Convex vNM–Stable Sets for a Semi Orthogonal Game. Part III: A Small Economy - Uniqueness and Multiple Solutions

Center for

Mathematical Economics

Working Papers 500

2014

Convex vNM–Stable Sets for a

Semi Orthogonal Game

Part III:

A Small Economy - Uniqueness and Multiple Solutions

Joachim Rosenm¨ uller

Center for Mathematical Economics (IMW) Bielefeld University

Universit¨ atsstraße 25 D-33615 Bielefeld · Germany e-mail: imw@uni-bielefeld.de http://www.imw.uni-bielefeld.de/wp/

ISSN: 0931-6558

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^{'('& &(}

η ∈ I

^{& (('& (}

m

^{( & (' '}

η

&!

$

#'

h 2 + h 3 ≤ m 2 + m 3 ,

^&

m 2 < h 2 , h 3 < m 3 ,

(

&

'

&

(

ε > 0

^{( &(}

ε 234

^'('&

S ⊆ T

^'

&

(

ε 23

^'('&

S ⊆ T

^{& '&('&}

x b

^'

x ¯

e ¹²

&

e ³⁴

^(&

ϑ ^{x b} dom S η

⁽⁽

m 2 > h 2 , h 3 > m 3

((

&

'

'

3

23

^{7 &!$*!' 0}

)

J

((

!$$

#

7

*

$

$

&!

$

5'

m 2 (1 − h 3 ) ≥ h 2 (1 − m 3 ).

m 2 [1 − (h 3 + h 3 )] ≥ h 2 [1 − (m 3 + m 3 )],

α 2 := m 2 − h 2

1 − (m 3 + m 3 ) 1 − (h 3 + h 3 ) ≥ 0

0

7

α ₂ , α ₃ , α

^{) ) J(( !$*}

(

) )

$

(

λ 1 + λ 3 < 1

^&

h 2 + h 3 < 1

^&

H

^(&

(

&

(

(

1 ^st STEP :

J

η ∈ I \ H

m

⁽⁽⁾

η

$

)

m

^{) 7}

14

24

^-

((

*

(

η

^{& * '}

)

*

!$#$

m 1 + m 4 ≥ 1, m 2 + m 4 ≥ 1

^$

(

&!

$

5

*

'

m 2 + m 3 ≤ 1

)

*

!

$

5

$

3 ^rd STEP :

^K(

&!$5!'

m 2 + m 3 < h 2 + h 3 .

* J

((

!$

(

(

x b

⁾

x ¯

e ¹²

)

*

(

ε > 0

⁽

ϑ ^{x b}

⁽

η

$

$

$

(

ε

23

S

^$

4 ^th STEP :

&!

$

5#'

h τ < m τ (τ = 2, 3)

$

* J

((

!

$*

(

(

x b

⁾

x ¯

e ¹²

e ³⁴

)

*

(

ε > 0

⁽

ϑ ^{x b}

⁽

η

$$

(

e

234

S

5 ^th STEP :

^{2 *}

&!$55'

h 2 + h 3 ≤ m 2 + m 3 ≤ 1,

m 2 < h 2 , h 3 < m 3 .

23

234

^$

)

((

* )

( J

((

!$*$

*((

m 2 >

h 2 , h 3 > m 3

*

$

?

_?

2

/

(

H

^{* 7 )}

$

(

H b

(

(

&

ϑ ¯ = ϑ ^{x ¯} ∈ H b

1)

ϑ ¯ ∈ / H b

⁽

ϑ b ∈ H b

^{&& ( &'(}

'

&

(

(

'

'

S b

&!

$

56'

ϑ b dom _{S b} ϑ ¯

$

J

(

(

)

ϑ b

^*

c m = ( m b 1 , . . . , m b 4 )

&!

$

5'

m b τ := ess inf D ^τ ϑ b (τ = 1, 2, 3, 4).

1 ^st STEP :

&!$5' %

(

$$$

( m b 2 , m b 3 ) ≥ (h 2 , h 3 ).

1

ϑ b

^{( * &J((!}

$

6'

a ²³ c m ≥ h 2 , a ³² c m ≥ h 3 .

1)

)

2

3

⁾

m c

2 !$!$

#

*

(

)

2, 3

⁽

*

)

x ¯

$

$

)

(h 2 , h 3 )

$ K

(

m b 3 ≤ h 3

* 2

!

$

!

$

(

ϑ ⁰ := (1 − t) ϑ b + te ²³ ∈ H b

)

t := ¯ x 3 − m b 3

$

J

(

( )

ϑ ⁰

^*

m ⁰

&!

$

5'

(m ⁰ ₂ , m ⁰ ₃ ) ≥ (¯ x ₂ , x ¯ ₃ ).

)

( )

(

!

$

5

$

2 ^nd STEP :

^{# / (}

&!

$

5'

x ¯ 4 ≥ m b 4 .

@ )

&!

$

5'

¯

x 1 λ 1 + ¯ x 2 λ 2 + ¯ x 3 λ 3 + ¯ x 4 λ 4 = 1, b

m 1 λ 1 + m b 2 λ 2 + m b 3 λ 3 + m b 4 λ 4 ≤ 1

a ²³ x = h 2

a ³² x = h 3

h ₂ h 3

x 2

x 3

x ¯ m ⁰

c m

{x a ²³ x ≥ h 2 , a ³² x ≥ h 3 }

2

!

$

!4

2

3

⁾

m c

x ¯

b

m ₁ λ ₁ + m b ₄ λ ₄ ≤ x ¯ ₁ λ ₁ + ¯ x ₄ λ ₄ = 1 − (¯ x ₂ λ ₂ + ¯ x ₃ λ ₃ )

= 1 − (h 2 λ 2 + h 3 λ 3 ) = 1 − λ ⁰ 23 .

#

ϑ ⁰

^{( *}

*

b

m 1 + m b 4 ≥ 1 = ¯ x 1 + ¯ x 4 .

K

− m b 1 λ 1 − m b 4 λ 1 ≤ −¯ x 1 λ 1 − x ¯ 4 λ 4

b

m 1 λ 1 + m b 4 λ 4 ≤ x ¯ 1 λ 1 + ¯ x 4 λ 4

(

(λ 4 − λ 1 )m ⁰ ₄ ≤ (λ 4 − λ 1 )¯ x 4

$

&!$5'

$$

(

*

0

¯

x ₁ ≤ m b ₁

^{$ K * (* ) -}

1, 4

⁾

x ¯

m ⁰

2 !$#$

3 ^rd STEP :

%

(

$$$

&!$6'

b

m 1 + m b 4 = 1 .

1

(

ϑ ¹ := ϑ b − tλ ₄ λ _D ¹ + tλ ₁ λ _D ⁴

?

_?

x 1 + x 4 = 1

λ 1 x 1 + λ 4 x 4 = 1 − λ ⁰ 23

1

1

x 1

x 4

1−λ ⁰ 23

λ ₄

1−λ ⁰ 23

λ 1

(¯ x 1 , x ¯ 4 )

( m b 1 , m b 4 )

{x λ 1 x 1 + λ 4 x 4 ≤ 1 − λ ⁰ 23 , x 1 + x 4 ≥ 1}

2

!

$

#4

1

4

⁾

m c

x ¯

)

t := ( m b 1 + m b 4 ) − 1 λ 4 − λ 1

(

(

)

(

(

m ¹

⁾

ϑ ¹

&!$5'

&!

$

6

*

'

(m ¹ ₂ , m ¹ ₃ ) ≥ (¯ x 2 , x ¯ 3 ).

*

)

t

&!$6!'

(m ¹ ₁ + m ¹ ₄ ) = 1

@

(m ¹ ₁ , m ¹ ₄ )

^7*

&!$6#'

λ 1 x 1 + λ 4 x 4 ≤ 1 − λ ⁰ 23

)

(

( m b 1 , m b 4 )

$

m b ¹ ₄ ≥ m b 4

b

m ¹ ₁ ≥ m b 1

)

*

)

2

!

$

# &!

$

6!' &!

$

6#'

(

*

*

x

^{) 7 * *}

x 1 ≥ x ¯ 1

$

ϑ ¹

(

$

@

*

(

)

ϑ ¹

⁽

ε

234

^{( '(' ( )}

ϑ b

⁷

D ² ∪ D ³

ϑ ¹

^/

ϑ b

D ⁴

$

7

*

ϑ ¹ ∈ H b

^(*

ϑ b

^*

ϑ ¹

⁰

&!$6'$

4 ^th STEP :

/

(

(

)

ϑ b

D ² ∪ D ³

(h 2 , h 3 )

^{$ /}

(

$$$

&!$65'

( m b 2 , m b 3 ) = (h 2 , h 3 )

$

J

((

!

$*

c m

α ₁ , α ₂ , α ¯

^{* (($ .}

x ^? := α 1 e ¹² + α 2 e ³⁴ + ¯ αx

0

(x ^? ₂ , x ^? ₃ ) = ( b x 2 , b x 3 ) = (α 1 , α 2 ) + ¯ α(h 2 , h 3 ) .

(x ^? ₂ , x ^? ₃ ) − (α ₁ , α ₂ ) = ¯ α(h ₂ , h ₃ ) .

7

*

x ^?? := x b − (α 1 e ¹² + α 2 e ³⁴ )

¯ α

x ^?? := (x ^? ₂ , x ^? ₃ ) − (α 1 , α 2 )

¯ α

0

(x ^?,? ₂ , x ^?,? ₃ ) = (h 2 , h 3 ) .

%

(

x ^?? ≥ 0

^{$ )}

0 ≤ t ≤ 1

b

x − t(α 1 e ¹² + α 2 e ³⁴ )

7

x 1 + x 4 = 1

x 2 + x 4 ≥ 1

⁾⁽

x b

e ¹²

e ³⁴

^{* 7 7$ ) ) (}

t < 1

x ^t ₄ = 0

x ^t ₁ = 1

x ^t ₂ ≥ 1 λ ₁ x ^t ₁ + . . . + λ 4 x ^t ₄ > λ 1 + λ 2 = 1

^{$ 7*}

x ^t ₄ = x ^?? ₄ ≥ 0

^{$ /}

x ^?? ₁ > 0

^{(( * * 2 !$# 7}

x ₁ + x ₄ = 1, λ ₁ x ^t ₁ + λ ₄ x ^t ₄ ≤ 1 − λ ⁰ ₂₃

^(*

x

x ₁ ≥ x ₁

^$

(

ϑ ^?? := ϑ b − (α 1 e ¹² + α 2 e ³⁴ )

¯ α

(

(

(

(h 2 , h 3 )

D ²

D ³

$

ϑ b = αϑ ^?? + (α 1 e ¹² + α 2 e ³⁴ ) ,

*

(

)

ϑ ^??

^{* (}

ϑ e

⁽

ε − 234

(

)

ϑ b

^*

α ϑ e + (α 1 e ¹² +α 2 e ³⁴ )

⁽

ε −

234

^$

ϑ ^?? ∈ H b

$

)

(

* )

*

ϑ b

^*

ϑ ^?,?

^{0 (}

&!

$65'$

5 ^th STEP :

?

_?

%

ϑ b

^*(

D ² ∪ D ³

$

$$$

&!$66'

ϑ b = h 2

D ² , ϑ b = h 3

D ³ ,

ϑ ^?

^{* ) ( )}

ϑ b

(h 2 , h 3 )

⁾⁽

D ² ∪ D ³

D ⁴

⁰

ϑ ^? := ϑ b

D 1 + h _{2 D} ² + h _{3 D} ³ + ϑ b

D 4 + ( Z

D ² ∪D ³

ϑdλ b − λ ⁰ ₂₃ ) _D ⁴ .

ϑ ^?

^{( * (}

ϑ e

⁽

ε − 234

◦

T

234

)

(

ε > 0

^{$ (}

λ ^h := h 2

h ₂ + h ₃ λ ¹ + h 3

h ₂ + h ₃ λ ²

/

ϑ b

⁽

T b ²³ ⊆ D ²³

^{)*(( ((}

)

(

/

(h 2 , h 3 )

^$

ϑ e

^/

ϑ b

T ^{◦ 4}

ϑ ^?

^/

ϑ b

T ^{◦ 4}

^$

)

*

(

ε > e 0

⁰

e ε − 123

T e ²³⁴ ⊆ T b ²³ ∪ T ^{◦ 4}

^{)* (}

e δ > 0

⁽

ϑ e := (1 − e δ) ϑ b + e δ λ ^h + ϑ e 2

/

ϑ b

T e ²³⁴

^$

ϑ( b T e ²³⁴ ) < e ε(1 − h 3 ) = v( T e ²³⁴ )

λ ^h ( T e ²³⁴ ) ≤ e

ε(1 − h 3 ) = v( T e ²³⁴ )

^{) *}

e δ

⁽

ϑ( e T e ²³⁴ ) < e ε(1 − h 3 ) = v( T e ²³⁴ )

$

ϑ e dom _{T e} 234 ϑ b

$

7

*

ϑ ^? ∈ H

^(*

ϑ b

^*

ϑ ^?

$

0

(

&!$66'$

6 ^th STEP :

K

)

ϑ b ∈ H b

ϑ b = x b 1 ≥ x 1

D ¹

ϑ b = h τ

D ^τ (τ = 2, 3)

^$

x b ₁ + x b ₄ = 1

D ⁴

^*

b x 4 := 1 λ(D ⁴ )

Z

D ⁴

ϑdλ b .

*

λ 1 x b 1 + λ 2 h 2 + λ 3 h 3 + λ 4 b x 4 = 1

λ 1 b x 1 + λ 4 b x 4 = 1 − λ ⁰ ₂ 3

b

x 1 + b x 4 ≥ 1 .

)

2

!

$

#

(x 1 , x 4 )

^*

) 7

*

7$

x b 1 = x 1

)

b

x 1 + b x 4 = 1

)

x b 4 = x 4

$

7

*

ϑ b ≥ ϑ ^{x ¯}

⁽

ϑ b = ϑ ^{¯ x}

(

$

0

*

(

$

H

^{( & ((}

J

H b

$ 2

)

b

H

⁽

*

(

$

@

* (

!$*#

ϑ = ϑ ^{b x} ∈ H b

$

7

*

H b ⊆ H

H b = H

?

_?

!

"

2 × 2

^{3/( & !" #"' /*}

(

&#$*'

h 1 = 0, h 2 + h 3 ≥ 1 h 4 = 1

&#$!'

λ 1 + λ 3 ≥ 1 .

( b τ 1 , τ b 2 ) = (1, 3)

^{* 7 *}

h _b τ 1 + h τ _b 2 = 0 + h 3 < 1

^$

)

λ ⁰

^{/ 4}

11

& #

"'$

1

/

(

h 1 , h 2

)

ε

^(.

3.6

^3/(

3.7

⁾

!

"$

)

(

)

$

&

(

(

'&

&#

$*'&#$

!'

(

&

(

(

'

&

(

&

'

'

&#

$

#'

a ¹⁴ := (1, 0, 0, 1) ;

v(a ¹⁴ ) = 1 = e ¹² a ¹⁴ = e ³⁴ a ¹⁴ = c ⁰ a ¹⁴ a ²³ := (0, 1, 1, 0)

v(a ²³ ) = 1 = e ¹² a ²³ = e ³⁴ a ²³ < c ⁰ a ²³ = h 2 + h 3

a ²⁴ := (0, 1, 0, 1) ;

v(a ²⁴ ) = 1 = e ¹² a ²⁴ = e ³⁴ a ²⁴ < c ⁰ a ²⁴ = h ₂ + 1 a ¹³ := (h 3 , 0, 1, 0)

v(a ¹³ ) = h 3 = e ¹² a ²³ = c ⁰ a ²³ < e ³⁴ a ²³ = 1 a ¹²³ := ((h 2 + h 3 ) − 1, 1 − h 3 , h 2 , 0)

v(a ¹²³ ) = 1 − h 3 = e ¹² a ¹²³ = e ³⁴ a ¹²³ = c ⁰ a ¹²³

K

*

)

(

)

ε

^-

(

+

((

* ,$

)

(

ε > 0

⁰

→

λ(T ¹³ ) = εa ¹³ = ε(h 3 , 0, 1, 0)

v(T ¹³ ) = λ ¹ (T ) = λ ³ (T ) = εh 3 < λ ² (T ) = ε,

&#

$5'

T ¹³

ε

13

^>:<:8ED

$

(

* )

T ¹²³ ⊆ D ¹²³

0

→ λ(T ¹²³ ) = εa ¹²³ = ε((h 2 + h 3 ) − 1, 1 − h 3 , h 2 , 0),

v(T ¹²³ ) = λ ¹ (T ) = λ ² (T ) = λ ³ (T ) = ε(1 − h 3 ) ,

&#$6'