
t
)L.
in charge of
the item, while producing a high-quality unit involves a disutility of
effort.
There are three types of workers. Using the same notation and terminology
as earlier, a
worker
is
"honest" with probability a, meaning that this worker
has no disutility of effort and always produces a high-quality item.
probability
/?,
the worker
is
With
"dishonest"; he then has a very high disutility
of effort for producing high-quality
and always produces a low-quality item.
G
Last, with probability 7, the worker has disutility of effort
of producing
high-quality and behaves opportunistically.
Keeping with the notation of the paper,
that the firm,
i.e.,
let
x* denote the probability
the workers, find out that a worker has produced at least
one low-quality item
in
the past
quality items in the past.
The
when the agent has
firm
may
in fact
produced k low-
find out either directly
word of mouth or observation of past decisions of the worker, or
through
indirectly
through consumers' complaints about the durability of the product.
sequence {it}
section,
satisfies
assumption
1.
For consistency with the previous
we assume that the firm does not know the
25
The
"age" of
its
workers
(or,
more
realistically,
the
number
of times they
had a good opportunity to
however the analysis would not be affected
knew the
the firm
if
age, because, in the derivations below, the firm fires a worker
we
has evidence of a low-quality production. As before,
each period, conditional on their not being
and we
probability A (the survival rate),
namely A times the workers' discount
factor,
We
workers'
anytime
assume
that, in
denote the relevant discount
factor.
look at steady states and define a high- (low-) reputation firm as a
firm in which opportunists always produce high- (low-) quality items.
follow section 3 in proving the possibility of existence of a
equilibria. In
both cases, the firm keeps workers for
wrongdoing and
of
a)
it
workers stay in the firm with
fired,
let 6
also
shirk);
fires
The expected
who produces
Y=
it
good and a bad
has no evidence
the others.
High-reputation firm.
firm of a worker
whom
We
-
(1
f) A
A)
present discounted tenure in the
low-quality in each period
f
(l
-*
)(1
-
ix )
•
•
•
(1
-
x
t
is:
),
t=o
=
(where z
t,
and
that
—
(1
io)
Y differs
exclusion
ing that
the probability of not quitting the firm before age
0): A' is
•
•
•
(1
from
when
it
—x
Y=
is
t )
(1
happens
new workers
are
the probability of not being caught. Note
- A) £~
is
A'(l
-x)
t
only to the extent that
permanent and not temporary. Assum-
drawn
in
a pool with proportions (a,
honest, dishonest and opportunistic workers 19
,
/?,
7) of
the steady-state pro-
portions in a high-reputation firm are (a,/?}', 7). Consumers therefore
19
In a general equilibrium context, this assumption
means
that, once fired
by a firm,
workers are not rehired by another firm; otherwise dishonest and (depending on the equilibrium) opportunistic workers would be overrepresented in the labor market compared
to the prior distribution.
We
could alternatively have conducted the analysis under the
assumption that firms cannot learn (or
infer the) previous behavior of
firms.
26
workers in other
pay per
unit:
a+i+
A
a+f
/3Y
+ PY
necessary condition for an opportunist to produce high quality
is
that
he prefers to always produce high quality rather than always producing
low quality:
G >
i-«
PH
where SZ
PH f:s (i-x )---(i-x
1
t
t
)=p H 1-* -6Z
the expected present discounted reduction in tenure due
is
to producing low quality.
that exclusion
Assumption
is
3':
Again,
permanent.
(7/(1
—
6)
We
Z
differs
Z
only to the extent
thus make:
< SpuZ.
Following the reasoning in Appendix
gether with assumption
from
1 ) is
1
shows that assumption
3' (to-
also sufficient for the existence of a high-
reputation equilibrium.
b)
Low-reputation firm. In a low- reputation firm only the honest workers
produce high
because opportunists don't produce
quality. In particular,
high quality, there
is
more
firing
and therefore more turnover
reputation firm than in a high-reputation firm.
in a low-
Consumers pay per
unit:
»
PL
=
a+
(/?
H
—n
+ 7 )r
I
m
+ 7)f — LI <
a + (/? + 7 )y
(/?
H
Ph.
A necessary and sufficient condition for the existence of a low- reputationfirm equilibrium
is:
Assumption
6':
Assumption
6' states
firm
is
G/(\
—
S)
>
SpiZ.
that the rent attached to working in a low-reputation
too small to dissuade the worker from shirking. Because assump-
27
tions 3'
and
6'
are not mutually inconsistent, there
may
exist multiple
equilibria.
Remark
1
(hysteresis):
The
analysis of section 4 suggests that reputation
a very valuable asset in the sense that
rebuild
its
may be
it
reputation after having lost
in
impossible for the firm to
Things however depend on the
it.
existence of costs of firing the whole labor force.
through a period of lax management
is
Suppose that a firm goes
which the opportunists produce low
quality and that, as in section 4, these are locked into producing low quality
in the future. If the cost of
and
firings
mass
on quits
firing is large, so that the firm relies
based on evidence to renew
its
labor force,
we know from
section 4
that the firm will never be able to (re)build a reputation for high quality even
long after the period of negligent management. Mass firing (implying firing
without evidence and therefore
only chance to recover,
if it is
Remark 2 (Labor market
firing
even the honest workers)
j3,
the firm's
doable at a reasonable cost.
externalities):
to take the proportions (a,
is
As we already noted, we were able
7) in the population of unemployed workers as
given for our partial equilibrium analysis.
It
would be interesting to extend
the analysis to study the labor market equilibrium. Indeed the proportions
pool of unemployed workers depend on
in the
reputation, and therefore
7
We
fare
fire
how many
firms have a high
only dishonest workers.
Conclusion.
all
belong to several organizations, cultures, and racial groups. Our wel-
and our incentives depend not only on our own reputation but
that of the groups
we
are associated with. This paper has
also
shown that
on
indi-
vidual reputations are determined by collective reputations, and vice versa.
Technically, collective reputations
seem to often give
28
rise to "strategic
com-
plementarities":
is
A
member's incentive
to maintain
stronger, the better the group's reputation.
an individual reputation
When
discipline
sustained
is
by the threat of exclusion from the group, low rents attached to being
in
a
low- reputation group create low individual incentives to remain in that group
and therefore perpetuate the group's bad reputation.
ing
is
an unalterable
trait, distrust in
the group
may be
a low-yield investment and thus distrust
Unsurprisingly, there
may
When
group belong-
may make good
a
self-fulfilling
behavior
prophecy.
(although there need not) exist multiple, Pareto-
ranked equilibrium collective reputations 20
Perhaps even more fascinating
is
.
the history dependence of collective rep-
utations. In our view, stereotypes are long lasting because
new members
a group at least partially inherit the collective reputation of their elders.
of
We
have seen that a one-time, non recurrent shock on the behavior of a population can prevent the population
from ever returning to a satisfactory state
even long after the members affected by the original shock are gone.
A more
general study of history dependence involves non-steady-state statistical inference techniques, but should be high on our research agenda.
The
genesis of collective reputations
modest object of
this
hope that the topic
it
is
paper has been to shed
will
a complex phenomenon.
light
on some of
its facets.
The
We
soon receive from economic theorists the attention
deserves.
20
See, e.g.,
Milgrom-Roberts (1990) and Vives (1990)
strategic complementarities.
29
for general results for
games with
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Appendix
1
(Incentives to cheat in a low corruption equilibrium)
Let
Vk
denote an agent's expected present discounted value of present and
future payoffs
when the agent has cheated k times
An
"continuation valuations".
will cheat
again only
G + SVk+1 >6Vk
Suppose that the agent
(A.2)
who has cheated
agent
to cheat
Vk =
(1
-
finds
it
fc
)
k times in the past
.
optimal to cheat when he has cheated k times,
when he has cheated
z
These are
if
(A.l)
and not
in the past.
(k
-f 1)
times.
Then
B + x k b + G + SVk+l >(l-x k )B + x k b + 6Vk
,
and
Vk+1 =
(A.3)
(1
-
x k+1 )
B + x k+1 b + 6Vk+1
.
[A2) and (A3) yield
G>S(x k+1 -x k )(B-b)/(l-6).
(A.4)
On
the other hand, the agent prefers stopping to cheat with record (k
to cheating once
+
1)
more and then stopping. So
G + 6Vk+ 2 <6Vk+u
(A.5)
where
Vk+J = (l-x k+2 )B + x k+2 b + 6Vk+2
(A.6)
(A.7)
Equations
(Vi+a
Ab and A6
(A.8)
Inequahties
< Vk+2 )
.
yield
G<
AA and Al
to cheat with record k,
8 (x k+3
-
x k+1 ) (B
-
6)/(l
-
6).
are inconsistent with assumption
it is
also optimal to cheat with
33
1.
So
optimal
if it is
any record
k'
>
k.
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