9 Union stable systems

A UNIFIED APPROACH

OF VALUES FOR GAMES ON UNION STABLE SYSTEMS

A UNIFIED APPROACH

OF VALUES FOR GAMES ON UNION STABLE SYSTEMS

E. Algaba

Department of Applied Mathematics II and

Mathematics Research Institute of Seville University

Algaba E, Bilbao JM, Borm P, López JJ (2000) The position value for union stable systems. Math. Meth. Oper. Res. 52:221-236

Algaba E, Bilbao JM, Borm P, López JJ (2001) The Myerson value for union stable structures. Math. Meth. Oper. Res. 54:359-371

Algaba E, Bilbao JM, Brink R van den, López JJ (2012) The Myerson value and superfluous supports on union stable systems. JOTA 155:650- 668

Algaba E, Bilbao JM, Brink R van den (2015) Harsanyi power solutions for games on union stable systems. ANOR 225:27-44

9 Union stable systems

9 Restricted and conference games

9 Myerson and position values

9 Characterizations

9 Harsanyi power solutions

9 Characterizations

9 Union stable systems

9 Restricted and conference games

9 Myerson and position values

9 Characterizations

9 Harsanyi power solutions

9 Characterizations

( )

/

M

Rectricted game by a communication graph

yerson (

value

, ( )

: 2

( )

, ( , )

),

i

N

i S S G

G

G G

G

N v v

v S

v S

N v,E N v

∈

•

•

⎯⎯→

=

= Φ

∑

μ

R

COMMUNICATION SITUATION

( , , )N v E ( , ), N v G = ( , )N E

Myerson (1977)

Graphs and Cooperation in Games

•Myerson (1977),

•Owen (1986)

•Borm, Owen and Tijs (1992)

•Van den Nouweland, Borm, Tijs (1992)

( )

A1.

A

A set system , is if

For , with , we have

2.

N

S T S T S

sta e

T bl

∅ ∈

∈ ∩ ≠ ∈

∪

∅ ∪

F F

F F

Comunication situations

( , , )N v E

Union stable cooperation structures ( , , )N v ^F

{1, 2,3, 4,5}

{{5},{2,3},{2,3, 4},{3, 4,5},{2,3, 4,5}}

Example

=

= F

{5}

{3, 4,5}

{2,3, 4,5}

{2,3, 4}

{2,3}

4

5 3

2

(_F ^,⊆)

{ , ,3, 4}

{{ , ,3},{ , , 4},{ , ,3, 1

1 1 1

1 /

2

2 2 2

2 / Stage

4}}

3, 4 / Bu S

ag el

ent yer

e s l r

Example

=

= F

(_F ^,⊆)

{1,2,3} {1,2,4}

{ , ,1 2 3,4}

4 1 2

3

⊆ _F G

{ }

(0)

( ) ( 1)

(0) ( )

( 1) ( )

( )

: , ,

If is the smallest integer such that ( )=

( ) is a union stable system

n n

n

k k

k

S T S T S T

k

SG SG

−

+

=

= ∪ ∈ ∩ ≠ ∅

⊆ ⊆ ⊆ ⊆

=

G G

G G

G G F

G G

G G

G

( )

( )

1

2

{{1,2},{2,3},{2,4}}

{{1,2},{2,3},{2,4},{1,2,3},{1,2,4},{

{1,2,3,4}

{{1},{3},{1,2},{2,3},{2,4},{1,2,3},{1,2,4},{2,3,4}, }

{{1,2},{2,3},{2,4},{1,2,3},{1,2,4},{2,3,4}}

Example

= N

⊆

=

=

=

=

F

F

G

G G

( ) ( )

( )

2 3

2

2,3,4}, }

{{1,2},{2,3},{2,4},{1,2,3},{1,2,4},{2,3,4}, } N

SG ⁼ N

=

=

G G

G

( ) ( ) ( ) ( ) ( )

( ( ) ) ^{( )}

( )

( )

is a closure operator ( )

( )

( ) ,

is a closure space

is a closed set is a union stable system

:2 2 , = = :2 2

2

2 ,

2 , ,

a b c

SG

N

ϕ ϕ

ϕ

ϕ

ϕ ϕ

ϕ ϕ ϕ

→

→

∀ ⇒

⇒

∀ =

⇔

∈ ⊆

⊆ ⊆ ⊆

∈

−

∈

F F

F F

F

F

F

G G G

G G G

G R F G R

G G G

F

G G

( N , F )

{ }

( ) : , , , , ,

( ) su

pport

Basis of

( ) \ ( )

D F F A B A B F A B

B B

A B

B D

= ∈ = ∪ ≠

∈

∈ ∩ ≠ ∅

=

F F

F F

F

F F F

{1,2,3,4}

{{1},{2},{3},{4},{1,2},{1,3},{3,4}, {1,2,3},{1,3,4},{2,3,4},{1,2,3,4}}

( ) {{1},{2},{3},{4},{1,2},{1,3},{

E

3,4},{2 xampl

, e

3,4}}

N

B

=

=

= F

F

( ) ( )

{ }

( ) { { } { } } { ^{{ }} }

cooperation structure

: , ( ) connected subgraph of ,

, : , :

( , , ) -

S N S E S N E

i

stab

j i j E i

e

i l

N

N v E

∪

= ⊆

∈ ∈

∪

= F

B F

•Myerson (1977),

•Owen (1986)

•Borm, Owen and Tijs (1992)

•Van den Nouweland, Borm, Tijs (1992)

C o m u n ic a tio n s itu a tio n s

( N v E, , )

U n io n s ta b le c o o p e ra tio n s tru c tu re s ( N v, , F )

•Myerson (1977),

•Owen (1986)

•Borm, Owen and Tijs (1992)

•Van den Nouweland, Borm, Tijs (1992)

Comunication situations

( , , )N v E

Union stable cooperation structures ( , , )N v F

( {{1,2},{1,3},{3,4},{2,3,4}}

{1,2,3,4}

{{1,2},{1,3},{3,4},{1,2,3},{1,3,4},{2,3,4}, }

{3,4} {2

)

Example

N

= N

=

=

⊆ F

B F

,3,4}

{ }

Propositi

( ) is the minimal subset of such that on

Proposition

( ( ))

( )= : \ is union stable basis of

(

( ) ( ) )

B SG B

ex F F

B

B ex

=

∈

=

F F F F

F F F F

F F

F

(

)

-com ponent of :

and there exists no ´ , ´

T S S

T T T T S

⊆

∈ ∈ ⊂ ⊆

F

F F

( , ) U-stable , ( ) partition of ' Proposition

N F ⇔ S ⊆ N C_F S S ⊆ S

(

)

{ }

( ) ( )

{ }

=

) ) )

( ) ( )

:

( ) ( , ) is a union stable system

( ( : (

S

S

S

S

S N

b C C

c

F F S

a N

B B S

= ⊆

= ⊆

∈ ⊆

∈ ⊆

F F

F F F

F

B F B F B F

Proposition

(

)

, with 2, we have

satisfying:

(1) For all

(2) all non-unitary feasible coalition can be expressed in a unique way as union of non-unitary

supports

N N

N

S T

S T S T

USI ⊆US

∈ ∩ ≥ ∩ ∈

F

F F

Example { }

{ } { } { }

{ }

{ } { } { }

{ }

1,2,3,4

1,2 , 1,2,3 , 2,3,4 , 1,2 , 1,2,3 , 2,3,4

N

N

=

= =

=

F

B C

{ } { } { }

{ } { } { } { }

1,2,3 2,3,4 2,3

1,2 2,3,4 = 1,2,3 2,3,4

N

∩ = ∉

∪ ∪

=

F

, USI

⎛

N v,

⎜ ⎟

⎝

_F

∉

(

)

, with 2, we have

satisfying:

(1) For all

(2) all non-unitary feasible coalition can be expressed in a unique way as union of non-unitary

supports

N N

N

S T

S T S T

USI ^⊆ US

∈ ∩ ≥ ∩ ∈

F

F F

2 2 2

{ , ,3} { , ,4} { , }1 1 1

∩ = ∉ F

, USI

⎛

N v,

⎜ ⎟

⎝

_F

∉

{ , ,3, 4}

{{ , ,3},{ ,

1 2 2 2, 4

1 1 },{ , ,3 4}1 2 , }

Example

=

= F

2/ Stage agen 1/

3,4/

S B

t u

eller yers

4 1 2

(F^,⊆)

3 {1,2,3} {1,2,4}

{ , ,123,4}

RESTRICTED GAME

( )

: 2 ,

( ) ( )

N

T C S

v

v S v T

∈

⎯ ⎯→

=

∑

F

F F

R

CONFERENCE GAME

( )

( )

{ }

2

:

, conference gam e

-stable cooperation structure

, ( )

,

( ) ( )

: 2 ,

C C

N v , v

N v ,

v v N

v

≥

⎯ ⎯→

= ∈

→

⊆ → ∪

=

C

C A

C C

C C

C A

F

A C

R

MYERSON VALUE

( ^{N v, ,} ) (

)

μ

F

CONFERENCE GAME

( )

( )

, conference game

-stable cooperation structure

, ( )

,

( ) ( )

:2 ,

N v, v

N v,

v v N

v ^⎯⎯→

→

⊆ → ∪

=

C

C A

C C

C A F

A C

R A POSITION VALUE

( ) ( )

{ }

1 , if

0, otherwise

:

, C _i ^C , ⁱ

i

i

C v

C i C

π N v, ∈ Φ ≠ ∅

=

= ∈ ∈

⎧⎪

⎨⎪

⎩

∑

C

C C

C C

F

RESTRICTED GAME

( )

( )

: 2 ,

( )

T C S N

v T v

v S

∈

⎯ ⎯→

= ∑

F

F F

R

{ , ,3, 4}

{{ , ,3},{ , , 4},{

1

1 1 1

2

2 2 , ,3, 4}}, 2 {{ , ,1 2 3},{ ,1 2, 4}}

Example

=

=

F B = =C

2/ Stage agen 1/

3,4/

S B

t u

eller

yers

( ) ( )

1 if

( ) 0 o th e rw is e

1, 0

S S

v S

v S S v

⎧⎪⎨

⎪⎩

− ∈

=

= − ∅ =

F F

( )

{ , , 3} { , , 3} { , , 3}

{ ,

( ) ( )

{ 1 } { 1 } { 1 } 2

{ 1 , 4}} { 1{ , , 4} } { 1{ , } 2

{

, 4}

{ ,1 , 3} ,{ , , 4}1 } { 1{ , , 3} 1 2

1 } { 1 } 3

,{ , , 4} ,{ , , 3, 4} { , ,

2 2

2 2 2

2 2 2 2 2 2 3 4},

1 3 1 3 5 5

, , , , ,

1 2 1 2 1 1

2 2

C N v

μ

⊆

=

C

A C A A A

F

( )

, , 1,1, ,1 1 N v 2 2

π ^⎛⎜_⎜ ^⎞⎟_⎟

⎝ ⎠

F =

4 1 2

(F^,⊆)

3 {1,2,3} {1,2,4}

{ , ,123,4}

Theorem

( )

( )

( ) ( )

{ }

, , ,

= C :

Let be a union stable cooperation structure and the associated conference game. Then

where

N v

S C S

v v

v

∈ ⊆

=

S

F C

C

F C

C

C C

RESTRICTED GAME

( )

( )

: 2 ,

( )

T C S N

v T v

v S

∈

⎯ ⎯→

= ∑

F

F F

R

CONFERENCE GAME

( )

( )

, conference game

-stable cooperation structure

, ( )

,

( ) ( )

:2 ,

N v, v

N v,

v v N

v ^⎯⎯→

→

⊆ → ∪

=

C

C A

C C

C A F

A C

R A

( ) { ( ) ( ) ( ) ( ) }

( ) ( )

( )

( )

: , ,

Bondareva (1963), Shapley (1967):

A game , is balanced if and only if A game , is superadditive if

( ) ( ) ( ), , 2 ,

A game , is conve

n

N

C v x x N v N x S v S S N

N v C v

N v

v S T v S v T S T S T N v

• = ∈ = ≥ ∀ ⊂

•

≠ ∅

•

∪ ≥ + ∀ ∈ ∩ = ∅

•

R

x if

( ) ( ) ( ) ( ), , 2^N

v S ∪T + v S ∩T ≥ v S + v T ∀S T ∈

Theorem

( )

( )

( )

, , ,

,

Let be a union stable cooperation structure. If non-negative and balanced, then is balanced

N v

N

v v

C F

F C

Theorem

( )

( )

( )

, , ,

,

Let be a union stable cooperation structure. If

non-negative and superadditive, then is superadditive

N v

N

v v

C F

F C

Theorem

( )

( )

( )

, , ,

,

Let be a union stable cooperation structure. If non-negative and convex, then is convex

N v

N

v v

C F

F C

{ , ,3, 4}2 {{ , ,3},{ , , 4},{ , ,3, 4}}2

1 2

1 1 2 1

Example

=

= F

2/ Stage agen 1/

3,4/

S B

t u

eller yers

if 2

( ) 0 o th e rw is e

( , ) is s u p e ra d d itiv e , s o is ( , )

S S

S N v v

N v

⎧⎪⎨

⎪⎩

= ≥

F

{ }

(

) ( ^{

^} ) ( ^{

^} )

T h e c o n fe re n c e g a m e ( , ) is n o t s u p e ra d { {1, , 3} ,{ , , 4} }2 1 2

d itiv e :

v

v ^≤ v ⁺ v

= =

C

C C C

C

B C

4 1 2

(_F^,⊆)

3 {1,2,3} {1,2,4}

{ , ,123,4}

{ , ,3, 4}2 {{ , ,3},{ , , 4},{ , ,3, 4}}2

1 2

1 1 2 1

Example

=

= F

2/ Stage agen 1/

3,4/

S B

t u

eller yers

if 2

( ) 0 o th e rw is e

( , ) is to ta lly b a la n c e d , s o is ( , )

S S

S

N v N v

v ^⎧⎪_⎨

⎪⎩

= ≥

F

{ , ,3}1 2 { , ,4}1 2 { , ,3}1 2 { , ,1 2 4}

T h e c o n fe re n c e g a m e ( ) is n o t b a la n c e d :

: 4 , 3, 3

1

,

{ { , , 3} ,{ , , 42 2 (

1 )

} } v

y y y y y

C v ^⎧_⎨ ^{⎫ =}_⎬

⎩ ∈ + = ≥ ≥ ⎭

= =

=

R

C

C

C B C

4 1 2

(F,⊆)

3 {1,2,3} {1,2,4}

{ , ,123,4}

Corollary

( ) ( )

( ) ( )

, , ,

, ,

Let be a union stable cooperation structure. If non-negative and convex, then

N v

N v v

v

μ ^∈C ^F

C C

F

F

( ) v ^{

^}

C

Theorem

(

) (

) ( )

Let . If is convex, then

is (non-negative) convex

N v F ∈USI N N v _C v^C

Corollary

( ) ( )

( ) ( )

( ) ( )

, , ,

, , , ,

Let . If is convex, then

N v USIN N v

N v v

N v v

C

π C

μ ^∈

∈

∈

F F

F

F F

1 1 1 1

( , ) if and only if ,

( ) ( ) ( ) ( ),

, , , 1, ,

k k

k k

i i i i i i i i

i j i i

N v co nvex

v T v T v S v T

T T i j S T i k

= − = ≥ = − =

∩ = ∅ ≠ ⊆ =

∑ ∑

∪ ∪

… Lemma

Shapley (1971)

( )

{

union stable cooperation structures }

N N v

US

( )

:

) , ( ),

( )

, 0

γ :

( , ,

,

,

N

k M k

M C N i

N n

US M C N

v M

i M

US

N v

N v,

N v,

∈

∈

⎛ ⎞

⎜ ⎟

⎝ ⎠

⎛ ⎞

⎜ ⎟

⎝ ⎠

∈ ∀ ∈

γ =

∀ ∉ γ =

∀

→

∑

∪

R Com ponent - efficiency

Com ponent - dum m y Allocation rule

F

F F

F

F

SUPERFLUOUS SUPPORT

( ) ( ^{{ }} )

( ) ( { } )

\

\

, ,

H H

N v, N v,

v v

=

=

γ γ

∀ ∀ ∈

⊆

C C

F B

A A A C C

INFLUENCE

( )

( ) ( )

1 , if

0, otherwise

i

i C

f

I N , v

= ⎧ ∈ ≠ ∅

⎪⎨

⎪⎩

=

∀

∑

C

C

C

A A

F

A C γ :U S ^N

→ R

ADDITIVITY

( ^{N v , w ,} ) (

^{N v, ,} ) (

^{N w , ,} )

γ γ γ

+

( ^{N v, ,} )

( ^{N ,} )

additivity

superfluous support influence

The posit

ion value satis efi s

•

•

•

Theorem

additivity

superfluous suppor The Myerson value satis

fies

t

•

•

Theorem

( )

superfluous sup

The position value is the unique allocation rule on satisfyi

port

influe

n

nce

g

additivity

N

v

USI

⎛α ⎞

⎜ ⎟

⎜ ⎟

⎝

•

•

• = ^{C C} ⎠

C

Theorem

(

)

, with 2, we have

satisfying:

(1) For all

(2) all non-unitary feasible coalition can be expressed in an unique way as union of non-unitary s

upports

N N

N

S T

S T S T

USI ⊆ US

∈ ∩ ≥ ∩ ∈

F

F F

{ , ,3, 4}

{{ , ,3},{ , , 4},{

1

1 1 1

2

2 2 , ,3, 4}}, 2 {{ , ,1 2 3},{ ,1 2, 4}}

Example

=

=

F B = =C

2/ Stage agen 1/

3,4/

S B

t u

eller yers

( ) ( )

1 if

( ) 0 o th e rw is e

1, 0

S S

v S

v S S v

⎧⎪⎨

⎪⎩

− ∈

=

= − ∅ =

F F

( )

( )

2 2 1 1 , , , ,

3 3 3 3

2 2 1 1, , , 1 3 1 3 5

, , , , , 5

1 2 1 2 1 2 1 2

1 3 1 3 5, , , 5

1 2 1 2 1 2 1 2 3 3 3 3

I

N v

N ^⎛_⎜⎜ ^⎞_⎟⎟

⎝

⎛ ⎞

⎜ ⎟

⎠

⎛ ⎞

⎜ ⎟

⎜ ⎟

⎝ ⎠

⎛ ⎞

⎜ ⎟

⎜ ⎟

⎝ ⎠ ⎜ ⎟

⎝ ⎠

≠

=

=

α

μ

F F

4 1 2

(_F^,⊆)

3 {1,2,3} {1,2,4}

{ , ,123,4}

SUPERFLUOUS SUPPORT

( ) ( ^{{ }} )

( ) ( { } )

\

\

, ,

H H

N v, N v,

v v

=

=

γ γ

∀ ∀ ∈

⊆

C C

F B

A A A C C

γ :U S ^N

→ R

ADDITIVITY

( ^{N v , w ,} ) (

^{N v, ,} ) (

^{N w , ,} )

γ γ γ

+

POINT ANONYMITY

( )

( ) ( )

{ }

, if

0, otherwise

: 0

,

i

i

i D

S f S D D i N

N v, v

= ⎧

α

⎨⎩

∈

= ∩ = ∈ >

γ

F C

F

SUPERFLUOUS SUPPORT

( ) ( ^{{ }} )

( ) ( { } )

\

\

, ,

H H

N v, N v,

v v

=

=

γ γ

∀ ∀ ∈

⊆

C C

F B

A A A C C

γ :U S ^N

→ R

ADDITIVITY

( ^{N v , w ,} ) (

^{N v, ,} ) (

^{N w , ,} )

γ γ γ

+

POINT UNANIMITY

( )

{ }

,

, if

0, otherwise

: 0

,

i

i

D

i D

D i N

u

N v, v

α β

= ⎧

⎨⎩

∈

= = ∈ >

γ

F C

F

( )

additivity superfluou

The Myerson value is the unique allocation rule on

s support point unanimity

satisfy

ing

N

v

USI

α

•

⎛ ⎞

⎜ ⎟

⎜ ⎟

⎝ ⎠

•

• = ^{F D}

D

Theorem

(

)

, with 2, we have

satisfying:

(1) For all

(2) all non-unitary feasible coalition can be expressed in an unique way as union of non-unitary s

upports

N N

N

S T

S T S T

USI ⊆ US

∈ ∩ ≥ ∩ ∈

F

F F

γ :U S ^N

→ R

ADDITIVITY

( ^{N v , w ,} ) (

^{N v, ,} ) (

^{N w , ,} )

γ γ γ

+

SUPERFLUOUS PLAYER

( ) (

)

( ) ( ^{{ }} )

\

\

, ,

S S

S N

N v, N v,

v v

=

=

γ γ

∀ ⊆

F F

F F

{ }

{ { } }

\ : \

N i = F ∈ F F ⊆ N i

F

POINT UNANIMITY

( )

{ }

,

, if

0, otherwise

: 0

,

i

i

D

i D

D i N

u

N v, v

α β

= ⎧

⎨⎩

∈

= = ∈ >

γ

F C

F

Theorem

additivity superfluo

The Myerson value is the unique allocation rule

us player point unanimity on sat

isfy

ing

US N

•

•

•

Lemma

superfluous playe

γ : . If γ is an additive allocation rule satisfyin

g r th n e

, , , ,

N n

N

US

N v,

N v ,

N v, US

⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

→

γ = γ ∀ ∈

R

F

F F

γ :U S ^N

→ R

ADDITIVITY

( ^{N v , w ,} ) (

^{N v, ,} ) (

^{N w , ,} )

γ γ γ

+

STRONG SUPERFLUOUS SUPPORT

( ) ( ^{{ }} )

( )

^{{ }} ( )

, ,

H

H

S S S N H

N v, N v,

v v

=

=

γ γ

∀ ⊆ ∀

B

∈

F

F

C

POINT UNANIMITY

( )

{ }

,

, if

0, otherwise

: 0

,

i

i

D

i D

D i N

u

N v, v

α β

= ⎧

⎨⎩

∈

= = ∈ >

γ

F C

F

Theorem

additivity strong su

The Myerson value is the unique allocation rule

perfluous support point un

animity on sat

isfying

US N

•

•

•

Lemma

γ : . I

f γ is an additive allocation rule satisfying point unanimity then

, , , ,

N n

N

US

N v,

N v ,

N v, US

⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

→

γ = γ ∀ ∈

R

F

F F

( )

( ) ( )

( )

.

non-negative power to player in

: ,

, A power measure is a function

,

,

N

N N

i

N

US

i S

N

U

N

N

S N

σ

σ σ

+

+

⎯⎯→

⎯⎯→ ∈

∈

∈

F F

F

F F

R

R

( )

A power measure is positive if

σ_i N ,_F ^> i ^∈ C C ^∈ _C

U-STABLE SYSTEM

a positive pow er m easure

: 2 ,

v N

σ

⎯ ⎯→ R

F

HARSANYI POWER SOLUTION

( ) ( )

( ) ( )

( )

( )

,

, 0

:

( )

, , , ,

, , ,

,

j T

j T

N N

N N

i

i T

i v

T N i T j T

N j T

T

US

N v US N v

N v N

N

σ

σ

σ

σ

ϕ

ϕ

ϕ σ σ

∈

⊆ ∈

> ∈

∈ ∈

⎯⎯ →

⎯⎯ →

= Δ

∑

∑ ∑

F

F F

F F

F

R

R

Vasil'ev (1982)

Vasil'ev (2003) Derks, Haller and Peters (2000)

van den Brink, van der Laan and

Pruzhansky (2011)

( )

{

union stable cooperation structures }

N N v

US

Theorem

:

Let be a positive power measure. The Harsanyi power solution is an allocation rule satisfying

additivity

N n

σ US

σ

ϕ

SUPERFLUOUS SUPPORT

( ) ( ^{{ }} )

( ) ( { } )

\

\

, ,

H H

N v, N v,

v v

=

=

γ γ

∀ ∀ ∈

⊆

C C

F B

A A A C C

γ :U S ^N

→ R

( )

{

union stable cooperation structures }

N N v

US

Theorem

:

Let be a positive power measure. The Harsanyi power solution satisfies the superfluous support

property on

N n

N

USI USI

σ

σ

ϕ

w ith 2 w e h a v e

2 . A ll n o n -u n ita ry fe a sib le c o a litio n s c a n b e w ritte n in a u n iq u e w a y a s a u n io n o f n o n -u n ita ry su p p o rts

1 . ,

N U S N

S T S

U S I

S T

⊂

∩ ∩ ∈

∀ ∈

INFLUENCE

( )

( ) ( )

1 , if

0, otherwise

i

i C

f

I N , v

= ⎧ ∈ ≠ ∅

⎪⎨

⎪⎩

=

∀

∑

C

C

C

A A

F

A C γ :U S ^N

→ R

( ^{N v, ,} )

( ^{N ,} )

( )

C i

i C

I N ,

∈

=

∑

C

F

( )

( )

1 if 0 if

i

i

i

i

E E

N , N ,

≠ ∅

= ∅

=

=

C C

F F

{

}

i = C ∈ i ∈ C

C C