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

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

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^{8# ( %" .$. *+,}

F ^(t)

v ^(t)

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a ^(t)

ε 0 > 0

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ε < ε 0

&

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^{# !}

ξ dom ^{v (t)}

T ^{ε a(t)} η

^{$ D #&}

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^(t) D

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, β r = ^(t) a _{τ r} = 1 − (h τ 1 + . . . + h τ r )

h _{τ r} − h τ r

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8

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λ( ^(t) D ^{τ b} ∩ {ξ > η}) > 0

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(t) D ^{b τ ρ} ⊆ {ξ > η}

^{$ D9#&}

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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

!

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8! &

T ^{εa (t)} ∩ C ^r ⊆ ^(t) D ^{b τ ρ} ∪ ^(t) D ^{τ r}

λ ( ^(t) D ^τ ∩ {ξ > η}) > 0

^"

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%

+$-/

(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)} )

?

_?

+$-F

v(T ^{1 t a (t)} ) = 1

tε v(T ^{εa (t)} ) = 1 t .

8

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ξ(T ^{1 t a (t)} ) = _tε 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}

#

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>

a ^(t)

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^(t) D

^{?BC ?}

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T ^{1 t a (t)}

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+$.

→

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

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v

v ⁰

^{$ 4}

C ^r ∩ T ^{1 t a (t)} v

v ⁰

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3

!

!

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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 :

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5

(t)

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(&

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 dλ

= 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 )

dλ

= εh ^(t) _{( b τ}

ρ ) dλ .

+$.F

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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) _(τ

r ) = 1

t

(t) a _τ

r h ^(t) _(τ

r ) = 1

t x ¯ ^(t)

τ r

(t) a _τ

r

?

_?

%

#

+$. - $

S

^"

λ(S) = ^p _t

^{$ %}

""

(t) D

S ∈ F ^(t)

^{G# !}

S q

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!

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1 t

!

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5

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ξ

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η

@<=>?>@BA ?BC

>

S

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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

""

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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 ∈

),

?

_?

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 _? .

?

_?

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)

$

?

_?

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 ^tρ ) = 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

?

_?

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

?

_?

{ 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 .

?

_?

#

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 .

?

_?

'!

+

+

' ,

'

(

"

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 .

?

_?

"

(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) .