Convex vNM-Stable Sets for a Semi Orthogonal Game. Part V: all games have vNM-Stable Sets

Volltext

(1)

Center for

Mathematical Economics

Working Papers 552

February 2016

Convex vNM–Stable Sets for a

Semi Orthogonal Game

Part V:

All Games have vNM–Stable Sets Joachim Rosenm¨ uller

Center for Mathematical Economics (IMW) Bielefeld University

Universit¨ atsstraße 25 D-33615 Bielefeld · Germany e-mail:

ISSN: 0931-6558

(2)

? ?

!

"

"

#

$ %

"

&' (&((&(((&()& *+

,& *-,& *., */,

%

"

0

#

$ %

!

#

r + 1

1 2 "

"

$

r

! # ! 13

2

! !

"

$ 4

3

r + 1

1 2 #

" "

1

2

"

3

0

$

(

$$&

5

!

"

"

!

"

$

6 7 !

3

&

"

1

2

3

#

#

"

3

#

$ %

5 "

#

5 "

$

8!

#

9

"

&! !

3

5

$

! :

#

&

"

1

2

& $$&

r + 1 st

$ ;&

"

0

"

#

<=>?>@A <=>?BC@A

! ! D

3

7

%

"

1

#

! 2

*E ,

& *F ,

$ % G

:

"

1

2 $

(3)

?

?

!!" #"$ % &

'( )*+,"

5

"

- !

6

"

*+,& *-,& *.,& */, $

- !

#

$ %&

./

01

"

#

& $$&

(I , F , v)

I := [0, r)

2 34/5167&

F

σ−

: 8

"

9

: /4; <;

:=7 &

v

9:/4; <;:=/4 >?=9<;:=

"

!

v : F →

@

+

# $$$ A!

"

λ

$ 1 ! "2& &

v

#

:

# "

# "

λ ρ , (ρ ∈ {0, 1, . . . , r})

+$+

v(S) := min {λ ρ (S) | ρ ∈ {0, 1, . . . , r}} (S ∈ F ),

&

+$-

v = ^

λ 0 , λ 1 , . . . , λ r = ^

ρ∈R 0

λ ρ ,

R = {1, . . . , r}

R 0 = R ∪ {0}

$ % "

λ 1 , . . . , λ r

! A!

"

C ρ = (ρ − 1, ρ] (ρ = 1, . . . , r)

$ %1 " 2

λ 0

# $$$ A!

"

λ 0 (I ) > 1

$

%

"

! !

v

36:B?9<;:=>/ 9<:679:00:B; <;17

v

! " "0 ! " $ ;

ρ } ρ∈R

12 C""

λ 0

1

2

C

"" #

$

+$.

l 0 ρ := ess inf C ρ

λ 0 (ρ ∈ R), l σ ? := X

ρ∈R\{σ}

l ρ 0

9

+$/

l ρ 0 > 0 (ρ ∈ R) , X

ρ∈R

l 0 ρ < 1 ,

"

+$E

l σ 0 < 1 − l σ ? < 1 (σ ∈ R ) .

"

#

$

4& +$/ & 9

!

"

$ % :

"

#

"

0

"

!

/

5

$

%

#

λ 0

# 3

$

"

#

& &

(4)

?

?

9

:=

<;

=

?

:

?

7 0:

B

1 4$ %

"

"

#

B@>@B

? >

?

=

;>

:60 0:

B

1 4$

#

"

""

"

3

λ (t)

"

λ 0

5

λ 0

$$ "

:

8 :

2

*/, $

#

&

!

"

"

λ 0

&

t ∈

=B@

< "

$

A

T = T (t) := {1, . . . , rt})

+$F

(t) D = (t) D τ

τ∈T

" #

I

+$

C ρ =

[ ρt τ = (ρ−1)t+1

(t) D τ

t

9 A! " &

+$

λ t τ = λ( (t) D τ ) = 1

t (τ ∈ T = T (t) ) .

A

F (t)

σ

! ! # " 102

(t) D

$ %

""

"

#

!

"

λ (t)

3 #& G " #

(t) D

&

"

$$

$

F (t)

$ "

+$

λ (t) := P

λ 0 F (t) (•) ,

#

5

λ 0

$$$

F (t)

& $$&

+$+

λ (t) (?) = E •

λ 0 F (t)

(?) = X

τ∈T

λ 0 ( (t) D τ ) λ( (t) D τ )

(t) D τ (?) ,

$

ρ } ρ∈R

λ (t)

!

+$++

v (t) := λ (t) ∧ ^

ρ∈R

λ ρ

!

"

(I, F , v (t) )

$ 8 !& "

! &

#

!

9

"

$

8

#

: &

λ (t)

λ 0

F (t)

$ % "

λ ρ (ρ ∈ R)

!

#

F (t)

9#

+$+-

v (t) (S) = v(S) (S ∈ F (t) ) .

(5)

?

?

%

#

&

"

"

"

"

(t) D

B@>@B ? > $

6

"

" "

:

61 4

1 /

=

<

1 9<

:67

. *+

,

a , a

&

a

$ !" !

(t) a , (t) a ,

(t) a .

0

1

$$

(t) D

2$ 8& "# #

!

"

! !

v

! !

v (t)

$

ξ dom v S η

ξ dom v S (t) η

$

>

t ∈

?BC>

(t) D

BA>@>=>

?=B@

< <C

>

ξ, η

@<=>?>@BA ?BC

>

ξ (t) := P

ξ F (t) (•)

>

S

? ?@>@B

S ∈ F (t)

{ξ > η} ∈ F (t)

>

B >

?>@BA

ξ dom v S η , ξ dom v S (t) η

?

nd ξ (t) dom v S (t) η

?

=@?

B>

ξ

ξ (t)

F (t) ,

# +$+- 0

+$+.

ξ(S) ≤ v(S)

#

ξ (t) (S) ≤ v (t) (S) .

T > := { ξ > η}

T := { ξ ≤ η}

$ %&

T > ∈ F (t)

&# "

(t) D τ

F (t)

T >

G & $$&

+$+/

(t) D τ ⊆ T >

(t) D τ ⊆ T

8

"

ξ (t) > η

1

S

2$ A

(t) D τ

"

S

&

$$

(t) D τ ⊆ S

$ % #

λ (t) D τ ∩ {ξ > η}

> 0

$ 4" +$+/

(t) D τ ⊆ {ξ > η}

#

ξ > η

(t) D τ

%&

ξ > η

1

S

2 & &

+$+E

ξ (t) > η

S

"

ξ > η

S.

- &

ξ > η

1

S

2 & #

ξ > η

#

"

(t) D τ

S

ξ (t) > η

# "& "

ξ (t) > η

1

S

2$ %

+$+F

ξ > η

S

"

ξ (t) > η

S.

(6)

?

?

+$+. & +$+F & +$+E +$+- A

""

$

λ

"

(λ(•∩C 1 ), . . . , λ(•∩C r ))

$ %&

!

"

"

#

&

"

1

ε

2 $

ε

>

t ∈

?BC>

(t) D

BA>@>=>

? =B@

< <C

>

ξ

?BC

η

@<=>?>@BA ?BC

>

S ∈ F (t)

?

?@>@B A=

>

?>

ξ dom v S (t) η

AA=<>?>

ξ

@A

F (t)

<?A= ?

T > := { ξ > η} ∈ F (t)

B

>

@A>A

?

>

a (t)

?B>

v (t)

?BC ? ?@>@B

T 1 t a (t)

A=

>

?> >

@B

CA > =

+$+

λ(T 1 t a (t) ) = 1 t a (t) ,

?BC

+$+

ξ dom v (t)

T 1 t a(t) η

CA > =

B

>

CA

B

?B C@AA

>

ε

?BC @>A =?B>@ A ?A CA @C @B

>

B

@>?B

<

? >

<

1 st STEP :

8# ( %" .$. *+,

F (t)

v (t)

$ %# :

a (t)

ε 0 > 0

&#

ε < ε 0

&

T ε = T εa (t) ⊆ S

# !

ξ dom v (t)

T ε a(t) η

$ D #&

ε ≤ 1 t

(t) D

"

$

2 nd STEP :

'

#

&

"

(t) a

#$ (

"

a

a

:

#

! %

"

.$E

? >

$

8

"

(t) a r

$

% &

(t) a

+$+

(t) a = (1, . . . , 1, . . . , 1, . . . , α r , β r )

α r = (t) a τ b r = (h τ 1 + . . . + h τ r−1 + h τ r ) − 1 h τ r − h τ r

, β r = (t) a τ r = 1 − (h τ 1 + . . . + h τ r )

h τ r − h τ r

;

+$-

(7)

?

?

!

:

%

"

.$E$

? >

$

"

0& 9 +$-

#

5

&

#

(t) h τ

$

α r + β r = 1

$ A

b

τ = ( b τ 1 , . . . , b τ r , τ r )

9 !

(t) a

$ 4

ρ ∈ R \ {r}

&

+$-+

T 1 t a (t) ∩ C ρ = (t) D b τ ρ (ρ ∈ R \ {r}) .

8

T > ∈ F (t)

& # "C0

(t) D τ

F (t)

T >

G

& $$&

+$--

(t) D τ ⊆ T >

(t) D τ ⊆ T ≤

4

"

T εa (t) ∩ C ρ(t) D b τ ρ

λ( (t) D τ b ∩ {ξ > η}) > 0

$ %&

! +$-- &

(t) D b τ ρ ⊆ {ξ > η}

$ D9#&

+$-.

T 1 t a (t) ∩ C ρ = (t) D b τ ρ ⊆ {ξ > η}

3 rd STEP :

5&

ρ = r

τ = τ b r , τ r

!

"

$

8! &

T εa (t) ∩ C r(t) D b τ ρ(t) D τ r

λ ( (t) D τ ∩ {ξ > η}) > 0

"

(t) D τ ⊆ {ξ > η} (τ = b τ r , τ r ).

;&

"

T 1 t a (t)

$ ( & ? @> ?

T 1 t α r

(t) D b τ

"

1 t α r

?

@> ?

T 1 t β r

(t) D τ

"

1 t β r

:

T 1 t a (t)

T 1 t a (t) ∩ C ρ := T 1 t α r ∪ T 1 t β r .

%

+$-/

(t) D τ ∩ T 1 t a (t)(t) D τ ⊆ {ξ > η} (τ = b τ r , τ r ) , 4 th STEP :

&

λ(T 1 t a (t) ) = 1

t (α r + β r ) = 1

tε (α r + β r )λ(T εa (t) )

λ 0 (T 1 t a (t) ) = 1

t (α r + β r ) = 1 t

1 − X

ρ∈R\{r}

h (t) b τ

ρ

 = 1

tε λ 0 (T εa (t) )

+$-E

λ(T 1 t a (t) ) = 1 tε

→ λ(T εa (t) ) , λ 0 (T 1 t a (t) ) = 1

tε λ 0 (T εa (t) )

(8)

?

?

+$-F

v(T 1 t a (t) ) = 1

tε v(T εa (t) ) = 1 t .

8

ξ

" $$

F (t)

$$& # " &

ξ(T 1 t a (t) ) = 1 ξ(T εa (t) )

$ ;&

+$-

ξ(T 1 t a (t) ) = 1

tε ξ(T εa (t) ) ≤ 1

tε v (t) (T εa (t) ) = v (t) (T 1 t a (t) ).

-

#

+$-. +$-/ &

+$-

ξ > η

T 1 t a (t) .

D

"

! +$- +$- &

ξ dom 1

t T a(t) η,

"

7

2 nd ST EP

3 rd ST EP

+$-

T 1 t a (t) ∩ C r = T 1 t α r ∪ T 1 t β r ⊆ {ξ > η}

#

! !

"

α r

β r

$ %

#

"

!

5

%

"

$

◦ eeeeee

>

t ∈

?BC >

(t) D

BA>@>=>

? =B@

< <C

>

ξ

?B

@<

=>?>@B ?BC

>

ξ (t) =

ξ F (t)

>

η

?B @<=>?>@B ?BC

S ∈ F (t)

?

?@>@BA=

>

?>

ξ (t) dom v S (t) η

AA=< >?>

T > := { ξ > η}

@A

F (t)

<?A= ?

B >

@A>A ?

>

a (t)

?B>

(t) D

?BC ?

?@>@B

T 1 t a (t)

A= >?>

+$.

λ(T 1 t a (t) ) = 1 t a (t) ,

?BC

+$.+

ξ dom v

T 1 t a(t) η

CA > =

1 st STEP :

!

"

$ 4

"

v

v 0

$ 4

C r ∩ T 1 t a (t) v

v 0

(9)

?

?

+$-E

3

!

!

"

$ 8

4 th ST EP

!

T 1 t α r

T 1 t β r

#

+$. -

ξ(T 1 t α r ) = ξ (t) (T 1 t α r )

ξ(T 1 t β r ) = ξ (t) (T 1 t β r )

:$ ;&

ξ (t)

#

ξ

$ %& #

0 ! 9 0

n

ξ > ξ (t) o

(t) D b τ ρ

n

ξ < ξ (t) o

(t) D b τ ρ

#

&

"

T 1 t α r

" #

T 1 t β r

$

2 nd STEP :

! !

"

0

"

"

"

"

!

5

(t)

# $

(&

1 t h (t) ( τ b

ρ )

#

λ 0

D b τ ρ

+$..

Z

λ 0 >h (t) (b

τρ )

λ 0 − h (t) ( τ b

ρ ) =

Z

λ 0 <h (t) (b

τρ )

h (t) ( b τ

ρ ) − λ 0 .

&

ε > 0

F > ε

λ 0 > h (t) ( b τ

ρ )

, F = ε

λ 0 = h (t) ( b τ

ρ )

, F < ε

λ 0 < h (t) ( τ b

ρ )

+$./

λ (F > ε ∪ F = ε ∪ F < ε ) = ε

+$. E

Z

F > ε

λ 0 − h (t) ( b τ

ρ )

dλ =

Z

F < ε

h (t) ( τ b

ρ ) − λ 0

dλ .

AG %

"

$ %

Z

F > ε

λ 0 dλ + Z

F = ε

λ 0 dλ + Z

F < ε

λ 0

= Z

F > ε ∪F = ε ∪F < ε

h (t) ( τ b

ρ ) dλ +

Z

F > ε

λ 0 − h (t) (b τ ρ )

d λ + Z

F = ε

λ 0 d λ − h (t) (b τ ρ )

d λ + Z

F < ε

λ 0 d λ − h (t) (b τ ρ )

d λ

= εh (t) ( b τ

ρ ) dλ + Z

F > ε

λ 0 − h (t) ( τ b

ρ )

dλ −

Z

F < ε

λ 0 dλ − h (t) ( b τ

ρ )

= εh (t) ( b τ

ρ ) dλ .

+$.F

(10)

?

?

9

!

"

+$. E $ %

&

ε = α t r

+$.

λ 0 (F > ε ∪ F = ε ∪ F < ε ) = Z

F > ε ∪F = ε ∪F < ε

λ 0 dλ = α r h (t) ( τ b

ρ )

t .

&

T 1 t α r := F > ε ∪ F = ε ∪ F < ε

&

+$.

λ 0 (T 1 t α r ) = α r h (t) (b τ

ρ )

t = x (t) τ b

ρ h (t) (b τ

ρ )

t = λ (t) (T 1 t α r ) .

%

λ 0 ( D b τ ρ ∩T 1 t (t) a ) = λ ( D b τ r )h (t) ( b τ r ) = α r

t h (t) ( b τ r ) = 1 t x ¯ (t) b τ

r

(t) a τ b r = 1 t

(t) a b τ r h (t) ( τ b r )

¯ ξ (t) ( (t) D b τ r ∩ T 1 t (t) a ) = ¯ ξ (t) ( (t) D τ b r ∩ T 1 t α r )

= α r λ (t) (D τ b ρ ) = α r λ 0 ( (t) D τ b r )

= 1 t x ¯ (t) b τ

r

(t) a τ b r = 1 t

(t) a b τ r h (t) (b τ

r )

= α r

t h (t) ( b τ r )

+$.

$$&

¯ ξ(T 1 t α r ) = ¯ ξ (t) (T 1 t α r ) = α r

t h (t) ( b τ

r ) . 3 rd STEP :

4

T 1 t β r

!" " $

4

#

τ r

(t) D τ r ⊆ E

&

¯ ξ • = 1 − l ? r = 1 − h (t)? r = x (t)

τ r

(t) D τ r .

%&

"

T 1 t β r

&

¯ ξ(T 1 t β r ) = ¯ ξ (t) (T 1 t β r ) = β r

1 − l r ?

t = β r

1 − h (t)? r

t = β r

1 t x ¯ (t) τ

r .

&

5

#

2 nd ST EP

A

T 1 t β r(t) D τ r

AG3

%

"

λ 0 (T 1 t β r ) = β r λ 0 (D τ r )

$ % !

+$/

λ 0 (D τ r ∩ T 1 t (t) a ) = β r

t h (t)

r ) = 1

t

(t) a τ

r h (t)

r ) = 1

t x ¯ (t)

τ r

(t) a τ

r

(11)

?

?

%

#

+$. - $

S

"

λ(S) = p t

$ %

""

(t) D

S ∈ F (t)

G# !

S q

"

!

"

1 t

!

"

I \S

#$ %

!!

#

%

"

"

5

$

>

ξ

?BC

η

@<=>?>@BA ?BC

>

S

? ?@>@B A= >?>

ξ dom v S η

AA=< >?>

λ(S) = p t

@A ?>@B?

T > := { ξ > η}

?A ?>@B? <?A=

λ(T > ) = q t

@> >

A?<

t

>

(t) D

? =B@

< <C

A=

>

?>

S ∈ F (t)

?BC

T > ∈ F (t)

?BC >

ξ (t) := P

ξ F (t) (•)

B > ?>@BA

+$/+

ξ dom v S η, ξ dom v S (t) η,

?BC

ξ (t) dom v S (t) η

C > =

A@<=?>?B

=A

>

@A>A ?

>

a (t)

?B>

v (t)

A= >?>

T 1 t a (t) ⊆ S

?BC

+$/-

ξ dom v

T 1 t a(t) η , ξ dom v (t)

T 1 t a(t) η , ξ (t) dom v (t)

T 1 t a(t) η .

%:

"

"

A

""

+$+

"

%

"

+$-&

#

5

"

"

$

(

! A

""

"

v

$

>

ξ

?BC

η

@<=>?>@BA

S

? ?@>@B ?BC >

ξ dom S η

B>

@A>A ?

?@>@B

T ⊆ S

A= >?>

λ(T ρ ) = λ(T ∩C ρ )

@A ?>@B?

ρ ∈ R

?BC

ξ dom T η .

1 st STEP :

4 "

ξ(S) < v(S)

$ 4

ρ ∈ R

9

"

r n ρ

r ρ n ↑ n λ (S ρ ) (n → ∞)

#

AG%

"

!9

S n ρ ⊆ S n+1 ρ ⊆ S ρ

λ(S n ρ ) = r n ρ

S n ρ ↑ S ρ (n ∈

)

&

S n ↑ S (n ∈

)

$ %

ξ(S n ) ↑ ξ(S)

λ ρ (S n ) ↑ λ ρ (S)

ρ ∈ R 0 (n ∈

),

(12)

?

?

v(S n ) ↑ v(S) (n ∈

)

$ D9#& # !

n ∈

ξ(S n ) < v(S n )

&

S n ⊆ S

&

ξ > η

S n

$ % &

ξ dom S n η

#

!

n ∈

$

2 nd STEP :

+$/.

ξ(S) = v(S) .

s ρ := λ(S ρ )(ρ ∈ R)

$ (&"

T ⊆ S

ξ(T ) < v(T )

&

"

#

:

$

%&

+$// 4

T ⊆ S

λ(T ) < λ(S),

ξ(T ) = v(T ).

%

0 ≤ t ≤ 1

S t,ρ ⊆ S ρ

λ(S t,ρ ) = ts ρ

$ %&

r ρ

t := r s ρ

ρ < 1

ρ ∈ R

&

λ(S t,ρ ) = ts ρ = r ρ s ρ

s ρ

= r ρ ;

#

λ(S t ) = v(S t )

# +$// $ 8

S t ⊆ S

0

ξ > η

S t

&

ξ dom S t η

$

3 rd STEP :

%&

"

ξ(S) = v(S)

T ⊆ S

λ(T ) < λ(S)

&

ξ(T ) > v(T )

$

A

+$/E

R ? := {ρ ∈ R ξ(S) = λ ρ (S)} = {ρ ∈ R λ ρ (S) = v(S)} 6= ∅

+$/F

ξ (S) < λ ρ (S) (ρ ∈ R \ R ? ) .

5

:

R := {ρ ∈ R

5

ε 0 > 0

0 < ε < ε 0

T ⊆ S

λ(S) − ε < λ(T ) < λ(S) ξ(T ) > λ ρ (T ) } 6= ∅ .

+$/

D

#

+$/

R ⊆ R ? .

(13)

?

?

ε •

G

#

ρ ∈ R

& $$& #

0 < ε < ε •

T ⊆ S

λ(S) − ε < λ(T ) < λ(S), ξ (T ) > λ ρ (T ) (ρ ∈ R )

ξ (T ) = λ ρ (T ) (ρ ∈ R ? \ R )

+$/

D

"

µ := (ξ, λ, λ 0 , . . . , λ r )

$ 4 "

ε < ε •

T ⊆ S

# !

λ (S) − ε < λ (T ) < λ (S)

ξ(S \ T ) = ξ(S) − ξ(T ) = λ ρ (S) − ξ(T )

< λ ρ (S) − λ ρ (T ) = λ ρ (S \ T ) (ρ ∈ R )

+$E

ξ(S \ T ) = ξ(S) − ξ(T ) = λ ρ (S) − ξ(T )

= λ ρ (S) − λ ρ (T ) = λ ρ (S \ T ) (ρ ∈ R ? \ R ) .

+$E+

5

& +$/F &

ε •

#

"

0

T ⊆ S

λ(S) − ε < λ(T ) < λ(S)

&

+$E-

ξ(T ) < λ ρ (T ) (ρ ∈ R \ R ? ) .

"

q, {q ρ } ρ∈R

+$E.

0 < q < s , s − q < ε ,

+$E/

0 < q ρ < s ρ , X

ρ∈R

q ρ = q ,

$

%

r

+$EE

0 < r < 1, s − rs < ε •

:$ 4

#

+$EF

r ρ := r

1 − q s ρ

ρ

1 − q s .

T ⊆ S

λ(T ) = rs

!#

T

: +$E

+$E+ $

D ! !

µ(S \ T )

µ(S)

$ 8# AG

%

"

:

#

0 ≤ t ≤ 1

"

S t

&

S \ T ⊆ S t ⊆ S

µ (S t ) = t µ (S \ T ) + (1 − t) µ (S)

$

(14)

?

?

S t

# :

ξ(S t ) < λ ρ (S t ) (ρ ∈ R ) ξ(S t ) = λ ρ (S t ) (ρ ∈ R ? \ R ) .

+$E

; :

+$/E +$E+ &

+$/E +$E $

(

t

# !

+$E

t = 1 r

1 − q s

=

1 − q ρ

s ρ

ρ ∈ R ;

#

+$EF $

%

λ(S t ) = tλ(S \ T ) + (1 − t)λ(S)

= t(s − rs) + (1 − t)s

= s − trs = s − s 1 − q

s

= q

+$E

λ(S ) = λ ρ (S t )

= tλ ρ (S \ T ) + (1 − t)λ ρ (S)

= t(s ρ − r r s ρ ) + (1 − t)s ρ

= s ρ − trs ρ = s ρ − s ρ

1 − q ρ

s ρ

= q ρ

+$F

"

$ 8

s − q < ε •

& 0

S t

: +$E- & $$&

+$F+

ξ(S t ) < λ ρ (S t ) (ρ ∈ R \ R ? ) .

D

"

!

+$E +$F+

ξ (S t ) ≤ v (S t )

S t ⊆ S

ξ > η

S

+$F -

ξ dom S t η ,

% !

""

$

>

ξ

?BC

η

@<=>?>@BA ?BC

>

T

? ?@>@B A= >?>

ξ dom v T η

B > @A>A ? ?@>@B

S

?BC ?B @<=>?>@B

ϑ

A= >?>

S ⊆ T

(15)

?

?

ϑ = η

B

S

{ ξ > ϑ} ⊆ { ξ > η}

λ(S ∩ C ρ )

@A ?>@B? ?

ρ ∈ R

λ({ ξ > ϑ}) ∩ C ρ

@A ?>@B? ?

ρ ∈ R

ξ dom S ϑ

8 !

A

""

+$F :

S ⊆ T

λ(S) ∩ C ρ

ρ ∈ R

ξ dom S η

$ 8

S ⊆ { ξ > η}

S = { ξ > η}

$ 8"& &

{ ξ > η} \ S

"

$ %

&

ξ

η

" &

{ ξ ≤ η} ⊆ I \ S

" $

T > := { ξ > η}

T := { ξ ≤ η}

$ D

R + ⊆ T >

R ⊆ T

# " "

λ (( T > \ R + ) ∩ C ρ )

ρ ∈ R

+$F.

Z

R +

( ξ − η )d λ = Z

R

( η − ξ )d λ

$ %

+$F/

ϑ := η + (ξ − η)

R + − (η − ξ)

R

"

$

ϑ = ξ + (η − ξ) = η > ξ

R +

$ " #&

ϑ = η + ( ξ − η ) = ξ

R

$

%

+$F E

{ ξ > ϑ} = T > \ R + ⊆ { ξ > η}

{ ξ > ϑ}∩C ρ

"

λ({ ξ > ϑ}∩C ρ ) = λ((T > \R + )∩C ρ )

ρ ∈ R

$ 4 #&

ϑ = η

S

& #

ξ dom S ϑ

$

>

ξ

?BC

η

@<=>?>@BA?BC

>

S

??@>@BA= >?>

ξ dom v S η

B > @A>A

t 0 ∈

A= >?> <=>@

t = rt 0

t 0

>

@A ? =B@

< <C

BA>@>=>

C

(t) D

?A ?A ?

?@>@B

R

?BC ?B

@<=>?>@B

ϑ

A= >?> >@B @A A?>@AC

R ⊆ S

ϑ = η

B

R

(16)

?

?

{ ξ > ϑ} ⊆ { ξ > η}

R ∈ F (t)

{ ξ > ϑ} ∈ F (t)

ξ dom R ϑ

4

""

#

"

A

""

+$

R

ϑ

!#

t 0

!

λ(R ∩ C ρ )

λ({ ξ > ϑ}) ∩ C ρ

"

1 t 0

ρ ∈ R

$ % "

C ρ (ρ ∈ R)

"

t D τ

#

R ∩ C ρ

{ ξ > ϑ} ∩ C ρ

$

%

"

(t) D

5

!

"

$

(

"

(t) D τ

9 "

1 t

S

{ ξ > ϑ}

! # A"" +$$

◦ eeeeee

D

"

!

"

$$

v

"

$$$

v (t)

$ %

!

"

#

$

>

ξ

?BC

η

@<=>?>@BA ?BC C

B>

ξ (t) := P

ξ F (t) (•)

A >

S

??@>@BA=>?>

ξ dom v S η

@A>A

t 0 ∈

?BC ? <=>@A

t = rt 0 ∈

A< =B@ < <C

(t) D

?A ?A ?

>

a (t)

?B> @> A> >

(t) D

A=

>

?>

T a (t)

A?>@A

A

T a(t) ⊆ S

ξ dom v T a(t) η , ξ dom v T (t) a(t) η ,

?BC

ξ (t) dom v T (t) a(t) η

D

t 0 ∈

" "

t = rt 0

(t) D

"

R ⊆ S

ϑ

! A"" +$$ % &

ξ = ϑ

R

{ξ > η}

R

F (t)

3" $ %

+$FF

v (t) (R) = v(R)

& A

""

+$+&

+$F

ξ dom v R ϑ, ξ dom v R (t) ϑ,

ξ (t) dom v R (t) ϑ .

(17)

?

?

#

1

ε

2 ( %" +$-& 5

a = a (t)

$$

v (t)

T a (t) ⊆ R

ξ dom v (t)

T a(t) ϑ .

8!

# A

""

+$+

+$F

ξ dom v T a(t) η , ξ dom v T (t) a(t) η ,

ξ (t) dom v T (t) a(t) η .

(18)

?

?

'!

+

+

' ,

'

(

"

0

" "

"

x ¯

3

"

"

$

(

5

$

"

7

< /

= B/

6 B <

6

?

= 9/

<;

:=

$

"

$

#

#

!

"

"

$ %

"

! :

5

$

"

#

!

"

"

$

% :

# !

"

(I , F , v)

v = ^

λ 0 , λ 1 , . . . , λ r = ^

ρ∈R 0

λ ρ ,

λ 0

# 1 2 $ %

#

"

"

#

(t) D = (t) D τ

τ∈T

T = T (t) := {1, . . . , rt})

%

#

λ (t)

$$ 3

!:

F (t)

$ % ! "

v (t)

: !#$

"

? >

$

4

T σ

T σ

# B@>@B ? > $

#

& !

"

5

(t)

(t) ∨

T σ

(t) ∨

T σ

$ % !

-$+

(t) ∨

T σ := [

τ∈ (t)

T σ

(t) D τ , (t) T σ := [

τ∈ (t)

T σ (t) D τ

T (t) := [

ρ∈R (t) ∨

T ρ , T (t) := [

ρ∈R (t) ∧

T ρ .

(

"

5

? >

h τ

#

9

-$-

h 0 ρ := min{h τ τ ∈ T ρ } ρ ∈ R

-$.

h ? σ = X

ρ∈R\{σ}

h 0 ρ .

(19)

?

?

"

(t) D

λ (t)

"

λ 0

& #

-$/

(t) h τ =

λ 0 (t) D τ (τ ∈ T (t) ).

8 !

#

-$E

(t) h 0 ρ := min{ (t) h τ τ ∈ (t) T ρ } ρ ∈ R

-$F

(t) h ? σ = X

ρ∈R\{σ}

(t) h 0 ρ .

:

-$

l σ 0 := ess inf C σ

λ 0 , l ? σ := X

ρ∈R\{σ}

l 0 ρ (σ ∈ R)

"

-$

X

ρ∈R

l 0 ρ ≤ 1

"

"

$ D

#

-$

(t) h 0 ρ ≥ l ρ 0 , (t) h ? ρ ≥ l ? ρ ,

1 − (t) h ? ρ ≤ 1 − l ? ρ (ρ ∈ R) .

:

σ ∈ R

E σ :=

 

 ω λ 0 (ω) + X

ρ∈R\{σ}

l 0 ρ < 1

 

 ∩ C σ =

ω λ 0 (ω) + l ? σ < 1

∩ C σ ,

E := [

ρ∈R

E ρ ,

-$+

E ρ := C ρ \ E ρ (ρ ∈ R) ,

E := [

ρ∈R

E ρ = [

ρ∈R

ω λ 0 (ω) ≥ 1 − l ? ρ

∩ C ρ ,

-$++

%

1 7

< /

= B/

6 B

<

6

?

= 9/<;

:=

2

x ¯

#

B@>@B

? >

!$

-$+-

x ¯ (t) =

x (t) τ∈T (t) .

Abbildung

Updating...

Referenzen

Updating...

Verwandte Themen :