Research Journal of Applied Sciences, Engineering and Technology 4(10): 1334-1342, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: December 23, 2011
Accepted: January 21, 2012
Published: May 15, 2012
The Interval Shapley Value for Type-2 Interval Games
1
Fanyong Meng and 2Feng Liu
School of Management, Qingdao Technological University, Qingdao 266520, China
2
Department of Information Management, the Central Institute for Correctional Police, Baoding,
071000, China
1
Abstract: In this study, we research the so called type-2 interval games, where the player participation levels
and the coalition values are both interval numbers. Two special kinds of type-2 interval games are studied. The
interval Shapley values for these two special classes of type-2 interval games are researched. By establishing
axiomatic systems, the existence and uniqueness of the given interval Shapley values are shown. When the
associated interval games are convex, the given interval Shapley values Are Interval Population Monotonic
Allocation Functions (IPMAF), and belong to the associated interval cores.
Key words: Cooperative game, interval core, interval number, interval Shapley value
INTRODUCTION
With the social development, the people realized the
application of classical games theory has been largely
restricted. Aubin (1974) introduced the concept of fuzzy
coalition where some players do not fully participate in a
coalition, but to a certain degree. Owen (1972) subtly
defined a kind of fuzzy games, which is called fuzzy
games with multilinear extension form. Later, Butnariu
(1980), Tsurumi et al. (2001) and Butnariu and Kroupa
(2008) researched a special kind of fuzzy games,
respectively. The same as classical game theory, the key
issue of cooperative fuzzy games is how to distribute the
payoffs. The Shapley value for fuzzy games is studied by
Butnariu (1980), Tsurumi et al. (2001), Butnariu and
Kroupa (2008), Li and Zhang (2009) and Meng and
Zhang (2010, 2011a, 2011b). Besides the Shapley value,
the fuzzy core for fuzzy games is researched by Tijs et al.
(2004) and Yu and Zhang (2009). The lexicographical
solution for fuzzy games is discussed by Sakawa and
Nishizaki (1994).
As we know, there are many uncertain factors during
the process of negotiation and coalition forming, so in
most situations players can only know imprecise
information regarding the values of the coalitions formed
by the players. The researches about this kind of fuzzy
games are discussed by Mares (2000), Mares and Vlach
(2001), Borkotokey (2008) and Yu and Zhang (2010).
Since interval numbers can conveniently describe the
lower and upper bounds of the player possible payoffs.
Alparslan Gök et al. (2009) focused on two-person
cooperative games with interval uncertainty, and
discussed the core for the given games. Later, Alparslan
Gök et al. (2010) studied cooperative games with interval
uncertainty, and researched the interval Shapley value.
Branzei et al. (2010) concerned the core of cooperative
games with interval uncertainty, and gave the definition
of convex games with interval uncertainty. Mallozzi et al.
(2011) researched the F-core for fuzzy interval
cooperative games.
All above mentioned researches only consider the
situation where the player participation is determined. As
above analysis, in most cooperation the payoffs are
uncertainty, so it is difficult for the players to decide the
participation levels exactly. In this study, we shall
research the situation where the player participation levels
and the coalition values are both interval numbers.
PRELIMINARIES
We first review some definitions about interval
numbers.
Definition 1: a = [a − , a + ] is said to be an interval number
if a − ≤ a + , where a − , a + ∈ R; a = [a − , a + ] is said to be
a positive interval number if aG # a+, where a − , a + ,R+.
From Definition 1, we know the interval number a
degenerates to be a real number when a − = a + In this
study, we use R and R+ to denote the sets of all interval
numbers and positive interval numbers on R and R+,
respectively.
Let a ,b ∈ R , from the extension principle on fuzzy
sets proposed by Zadeh (1973), we have:
a + b = [a − + b − , a + + b + ]
a − b = [a − − b + , a + − b − ]
Corresponding Author: Fanyong Meng, School of Management, Qingdao Technological University, Qingdao 266520, China
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Res. J. Appl. Sci. Eng. Technol., 4(10): 1334-1342, 2012
λa = [ λa − , λa + ]
λ ∈ R+
v0 ( S0 ∪ T0 ) + v0 ( S0 ∩ T0 ) ≥ v0 ( S0 ) + v0 (T0 )
a ∨ b = [a − ∨ b − , a + ∨ b + ]
for any
a ∧ b = [a − ∧ b − , a + ∧ b + ]
TWO SPECIAL KINDS OF TYPE-2
INTERVAL GAMES
Definition 2: For all a , b , R we write
C
C
a≥b
a =b
if and only if
if and only if
a − ≥ b−
a − = b−
and
and
a + ≥ b+
a + = b+
Definition 3: Let a , b ∈ R , if there exists c ∈ R such that
a = b + c , then c is called the Hukuhara difference
between a and b , denoted by a − H b .
Let the set of players N = {1, 2, ..., n}. The crisp
coalitions on N are denoted by S0, T0, .... The power set of
all crisp subsets on N is denoted by P(N). For any S00P
(N), the cardinality of S0 is denoted by the corresponding
lower case s. A function v0 : P(N)6 R , satisfying v0 (∅ ) = 0 ,
is called an interval characteristic function. The set of all
games with interval characteristic function on P(N) is
denoted by IG0 (N).
Definition 4: (Alparslan Gök et al., 2010) Let
v0 is said to be size monotonic if:
+
0
−
0
+
0
v0 ∈ IG0(N).
−
0
for any S0,T00 P(N) with T0fS0.
Property 1: Let a ,b ∈ R , there exists c ∈ R such that
a = b + c if and only if b + − b − ≤ a + − a − .
⇒
From a = b + c , we get
a + = b + + c + . Since c − ≤ c + we have:
a − = b− + c−
and
a + − a − = b + + c + − (b − + c − ) ≥ b + − b −
⇐
From
b+ − b− ≤ a + − a −
, we have:
a − − b− ≤ a + − b+
and c+ = a + − b+ , we get a − = b− + c− and
. Since c− ≤ c+ we know c = [c− , c+ ] is an
interval number, and a = b + c .
From Property 1, we know a size monotonic game,
given by Alparslan Gök et al. (2010), and a Hukuhara
difference existence game, introduced by Yu and Zhang
(2009), are equivalent.
Let
c− = a − − b−
a + = b + + c+
Definition 5: Let v0 ∈ ,
IG0(N), v0 is said to be convex if:
In this section, we shall study two special kinds of
type-2 interval games, which can be seen the extensions
of fuzzy games introduced by Owen (1972) and Tsurumi
et al. (2001). The fuzzy coalition values for these two
kinds of fuzzy games are written as:
v (U ) =
⎫⎪
⎧⎪
⎨ ∏ U (i ) ∏ (1 − U (i )) ⎬v0 (T0 )
⎪⎭
i ∈SuppU \ T0
T0 ⊆ SuppU ⎪
⎩ i ∈T0
∑
(1)
v (U ) = ∑ ql =(1U ) v0 ([U ]hl )(hl − hl − 1 )
(2)
where U is a fuzzy coalition as usual, and T0 is a crisp
coalition. Q (U) = {U(i)| U(i) > 0, i0 N} and q(U) =
|Q(U)|. The elements in Q (U) are written in the increasing
order as 0 = h0#h1#...# h q(U) and [U ]h = {i| U(i) ≥ hl, i0N,
l
l = 1,2, ..., q(U)}. v0 is a crisp game defined in N.
An interval fuzzy coalition
v (T0 ) − v (T0 ) ≤ v ( S 0 ) − v ( S 0 )
Proof:
S0 , T0 ⊆ N .
{
} ⎩[
[
]
S = S (i )i ∈N = ⎧⎨ Si− , Si+
]
⎫
⎬
i ∈N ⎭
on
N is an n-dimension vector, where Si− , Si+ ⊆ [0,1] for any
i0 N. The set of all interval fuzzy coalitions on N is
denoted by IL(N). For any S ∈ IL(N) and player i, S (i )
indicates the interval membership grade of i in S , i.e., the
interval rate of the ith player in S . For any S ∈ IL(N), the
support S − = {Si− }i ∈N is denoted by Supp S ={i 0 N |S-i >
0} and the cardinality is written as |Supp S |, which is the
same as S+ = {S+i}i 0 N. Namely, the support S+ = {S+i}i 0 N
is denoted by Supp S+= {i 0 N |S+i>0} and the cardinality
is written as |Supp S+|. We use the notation S ⊆ T if and
only if S (i ) = T (i ) or S (i ) = 0 for all i0 N. For all S , T 0
IL(N), S ∨ T denotes the union of interval fuzzy coalitions
S and T , where ( S ∨ T ) (i) = S (i) ∨ T (i) for any i0 N,
S ∧ T denotes the intersection of interval fuzzy coalitions
S and T , where ( S ∧ T ) (i) = S (i ) ∧ T (i ) for any i0 N.
A function v : IL ( N ) → R+ , satisfying v (∅ ) = 0 , is
called a type-2 interval characteristic function. The set of
all games with type-2 interval characteristic function on
IL (N) is denoted by IG (N).
For any S ∈ IL (N), v ( S ) = [v − ( S − ), v + ( S + )] denotes the
interval payoff of S , where v!(S!) and v+(S+) denote the
left and right extreme points, respectively.
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Res. J. Appl. Sci. Eng. Technol., 4(10): 1334-1342, 2012
Definition 6: Let v ∈ IG (U ),
v in U if:
v (S ∧ T ) = v (S )
T
⎧ ⎡
⎤
⎪ ⎢
⎥
CO ( vO ,U ) = ⎨ x | ⎢
xi− ,
xi+ ⎥ =
⎪ ⎢ i ∈SuppU −
i ∈SuppU + ⎥⎦
⎩ ⎣
is said to be a carrier for
∑
∀S ⊆ U
Definition 7: Let v ∈ IG (U ),
v
[v
−
−
+
+
O (U ), vO (U )
is said to be convex if:
[v
v ( S ∨ T ) + v ( S ∧ T ) ≥ v ( S ) + v (T )
=
−
+
+
O ( S ), vO ( S )
for any S , T ∈ U .
Similar to population monotonic allocation schemes
given by Sprumont (1990) for crisp case, we give the
following concept for type-2 interval games.
Definition 8: Let
∑
v ∈ IG (U ),
C
[
C
[ xi− , xi+ ] = 0 ∀ i ∈ N \ SuppU +
∑ x+i+ ⎥⎥ ≥
i ∈SuppS
⎥⎦
], ∀ S ⊆ U }
⎧ ⎡
⎤
⎪
CO (vO ,U ) = ⎨ x | ⎢⎢
xi− ,
xi+ ⎥⎥ =
⎪ ⎢ i ∈SuppU −
i ∈SuppU + ⎥⎦
⎩ ⎣
∑
cSuppU+}is said to be an imputation for v in U if
[ xi− , xi+ ] ≥ v (U (i )) ∀ i ∈ SuppU − ∪ SuppU +
⎢⎣ i ∈SuppS −
⎤
xi− ,
Theorem 1: Let vO ∈ IGO (U ) , if the associated game
v0 ∈ IG0 ( N ) of vO is convex, then CO (vO ,U ) ≠ ∅ , and can be
expressed by:
the vector x ={[x-I, x+i]i0SuppUG
C
⎡
], ⎢⎢ ∑
∑
⎡
⎧
⎞⎫
⎪
⎪
⎢
∑ − ⎨ ∏ U − (i ) ∏ − (1 − U − (i)⎟⎟ ⎬ yT−0
⎢
⎪
i ∈SuppU \ T0
⎠ ⎪⎭
⎢⎣ T0 ⊆ SuppU ⎩ i ∈T0
∑ xi− , ∑ xi+ ] = v (U )
,
i ∈SuppU − i ∈SuppU +
⎧
⎪
+
⎨ ∏ U (i )
∏ 1 − U + (i )
+ ⎪ i ∈S
i ∈SuppU + \ S0
S0 ⊆ SuppU ⎩
0
TYPE-2 INTERVAL GAMES WITH
MULTILINEAR EXTENSION FORM
⎥⎦
,
∀ yS+0 ∈ C0 (v0+ , S0 ), ∀ S0 ⊆ SuppU +
}
Proof: Let
⎡
⎫
⎧
⎪
⎪
⎢
=⎢
U − (i )
(1 − U − (i )) ⎬v0− (T0 )
⎨
⎪
−
⎢ T0 ⊆ SuppU − ⎪⎩ i ∈T0
i ∈SuppU \ T0
⎭
⎣
∏
⎭
⎤
+ ⎥
S0 ⎥
where C0( vG0,T0) and C0( v+0,S0) denote the core for vG0
in T0 and for v+0 in S0, respectively.
−
+
[vO
(U − ), vO
(U + )]
∑
⎫
)⎪⎬⎪ y
∀ yT−0 ∈ C0 (v0− , T0 ), ∀ T0 ⊆ SuppU − ,
Similar to Eq. (1), we give the interval values of
interval fuzzy coalitions in type-2 interval games with
multilinear extension form as follows:
vO (U ) =
(
∑
∏
⎤
⎧
⎫
⎪
⎪
∑ + ⎨ ∏ U + (i ) ∏ U+ + (1 − U + (i ))⎬ v0+ (T0 )⎥⎥
⎪⎭
T0 ⊆ SuppU ⎪
i ∈SuppU \ T0
⎥⎦
⎩ i ∈T0
−
CO− (vO
,U − ) = {x − = {xi− }i ∈SuppU − |
−
vO
(U − ),
(3)
∑
xi−
i ∈SuppT −
∑
xi−
i ∈SuppU −
=
−
≥ vO
(T − ), ∀ T − ⊆ U − }
(4)
and
+
CO+ (vO
,U + ) = {x + = {xi+ }i ∈SuppU + |
where U ∈ IL(N)
By IGO (U ) , we denote the set of all type-2 interval
games with multilinear extension form on U ∈ IL( N ) .
If there is not special explanation, we always mean the
associated interval game v0 ∈ IG0(N) of vO ∈ IGO (U ) is size
monotonic.
+
vO
(U + ),
∑
xi+
i ∈SuppS +
∑
xi+
i ∈SuppU +
+
≥ vO
( S + ), ∀ S + ⊆ U + }
=
(5)
From Proposition given by Yu and Zhang (2009), we
know Eq. (4) is equivalent to the following expression
Definition 9: Let vO ∈ IGO (U ) , the interval core
CO (vO , U ) for v O in U is defined by:
1336
{x
−
= {xi− }
i ∈SuppU −
|
∑
xi−
i ∈SuppU −
=
Res. J. Appl. Sci. Eng. Technol., 4(10): 1334-1342, 2012
⎧
⎫
⎪
⎪ −
−
−
U
(
i
)
(
1
−
U
(
i
))
⎨
⎬ yT0 ,
⎪
⎪
,
i ∈SuppU − \ T0
T0 ⊆ SuppU − ⎩ i ∈T0
⎭
∑
∏
∏
∀ yT−0 ∈ C0 (v0− , T0 ), ∀ T0 ⊆ SuppU −
+
= {xi+ }
i ∈SuppU
∑
S0 ⊆ SuppU
+
+
|
Definition
−
}
v O ∈ IGO (U ) ,
the
function
if it satisfies:
T
is a carrier for v O in
U,
⎡
⎤
⎢ ∑ fi − (vO− ,U − ), ∑ fi + (vO+ ,U + ) ⎥
⎢ i ∈SuppT −
⎥
i ∈SuppT +
⎣
⎦
)
−
SuppU !
(
)
+
SuppU !
Then ϕ (vO ,U ) is the unique interval Shapley function for
vO in U .
Proof. (Existence) Axiom 1: Since T is a carrier for
vO in U we have:
−
+
−
+
[vO
( S − ∧ T − ), vO
( S + ∧ T + )] = [vO
( S − ), vO
( S + )]
then
=[vGO(TG),v+O(T+)]
(
SuppS − ! SuppU − − SuppS − − 1 !
SuppS + ! SuppU + − SuppS + − 1 !
and βU+ + =
f : IGO (U ) → R + is said to be an interval Shapley function
Axiom 1: If
(7)
where βUS − =
i ∈SuppU
Let
(v O+ ( S + ∨ U + (i )) − v O+ ( S + ))
œi0SuppU+
⎫
⎧
⎪
⎪
+
⎨ ∏ U (i )
∏ +(1 − U + (i ))⎬ yS+0
i ∈SuppU \ S0
⎭⎪
⎩⎪ i ∈S0
10:
S+
U+
i ∉SuppS +
∑ xi++ =
∀ yS+0 ∈ C0 (v0+ , S0 ), ∀ S0 ⊆ SuppU +
S + ⊆U +
}
and Eq. (5) is equivalent to the following expression:
{x
∑β
ϕi+ (v O+ , U + ) =
for any
S ⊆ U,
and
−
+
[vO
( S − ∨ U − (i )), vO
( S + ∨ U + (i ))]
Axiom 2: For any I, j 0 SuppU+, if we have:
= [vO− ( S − ∨ U − (i )) ∧ T − , vO+ ( S + ∨ U + (i )) ∧ T + )]
vO ( S ∨ U (i )) = vO ( S ∨ U ( j ))
for any
S ⊆U
= [vO− ( S − ), vO+ ( S + )]
with I, j ó SuppS+, then:
for any i 0 SuppU+\SuppS+.
Hence, we have:
f i (vO ,U ) = f j (vO ,U )
[vGO (TG), v+O (T+)]
Axiom 3: For any vO , ωo ∈ IGO (U ), if we have:
[v!O (UG v TG), v+O (U+ v T+)]
(vO + wO )( S ) = vO ( S ) + wO ( S )
for any
S ⊆U
= [v!O (U-), v+O (U+)]
then:
=[
(
f (vO + w0 ,U ) = f (vO ,U ) + f w0 ,U
)
=[
Theorem 2: Let vO ∈ IGO (U ) , define the function:
ϕ = [ϕ , ϕ ]: IGO (U ) → R + as follows:
ϕi− (vO− ,U − ) =
∑ βUS
−
S ⊆U
−
−
−
−
−
(vO
( S − ∨ U − (i )) − vO
( S − ))
∑
∑
∑
ϕi− (vO− ,U − ),
ϕi+ (vO+ ,U + )]
i ∈SuppT −
i ∈SuppT +
From vO ( S ∨ U (i )) = vO ( S ∨ U ( j )) , we obtain:
−
+
[vO
( S − ∨ U − (i )), vO
( S + ∨ U + (i ))]
œi0SuppU!(6)
i ∉SuppS −
and
∑
ϕi− (vO− ,U − ),
ϕi+ (vO+ ,U + )]
i ∈SuppU −
i ∈SuppU +
= [v O− ( S − ∨ U − ( j )), v O+ ( S + ∨ U + ( j ))]
Thus, we have:
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Res. J. Appl. Sci. Eng. Technol., 4(10): 1334-1342, 2012
vo− ( S − ∨ U − (i )) − vo− ( S − ) = vo− ( S − ∨ U − ( j )) − vo− ( S − )
ϕ (vO ,U ) ∈ CO (vO ,U ) .
vo− ( S − ∨ U − (i ) ∨ U − ( j )) − vo− ( S − ∨ U − ( j ))
= vo− ( S − ∨ U − (i ) ∨ U − ( j )) − vo− ( S − ∨ U − (i ))
vo+ ( S + ∨ U + (i )) − vo+ ( S + ) = vo+ ( S + ∨ U + ( j )) − vo+ ( S + )
Proof: From Theorem 2, we have:
∑ ϕi− (vi− ,U − ), ∑ ϕi+ (vi+ ,U + )]
[
and
i ∈SuppU −
vo+
=
+
+
+
( S ∨ U (i ) ∨ U ( j )) −
vo+
+
+
vo+
+
( S ∨ U ( j ))
−
+
= [vO
(U − ), vO
(U + )]
( S ∨ U (i ) ∨ U ( j )) − vo+ ( S + ∨ U + (i ))
+
From Eq. (6) and (7), it is not difficult to get Axiom2;
It is obvious Axiom 3 holds; (Uniqueness) Since can be
uniquely expressed by:
vO =
∑c
i ∈SuppU +
+
From the convexity of
v O is convex. Thus:
v0 ∈ IG0 ( N )
and Eq. (3), we get
vO ( S ∨ U (i )) − vO ( S ) ≥ vO (T ∨ U (i )) − vO (T )
u
S S
∅≠S ⊆0
for any
for any vO ∈ IGO (U ), where
S ,T − ⊆ U
with
T ⊆ S
and i ∉ SuppS +
From Eq. (6) and (7), we have:
⎧1 S ⊆ T
uS (T ) = ⎨
⎩ 0 otherwise
ϕi− (vO− , S − ) ≥ ϕi− (vO− , T − ) ∀ i ∈ SuppT −
and
and
cS = [
∑ ( − 1)
SuppS − − SuppT−
T−⊆S−
∑ (− 1)
SuppS + − SuppT +
ϕi+ (vO+ , S + ) ≥ ϕi+ (vO+ , T + ) ∀ i ∈ SuppT +
vO− (T − )
where S , T ⊆ U with
Hence, we have:
vO+ (T + )]
T+⊆S+
From Axiom 3, we only need to show the uniqueness of
ϕ on unanimity game uS , where ∅ ≠ S ⊆ U . From Axiom
1, we get:
ϕi− (vO− , U − ) ≥ ϕi− (vO− , S − )
∀ i ∈ SuppS −
and
1 = uS ( S )
=[
T ⊆ S
ϕi+ (vO+ , U + ) ≥ ϕi+ (vO+ , S + ) ∀ i ∈ SuppS +
∑
∑
ϕi− (uS ,U − ),
ϕi+ (uS ,U + )]
i ∈SuppS +
i ∈SuppS −
where S ⊆ U .
Namely,
From Axiom 2, we obtain:
⎧
⎪
1
⎪
⎩
0
ϕi− (uS ,U − ) = ⎨ SuppS −
[
∑
∑
ϕi− (vO− , U − ),
ϕi+ (vO+ , U + )]
i ∈SuppS −
i ∈SuppS +
i ∈ SuppS −
≥[
otherwise
and
∑
∑
ϕi− (vO− , S − ),
ϕi+ (vO+ , S + )]
i ∈SuppS −
i ∈SuppS +
−
+
= [vO
( S − ), vO
( S + )]
⎧
⎪
1
⎪
⎩
0
ϕi+ (uS ,U + ) = ⎨ SuppS +
i ∈ SuppS +
for any
S ⊆U
.
otherwise
Theorem 3: Let vO ∈ IGO (U ) , if the associated interval
game v0 ∈ IG0 ( N ) of
vO is convex, then
Corollary 1: Let vO ∈ IGO (U ) , if the associated interval
game v0 ∈ IG0 ( N ) of v O is convex, then ϕ (vO ,U ) is an
imputation for vO in U .
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Res. J. Appl. Sci. Eng. Technol., 4(10): 1334-1342, 2012
Definition
11:
vO ∈ IGO (U ) ,
Let
x = {[ xi− , xi+ ]i ∈SuppU − ∪ SuppU + } is
the
CC (vC ,U ) = {x |[
vector
said to be an IPMAF for
[vC− (U − ),[vC+ (U + )],[
vO in U if:
C
∑
i ∈SuppS − ∪ i ∈SuppS +
xi = [
∑
∑
xi−
xi+ ]
i ∈SuppS − i ∈SuppS +
C
∑
v0 ∈ IG0 ( N )
game
∀S ⊆ U
of
v C is convex, then
xi+ (T )]
CC (vC ,U ) = {x |[
∑
i ∈SuppU −
[
Theorem 4: Let vO ∈ IGO (U ) , if the associated interval
game v0 ∈ IG0 ( N ) of vO is convex, then ϕ (vO ,U ) is an
IPMAF for vO in U .
,
y[−U − ] (hl
hl
∑
xi+ ]
i ∈SuppU +
=
q (U + )
− h− 1),
∑ y[+U
l =1
+
]hl
(hl − h−1)]
∀ y[−U − ] ∈ C(v0− [U − ]hl ), ∀ l = 1,2,..., q (U − ) ,
hl
∀ y+
[U + ]hl
∈ C (v0− ,[U − ]hl ), ∀ l = 1, 2, ..., q ([U − ]}
−
v 0− in [U ]hl and for
Similar to Eq. (2), we give the values of interval
fuzzy coalitions in type-2 interval games with Choquet
integral form as follows:
vC (U ) = [vC− (U − ), vC+ (U + )]
⎤
− hl − 1) ⎥
⎥
⎦
xi− ,
+
+
where C(v0− ,[U − ]hl ) and C(v0 ,[U ]hl ) denote the core for
TYPE-2 INTERVAL GAMES WITH CHOQUET
INTEGRAL FORM
⎡q (U − )
= ⎢ ∑ v0− ([U − ]hl (hl − hl −1 )
⎢ l =1
⎣
∑
l =1
Proof: From Theorem 2, we know the first condition in
Definition 11 holds. From Theorem 3, we get the second
condition in Definition 11.
l =1
CC ( vC ,U ) ≠ ∅
s.t. S ⊆ T
q (U − )
∑
∑
xi− ,
xi+ ,]}
i ∈SuppS − i ∈SuppS +
and can be expressed by:
∀ i ∈ SuppS − ∪ SuppS + , ∀ S , T ⊆ U
v0+ ([U + ]hl )(hl
=
Theorem 5: Let vC ∈ IGC (U ) , if the associated interval
xi ( S ) = [ xi− ( S ), xi+ ( S )] ≤ xi (T ) = [ xi− (T )],
q (U + )
∑
≥ [vC− ( S − ), vC+ ( S + )], ∀ S ⊆ U }
−
+
= [vO
( S − ), vO
( S + )]
= vO ( S )
∑
xi− ,
xi+ ,]
i ∈SuppU − i ∈SuppU +
respectively.
Proof: From Theorem 1 and by Yu and Zhang (2009), we
can easily get the conclusion.
Definition
13:
f : IGC (U ) → R + is
,
v 0+ in [U + ]hl ,
Let
vC ∈ IGC (U )
,
the
function
said to be an interval Shapley function
if it satisfies:
C
Axiom 1: If
T
is a carrier for
in
vC
U
, then:
(8)
where U ∈ IL(N), Q (UG) = {U- (i) |U- (i) > 0, i0 N} and
q(U-) = |Q (U+)| and Q (U+) = {U+(i) | U+(i) > 0, i0 N} and
q (U+) = Q(U+)|.
By IGC (U ), we denote the set of all type-2 interval
games with Choquet integral form on U ∈ IL (N). Similar
to IGO (U ) , if there is not special explanation, we always
mean the associated interval game v 0 ∈ IG0 (N) of
vC ∈ IGC (U ) is size monotonic.
Definition 12: Let vC ∈ IGC (U ) , the interval core
CC (vC ,U ) for vC in U is defined by:
⎤
⎡
⎢ ∑
f − (v − ,U − ), ∑ f i + (vC+ ,U + ) ⎥
⎥
⎢ i ∈SuppT − i C
+
i ∈SuppT
⎦
⎣
C
= [vC− (T − ), vC+ (T + )]
Axiom 2: Let l0{1, 2, ... ,q(U)}and i, j,0 [U ] h , if
l
we have v0 ( S0 Ui ) =
with i , j ∉ S0 , then:
v0 ( S0 U j )
for
any S0 ⊆ [U ]hl
f i ([U ]hl , v0 ) = f j ([U ]hl , v0 )
Axiom 3: For any
1339
vC , wC ∈ IGC (U )
(vC + wC )( S ) = vC ( S ) + wC ( S )
for any
, if we have
S ⊆ U,
then:
Res. J. Appl. Sci. Eng. Technol., 4(10): 1334-1342, 2012
v0− ([T − ]hl ) − v0− ([U − ]hl )
f (vC + wC ,U ) = f (vC ,U ) + f ( wC ,U )
Theorem 6: Let vC ∈ IGC (U ) , define the function
φ = [φ − , φ + ]: IGC (U ) → R + as follows:
∑φ
−
i
l =1
φi− (v0− ,([U − ]hl )
i ∈[U − ]hl
=
∑ φi− (v0− ,([U − ]h )
l
i ∈[ T − ]hl
and
q (U − )
φi− (v C− ,U − ) =
∑
=
(v 0− ,[U − ]hl )(hl − h−1 )
∀ i ∈ SuppU −
v0+ ([T + ]hl ) = v0+ ([U + ]hl )
∑+ φi+ (v0+ ,([U + ]hl )
=
(9)
i ∈[U ]h l
∑+ φi+ (v0+ ,([U + ]hl )
=
and
i ∈[ T ]h l
q (U + )
∑φ
φi+ (vC+ ,U + ) =
+
i
l =1
(v0+ ,[U + ]hl ) (hl − h−1 )
œ i 0 Supp U+
From Eq. (9), we get:
(10)
vC− (T − ) =
q (T − )
∑ v0− ([T − ]h )(hl − h−1)
l
l =1
where
q (U − )
=
φi− (v0− ,[U − ]hl ) =
∑
β S0 −
[U
S0 ⊆ [U − ]hl ,i ∉S0
∑ v0− ([T − ]h )(hl − h−1)
l
l =1
]hl ( v0− ( S0 ∪ i ))
q (U − )
=
∑ v0− ([U − ]h )(hl − h−1)
l
l =1
− v0− ( S0 )) ∀ i ∈ [U − ]hl
=
q (U − )
∑
∑
+
φi+ (v0+ ,[U + ]hl ) =
S 0⊆ [U ]hl i ∉S0
[ ]
∀i ∈ U +
-v+0 (S0))
S0
[U + ]hl
β
∑
φ0− (v0− ,[U − ]hl )(hl
l = 1 i ∈[U − ]h
l
and
− h− 1)
q (U − )
(v0+ ( S0 ∪ i )
=
∑ ∑ φi− (v0− ,[U − ]h )(hl − h−1)
l
l = 1 i ∈[ T − ]h
l
=
hl
∑
q (U − )
∑ φi− (v0− ,[U − ]hl (hl − h−1)
i ∈SuppT − l = 1
β S0−
[U ]h
l
=
s!(|[U − ]hl |− s − 1)!
|[U − ]hl |!
=
∑ φi− (vC− ,U − )
i ∈SuppT −
and
β[SU0 + ]
hl
=
s!(|[U + ]hl |− s − 1)!
Similarly, we have:
|[U + ]hl |!
Then φ is the unique interval Shapley function for vC in U .
Proof. (Existence) Axiom 1: From Theorem 4 introduced
by Tsurumi et al. (2001), we know T is a carrier in U for
vC if and only if [T − ]hl is a carrier for v0− in [U − ]h , and
l
+
[T ]hl
is a carrier for v+0in [U + ]h , where l = 1, 2, ..., q (U).
∑ φi+ (vC+ ,U + ) = vC+ (T + )
i ∈SuppT +
From Eq. (9) and (10), we can easily get Axioms 2,
3. (Uniqueness) Since v0 can be uniquely expressed by:
v0 =
∑
cT0 uT0
∅ ≠ T0 ⊆ N
for any v 0 ∈ IG0(N), where
l
Thus, we have:
1340
⎧ 1 T0 ⊆ W0
uT0 (W0 ) = ⎨
⎩ 0 otherwise
Res. J. Appl. Sci. Eng. Technol., 4(10): 1334-1342, 2012
and
φi+ (vC+ , S + ) ≥ φi+ (vC+ , T + ) ∀ i ∈ SuppT +
⎡
⎤
t−s
t−s
cT0 = ⎢ ∑ ( − 1) v0− ( S0 ), ∑ ( − 1) v0+ ( S0 ) ⎥
⎢S ⊆T
⎥
S0 ⊆ T0
⎣ 0 0
⎦
of
where S ⊆ U .
Namely,
From Axiom 3, we only need to show the uniqueness
on unanimity games uS0 and uT0 , where
⎡
⎤
⎢ ∑ φi− (vC− ,U − ), ∑ φi+ (vC+ ,U + ) ⎥
⎢ i ∈SuppS −
⎥
+
i ∈SuppS
⎣
⎦
φ
i…S0f
[U ]
−
hl
and i… T0f [U + ]
hl
for any l0{1, 2,...,q(U)}.
From Axioms 1, 2, we get:
φi− (uS0 ,[U − ]hl )
⎧1
⎪
= ⎨s
⎩⎪ 0
⎡
⎤
≥ ⎢ ∑ φi− (vC− , S − ), ∑ φi+ (vC+ , S + ) ⎥
⎢ i ∈SuppS −
⎥
i ∈SuppS +
⎣
⎦
i ∈ S0
otherwise
and
= [vC− ( S − ), vC+ ( S + )]
⎧1
⎪
φi+ (uS0 ,[U + ]hl ) = ⎨ s
⎪⎩ 0
Theorem 7: Let
game v0 ∈ IG0 ( N ) of
i ∈ S0
otherwise
, if the associated interval
is convex, then φ (vC ,U ) ∈ CC (vC ,U ) .
vC ∈ IGC (U )
vC
for any S ⊆ U .
Similar to IGO (U ) , when the associated interval game
v0 ∈ IG0 ( N ) of vC is convex, we get:
Proof: From Theorem 6, we have:
{[φi− (vC− ,U − ).φi+ (vC+ ,U + )]i ∈SuppU − ∪ i ∈SuppU + }
⎡
⎤
⎢ ∑ φi− (vC− ,U − ), ∑ φi+ (vC+ ,U + ) ⎥
⎢ i ∈SuppU −
⎥
i ∈SuppU +
⎣
⎦
[
= vC− (U − ), vC+ (U + )
]
From the convexity of
convex. Thus:
v0 ∈ IG0 ( N ) and
Eq. (8), we get
vC
is
is an IPMAF for vC in U , and an imputation for vC in U .
Since the type-2 interval games in IGO (U ) and
IGC (U ) are continuity and monotone node creasing with
respect to the player participation levels, we know every
player interval Shapley value, obtained by Eq. (6) and (7)
or Eq. (9) and (10), is an interval number, where their
associated interval games are size monotonic.
vC ( S ∨ U (i )) − vC ( S ) ≥ vC (T ∨ U (i )) − vC ( T )
NUMERICAL EXAMPLE
for any S , T ⊆ U with T ⊆ S and i ó Supp S
From Eq. (9) and (10), we have:
+
φi− (vC− , S − ) ≥ φi− (vC− , T − )
∀ i ∈ SuppT −
and
φi+ (vC+ , S + ) ≥ φi+ (vC+ , T + ) ∀ i ∈ SuppT +
There are 3 companies cooperate to complete a
project. Namely, the set of players N = {1, 2, 3}. Since
there exist many uncertainty factors in the process of
development. The players only know the scope of the
crisp coalition values, which are given in Table 1.
Furthermore, the players are only sure the lower and
upper participation levels in this cooperation, which are
given by:
U (1) = [U − (1), U + (1)] = [0.3, 0.7]
where S , T ⊆ U with
Hence, we have:
φi− (vC− ,U − )
and
≥
T ⊆ S
φi− (vC− , S − )
.
∀ i ∈ SuppS
−
Table 1: The crisp coalitions interval values
S0
S0
v ( S0 )
{1}
[2, 4]
{1, 3}
{2}
[1, 3]
{2, 3}
{3}
[2, 3]
{1, 2, 3}
{1, 2}
[5, 10]
1341
v (S0 )
[6, 11]
[5, 9]
[12, 20]
Res. J. Appl. Sci. Eng. Technol., 4(10): 1334-1342, 2012
U (2) = [U − (2), U + (2)] = [0.6, 0.9]
U (3) = [U − (3),U + (3)] = [0.8, 1]
Then this is a type-2 interval game. When the fuzzy
coalition values and their associated crisp coalition values
have the relationship given in Eq. (3). Namely, this game
belongs to IGO (U ) . From Eq. (6) and (7), we get the player
interval Shapley values as are:
ϕ1 (vO ,U ) = [123
. , 5.22]
ϕ2 (vO ,U ) = [163
. , 513
. ]
ϕ3 (vO ,U ) = [2.63, 5.75]
when the fuzzy coalition values and their associated crisp
coalition values have the relationship given in Eq. (8).
Namely, this game belongs to IGC (U ) . From Eq. (9) and
(10), we get the player interval Shapley values as are:
φ1 (vC , U ) = [1068
.
, 5145
. ]
φ2 (vC , U ) = [1.308, 4.995]
φ3 (vC , U ) = [2.368, 5.75]
CONCLUSION
We have researched two special kinds of type-2
interval games, where the player participation levels and
the fuzzy coalition values are both interval numbers. The
research in his paper extension the learning scope of fuzzy
games, and can better applied in practical problems. But
we only discuss type-2 interval games, and it will be
interesting to research the other fuzzy games with type-2
fuzzy payoffs, which combine the operations of fuzzy
sets.
ACKNOWLEDGMENT
This study was supported by the National Natural
Science Foundation of China (Nos 70771010, 70801064
and 71071018).
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