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Fuzzy Games under Possibility and Necessity Expectations



Fanyong Meng, Hao Cheng and Feng Liu
 
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ABSTRACT

In this study, fuzzy games with fuzzy payoffs that consider the player risk preferences are studied. Based on the possibility and necessity expectations, the Shapley function for this kind of fuzzy games is researched. An axiomatic system of the given Shapley function is defined. Meantime, some properties are also discussed which coincide with the classical case. Finally, a numerical example is given to explain the player Shapley values for fuzzy games under possibility and necessity expectations.

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  How to cite this article:

Fanyong Meng, Hao Cheng and Feng Liu, 2012. Fuzzy Games under Possibility and Necessity Expectations. Information Technology Journal, 11: 1024-1031.

DOI: 10.3923/itj.2012.1024.1031

URL: https://scialert.net/abstract/?doi=itj.2012.1024.1031
 
Received: January 12, 2012; Accepted: March 03, 2012; Published: June 05, 2012



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 practices-advergaming and product placement. Lo (2008) and Akramizadeh et al. (2009) studied multi-agent 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 so-called Hukuhara-Shapley 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 type-2 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 real-life 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 Image for - Fuzzy Games under Possibility and Necessity Expectations be (-∞, ∞), i.e., the set of all real numbers.

Definition 1: A fuzzy number, denoted by Image for - Fuzzy Games under Possibility and Necessity Expectations, is a fuzzy subset with membership function Image for - Fuzzy Games under Possibility and Necessity Expectations: Image for - Fuzzy Games under Possibility and Necessity Expectations→[0, 1] satisfying the following conditions:

Image for - Fuzzy Games under Possibility and Necessity Expectations is upper semi-continuous
There exists an interval number [a, d] such that Image for - Fuzzy Games under Possibility and Necessity Expectations (x) = 0 for any x∉ [a, d]
There exist real numbers b, c such that a≤b≤c≤d and (i) Image for - Fuzzy Games under Possibility and Necessity Expectations (x) is nondecreasing on [a, b] and nonincreasing on [c, d]; (ii) Image for - Fuzzy Games under Possibility and Necessity Expectations(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 Image for - Fuzzy Games under Possibility and Necessity Expectations Pos (Ai), where, Ai∈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 Image for - Fuzzy Games under Possibility and Necessity Expectations Nec (Ai), where Ai∈P (X) for all 1≤i≤+∞

Let Image for - Fuzzy Games under Possibility and Necessity Expectations with membership function Image for - Fuzzy Games under Possibility and Necessity Expectations. In the setting of creditability theory, the creditability measure for the fuzzy event given as follows:

Image for - Fuzzy Games under Possibility and Necessity Expectations
(1)

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 (i1), S (i2), …, S (in)} to denote S∈L(N). A function : Image for - Fuzzy Games under Possibility and Necessity Expectations, satisfying Image for - Fuzzy Games under Possibility and Necessity Expectations(ø) = 0, is called a fuzzy characteristic function. All fuzzy games with fuzzy number payoffs on L (N) are denoted by Image for - Fuzzy Games under Possibility and Necessity Expectations. 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 Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), Ep (Image for - Fuzzy Games under Possibility and Necessity Expectations) is defined by:

Image for - Fuzzy Games under Possibility and Necessity Expectations
(2)

If Ep ( Image for - Fuzzy Games under Possibility and Necessity Expectations (S)) exists for any S⊆U, then Eq. 2 is called Possibility Expectation (PE) for Image for - Fuzzy Games under Possibility and Necessity Expectations in U.

Definition 5: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), Ene (Image for - Fuzzy Games under Possibility and Necessity Expectations) is defined by:

Image for - Fuzzy Games under Possibility and Necessity Expectations
(3)

If ENe ( Image for - Fuzzy Games under Possibility and Necessity Expectations (S)) exists for any S⊆U, then Eq. 3 is called Necessity Expectation (NE) for Image for - Fuzzy Games under Possibility and Necessity Expectations in U.

Remark 1: In this study, without loss of generality, for any Image for - Fuzzy Games under Possibility and Necessity Expectations 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 wi∈[0, 1] to denote the player i’s NE weight for the approximate value of U∈L (N) and 1-wi to denote the player i’s PE weight for the approximate value of U∈L (N). By:

Image for - Fuzzy Games under Possibility and Necessity Expectations

we denote the weight of S⊆U with respect to ENe ( Image for - Fuzzy Games under Possibility and Necessity Expectations(S)) and WSP = 1-WSN indicates the weight of S⊆U with respect to EP ( Image for - Fuzzy Games under Possibility and Necessity Expectations(S)).

Definition 6: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), Image for - Fuzzy Games under Possibility and Necessity Expectations is said to be weighted NPE superadditivity if we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations

for any S, T⊆U with Image for - Fuzzy Games under Possibility and Necessity Expectations.

Remark 2: If there is no special explanation, for any Image for - Fuzzy Games under Possibility and Necessity Expectations, we always mean Image for - Fuzzy Games under Possibility and Necessity Expectations is weighted NPE supperadditive.

Definition 7: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), Image for - Fuzzy Games under Possibility and Necessity Expectations is said to be weighted NPE convex if we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations

for any S, T⊆U.

Definition 8: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), T⊆U is said to be a weighted NPE carrier for Image for - Fuzzy Games under Possibility and Necessity Expectations in U, if it satisfies Image for - Fuzzy Games under Possibility and Necessity Expectations for any S⊆U.

Definition 9: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), C (ENP (Image for - Fuzzy Games under Possibility and Necessity Expectations), U) is said to be the weighted NPE fuzzy core of in U which is defined by:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Definition 10: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), the vector y = (yi)i∈SuppU is said to be a weighted NPE participation monotonic allocation schemes for Image for - Fuzzy Games under Possibility and Necessity Expectations in U if it satisfies:

Image for - Fuzzy Games under Possibility and Necessity Expectations
yi (S)≤yi (T) ∀ i∈SuppS, ∀S, T⊆U s.t. S⊆T

Definition 11: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), the function is Image for - Fuzzy Games under Possibility and Necessity Expectations said to a weighted NPE Shapley function for Image for - Fuzzy Games under Possibility and Necessity Expectations in U it satisfies the following axioms:

Axiom 1 (weighted NPE efficiency): If T⊆U is a weighted NPE carrier for Image for - Fuzzy Games under Possibility and Necessity Expectations in U, then we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

where, WUNe and WUp are the weights of U with respect to ENe ( Image for - Fuzzy Games under Possibility and Necessity Expectations(U)) and EP ( Image for - Fuzzy Games under Possibility and Necessity Expectations(U)), respectively.

Axiom 2 (weighted NPE symmetry): For any i, j∈SuppU and any given weight WUNe, if we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations

for any S⊆U with i, j∉SuppS . Then, we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Axiom 3 (weighted NPE additivity): Let Image for - Fuzzy Games under Possibility and Necessity Expectations, Image for - Fuzzy Games under Possibility and Necessity Expectations if we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations

for any S⊆U.

Then we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations

THE WEIGHTED NPE SHAPLEY FUNCTION FOR FUZZY GAMES WITH FUZZY PAYOFFS

Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), we give the weighted NPE Shapley function for Image for - Fuzzy Games under Possibility and Necessity Expectations in U as follows:

Image for - Fuzzy Games under Possibility and Necessity Expectations
(4)

Where:

Image for - Fuzzy Games under Possibility and Necessity Expectations
(5)

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations
(6)

for any i∈SuppU and:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Theorem 1: Let Image for - Fuzzy Games under Possibility and Necessity Expectations, U∈L(N) and any given weight WUN, then the function Image for - Fuzzy Games under Possibility and Necessity Expectations, given in Eq. 4, is the unique weighted NPE Shapley function for Image for - Fuzzy Games under Possibility and Necessity Expectations in U.

Proof
⇒ Axiom 1:
Since T⊆U is a weighted NPE carrier, we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

where, i∉SuppT.

From Eq. 5, we get φNi (EN (Image for - Fuzzy Games under Possibility and Necessity Expectations), U) = 0 for any i∈SuppU\SuppT. Thus, we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Similarly, we obtain:

Image for - Fuzzy Games under Possibility and Necessity Expectations

From Eq. 4, we get:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Axiom 2: From Eq. 5, we obtain:

Image for - Fuzzy Games under Possibility and Necessity Expectations

where:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Similarly, we have φPi (EP (Image for - Fuzzy Games under Possibility and Necessity Expectations), U). Thus, φi (ENP (Image for - Fuzzy Games under Possibility and Necessity Expectations), U) = φj (ENP (Image for - Fuzzy Games under Possibility and Necessity Expectations), U).

From Eq. 3-5, we can easily get Axiom 3.

Uniqueness: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), it is not difficult to show ENe = (Image for - Fuzzy Games under Possibility and Necessity Expectations) can be expressed by:

Image for - Fuzzy Games under Possibility and Necessity Expectations

where:

Image for - Fuzzy Games under Possibility and Necessity Expectations

and us is a weighted NE unanimity game defined by:

Image for - Fuzzy Games under Possibility and Necessity Expectations

and EP (Image for - Fuzzy Games under Possibility and Necessity Expectations) can be expressed by:

Image for - Fuzzy Games under Possibility and Necessity Expectations

where:

Image for - Fuzzy Games under Possibility and Necessity Expectations

and EP (us) is a weighted PE unanimity game given as:

Image for - Fuzzy Games under Possibility and Necessity Expectations

From Axiom 3 and Eq. 3, we only need to show the uniqueness of φN and φP on unanimity games, respective.

For any uT with ø≠T⊆U, since T is a weighted NPE carrier for uT, from Axiom 1, we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations

From Axiom 2, we get:

Image for - Fuzzy Games under Possibility and Necessity Expectations

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Namely, φN and φP are unique in the unanimity games and the proof is finished.

Theorem 2: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), if Image for - Fuzzy Games under Possibility and Necessity Expectations is NPE convex, then we have φ (ENP (Image for - Fuzzy Games under Possibility and Necessity Expectations), U)∈C (ENP (Image for - Fuzzy Games under Possibility and Necessity Expectations), U).

Proof: From Theorem 1, we only need to show:

Image for - Fuzzy Games under Possibility and Necessity Expectations

for any S⊆U.

From the NPE convexity of , we get:

Image for - Fuzzy Games under Possibility and Necessity Expectations

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations

for any S, T⊆U with S⊆T, where i∉SuppT.

From Eq. 5, 6, we obtain:

Image for - Fuzzy Games under Possibility and Necessity Expectations

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations

for any i∈SuppS.

Thus:

Image for - Fuzzy Games under Possibility and Necessity Expectations

From Theorem 1, we obtain:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Property 1: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L(N), if Image for - Fuzzy Games under Possibility and Necessity Expectations is NPE convex, then n (ENP (Image for - Fuzzy Games under Possibility and Necessity Expectations), U) is a NPE participation monotonic allocation schemes.

Proof: From Theorem 1 and 2, we can easily get the result.

Property 2: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), if T⊆U is a weighted NPE carrier for Image for - Fuzzy Games under Possibility and Necessity Expectations in U, then we have φi (ENP (Image for - Fuzzy Games under Possibility and Necessity Expectations), U) = φi (ENP (Image for - Fuzzy Games under Possibility and Necessity Expectations), T) for any i∈SuppU.

Proof: From Theorem 1, we get:

Image for - Fuzzy Games under Possibility and Necessity Expectations

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:

Image for - Fuzzy Games under Possibility and Necessity Expectations

where, αUS and αUS+1 as given in Theorem 1 and:

Image for - Fuzzy Games under Possibility and Necessity Expectations

When, |SuppU| = |SuppT|+q, without loss of generality, suppose SuppU = SuppT∪{k1, k2,…, kq}. Let SuppT1 = SuppT∪{k1}, SuppT2 = SuppT1∪{k2},…, SuppTq = SuppTq-1∪{kq}.

From above, we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

for any i∈SuppT.

Similarly, we get:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Thus, we obtain:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Property 3: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), if we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations

for any S⊆U with i∉SuppS.

Then:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Proof: From Eq. 5, we get:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Similarly, we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Thus:

Image for - Fuzzy Games under Possibility and Necessity Expectations

Corollary 1: Let Image for - Fuzzy Games under Possibility and Necessity Expectations and U∈L (N), if we have:

Image for - Fuzzy Games under Possibility and Necessity Expectations

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations

for any S⊆U with i∉SuppS. Then:

Image for - Fuzzy Games under Possibility and Necessity Expectations

From Eq. 1-3, we know the Credibility Expectation (CE) is equal to:

Image for - Fuzzy Games under Possibility and Necessity Expectations
(7)

When each player’s weight for the approximate value of U∈L (N) is 0.5, then we have WSN = WSP = 0.5 for any S⊆U with respect to ENe ( Image for - Fuzzy Games under Possibility and Necessity Expectations(S)) and EP ( Image for - Fuzzy Games under Possibility and Necessity Expectations(S)). From Eq. 4, we get the weighted CE Shapley function for Image for - Fuzzy Games under Possibility and Necessity Expectations in U as follows:

Image for - Fuzzy Games under Possibility and Necessity Expectations
(8)

where:

Image for - Fuzzy Games under Possibility and Necessity Expectations
(9)

And:

Image for - Fuzzy Games under Possibility and Necessity Expectations
(10)

for any i∈SuppU and αSU as given in Eq. 5, 6.

Equation 9 and 10 are called the NE Shapley function and the PE Shapley function for Image for - Fuzzy Games under Possibility and Necessity Expectations 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, S0 = {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:

Image for - Fuzzy Games under Possibility and Necessity Expectations
(11)

Namely, this is a fuzzy game with multilinear extension form and fuzzy payoffs.


Table 1: The fuzzy payoffs of the crisp coalitions
Image for - Fuzzy Games under Possibility and Necessity Expectations

Table 2: The possibility expectations of the fuzzy coalitions
Image for - Fuzzy Games under Possibility and Necessity Expectations

Table 3: The necessity expectations of the fuzzy coalitions
Image for - Fuzzy Games under Possibility and Necessity Expectations

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:

Image for - Fuzzy Games under Possibility and Necessity Expectations

for any S⊆U, where wi 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, w1 = 1, w2 = 0 and w3 = 0.5. From Eq. 4, we get the player weighted NPE Shapley values are:

φ1 (ENP (Image for - Fuzzy Games under Possibility and Necessity Expectations), U) = 2.85
φ2 (ENP (Image for - Fuzzy Games under Possibility and Necessity Expectations), U) = 5.94
φ3 (ENP (Image for - Fuzzy Games under Possibility and Necessity Expectations), 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).

REFERENCES

1:  Ayanzadeh, R., A.S.Z. Mousavi and H. Navidi, 2011. Honey bees foraging optimization for mixed Nash equilibrium estimation. Trends Applied Sci. Res., 6: 1352-1359.
CrossRef  |  Direct Link  |  

2:  Akramizadeh, A., A. Afshar and M.B. Menhaj, 2009. Multiagent reinforcement learning in exrensive form games with perfect information. J. Applied Sci., 9: 2056-2066.

3:  Amiri, M., N.E. Nosratian, A. Jamshidi and A. Kazemi, 2008. Developing a new ELECTRE method with interval data in multiple attribute decision making problems. J. Applied Sci., 8: 4017-4028.
CrossRef  |  Direct Link  |  

4:  Borkotokey, S., 2008. Cooperative games with fuzzy coalitions and fuzzy characteristic functions. Fuzzy Sets Syst., 159: 138-151.
CrossRef  |  Direct Link  |  

5:  Banks, H.T. and M.Q. Jacobs, 1970. A differential calculus for multifunctions. J. Math. Anal. Appl., 29: 246-272.
CrossRef  |  Direct Link  |  

6:  Cheheltani, S.H. and S.M. Ebadzadeh, 2010. Immune based approach to find mixed Nash equilibrium in normal form games. J. Applied Sci., 10: 487-493.
CrossRef  |  Direct Link  |  

7:  Chao, C.H., M.Y. Hsiao, S.H. Tsai and T.H.S. Li, 2010. Design of an interval type-2 fuzzy immune controller. Inform. Technol. J., 9: 1115-1123.
CrossRef  |  Direct Link  |  

8:  Dubois, D. and H. Prade, 1987. The mean value of a fuzzy number. Fuzzy Sets Syst., 24: 279-300.
CrossRef  |  

9:  Dubois, D. and H. Prade, 1988. Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press, New York

10:  Daghistani, B.I., 2011. Effective use of educational games in the development of some thinking skills of kindergarten children. Trends Applied Sci. Res., 6: 656-671.
CrossRef  |  Direct Link  |  

11:  Dhar, S., A. Ray, S. Chakravorty and R.N. Bera, 2011. Intelligent vertical handover scheme for utopian transport scenario. Trends Applied Sci. Res., 6: 958-976.
CrossRef  |  Direct Link  |  

12:  Isin, S., and B. Miran, 2005. Farmers attitudes toward crop planning in Turkey. J. Applied Sci., 5: 1489-1495.
CrossRef  |  

13:  Kalliny, M. and L. Gentry, 2010. Marketing in the 22nd century: A look at four promising concepts. Asian J. Market., 4: 94-105.
CrossRef  |  Direct Link  |  

14:  Liu, B. and Y.K. Liu, 2002. Expected value of fuzzy variable and fuzzy expected value models. IEEE Trans. Fuzzy Syst., 10: 445-450.
CrossRef  |  

15:  Liu, Y.K. and B.D. Liu, 2003. Expected value operator of random fuzzy variable and random fuzzy expected value models. Int. J. Uncertain. Fuzz., 11: 195-215.

16:  Lo, C.Y., 2008. Multi-agent conflict coordination using game bargain. Inform. Technol. J., 7: 234-244.
CrossRef  |  Direct Link  |  

17:  Mares, M., 2000. Fuzzy coalition structures. Fuzzy Sets Syst., 114: 23-33.
CrossRef  |  Direct Link  |  

18:  Mares, M. and Vlach, 2001. Linear coalition games and their fuzzy extensions. Int. J. Uncertainty Fuzziness Knowledge-Based Syst., 9: 341-354.
Direct Link  |  

19:  Meng, F. and Q. Zhang, 2010. Fuzzy cooperative games with Choquet integral form. Syst. Eng. Electron., 32: 1430-1436.

20:  Meng, F.Y., J.X. Zhao and Q. Zhang, 2012. The Banzhaf value for fuzy games with fuzzy payoffs. Inform. Technol. J., 11: 262-268.

21:  Tsurumi, M., M. Inuiguchi and A. Nishimura, 2005. Pseudo-Banzhaf values in bicooperative games. Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, October 9-12, 2005, Waikoloa, HI., USA., pp: 1138-1143
CrossRef  |  

22:  Tan, M., H. Xu, L. Zeng and S. Xia, 2011. Research on fuzzy self-adaptive variable-weight combination prediction model for IP network traffic. Inform. Technol. J., 10: 2322-2328.
CrossRef  |  

23:  Yu, X. and Q. Zhang, 2010. An extension of cooperative fuzzy games. Fuzzy Sets Syst., 161: 1614-1634.
CrossRef  |  

24:  Zadeh, L.A., 1978. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst., 1: 3-28.
CrossRef  |  Direct Link  |  

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