Research journal of Applied Sciences, Engineering and Technology 4(8): 919-928, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: October 15, 2011
Accepted: November 18, 2011
Published: April 15, 2012
The Banzhaf Value for Fuzzy Games with a Coalition Structure
Fanyong Meng and Dong Tian
School of Management, Qingdao Technological University, Qingdao, 266520, China
Abstract: In this study, a general case of fuzzy games with a coalition structure is studied, where the player
can participate in different unions. The expression of the Banzhaf value for this kind of fuzzy games is given,
and its existence and uniqueness is shown. In order to better understand the Banzhaf value for this kind of fuzzy
games, we discuss two special cases. The Banzhaf values on the associated fuzzy games are researched.
Furthermore, some properties are studied. Finally, a numerical example is given.
Key words: Banzhaf value, choquet integral, fuzzy game, multilinear extension
Shapley function. Meng and Zhang (2010) researched the
Shapley function for fuzzy games with multilinear
extension form given by Owen (1971), and showed its
existence and uniqueness. Li and Zhang (2009) studied
the Shapley value for the general case of fuzzy games,
which can be used in all kinds of fuzzy games. More
researches refer to Sakawa and Nishizalzi (1994), Branzei
et al. (2003), Tijs et al. (2004), Hwang (2007), Hwang
and Liao (2008) and Butnariu and Kroupa (2009).
In this study, we first propose the model of fuzzy
games with a coalition structure, which is different to the
coalition structure introduced by Aumann and Drèze
(1974) and Owen (1977, 1978). The Banzhaf value for
fuzzy games with a coalition structure is studied. Some
properties are researched. Furthermore, we discuss two
special kinds of fuzzy games with a coalition structure,
and the associated Banzhaf values are given. Since the
relationship between the fuzzy coalition values and that of
their associated crisp coalitions is determined, their
properties can be obtained by studying the associated
games.
INTRODUCTION
As we know, in some cooperative games, not all
coalitions can be formed. Aumann and Drèze (1974) gave
a model of such games, where the players in a union are
independent to other players. Different to Aumann and
Drèze (1974), Owen (1977) researched games with a
coalition structure where the probability of cooperation
among unions is considered, and provided the Owen
value, which is an extension of the Shapley value. The
axiomatic systems of the Owen value are considered by
Hamiache (1999), Khmelnitskaya and Yanovskaya (2007)
and Albizuri (2008). Later, Owen (1978) proposed the
Banzhaf- Owen value for games with a coalition structure,
which is an extension of the Banzhaf value. The
axiomatic systems of the Banzhaf- Owen coalition value
is studied by Alonso-Meijide et al. (2007). Alonso-Meijde
and Fiestras-Janeiro (2002) pointed the Banzhaf-Owen
value dissatisfy symmetry in quotient game, and gave
another solution concept for games with coalition
structures, which is known as the symmetric coalitional
Banzhaf value.
On the other hand, there are some situations where
some players do not fully participate in a coalition, but to
a certain degree, this kind of games is called fuzzy games,
which is first introduced by Aubin (1974). Owen (1971)
defined a kind of fuzzy games, which are called fuzzy
games with multilinear extension form. Tsurumi et al.
(2001) defined a kind of fuzzy games with Choquet
integral form, and the Shapley function defined on this
class of games is given. Butnariu (1980) defined a class of
fuzzy games with proportional value, and gave the
expression of the Shapley function on this limited class of
games. Recently, Butnariu and Kroupa (2008) expanded
the fuzzy games with proportional value to fuzzy games
with weighted function, and gave the corresponding
PRELIMINARIES
Let N = {1, 2, …, n} be a finite set, and P(N) denote
the class of all subsets in N. The coalitions in P(N) are
denoted by S0, T0, …. For any S0 , P(N), the cardinality of
S0 is denoted by the corresponding lower case s.
A coalition structure '={B10, B20,…,Bm0}on player
set N is a partition of N, i.e., c1#h#m B h0 = N and Bh01Bl0
=i for all h,l0 M = {1, 2, ..., m} with h…l. A coalition
structure on player set N is denoted by (N,'). For any S0
0 P(N,'). S0 is called a feasible coalition, where P(N,')
denotes the set of all feasible coalitions in (N,'). A
function v0 :P{N,'}6R+, such that v0 (i) = 0, is called a
set function. The set of all set functions in (N,')is denoted
by G0(N,').
Corresponding Author: Fanyong Meng, School of Management, Qingdao Technological University, Qingdao, 266520, China
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Res. J. Appl. Sci. Eng. Technol., 4(8): 919-928, 2012
Furthermore, when every union has the same
cardinality, then Eq. (2) degenerates to be the Banzhaf
value for fuzzy games on matroids.
Owen (1978) proposed the Banzhaf-Owen value on
G0(N,')as follows:
i ( N , v0 ,  ) 
 
R  M \ k i S0  B0k
 v0 ((Q0  S0 ) \ i )
1 1
(v (Q  S0 ))
2m1 2bk 1 0 0
i  N
Definition 1: Let v0 G U(M,B k ), v is said to be
superadditive if: v(K) + v(T)#v(K wT),œ K
T0 LU(M,Bk),
(1)
where Supp K  SuppT = i.
where Q0=cl,R Bl0. m and r denote the cardinalities of M
and R, respectively.
If there is no special explanation, for any v0
GU(M,Bk), we always mean v is superadditive.
THE BANZHAF FUNCTION FOR FUZZY
GAMES WITH A COALITION STRUCTURE
Definition 2: Let v0 GU(M,Bk), the fuzzy coalition TfBk
is said to be an f-carrier for v Bk in Bk, if we have v(KvT)
= v(K) for any KfBk.
Similar to Lehrer (1988), we give the concept of the
“reduced” fuzzy game for v RBk in Bk. For any different
indices i, j0 SuppBk, put R= {U(i), U(j)} and consider the
“reduced” game VRBk in Bk, (with (Bk\R) v{R} as the set of
players) defined by:
In this section we shall research fuzzy games with a
coalition structure. Different to the coalition structure
given by Aumann and Drèze (1974) and Owen (1977,
1978), the coalition structure given in here does not
required the intersection of different unions is empty set.
A fuzzy coalition structure 'F = {B1, B2, ..., Bm} on fuzzy
coalition U0L(N) is a set of unions on U, where w1#h#m
Bh = U. Here, the players in a union are independent to
other players, and the players can participate in different
unions. A fuzzy coalition structure on player set U is
denoted by (U,'F). Let P('F) denote the set of all
probability distributions in 'F. For any P 0 P('F) and any
Bk 0 'F, we have P(Bk) $0 and  P( Bk )  1 .
Bk
B
B
B
v R k (T) = v k (T) and v R k (Tw{R}) = v (TwR)
for any TfBk\R
From above, we know the “reduced” fuzzy game
v RBk has index |SuppBk|-1.
Let f be a solution on v 0 GU(M,Bk). Similar to the
crisp case, we give the following properties:
Bk F
By LU(M,Bk), we denote the set of all feasible
coalitions in (U,'F) where M = {1, 2, …, m}. A function:
: LU(M,Bk)6R+, such that v(i) = 0, is called a set function.
The set of all set functions in LU(M,Bk) is denoted by
GU(M,Bk).
Let v0 GU(M,Bk) and P , P('F), we give the Banzhaf
value on LU(M,Bk) as follows:
i ( M , Bk , v ) 


P( Bk )
|Supp Bk |
1
k M S  Bk ,iSuppS 2
 v ( S )) i  SuppU
Probabilistic efficiency in unions (2-PEU): Let v 0
GU(M,Bk), P, P('F) and any Bk 0 'F, we have:
f i ( Bk , v Bk )  f j ( Bk , LB k )  f R ( Bk , v RBk )
where i, j , SuppBk, and R = {U(i), U(j)}.
( v ( S  U ( i ))
(2)
Null property (NP): Let v 0 GU(M,Bk) and P , P('F), if
we have v (SvU(i)) = v (S) for any SfBk with I f Supp S
and any k 0 M, then fi (M, Bk, v) = 0, where i0 SuppU.
(3)
Symmetry in unions (SU): Let v 0 GU(M,Bk) and P,
P('F), if we have v (S w U(i)) = v (S w U(j) ) for any SfBk
with if SuppS and any k 0 M, then we have fi(M,Bk,v) =
0, where i0 SuppU.
From Eq. (2), we get:
i ( M , Bk , v ) 
 ( B ,v
k M
i
Bk
k
)  i  SuppU
where
i ( Bk , v Bk ) 

P( Bk )
Additivity (ADD): Let v1,v2 0 GU(M,Bk) and P , P('F),
if we have:
(v1+v2)(S) = v1(S)+ v2(S)
for any S 0 LU(M,Bk), then:
f(M,Bk,v1+v2) = f(M,Bk,v1) + f(M,Bk,v2)
|SuppBk |1
s Bk 2
i SuppS
( v Bk ( SvU ( i ))  v Bk ( S ))
and v Bk denotes the restriction of v in Bk.
From Eq. (2), we know when there is only one union
in 'F, then Eq. (2) degenerates to be the Banzhaf value for
traditional fuzzy games.
Definition 3: Let v 0 GU(M,Bk) and P,P('F), the function
f: GU(M,Bk)6{R+}|SuppU| is called a Banzhaf value for v in
LU(M,Bk) if it satisfies 2-PEU, NP, SU and ADD.
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Res. J. Appl. Sci. Eng. Technol., 4(8): 919-928, 2012
Theorem 1: Let v 0 GU(M,Bk) and P,P('F), a value f
satisfies 2-PEU, NP, SU and ADD if and only if f = n.
From NP, we obtain
f i ( Bk , v Bk )  P( Bk )
Proof: From Eq. (3), it is not difficult to get the existence.
In the following, we shall prove the uniqueness holds.
Let v 0 GU(M,Bk), we first show v can be expressed by:
v
 P( B )  c u
k
k M
cS

|SuppS |
 P( Bk ) 
S  Bk 2
1
f i ( Bk , v )  
i SuppS

0

Bk
where
cS 
 ( 1)
|SuppS | |SuppT |
v (T )
Namely, f(Bk,
TS
ST
1
u S (T )  
0 otherwise
w U(i))- v Bk (S) = w Bk (S w U(i))  w Bk (S) for all SfBk
with i 0 Supp Bk\Supp S, then we have:
fi (Bk, v Bk ) = fj (Bk, v Bk ).
Theorem 2: Let v 0 GU(M,Bk) and P , P('F) a value f
satisfies 2-PEU, NP, SU and MU if and only if f = n.
Proof: From Theorem 1 and Eq. (2), we know the
existence holds. In the following, we will show the
uniqueness.
Define the index I of v to be the minimum number of
non-zero terms in some expression for v of form (4).
When Eq. (4) has only one fuzzy coalition i…TfBk
such that cT…0, where kfM. From Theorem 1, we get.


  P( B k )  c S u S  (T )


  S  Bk
(T )
(5)
From the uniqueness proof for traditional games, we
get Eq. (5). Thus, Eq. (4) holds.
From ADD, we only need to show the uniqueness of
v in Bk for any k 0 M. Namely, the uniqueness of Eq. (3)
for v Bk . Sinces:
c u
cT

 P( Bk ) |SuppT |1 i  SuppT
f i (U , cT uT )  
2

0
otherwise
Thus, the result holds.
Hypothesis, when I = m, we have the conclusion. In
the following, we shall show the result holds when I =
m+1. Without loss of generality, suppose:
(6)
S S
 S  Bk
We get:
f i ( Bk , v Bk )  P( Bk )
B
) = n(Bk, v k ).
GU(M,Bk), P , P('F), and any Bk 0 'F, if we have v Bk (S


v (T )    P( Bk )  cS uS  (T )
 k M

 S  Bk
v Bk  P( Bk )
otherwise
Marginal contributions in unions (MU): Let v 0
For any T 0 LU (M,Bk), without loss of generality,
suppose TfBk, then:
 cS uS
  S  Bk
v Bk
i  SuppS
In the following, we give another axiom system of
Eq. (2), which is inspired by Young (1985), Lehrer (1988)
and Nowak (1997).
and
 P( B k )
f i ( Bk , cS uS )
From 2-PEU and SU, we get:
(4)
S S
 S  Bk

 S  Bk
m1

 S  Bk
v   cTq uT
q 1
q
f i ( Bk , cS uS )
1
Let T =  m
q 1 Tq, for any i 0 SuppU\SuppT, construct the
for any i 0 Supp Bk
game:
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Res. J. Appl. Sci. Eng. Technol., 4(8): 919-928, 2012
for any i0SuppBk\ SuppTk.
when i , SuppTk and |SuppTk| = |SuppBk | -1, without loss
of generality, suppose SuppTkcj = Supp Bk.
From Eq. (7), we have:
w   cTq uTq
q:i Tq
The index of wis at most I = m. Since:
v(S w U(i))-v(S) = w(S w U(i))-w(S)
i ( Bk , v B )
k
for all S0 LU(M,Bk) with i ó Supp S.
From MU and hypothesis, it follows that:
f i (U , w)  f i (U , v )   P( Bk )
k M


q:i SuppTq
2
cT q
|SuppTq |1

When i0 SuppT, amalgamate i with any other coalition j
0 SuppT, put R = {U(i), U(j)} and consider the “reduced”
fuzzy game vR. From hypothesis, we get:
f R (U , v R )   P( Bk ) 
k M
qi Tq

From 2-PEU and SU, we have:
k M
  P( Bk )
k M

cTq

q:i SuppTq
2
|SuppTq |1
Tq U ,1 q  m1 2
P ( Bk )

S  Bk , i , j SuppS
2 |Supp Bk |1
( S  U (i ))  v Bk ( S ))
(v Bk ( S  U (i ))  v Bk ( S ))
2 P ( Bk )

S  Bk ,i , j SuppS
2 |Supp Bk |1
k
(v Bk ( S  U (i ))  v B ( S ))

S  Tk i SuppS

S  Tk i SuppS
2
P ( Bk )
|Supp Bk |2
2
P ( Bk )
Bk
|Supp Tk |1 (v
(v Bk ( S  U (i ))  v Bk ( S ))
( S  U (i )  v Bk ( S ))
 i (Tk , v Bk )
cT q

2
|SuppTq |1

f i (U , v )   P( Bk )
S  Bk , i , j Supps
P ( Bk )
Bk
|SuppBk |1 ( v
 v Bk ( S  U (i )  U ( j ))  v Bk ( S  U ( j ))
cTq
2

when i 0 SuppTk and |SuppTk| = |SuppBk| - p, without loss
of generality, suppose Supp Tkc{j1, j2, …, jp}= SuppBk.
Let T1=TkwU(k1), T2=T1 wU(k2),..., Tp= Tp-1 wU(kp). From
above, we get:
|SuppTq |1
Thus, we get f (U,v) = n (U,v).
B
ni(Tk, v k ) = ni (T1, v Bk ) =...= ni (Bk, v Bk )
Theorem 3: Let v 0 GU(M,Bk) and P , P('F), if TkfBk is
Namely, n(Bk, v Bk ) = n(Tk, v Bk )
B
an f-carrier for v k in Bk, where k0M, then:
 i ( M , Bk , v ) 
  (T , v
k M
i
k
Bk
)  i  SuppU
From Eq. (3), we get the conclusion.
(7)
TWO SPECIAL KINDS OF FUZZY GAMES
WITH A COALITION STRUCTURE
where
i (Tk , v Bk ) 

S  Tk
i SuppS
P ( Bk )
2 |SuppS Tk |1
(v Bk ( S  U (i ))  v Bk ( S ))
In this section, we shall research two special kinds of
fuzzy games with a coalition structure, which are
extensions of fuzzy games given by Owen (1971) and
Tsurumi et al. (2001), respectively.
Owen (1971) introduced fuzzy games with
multilinear extension form. The fuzzy coalition value for
this kind of fuzzy games is expressed by:
Proof: Since Tk is an f-carrier for v Bk in Bk, we get:
i ( Bk , v B )  i (Tk , v B )  0
k
k
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Res. J. Appl. Sci. Eng. Technol., 4(8): 919-928, 2012
vO (U ) 


From Eq. (10), we get:
  U (i )  (1  U (i )) v (T )
T0  SuppU
i T0
0
i SuppU \ T
0
œU0L(N)
i ( M , Bk , vO )
(8)
 
where v0 is the associated crisp game with respect to vO.
Let GoU (M,Bk) denote the set of all fuzzy games with
multilinear extension form for GoU (M,Bk).
Later, Tsurumi et al. (2001) proposed fuzzy games
with Choquet integral form. The fuzzy coalition value for
this kind of fuzzy games is written as:
q (U )
v C (U )   v0 ([U ]hl )(hl  hl 1 )
l 1
 U  L( N )
i SuppS
œi0 Supp U
i ( M , Bk , vO ) 
(9)




P ( Bk ) 
  U ( j)


|SuppBk | 1 
2
 T0   j T0
j Supp ( S  U ( i ))\ T0
k  M S  Bk
 Supp ( S U ( i ))
i SuppS


  U ( j) 

j Supp ( S )\ T0 
T0  Supp ( S  ( i ))  j T0
(10)
Supp ( S  U ( i ))
j T0
(12)

S  Bk ,i SuppS

j SuppS / T0
2
P ( Bk )
|SuppBk |1

  (U (i )  U ( j )
 T0  SuppS
j T0

(1  U ( j ))) (v 0 (T0  i )  v 0 (T0 ))

Proof:YFor any S 0 LU(M,Bk)such that ióSuppS, without
loss of generality, suppose SfBk. When ió SuppBk, we
easily get i is a null player for v0 in SuppBk. When i0 Supp
Bk, we have:
vo ( S  U (i ))  vO ( S )
(  U ( j)
)  i  SuppU
Lemma 1: Let vO 0 GOU (M,Bk) , i is a fuzzy null player
for vO 0 GOU (M,Bk) if and only if i is a null player for the
associated crisp game v0 in SuppBk and any k0M.
Since

Bk
O
Definition 4: Let v0 be a crisp game defined in N, if we
have v0 (S0 c i) = v0(S0) for any S0-fN, then i is said to be
a null player for v0 in N.
Similar to Definition 4, we can have the definition of
fuzzy null player for v0 0 GOU (M,Bk) . Here, we omit.

i T0 
k
when we restrict GU(M,Bk)in the setting of GOU(M,Bk),
from Definition 3, we get the definition of the Banzhaf
value on GOU(M,Bk) .Furthermore, from Theorem 1, we
can show Eq. (10) is the unique Banzhaf value for v0 0
GOU(M, Bk). Here, we on longer discuss in detail.
 (1  U ( j )))v 0 (T0 )

i
k

 i  SuppU
 ( B ,v
k M
i ( Bk , vOB )
ni(M,Bk ,vo)
 (1  U ( j )))v0 (T0 ))
(11)
B
where vo k denotes the restriction of vO in Bk, and
The Banzhaf value for GOU(M,Bk): Let vO 0 GOU (M,Bk)
and P-P('F), we give the Banzhaf value on LU(M,Bk)as
follows:

P( Bk ) 


 U ( j)
2|SuppBk |1  i T0  Supp( S U (i )) j T0
and
where [U]h1={i0N|U(i)$hl} and Q(U) ={U(i)|U(I)>0, i0
N}. q(U) denotes the cardinality of Q(U). The elements in
Q(U) are written in the increasing order as 0 = h0 #h1
#…#h q(U). v0 is the associated crisp game with respect to
vC.
By GCU(M,Bk), we denote the set of all fuzzy games
with Choquet integral form for GU(M,Bk).


k M , S  Bk
 (1  U ( j )))
v O ( S  U (i ))  v O ( S )
j Supp ( S U ( i ))\ T0
=
 (v0 (T0 )  v0 (T0 \ i ))
923


  U ( j)
(1  U ( j ))


j Supp ( S U ( i ))\ T0
i T0  Supp ( S U ( i ))  j T0

Res. J. Appl. Sci. Eng. Technol., 4(8): 919-928, 2012
 (v0 (T0 )  v0 (T0 \ i ))  0
i ( M , Bk , vO ) 
  (T , v
k M
i
k
Bk
O
)
 i  SuppU
(13)
Since
where


  U ( j)

(1  U ( j ))  0

j Supp ( S U ( i ))\ T0
i T0  Supp ( S U ( i ))  j T0

i (Tk , v Bk )

 U (i ) U ( j )  


P ( Bk ) 
j T0
  |SuppTk |1   

 T0  SuppS   (1  U ( j )) 
S  Tk 2
i SuppS

 j SuppS \T0

we have:
v0 (T0 )  v0 (T0 \ i )  0  T0  Supp( S  U (i ))
 (v 0 (T0  i )  v 0 (T0 ))
where i0 T0.
From the randomicity of S. we know i is a null player
for v0 in SuppBk and any k 0 M.
Z
Proof: For any i0 SuppBk\SuppTk, we know i is a null
player for the associated crisp game v0 in SuppBk. From
Lemma 1, we have i is a fuzzy null player for voBk .
To prove i is a fuzzy null player for vO 0 GOU (M,Bk),
we only need to show i is a fuzzy null player for
voBk and any k0M. When ió SuppBk, it is obviously
Thus,ni (Bk , v k ) = 0. When i 0 SuppTk and |SuppTk| =
|SuppBk | -1, without loss of generality, suppose SuppTkcj
= SuppBk. From Eq. (12), we have:
B
that i is a fuzzy null player for vo k . When i0 SuppBk,
for any SfBk with i ó SuppS, we have:
B
 i ( Bk , vo k )
B
vO ( S  U (i ))  vO ( S )
 

S  Bk


 (1  U ( j ))
  U ( j)

i T0  Supp ( S  U ( i ))  j T0
j  Supp ( S  U ( i ))\T0
i SuppS
P( Bk ) 


   U (i )  U ( p)  (1  U ( p) 

pT0
pSuppS \T0
2 |SuppBk |1  T0  SuppS 

 (v0 (T0  i )  v0 (T0 ))
 (v0 (T0 )  v0 (T0 \ i ))
Since v0(T0)-v0(T0\I) = 0, we have:

vO(SwU(i))-vO(S) = 0

S  Bk
i , j SuppS


P( Bk ) 
(1  U ( p))
   U (i )  U ( p) 
2 |SuppBk |1  T0  SuppS 

p T0
pSuppS \ T0
Thus, i is a fuzzy null player for voBk

Lemma 2: Let vO 0 G (M,Bk), Tk 0 Bkis an f- carrier for v
if and only if SuppTkfSuppBk is a carrier for the
associated crisp game v0 in SuppBk.
O
U
Bk
o

T0  SuppS


(1  U ( p)) 
 U (i )1  U ( j ))  U ( p) 


pT0
pSuppS \ T0
 (v0 (T0  i )  v0 (T0 )) 
Proof: From the definitions of the f-carrier for fuzzy
games and the carrier for crisp games, we know it is
sufficiency to show i is a fuzzy null player for v Bk if and
only if i is a null player for the associated crisp game v0 in
SuppBk. From Lemma 1, we get the conclusion.

Theorem 4: Let vO 0 GOU (M,Bk) and P0P('F), if
SuppTkfSuppBk is a carrier for the associated crisp game
v0 in SuppBk, where k0M. Then,


p  SuppS / To

S  Bk
i . j SuppS
924

T0  SuppS
(U (i )U ( j )  U ( p))
pT0
(1  U ( p))(v0 (T0  {i , j})  v0 (T0  j ))

P( Bk )  
   U (i )  U ( p)  (1  U ( p))
2 |SuppBk |1   T0  SuppS

pT0
pSuppS \ T0
Res. J. Appl. Sci. Eng. Technol., 4(8): 919-928, 2012

 (v0 (T0  i )  v0 (T0 )) 

i ( Bk , vOB )


 U (i )(1  U ( j ))  U ( p)  (1  U ( p))

p T0
pSuppS \ T0
T0  SuppS 
k


 U (i )U ( j )  U ( p)
pT0
T0  SuppS 

S  Bk
i SuppS

(1  U ( p)) (v 0 (T0  i )  v 0 (T0 ))

pSuppS \ T0
 U ( j ))))(v0 (T0  i )  v0 (T0 ))




S  Bk
i , j SuppS
 U ( j ))))(v0 (i )


S  Tk
i SuppS
2
P ( Bk )
|SuppTk |1


  (U (i ))  U ( p)
 T 0 SuppS
pT0

S  Bk ,i SuppS
2
P ( Bk )
|Supp Bk |1
Thus, we have:
i ( M , Bk , vO )
= i (Tk , vO k ) .
B
   i ( Bk , v O k ) 
B
When i0SuppTk and |SuppTk| = |SuppBk| - q, without
loss of generality, suppose SuppTkc{j1, j2,…, jq} =
SuppBk. Let T1 = Tkw U(k1),T2=T1 wU(k2),...,Tq=Tq-1wU(kq).
From above, we get:
k M
k
k
k
 
i SuppS
2
|SuppBk |1
q(S)
 (v0 [ S ]hl )  v0 ([ S \ U (i )]hl )
l 1
œi 0SuppS
(14)
From Eq. (14), we further have:
i ( M , Bk , vC ) 
Proof: From the superadditivity of v0, we have v0(S0 c i)v0(S0 )$v0(i) for any S0fN with ióS0.


 U (i )  U ( j )  (1  U ( j ))  U (i )


j T0
j SuppS \ T0
P( Bk )
×(hl-hl-1)
Theorem 5:Let vO 0 GOU (M,Bk) and P0P('F), if the
associated crisp game v0 is superadditive, then we have
ni (M,Bk,vO)for any i0 SuppU.


k M S  Bk
k M
The result is obtained.
T0  SuppS
P( Bk )v0 (i )U (i )
i ( M , Bk , vC )
Thus,  i ( M , Bk , vO )    i ( Bk , vOBk )    i (Tk , vOBk )
k M

k M ,i SuppBk
The Banzhaf value for: GCU(M,Bk): Let Vc0GCU (M,Bk)
and P0P('F), we give the Banzhaf value onLU(M,Bk)as
follows:
i (Tk , vOB )  i (T1 , vOB )  ,...  i ( Bk , vOB ) .
Since,
U (i )v 0 (i )
= P(Bk)U(i)v0 (i)
U ( p))))(v 0 (T0  i )  v 0 (T0 ))
pSuppS \T0
P( Bk ) 
  (U (i ) (U ( j )  (1
|SuppBk |1
 T0  SuppS
j T0
j SuppS \ T0
2
S  Bk
i SuppS
2 P( Bk ) 
 
2 |SuppBk |1  T0  SuppS

 U (i )  (1  U ( p)  (1  U ( p))))(v0 (T 0  i )  v0 (T0 ))

pT0
pSuppS \ T0

P( Bk ) 
  (U (i )  U ( j )  (1
j T0
j SuppS \ T0
2 |SuppBk |1  T0  SuppS
 
q ( Bk )
 P( B )   ([ B ]
k M
k
l 1
i
k hl
, v0 )(hl  hl 1 ) (15)
where
for
i ([ Bk ]h , v0 ) 
l

S0  [ Bk ] hl
1
2
|[ Bk ]hl | 1
(v0 ( S0 )  v0 ( S0 \ i ))
iS0
any SfBk.
From Eq. (12), we get:
and|[Bk]hl| denotes the cardinality of [Bk]hl.
925
Res. J. Appl. Sci. Eng. Technol., 4(8): 919-928, 2012
Theorem 7: Let vC 0 GCU (M,Bk) and P0P('F), if TkfBk
is an f-carrier for vC in Bk, then
Similar to GOU (M,Bk), when we restrict GU (M,Bk) in
the setting of GCU (M,Bk), from Definition 3, we get the
definition of the Banzhaf value on GCU (M,Bk).
Furthermore, from Theorem 1, we can show Eq. (14) is
the unique Banzhaf value for vC 0 GCU (M,Bk). Here, we
also on longer discuss in detail.
i ( M , Bk , vC ) 
Theorem 6: Let vc0 GCU (M,Bk) and any i,j 0 SuppBk,
then vC(S wU(i)) = vC(S wU(j)) for any SfBk with i,j ó
SuppS if and only of v0(S0 c i) = v0 (S0 c j) for any
S0f[Bk]hl with i,j óS0 and i,j 0[Bk]hl wherev0 is the
associated crisp game of vC and #l #q (Bk).
q ( Tk )
 P( B )   ([T ]
k
k M
l 1
i
k hl
, v0 )(hl  hl 1 )
 i  SuppU
(16)
where
i ([Tk ]h , v0 ) 
l
Proof: From Eq. (9), it is not difficult to get the result.
S0  [ Tk ]he ,i S0
Lemma 3: Let vc0 GCU (M,Bk) is an f- carrier for vC in Bk
if and only if [Tk]klf[Bk]he is a carrier for the associated
crips game v0 in [Bk]he for any 1#l #q(Bk).
= ni ([Tk]hl,v0)
l1
1


vC(SwU(i))

(v0 ( S0 )  v0 ( S0 \ i ))
ni ([Bk]hl,v0)
YFor any SfBk with i ó SuppS, since:
q ( S  U ( i ))
2
|[ Tk ]he | 1
Proof: From Theorem 6, we know Tk]kl f [Bk] is a carrier
for v0 in [Bk]kl. Similar to Theorem 4, we get:
Proof: In order to show the conclusion, we only need to
prove i is a fuzzy null player for vC in Bk if and only if i
is a null player for v0 in [Bk] and any1#l #q (Bk) .

1

S0  [ Tk ]kl ,i S0
2
|[ Tk ]he | 1
(v0 ( S0 )  v0 ( S0 \ i ))
 i  [ Bk ]hl
v0 ([ S  U (i )]hl )(hl  hl  1 )
Thus,
q ( Bk )
  ([ B
q ( S  U ( i ))
 q (U (i ))

   v0 ([ S ]hl  i )  
v0 ([ S ]hl ) (hl  hl  1 )
 l1

l  1(U ( i ))  1
l 1

i
] , v0 )(hl  hl 1 )
k hl
q ( Bk )
  ([T ]
l 1
i
k hl
, v 0 )(hl  hl 1 )
and
Since
vC (SwU(i)) = vC(S)
we get
 (v ([ S ]
l 1
0
hl
q ( Bk )
q ( Tk )
l 1
l 1
 ) i ([Tk ]hl , v0 )(hl  hl 1 )    i ([Tk ]hl , v0 )(hl  hl 1 )
q (U ( i ))
 i )  v0 ([ S ]hl )(hl  hl 1 )  0 .
We get the conclusion.
Theorem 8: Let vC 0 GCU (M,Bk) and P0P('F),if the
associated crisp game v0 is superadditive, then we have
 i ( M , Bk , v c ) 
 P( Bk )v0 (i)U (i) for any i0 SuppU.
Namely, v0([S]hl c i) = v0 ([S]hl c i)-for any1#l #q
(U(i)). From the randomcity of S, we get i is a null player
for v0 in [Bk]hl , where1#l #q (U(I)).
Furthermore, it is obvious that i is a null player for v0 in
[Bk]hl , where q (U(i)) <l #q (U(Bk)).
Z If i is a null player for v0 in[Bk]kl for any1#l#q
(U(Bk)), from Eq.(9), we know i is a fuzzy null player
for vC in Bk.
k  M , i SuppBk
Proof: The proof of Theorem 8 is similar to that of
Theorem 5.
926
Res. J. Appl. Sci. Eng. Technol., 4(8): 919-928, 2012
Table 1: The crisp coalition values
v0(S0)
S0
{1}
2
{2}
2
{3}
1
{4}
2
{5}
1
{1,2}
6
{1,3}
6
{1,4}
5
{2,3}
5
{2,4}
7
S0
{3,4}
{2,5}
{4,5}
{1,2,3}
{1,2,4}
{1,3,4}
{2,3,4}
{2,4,5}
{12,3,4}
learning scope of game theory. The Banzhaf value for this
kind of fuzzy games is studied. In order to better
understand this class of fuzzy games, we further study
two special cases, which can be seen as the extensions of
fuzzy games given by Owen (1971) and Tsurumi et al.
(2001).
v0(S0)
6
5
6
15
18
16
16
18
30
ACKNOWLEDGMENT
This study was supported by the National Natural
Science Foundation of China (Nos 70771010, 70801064
and 71071018).
NUMERICAL EXAMPLE
There are 5 companies (which represent the players
1, 2, 3, 4 and 5), who will possibly cooperate. As a result
of the differences in technical and fund among them, the
players 1, 2, 3 and 4 decide to cooperate and the players
2, 4 and 5 decide to cooperate. As is in the real life, each
company is not willing to supply all its resources to a
particular cooperation. Thus, we have to consider a fuzzy
game. Consider a fuzzy coalition U defined by:
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U(1) = 0.6, U(2) = 0.3, U(3) = 0.8
U(4) = 0.7, U(5) = 0.5
From above, we know N = {1, 2, 3, 4, 5}, B1 ={U(1),
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When the fuzzy coalition values and their associated
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(8). Namely, this fuzzy game belongs to GOU(M,Bk). From
Eq. (10) , (12), we obtain the player Banzhaf values are
n1 (M,Bk,v0) = 1066, n2(M,Bk,v0) = 1.6206
n3 (M,Bk,v0) = 1.1603, n4(M,Bk,v0) = 2.5424
n5 (M,Bk,v0) = 0.793
When the fuzzy coalition values and their associated
crisp coalition values have the relationship given by Eq.
(9). Namely, this fuzzy game belongs to GcU (M,Bk). From
Eq. (14), (15), we obtain the player Banzhaf values are
n1 (M,Bk,vC) = 1.3922, n2(M,Bk,vC) = 1.9359
n3 (M,Bk,vC) = 1.4297, n4(M,Bk,vC) = 3.4391,n5
(M,Bk,vC) = 1.2031
CONCLUSION
We have researched a more general case of
cooperative fuzzy games, which allows the player to
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927
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928