【正文】 ? ?? ? .0?Avgi ? Consequently, ? ?? ? .0?Avgi ? Proposition 3 Let normalized games ? ?NGv ?? ??? and ? ?.NPA? Then the explicit form of the Shapley function on ? ?NG?? is as follows: ? ?? ? ? ?? ? ? ? ? ?? ?? ?? ?? ?????? ????????? ????? ? ?? ? ,0,1。iP \o t h e r w i s eAijvABivAvgAB iBji??? ?? (4) where ? ? ? ?? ? ? ? ? ? ? ? !/!!1。| ABABABandBiAPBPAP i ?????? ?. Proof ? ? ? ?? ? ? ?? ? ? ?? ?? ?? ? ???????? ??????? ?? iBjjviviBvBv\1\ ???? ?by the deduction of Eq.(1). By substituting the above formula into Eq.(3), we get Eq.(4). Proposition 4 Let normalized games ? ?NGv ?? ??? , and ?v be defined by Eq.(2), then ? ?NGv ?? ? and ? ?? ? ? ?? ? NibAvcgAvg iii ????? ,?? (5) where ??? ?? otherwiseAiab ii ,0 , Proof If ,Ai? then ? ?? ? ,0?? Avgi ? ? ?? ? ,0?Avgi ? .0?ib According to Eq.(3), if ,Ai? then ? ?? ? ? ? ? ? ? ?? ?? ?? ? ? ?? ? ? ?? ? iiAPBiAPBi aAvcgABaiBvBvABcAvgii???????? ?? ?? ???? ?? 。\。 2 λFuzzy Cooperative Games 22 Given ? ?NFS? , let ? ? ? ? ? ?? ?NiiSiSSQ ??? ,0| and ??Sq be the cardinality of ??.SQ We write the elements in Q(S) in the increasing order as ? ?.21 Slll q??? ? Definition 4 let normalized games ? ?.NGv ?? ??? The game ?v? is said to be a normalized λfuzzy game if and only if the following equation holds: ? ? ? ? ? ?? ?? ?11 ?? ???? ? iilSqi llSvSv i?? (6) For any ? ?NFS? , where 00?l 。 a fuzzy game ?v defined on F(N) is called a general λfuzzy game if for any ? ?NFS? , ? ? ? ? ? ? ? ?iSaSvcSv SS up pi i????? ?? (7) Where 。0?c ia are constant satisfying the same constraint as in Eq.(2) for any ? ? ,ilSA? ? ?.1 Sqi lll ?? The set of all normalized λfuzzy games is denoted by ? ?,NGF? and the set of all general λfuzzy games denoted by ? ?.NGF Obviously, we have ? ? ? ?.NGNG FF ?? It is apparent that Eq.(6) is a Choquet integral of the function S with regard to λfuzzy measure ?v . By Eq.(2) and Eq.(6) , Eq.(7) is equal to ? ? ? ? ? ?? ?? ?11 ?? ?? ? iilSqi llSvSv i?? , where ? ?? ?ilSv? and ? ?? ?ilSv?? satisfy Eq.(2). For S? , ? ?NFT? , union, intersection and inclusion of two fuzzy sets are defined as usual , . , ? ?? ? ? ? ? ?? ? ,max NiiTiSiTS ???? ? ?? ? ? ? ? ?? ? ,m i n NiiTiSiTS ???? ? ? ? ? ., NiiTiSTS ????? Then the definitions about crisp games can be applied to fuzzy games directly , which are superadditivity , subadditivity , supermodularity , and submodularity respectively. Proposition 5 Let ? ?.NGv F?? If 0?? , then ?v is superadditive。 otherwise it is subadditive. 23 Proof Note that if ???TS then ? ? ? ? ??? aa TS for any ? ?1,0?a . Hence, the proposition is apparent from the definition of ? ?NGv F?? since any crisp game ? ?NGv ?? ? is superadditive (resp. subadditive ) when 0?? (resp. 0?? ). Proposition 6 Let ? ?.NGv F?? Then the following holds ? ? ? ? ? ? ..., TStsNFTSTvSv ???? ?? Proof Note that TS? if and only if ? ? ? ?aa TS ? for any ? ?1,0?a . According to Proposition 1, ? ?? ? ? ?? ?,aa TvSv ?? ? for ? ? ? ?aa TS ? . 3 Shapley Value for Cooperative Games with Fuzzy Coalition Definition 5 Let ? ?NGv F?? and ? ?NFS? . ? ?SFT? is called a fuzzy carrier in coalition S for a game ?v if it satisfies ? ? ? ?MvMTv ?? ?? for any ? ?.SM? We denote the set of all carriers in S for ?v by ? ?.| ?vSCF Let ? ?NFS? and ., Nji ? For any ? ?SFT? , define ? ? ? ?SandpSFT ijSi ? by ? ? ? ? ? ?? ?? ? ? ?? ?? ?????? ???,m in,m ino th e r w isekTjkiSjTikjSiTkT Sij ? ?? ? ? ?? ?? ?????? ???o th e r w isekSjkiSikjSkSp ij, Clealy, ,SijT ? ? ? ?.SFTp Sij ? Now, we introduce Shapley function as follows. Definition 6 A function ? ? ? ? ? ?NFnRNGg ???: is said to be Shapley function on ? ?NGF if it satisfies the following four axioms. Axiom 1: If ? ?NFS? then ? ?? ? ? ? ? ?? ? 0。 ???? SvfSvSvf iNi i ??? for ? ?,SSuppi? where ? ?? ?Svfi ? is the ith element of ? ?? ? .nRSvf ??? 24 Axiom 2: If ? ?NFS? and ? ??vSCT F |? then ? ?? ? ? ?? ?TvfSvf ii ?? ? for any .Ni? Axiom 3: If ? ?NFS? , ? ? ? ? ? ?? ?TpvTan d vvSCS ijFSij ??? ?? | for any ? ?SijSFT? then ? ?? ? ? ?? ?.SvfSvf ji ?? ? Axiom 4: For any two games ? ?NGvv ??? ?21 , define ? ?? ? ? ? ? ?TvTvTvvbyvv 212121 ?????? ???? for any ? ?.NFT? If ? ? ? ? ? ?? ? ? ?? ? ? ?? ?SvfSvfSvvt h e n fNFandSNGvv iiiF 212121 , ?????? ?????? for any Ni? . Define a function ? ? ? ? ? ?NFnF RNGf ??: by ? ?? ? ? ? ? ?? ?? ?? ??? ??? Sqj jjlii llSvgSvf j1 1?? (8) where ig is the function given in the Eq.(3) and .00?l Theorem 2 The function defined by Eq.(8) is a Shapley function on ? ?.NGF Proof Note that Eq.(8) is the same with the Shapley function for the superadditive games in Theorem 8 by Tsurumi. Here, we do not give unnecessary details about the proof process that Eq.(8) satisfy the four axioms in Definition 6, but we should make sure that Eq.(8) is not less than zero. By Proposition 2, we know that ? ?? ? 0?Avgi ? . Thus, ? ?? ? ? ? ? ?? ?? ?? ? 01 1 ??? ?? ?Sqj jjlii llSvgSvf j?? since ? ?.21 sqlll ??? ? Proposition 7 Let ? ? ? ? ? ? ? ?.10, 0 SqmandlliSNFSNGv mF ????????? Then Shapley function on ? ?NGF? is ? ?? ? ? ?? ? ? ? ? ?? ? ? ?? ?? ?? ?? ?????? ????????? ?????? ? ? ?? ??o t h e r w i s elkvSBllivSvf mj lSPBmljjijij,00,1。11 ??? ?? (9) where ? ?? ? ? ? ? ?? ? ? ? ? ?? ? ? ?? ?? ?.|,!/!!1。 TiSPTPSPSTSTSTjjjjj llilll ??????? Proof By substituting Eq.(4) into Eq.(8), we can get this conclusion. Proposition 8 Let normalized fuzzy games ? ?.NGv F???? Let ?v be defined as in Eq.(7), 25 then ? ? ? ?? ?