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Fukuzumi presen18 0309

Corruption in a repeated psychologi
cal game with imperfect monitoring

Masakazu Fukuzumi
University of Tsukuba, Japan


Motivation
We consider the effect of imperfect
monitoring the bureaucrat on bribery and
corrupt practices in public administration.
Our findings suggest that if we reconstruct
public service system to exterminate its
corrupt practices, the rate of turnover of the
bureaucrat and the amount of the noise in
monitoring the bureaucrat’s behavior should
be in the suitable regions.


The stage game without noise
Assumpto

ns:

Balafoutas (2011)

aH  bH  cH  aL  bL  cL
a H  a L , bL  bH

payoffs


Policy
H p
The lobby

k

bureaucra
t
lobby
public

The bureaucrat
1-p

0

q

 aH  k 


 bH  k 
 c 
 H 

1 q

Policy L

k : an amount of transfer; bribe


q : the bureaucrat’s belief about the public

payoffs

 aL  k  q  bureaucra

t
 bL  k  lobby

 public
c
L



’s belief about the bureaucrat’ s
behavior
p

0 : the parameter
of the bureaucrat’s sense of
guilty


The stage game with noise

k

The
lobby

 aH  k 


 bH  k 
 c 
 H 

p
Policy
H
The bureaucrat

q

1 q

0

Policy L
1-p



bureaucra
t
lobby
public

 aH  k 


 bH  k 
 c 
 H 

bureaucra
t
lobby
public

N

k : an amount of transfer; bribe
1 
q : the bureaucrat’s belief about the public

’s belief about the bureaucrat’ s
behavior p
 0: the parameter of the bureaucrat’s sense of
guilty

 aL  k  q 


 bL  k 
 c

L



bureaucra
t
lobby
public


The subgame perfect outcome of
the stage game
Assumptio a H
n:

 a L , bL  bH



The
lobby

 aH  k 


 bH  k 
 c 
 H 

P=1
Policy
H
The bureaucrat

q k 0

1 q

k 0
0

bureaucra
t
lobby

Policy L
1-p=0



public

 aH  k 


 bH  k 
 c 
 H 

bureaucra
t
lobby
public

N

1
No
corruption
q: the bureaucrat’s belief about the public



’s belief about the bureaucrat’ s
 behavior
0: the parameter
of the bureaucrat’s sense of
p
guilty

 aL  k  q 


 bL  k 
 c

L



bureaucra
t
lobby
public


Infinitely repeated game
Two-phase strategy
k*  0

star
t

Corruption
phase

k*  0

policy L

policy
L
other actions
( and/or H
realized (
after T
periods

for T
periods

Punishment
phase

k 0

policy H

Balafoutas (2011) examines only the grim-trigger strategy.


Proposition
Given a q . Our two-phase strategy
constitutes the subgame perfect equilibrium
of our infinitely repeated game
if and only if
*

in the corruption phase the lobby pays a bribe
k
such that

1   T 1
*
T


(
a

a
)


q

k

(
1


)(
b

b
)


bH .
H
L
L
H
T 1

   0,1: a common discount factor among players


An equilibrium
play
the social welfare

the lobby’s
payoff

T periods

T

period

periods

0
Corruption
phase

Punishment
phase

Punishment
Corruption phase
phase

Corruption
phase


Proof (in brief)
Let’s check the optimality of the two-phase strategy
for the bureaucrat in the corruption phase.

V:bthe expected present value of the bureaucrat’s total payoff
: the
expected present value of the bureaucrat’s total payoff
W
b
at the beginning of the punishment phase

Vand
b

the solutions of the following recursive system.
Ware
b


If the bureaucrat deviates from the two-phase strategy, then he gets
his total payoff

By the one-shot deviation principle, the two-phase strategy is optimal iff

・ In the corruption phase, we can check the optimality of the two-phase strat
for the lobby with the same line with the above argument.
・ In the punishment phase, we can easily check the optimality of
the two-phase strategy for both players .



Implications
k : the amount of
bribe

1   T 1
 (aH  aL )  q  k * (1   )(bL  bH )   T bH
T 1
  

Corruption
region

T

1

: the length of
punishment
periods


Implications
k : the amount of
bribe

1   T 1
*
T

(
a

a
)


q


k

(
1


)(
b

b
)


bH
H
L
L
H
T 1


k*
Corruption
region

 : the level of noise
0
Too noisy, no corruption


Corollary
T 1
1


T
If


(
a

a
)



(
1


)(
b

b
)


bH
H
L
L
H
T 1

, then

our two-phase strategy composes a
psychological Nash equilibrium where
q 0

in every period of the corruption phase,
q 1

in every period of the punishment phase.
* Psychological equilibria require the consistency of beliefs
・ Geanakoplos, Pearce and Stacchetti (1989) ・ .

consistency ≒ ( rational expectation


An equilibrium play with the consistent belief

the social welfare

the lobby’s
payoff

q 0

q 1
T periods

q 0

q 1
T

q 0
periods

periods

0
Corruption
phase

Punishmen
t
phase

CorruptionPunishmen
phase

t
phase

Corruption
phase


Essence
k : the amount of

Two-phase
strategy

bribe

Corruption
phase
policy L

(1   )(bL  bH )   T bH

q 1

Return
possibl
e

k*

1   T 1
 (aH  aL )   
   T 1

 0 : the parameter of the bureaucrat’s sense of
guilty

Punishment
phase
policy H


An a ((((((( second order
belief q
(ongoing) ・
1   T 1
*m
 (aH  aL )  q  : the minimum amount of the bribe
k :
T 1


to sustain corruption with our strategy

q
: 0an initial level of the second order belief

A monotonic adaptation of the belief q
 qt 1 :qift the policy H is realized at t

 qt 1  :qift the policy L is realized at t

kt

*m

and qt 1
andqt 0

1   T 1
 (aH  aL )  qt 
(qt ) :
T 1


.
.


k

*

kt

*m

1   T 1
 (aH  aL )  qt 
(qt ) :
T 1


The case that corruption can be sustained

(1   )(bL  bH )   T bH

k * (1)

k * ( q0 )

0

CorruptionPunishmen CorruptionPunishmen
phase
phase
t
t
phase
phase

T periods

T periods

Corruption
phase
periods


k

*

kt

*m

1   T 1
 (aH  aL )  qt 
(qt ) :
T 1


The case that corruption can not be sustained

k * (1)
(1   )(bL  bH )   T bH

k * ( q0 )

0

CorruptionPunishmen CorruptionPunishmen Punishmen
phase
phase
t
t
t
phase
phase
phase
periods

T periods

T periods


Remark
With the constant second order belief q
The length of punishmentT

k

*m

With the adaptive (varying) second order belief q
The length of punishmentT

k *m


For MPP Students 2018
Game theory is a mathematical and highly abstract
language to describe social situations.
So, it is applicable to many fields of social
sciences, not only economics but other fields.
Using this scientific language-game theory,
try to construct your original model describing
the social phenomena that you are interested in.
(specifying the players, the set of strategies,
payoffs, the timing of decision making, types, prior
distribution)
Moreover, finding the equilibrium of the model and
analyzing it, you could be able to propose new
policy or social systems to improve our civilian
society.



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