INTRODUCTION
With the social development, game theory is used in many fields. Kalliny
and Gentry (2010) introduced the online games and applied it to advertising
practicesadvergaming and product placement. Lo (2008)
and Akramizadeh et al. (2009) studied multiagent
model by using Nash equilibrium. Cheheltani and Ebadzadeh
(2010) further researched the Nash equilibrium. Recently, Ayanzadeh
et al. (2011) discussed the mixed Nash equilibrium of honey bees
foraging optimization. Daghistani (2011) studied the
thinking skills of kindergarten children by using educational games. Isin
and Miran (2005) researched agriculture in Turkey by using game theory.
Since there are many uncertain factors during the cooperation of the players
and they can only know the possible payoffs for the cooperation. For this problem,
Mares (2000) and Mares and Vlach
(2001) studied games with fuzzy payoffs, where the coalitions are crisp
and the coalition values are fuzzy numbers. Fuzzy games of which the coalition
and the characteristic function are both fuzzy information are researched by
Borkotokey (2008). But the given Shapley function does
not satisfy the efficiency. Based on the Hukuhara difference on fuzzy numbers
(Banks and Jacobs, 1970). Yu and
Zhang (2010) concerned games with fuzzy payoffs, where the coalitions are
crisp and gave the socalled HukuharaShapley value. Furthermore, the authors
researched a special kind of fuzzy games with fuzzy payoffs which can be seen
an extension of fuzzy games given by Tsurumi et al.
(2005). Furthermore, the Shapley function for fuzzy games with Choquet integral
and fuzzy payoffs is researched by Meng and Zhang (2010).
Meng et al. (2012) researched the Banzhaf value
for fuzzy games with fuzzy payoffs. Dhar et al. (2011)
and Amiri et al. (2008) studied Utopian transport
scenario and multiple attribute decision making problems by using fuzzy sets,
respectively. Tan et al. (2011) researched IP
network traffic by introducing fuzzy decision mechanism. Furthermore, Chao
et al. (2010) researched type2 interval fuzzy immune controller.
Possibility and necessity measures play a key role in possibility theory. The
necessity and possibility constraints are introduced by Zadeh
(1978) and Dubois and Prade (1987, 1988)
which are very relevant to the reallife decision making problems and presented
the process of defuzzification for these constraints. Later, Liu
and Liu (2002, 2003) proposed the creditability
theory and gave the creditability measure. These three measures respect the
risk attitudes of the decision makers or the players for uncertainty information.
Based on above analysis, we shall research the fuzzy games with fuzzy payoffs
under possibility and necessity measures.
PRELIMINARIES
Here, we recall some basic concepts for possibility/necessity/credibility measures and give the model of fuzzy games under possibility and necessity measures.
Some concepts for possibility/necessity/credibility measures: Let us
start by recalling the most general definition of a fuzzy number. Let
be (∞, ∞), i.e., the set of all real numbers.
Definition 1: A fuzzy number, denoted by ,
is a fuzzy subset with membership function :
→[0,
1] satisfying the following conditions:
• 
is upper semicontinuous 
• 
There exists an interval number [a, d] such that
(x) = 0 for any x∉ [a, d] 
• 
There exist real numbers b, c such that a≤b≤c≤d and (i)
(x) is nondecreasing on [a, b] and nonincreasing on [c, d]; (ii) (x)
= 1 for any z∈[b, c] 
Definition 2: Let X be a nonempty set and P(X) be the power set of X. Pos: P(X)→[0, 1] is called possibility measure, if:
• 
Pos (ø) = 0, Pos(X) = 1 
• 
Pos
Pos (A_{i}), where, A_{i}∈P (X) for all 1≤i≤+∞ 
Definition 3: Let X be a nonempty set and P(X) be the power set of X. Nec: P(X)→[0, 1] is called necessity measure, if:
• 
Nec (ø) = 0, Nec(X) = 1 
• 
Nec
Nec (A_{i}), where A_{i}∈P (X) for all 1≤i≤+∞ 
Let
with membership function .
In the setting of creditability theory, the creditability measure for the fuzzy
event given as follows:
Some concepts for fuzzy games with fuzzy payoffs: Let N = {1, 2,…,n} be the set of the players. By L (N), we denote the set of all fuzzy coalitions in N. The fuzzy coalitions in L (N) are denoted by S, T, …. Let S∈L (N) and player i, S (i) indicates the membership grade of i in S, i.e., the rate of the ith player in S. For any S∈L(N), the support is denoted by SuppS = {i∈N  S (i)>0} and the cardinality is written as SuppS. We use the notation S⊆T if and only if S(i) = T(i) or S (i) = 0 for any i∈N. Let S, T∈L(N), S∨T denotes the union of fuzzy coalitions S and T, namely, i∈Supp (S∨T) if and only if i∈SuppS∪SuppT and (S∨T) (i) = S(i)∨T(i); S∧T denotes the intersection of fuzzy coalitions S and T, namely, i∈ Supp (S∨T) if and only if i∈SuppS∩SuppT and (S∧T) (i) = S(i)∧T(i).
In the following, we use S = {S (i_{1}), S (i_{2}), …,
S (i_{n})} to denote S∈L(N). A function : ,
satisfying (ø)
= 0, is called a fuzzy characteristic function. All fuzzy games with fuzzy number
payoffs on L (N) are denoted by .
We will omit braces for singletons, e.g., by writing S, S∨(∧)T, S (i)
instead of {S}, (S)∨(∧)(T), {S(i)} for any {S}, {T}, {S(i)}∈L(N).
Definition 4: Let
and U∈L (N), E_{p} ()
is defined by:
If E_{p} (
(S)) exists for any S⊆U, then Eq. 2 is called Possibility
Expectation (PE) for
in U.
Definition 5: Let
and U∈L (N), E_{ne} ()
is defined by:
If E_{Ne} (
(S)) exists for any S⊆U, then Eq. 3 is called Necessity
Expectation (NE) for
in U.
Remark 1: In this study, without loss of generality, for any
and U∈L (N), we always mean the Necessity and Possibility Expectations
(NPE) exist. Since NE and PE reflect the player risk preferences for the approximate
values of fuzzy coalitions, we shall use w_{i}∈[0, 1] to denote
the player i’s NE weight for the approximate value of U∈L (N) and
1w_{i} to denote the player i’s PE weight for the approximate
value of U∈L (N). By:
we denote the weight of S⊆U with respect to E_{Ne} ( (S))
and W^{S}_{P} = 1W^{S}_{N} indicates the weight
of S⊆U with respect to E_{P} ( (S)).
Definition 6: Let
and U∈L (N),
is said to be weighted NPE superadditivity if we have:
And:
for any S, T⊆U with .
Remark 2: If there is no special explanation, for any ,
we always mean
is weighted NPE supperadditive.
Definition 7: Let
and U∈L (N),
is said to be weighted NPE convex if we have:
And:
for any S, T⊆U.
Definition 8: Let
and U∈L (N), T⊆U is said to be a weighted NPE carrier for in U, if it satisfies
for any S⊆U.
Definition 9: Let
and U∈L (N), C (E_{NP} (),
U) is said to be the weighted NPE fuzzy core of in U which is defined by:
Definition 10: Let
and U∈L (N), the vector y = (y_{i})_{i∈SuppU} is said
to be a weighted NPE participation monotonic allocation schemes for
in U if it satisfies:
• 

• 
y_{i} (S)≤y_{i} (T) ∀ i∈SuppS,
∀S, T⊆U s.t. S⊆T 
Definition 11: Let
and U∈L (N), the function is
said to a weighted NPE Shapley function for in U it satisfies the following
axioms:
Axiom 1 (weighted NPE efficiency): If T⊆U is a weighted NPE carrier for in U, then we have:
where, W^{U}_{Ne} and W^{U}_{p} are the weights
of U with respect to E_{Ne} ( (U))
and E_{P} ( (U)),
respectively.
Axiom 2 (weighted NPE symmetry): For any i, j∈SuppU and any given weight W^{U}_{Ne}, if we have:
And:
for any S⊆U with i, j∉SuppS . Then, we have:
Axiom 3 (weighted NPE additivity): Let ,
if we have:
And:
for any S⊆U.
Then we have:
And:
THE WEIGHTED NPE SHAPLEY FUNCTION FOR FUZZY GAMES WITH FUZZY PAYOFFS
Let
and U∈L (N), we give the weighted NPE Shapley function for
in U as follows:
Where:
And:
for any i∈SuppU and:
Theorem 1: Let ,
U∈L(N) and any given weight W^{U}_{N}, then the function
,
given in Eq. 4, is the unique weighted NPE Shapley function
for
in U.
Proof
⇒ Axiom 1: Since T⊆U is a weighted NPE carrier, we have:
where, i∉SuppT.
From Eq. 5, we get φ^{N}_{i} (E_{N}
(),
U) = 0 for any i∈SuppU\SuppT. Thus, we have:
Similarly, we obtain:
From Eq. 4, we get:
Axiom 2: From Eq. 5, we obtain:
where:
Similarly, we have φ^{P}_{i} (E_{P} (),
U). Thus, φ_{i} (E_{NP} (),
U) = φ_{j} (E_{NP} (),
U).
From Eq. 35, we can easily get Axiom 3.
Uniqueness: Let
and U∈L (N), it is not difficult to show E_{Ne} = ()
can be expressed by:
where:
and u_{s} is a weighted NE unanimity game defined by:
and E_{P} ()
can be expressed by:
where:
and E_{P} (u_{s}) is a weighted PE unanimity game given as:
From Axiom 3 and Eq. 3, we only need to show the uniqueness of φ^{N} and φ^{P} on unanimity games, respective.
For any u_{T} with ø≠T⊆U, since T is a weighted NPE carrier for u_{T}, from Axiom 1, we have:
And:
From Axiom 2, we get:
And:
Namely, φ^{N} and φ^{P} are unique in the unanimity games and the proof is finished.
Theorem 2: Let
and U∈L (N), if
is NPE convex, then we have φ (E_{NP} (),
U)∈C (E_{NP} (),
U).
Proof: From Theorem 1, we only need to show:
for any S⊆U.
From the NPE convexity of , we get:
And:
for any S, T⊆U with S⊆T, where i∉SuppT.
From Eq. 5, 6, we obtain:
And:
for any i∈SuppS.
Thus:
From Theorem 1, we obtain:
Property 1: Let
and U∈L(N), if
is NPE convex, then n (E_{NP} (),
U) is a NPE participation monotonic allocation schemes.
Proof: From Theorem 1 and 2, we can easily get the result.
Property 2: Let
and U∈L (N), if T⊆U is a weighted NPE carrier for
in U, then we have φ_{i} (E_{NP} (),
U) = φ_{i} (E_{NP} (),
T) for any i∈SuppU.
Proof: From Theorem 1, we get:
for any i∈SuppU\SuppT.
When i∈SuppT. The property is proved by induction on SuppT.
Case 1: When SuppU = SuppT+1, without loss of generality, suppose SuppU = SuppT∪k. From Eq. 5, we get:
where, α_{U}^{S} and α_{U}^{S+1} as given in Theorem 1 and:
When, SuppU = SuppT+q, without loss of generality, suppose SuppU = SuppT∪{k_{1}, k_{2},…, k_{q}}. Let SuppT_{1 }= SuppT∪{k_{1}}, SuppT_{2 }= SuppT_{1}∪{k_{2}},…, SuppT_{q }= SuppT_{q1}∪{k_{q}}.
From above, we have:
for any i∈SuppT.
Similarly, we get:
Thus, we obtain:
Property 3: Let
and U∈L (N), if we have:
And:
for any S⊆U with i∉SuppS.
Then:
Proof: From Eq. 5, we get:
Similarly, we have:
Thus:
Corollary 1: Let
and U∈L (N), if we have:
And:
for any S⊆U with i∉SuppS. Then:
From Eq. 13, we know the Credibility Expectation (CE) is equal to:
When each player’s weight for the approximate value of U∈L (N) is
0.5, then we have W^{S}_{N} = W^{S}_{P} = 0.5
for any S⊆U with respect to E_{Ne} ( (S))
and E_{P} ( (S)).
From Eq. 4, we get the weighted CE Shapley function for
in U as follows:
where:
And:
for any i∈SuppU and α^{S}_{U} as given in Eq.
5, 6.
Equation 9 and 10 are called the NE Shapley
function and the PE Shapley function for
in U, respectively.
Similar to the above given definitions, we can get their concepts with respect
to credibility expectation. Furthermore, all above given theorems and properties
still hold for the weighted CE Shapley function.
NUMERICAL EXAMPLE
With the increasing competition among manufacturers, there are three electrical appliances enterprises, named 1, 2 and 3, decide to cooperate with their resources. They can cooperate freely. For example, S_{0 }= {2, 3} denotes the cooperation of the enterprises 2 and 3. Since there are many uncertain factors during the process of cooperation, it is impossible for the player to know the accurate payoffs of the coalitions. Here, we use the trapezoidal fuzzy numbers to denote the possible payoffs (millions RMB) of the crisp coalitions which are given by Table 1.
From Table 1, we know when the companies 1 and 2 cooperates with all their resources, their fuzzy payoff is (6, 9, 13, 18) millions RMB.
As above pointed, since there are many uncertain factors during the process of cooperation, each company is not willing to offer all its resources to a particular cooperation. Thus, we have to consider a fuzzy game. For example, the company 1 has 1000 unit resources and it supplies only 100 units to cooperate, then we think the 1th player's participation level is 0.1 = 100/1000. In such a way, a fuzzy coalition is explained. Consider a fuzzy coalition U defined by U (1) = 0.6, U(2) = 0.8 and U(3) = 0.3.
When the fuzzy coalition values and that of their associated crisp coalitions have the relationship:
Namely, this is a fuzzy game with multilinear extension form and fuzzy payoffs.
Table 1: 
The fuzzy payoffs of the crisp coalitions 

Table 2: 
The possibility expectations of the fuzzy coalitions 

Table 3: 
The necessity expectations of the fuzzy coalitions 

From Eq. 2 and 11, we get the possibility
expectations of the fuzzy coalitions as given in Table 2.
From Eq. 3 and 11, we get the necessity
expectations of the fuzzy coalitions as given in Table 3.
Let:
for any S⊆U, where w_{i} denotes the weight of the i’s with respect to NE. If the players 1, 2 and 3 are risk averse, risk pursuit and risk neutral, respectively. Namely, w_{1} = 1, w_{2} = 0 and w_{3} = 0.5. From Eq. 4, we get the player weighted NPE Shapley values are:
• 
φ_{1} (E_{NP} (),
U) = 2.85 
• 
φ_{2} (E_{NP} (),
U) = 5.94 
• 
φ_{3} (E_{NP} (),
U) = 2.49 
CONCLUSIONS
We have researched a general case of fuzzy games with fuzzy payoffs under possibility and necessity measures which can be used in all kinds of games with fuzzy payoffs, where the possibility and necessity expectations exist. Since the possibility and necessity measures reflect the players’ risk attitudes, we give the weighted NPE Shapley function for fuzzy games with fuzzy payoffs. Some properties are also discussed.
However, we only research the weighted NPE Shapley function for fuzzy games with fuzzy payoffs and it will be interesting to study other payoff indices for this kind of fuzzy games under possibility and necessity measures.
ACKNOWLEDGMENT
This study was supported by the National Natural Science Foundation of China (Nos. 70771010, 70801064 and 71071018).