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.

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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