# Chapter 4

Chapter 4
Dynamic Games of Incomplete Information
An example: a signaling game model of a labor market

A model of a labor employment market model
Suppose that there are a firm and an applicant (a labor).
The firm is not sure about the abilities of the applicant, i.e., the
firm does not know whether the labor is a skilled labor or not.
-1

0

Not employ

0

Employ

1

Not employ

Employ

The firm’s
information set
The applicant has Low ability
( be Unskilled)

The applicant has High
ability
1/3 (be Skilled)

2/3

N

The firm’s expected payoff
Employ : (2/3) ×(-1) + (1/3) ×(1) ＝－1/3.

Not Employ : (2/3) ×0 + (1/3) ×0 ＝0.
Hence, the firm does not employ the applicant (labor) and get its payoff of 0.
However, If the firm could hire the applicant with high ability,
it could have got the payoff of 1and the applicant could have got the job.
So, “NOT Employ” is inefficient.
( This phenomenon is called an “adverse selection” in economics. )

Introducing a qualification (or a career) system to the model
A qualification (or a career) is called a signal in game theory.
Notations
Type-H : the applicant with high ability
Type-L : the applicant with low ability

Assumptions
It cost 2 for Type-H to acquire the qualification (the signal).
It cost 5 for Type-L to acquire the qualification (the signal).
If the applicant is employed by the firm, then he (or she) gets wages of 3.
It is assumed that this wage level does not depend on the ability of the applicant.

If the applicant is not employed, then he (or she) gets wage of zero.

The game tree with a qualification system (a signaling game)
3, 1
applicant, firm

0, 0

Employ

Not

Type-H applicant
Not
acquire

H

acquisition
of the qualification

Employ
Not

-2, 0

1/3
The firm’s information set:
the applicant does not
have the qualification

1, 1
applicant, firm

The firm’s information set:
the applicant has the
qualification

N
2/3

3, - 1
Employ

0, 0

Not

Not
acquire

L

acquisition
of the qualification

Type-L applicant

-2, - 1
Employ
Not
-5, 0

Let’s consider the equilibrium of this dynamic incomplete information game.
The equilibrium is referred to as a perfect-Bayesian equilibrium (P.B.E.).
The equilibrium concept requires following conditions.
Requirement 1 (consistency) The firm’s beliefs in its information sets must be
consistent with the behaviors of both types of the applicant.
Requirement 2 (rationality) Given the consistent beliefs above, the firm makes
its best choice in each information set. Given the choices above made by the firm,
each type of the applicant makes its best choice.

Can ( Type-H acquires the qualification, Type-L does not )
be a perfect Bayesian equilibrium strategy?

In this case, different types choose different action, respectively,
which is called a separating equilibrium.

Consistent belief with the separating equilibrium strategy of each applicant’s type
3, 1
applicant, firm

0, 0

Employ

0

Type-H applicant
Not
acquire

Not

H

1

acquisition
of the qualification

Employ

1, 1
applicant, firm

Not
-2, 0

1/3
The firm’s information set:
the applicant does not
have the qualification

The firm’s information set:
the applicant has the
qualification

N
2/3

3, - 1
Employ

0, 0

Not

Not
acquire

1

L

-2, - 1

acquisition
of the qualification

Type-L applicant

Employ

0

Not
-5, 0

The firm’s best choices when the consistent beliefs above are given.
3, 1
applicant, firm

0, 0

Employ

0

Type-H applicant
Not
acquire

Not

H

1

acquisition
of the qualification

Employ

1, 1
applicant, firm

Not
-2, 0

1/3
The firm’s information set:
the applicant does not
have the qualification

The firm’s information set:
the applicant has the
qualification

N
2/3

3, - 1
Employ

0, 0

Not

Not
acquire

1

L

Employ

acquisition
of the qualification

Type-L applicant

0

-2, - 1

Not
-5, 0

The best choice of each type of the applicant
when the firm’s choices above are given.
3, 1
applicant, firm

0, 0

Employ

Not

Type-H applicant
Not
acquire

H

acquisition
of the qualification

Employ
Not

-2, 0

1/3
The firm’s information set:
the applicant does not
have the qualification

1, 1
applicant, firm

The firm’s information set:
the applicant has the
qualification

N
2/3

3, - 1
Employ

0, 0

Not

Not
acquire

L

acquisition
of the qualification

Type-L applicant

-2, - 1
Employ
Not
-5, 0

Therefore, we have confirmed that the following pair of strategies
(and beliefs) constitutes a perfect Bayesian equilibrium.
Type-H acquires the qualification.

The applicant
Type-L does not acquire the qualification.

If the applicant has the qualification, then the firm employs her (him).
The firm
If the applicant does not have the qualification,
then the firm does not employ her (him).

A perfect-Bayesian Nash Equilibrium (the separating equilibrium)
3, 1
applicant, firm

0, 0

Employ

0

Type-H applicant
Not
acquire

Not

H

1

acquisition
of the qualification

Employ
Not

-2, 0

1/3
The firm’s information set:
the applicant does not
have the qualification

1, 1
applicant, firm

The firm’s information set:
the applicant has the
qualification

N
2/3

3, - 1
Employ

0, 0

Not

Not
acquire

1

L

Employ

acquisition
of the qualification

Type-L applicant

0

-2, - 1

Not
-5, 0

Summary
Consider a situation where an asymmetric information among players
might induce inefficiency.
Introducing some qualification system that is costly for a part of players,
the economic welfare of the situation could be improved.
Applications of signaling games: a school career (in labor economics);
foreign direct investments (in international economics), a peacock in his
pride (in theoretical biology) , a military parade or saber-rattling (in
political science or international relationships), etc.
Dynamic incomplete information

Key words: perfect Bayesian equilibrium, consistent belief, rationality,

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