The Interval Shapley Value for Type-2 Interval Games

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1 Research Journa of Appied ciences, Engineering Technoogy 4(): , 22 IN: Maxwe cientific rganization, 22 ubmitted: December 23, 2 Accepted: January 2, 22 Pubished: May 5, 22 The Interva hapey Vaue for Type-2 Interva Games Fanyong Meng 2 Feng Liu choo of Management, Qingdao Technoogica University, Qingdao 26652, hina 2 Department of Information Management, the entra Institute for orrectiona Poice, Baoding, 7, hina Abstract: In this study, we research the so caed type-2 interva games, where the payer participation eves the coaition vaues are both interva numbers. Two specia kinds of type-2 interva games are studied. The interva hapey vaues for these two specia casses of type-2 interva games are researched. By estabishing axiomatic systems, the existence uniqueness of the given interva hapey vaues are shown. When the associated interva games are convex, the given interva hapey vaues Are Interva Popuation Monotonic Aocation Functions (IPMAF), beong to the associated interva cores. Key words: ooperative game, interva core, interva number, interva hapey vaue INTRDUTIN With the socia deveopment, the peope reaized the appication of cassica games theory has been argey restricted. Aubin (974) introduced the concept of fuzzy coaition where some payers do not fuy participate in a coaition, but to a certain degree. wen (972) subty defined a kind of fuzzy games, which is caed fuzzy games with mutiinear extension form. Later, Butnariu (98), Tsurumi et a. (2) Butnariu Kroupa (28) researched a specia kind of fuzzy games, respectivey. The same as cassica game theory, the key issue of cooperative fuzzy games is how to distribute the payoffs. The hapey vaue for fuzzy games is studied by Butnariu (98), Tsurumi et a. (2), Butnariu Kroupa (28), Li Zhang (29) Meng Zhang (2, 2a, 2b). Besides the hapey vaue, the fuzzy core for fuzzy games is researched by Tijs et a. (24) Yu Zhang (29). The exicographica soution for fuzzy games is discussed by akawa Nishizaki (994). As we know, there are many uncertain factors during the process of negotiation coaition forming, so in most situations payers can ony know imprecise information regarding the vaues of the coaitions formed by the payers. The researches about this kind of fuzzy games are discussed by Mares (2), Mares Vach (2), Borkotokey (28) Yu Zhang (2). ince interva numbers can convenienty describe the ower upper bounds of the payer possibe payoffs. Aparsan Gök et a. (29) focused on two-person cooperative games with interva uncertainty, discussed the core for the given games. Later, Aparsan Gök et a. (2) studied cooperative games with interva uncertainty, researched the interva hapey vaue. Branzei et a. (2) concerned the core of cooperative games with interva uncertainty, gave the definition of convex games with interva uncertainty. Maozzi et a. (2) researched the F-core for fuzzy interva cooperative games. A above mentioned researches ony consider the situation where the payer participation is determined. As above anaysis, in most cooperation the payoffs are uncertainty, so it is difficut for the payers to decide the participation eves exacty. In this study, we sha research the situation where the payer participation eves the coaition vaues are both interva numbers. PRELIMINARIE We first review some definitions about interva numbers. Definition : a = [ a, a ] is said to be an interva number if a a, where a a, R; a = [ a, a ] is said to be a positive interva number if ag # a, where a, a,r. From Definition, we know the interva number a degenerates to be a rea number when a = a In this study, we use R R to denote the sets of a interva numbers positive interva numbers on R R, respectivey. Let ab, R, from the extension principe on fuzzy sets proposed by Zadeh (973), we have: a b = [ a b, a b ] a b = [ a b, a b ] orresponding Author: Fanyong Meng, choo of Management, Qingdao Technoogica University, Qingdao 26652, hina 334

2 Res. J. App. ci. Eng. Techno., 4(): , 22 λa = [ λa, λa ] λ R a b = [ a b, a b ] a b = [ a b, a b ] Definition 2: For a ab,, R we write a b if ony if a b a = b if ony if a b a b = a = b Definition 3: Let a, b R, if there exists c R such that a b c, then is caed the Hukuhara difference between a b, denoted by a H b. Let the set of payers N = {, 2,..., n}. The crisp coaitions on N are denoted by, T,... The power set of a crisp subsets on N is denoted by P(N). For any P (N), the cardinaity of is denoted by the corresponding ower case s. A function v : P(N)6 R, satisfying v ( ) =, is caed an interva characteristic function. The set of a games with interva characteristic function on P(N) is denoted by IG (N). = c Definition 4: (Aparsan Gök et a., 2) Let v is said to be size monotonic if: v ( T ) v ( T ) v ( ) v ( ),T P(N) with T f. v IG (N). Property : Let ab, R, there exists c R such that a = b c if ony if b b a a. a b c Proof: From =, we get a = b c a = b c. ince c c we have: a a = b c ( b c ) b b From b b a a a b a b, we have: Let c a b = c = a b, we get a = b c a = b c. ince c c we know c = [ c, c ] is an interva number, a = b c. From Property, we know a size monotonic game, given by Aparsan Gök et a. (2), a Hukuhara difference existence game, introduced by Yu Zhang (29), are equivaent. Definition 5: Let v, IG (N), is said to be convex if: v v( T) v( T) v( ) v( T) T N,. TW PEIAL KIND F TYPE-2 INTERVAL GAME In this section, we sha study two specia kinds of type-2 interva games, which can be seen the extensions of fuzzy games introduced by wen (972) Tsurumi et a. (2). The fuzzy coaition vaues for these two kinds of fuzzy games are written as: vu ( ) = Ui ( ) ( Ui ( )) v( T) () T uppu i T \ T vu ( ) = v([ U] )( h h ) where U is a fuzzy coaition as usua, T is a crisp coaition. Q (U) = {U(i) U(i) >, i N} q(u) = Q(U). The eements in Q (U) are written in the increasing order as = h #h #...# h q(u) [ ] = {i U(i) h, in, U =,2,..., q(u)}. v is a crisp game defined in N. An interva fuzzy coaition = { ( i) i N } = [ i, i ] N is an n-dimension vector, where [ i i ] i N (2) on, [, ] i N. The set of a interva fuzzy coaitions on N is denoted by IL(N). For any IL(N) payer i, i () indicates the interva membership grade of i in, i.e., the interva rate of the ith payer in. For any IL(N), the support = { is denoted by upp ={i N - i } i N i > } the cardinaity is written as upp, which is the same as = { i} i N. Namey, the support = { i} i N is denoted by upp = {i N i>} the cardinaity is written as upp. We use the notation T if ony if i ( ) = Ti ( ) or i for a i N. For a ()= T, IL(N), T denotes the union of interva fuzzy coaitions T, where ( T) (i) = (i) T (i) i N, T denotes the intersection of interva fuzzy coaitions T, where ( T) (i) = i () Ti () i N. A function v: IL ( N) R, satisfying v ( ) =, is caed a type-2 interva characteristic function. The set of a games with type-2 interva characteristic function on IL (N) is denoted by IG (N). For any IL (N), v( ) = [ v ( ), v ( )] denotes the interva payoff of, where v! (! ) v ( ) denote the eft right extreme points, respectivey. 335

3 Res. J. App. ci. Eng. Techno., 4(): , 22 Definition 6: Let v IG( U), T is said to be a carrier for v in U if: v( T) = v( ) U Definition 7: Let v IG( U), v is said to be convex if: v( T) v( T) v( ) v( T) T, U. imiar to popuation monotonic aocation schemes given by prumont (99) for crisp case, we give the foowing concept for type-2 interva games. Definition 8: Let v IG( U), the vector x ={[x - I, x i] iuppug cuppu}is said to be an imputation for v in U if [ xi, xi ] v( U ( i)) i uppu uppu = [ xi, xi ] v( U) [ xi, xi ] = i N \ uppu TYPE-2 INTERVAL GAME WITH MULTILINEAR EXTENIN FRM imiar to Eq. (), we give the interva vaues of interva fuzzy coaitions in type-2 interva games with mutiinear extension form as foows: v( U) = [ v( U ), v( U )] ( v, U) = x xi, xi = v( U ), v( U ), xi, x i [ ] = [ ] v ( ), v ( ), U} Theorem : Let v IG( U), if the associated game v IG( N) of v is convex, then ( v, U), can be expressed by: ( v, U) = x xi, x i = U i U i () ( () y T T uppu i T T \ U i ( U i ) y () () uppu i \ yt ( v, T), T uppu, y ( v, ), uppu where ( vg,t ) ( v, ) denote the core for vg in T for v in, respectivey. },, = U () i ( U ()) i v( T) T uppu i T \ T Proof: Let ( v, U ) = { x = { xi } x i = U () i U ( U ()) i v( T) T uppu i T \ T (3) v( U ), xi v( T ), T U } i uppt (4) where U IL(N) By IG ( U ), we denote the set of a type-2 interva games with mutiinear extension form on U IL( N). If there is not specia expanation, we aways mean the associated interva game v IG (N) of v IG( U) is size monotonic. Definition 9: Let v IG( U) ( v, U) for v in U is defined by:, the interva core ( v, U ) = { x = { xi } x i = v( U ), xi v( ), U } (5) From Proposition given by Yu Zhang (29), we know Eq. (4) is equivaent to the foowing expression i i {x = { x } x = 336

4 Res. J. App. ci. Eng. Techno., 4(): , 22 U () i ( U ()) i yt, T uppu i T T, \ y ( v, T ), T uppu T Eq. (5) is equivaent to the foowing expression: {x = { xi } xi = U () i ( U ()) i y uppu i \ y ( v, ), uppu Definition : Let v IG( U), the function f: IG ( U) R is said to be an interva hapey function if it satisfies: Axiom : If T is a carrier for v in U, then fi ( v, U ), fi ( v, U ) i uppt i uppt Axiom 2: For any I, j uppu, if we have: v( U( i)) = v( U( j)) U with I, j ó upp, then: f ( v, U) = f ( v, U) i j Axiom 3: For any } } v, ωo IG( U), ( v w)( ) = v( ) w( ) U then: ( ) f ( v w, U) = f ( v, U) f w, U Theorem 2: Let v IG( U) =[vg (TG),v (T )] if we have:, define the function: ϕ ( v, U ) = β ( v ( U ( i)) v ( )) i U U i upp œiuppu (7) where ( ) upp! uppu upp! β U = uppu! ( ) upp! uppu upp! β U = uppu! Then ϕ ( v, U) is the unique interva hapey function for v in U. Proof. (Existence) Axiom : ince v in U we have: [ v( T ), v( T )] = [ v( ), v( )] U, [ v ( U ( i)), v ( U ( i))] T is a carrier for = [ v( U ( i)) T, v( U ( i)) T )] = [ v ( ), v ( )] i uppu \upp. Hence, we have: [vg (TG), v (T )] [v! (UG v TG), v (U v T )] = [v! (U - ), v (U )] = [ ( v, U ), ( v, U )] = [ ( v, U ), ( v, U )] i uppt i uppt ϕ = [ ϕ, ϕ]: IG ( U ) R as foows: From v ( U( i)) = v ( U( j)), we obtain: ( v, U ) = β ( v U ( U ( i)) v( )) U i upp œiuppu! (6) [ v ( U ( i)), v ( U ( i))] = [ v( U ( j)), v( U ( j))] Thus, we have: 337

5 Res. J. App. ci. Eng. Techno., 4(): , 22 vo( U ( i)) vo( ) = vo( U ( j)) vo( ) vo( U ( i) U ( j)) vo( U ( j)) = vo( U ( i) U ( j)) vo( U ( i)) vo( U ( i)) vo( ) = vo( U ( j)) vo( ) o o v ( U ( i) U ( j)) v ( U ( j)) o o = v ( U ( i) U ( j)) v ( U ( i)) ϕ ( v, U) ( v, U). Proof: From Theorem 2, we have: [ ( vi, U ), ( vi, U )] = [ v( U ), v( U )] From Eq. (6) (7), it is not difficut to get Axiom2; It is obvious Axiom 3 hods; (Uniqueness) ince can be uniquey expressed by: From the convexity of v is convex. Thus: v IG( N) Eq. (3), we get v = c u v IG ( U), where T u T ( ) = otherwise v( U( i)) v( ) v( T U( i)) v( T) T, U with T i upp From Eq. (6) (7), we have: i i ϕ ( v, ) ϕ ( v, T ) i uppt upp uppt c = [ ( ) v T ( ) T upp uppt ( ) v( T )] T i i ϕ ( v, ) ϕ ( v, T ) i uppt where T U with Hence, we have:, T From Axiom 3, we ony need to show the uniqueness of ϕ on unanimity game u, where U. From Axiom, we get: i i ϕ ( v, U ) ϕ ( v, ) i upp = u ( ) i i ϕ ( v, U ) ϕ ( v, ) i upp = [ ( u, U ), ϕ i ( u, U )] From Axiom 2, we obtain: i upp ϕ i ( u, U ) = upp otherwise i upp ϕ i ( u, U ) = upp otherwise Theorem 3: Let v IG( U), if the associated interva game v IG N of is convex, then ( ) v where U. Namey, [ ( v, U ), ( v, U )] [ ( v, ), ( v, )] = [ v( ), v( )] U. oroary : Let v IG ( U), if the associated interva game v IG ( N) of v is convex, then ϕ ( v, U) is an imputation for v in U. 338

6 Res. J. App. ci. Eng. Techno., 4(): , 22 Definition : Let v IG ( U), the vector x= {[ x, x ] } is said to be an IPMAF for v in i i uppu U if: xi = [ xi xi ] = [ v( ), v( )] = v( ) U xi( ) = [ xi ( ), xi ( )] xi( T) = [ xi ( T)], xi ( T)] i upp upp, T, Us.t. T Theorem 4: Let v IG ( U), if the associated interva game v IG( N) of v is convex, then ϕ ( v, U) is an IPMAF for v in U. Proof: From Theorem 2, we know the first condition in Definition hods. From Theorem 3, we get the second condition in Definition. TYPE-2 INTERVAL GAME WITH HQUET INTEGRAL FRM imiar to Eq. (2), we give the vaues of interva fuzzy coaitions in type-2 interva games with hoquet integra form as foows: v( U) = [ v( U ), v( U )] qu ( ) = v ([ U ] h ( h h ) v([ U ] h )( h ), (8) where U IL(N), Q (UG) = {U - (i) U - (i) >, i N} q(u - ) = Q (U ) Q (U ) = {U (i) U (i) >, i N} q (U ) = Q(U ). By IG ( U ), we denote the set of a type-2 interva games with hoquet integra form on U IL (N). imiar to IG ( U ), if there is not specia expanation, we aways mean the associated interva game v IG (N) of v IG( U) is size monotonic. Definition 2: Let v IG( U), the interva core ( v, U) for v in U is defined by: ( v, U) = { x [ xi, xi,] = [ v( U ),[ v( U )],[ xi, xi,]} [ v( ), v( )], U} Theorem 5: Let v IG( U), if the associated interva game v IG ( N) of v is convex, then ( v, U), can be expressed by: ( v, U) = { x [ xi, xi ] = [ y ( h h ), y ( h h )] [ U ] [ U ] y ( v [ U ] ), = 2,,..., q( U ), [ U ] y ( v U = q U [ U ],[ ] ), 2,,..., ([ ]} where v (,[ U ] ) v (,[ U ] ) denote the core for v [ U ] in for in [ U ], respectivey. v Proof: From Theorem by Yu Zhang (29), we can easiy get the concusion. Definition 3: Let v IG ( U), the function f: IG ( U) R is said to be an interva hapey function if it satisfies: Axiom : If T is a carrier for v in U, then: fi v U fi v U = [ v (, ), (, ) ( T ), v( T )] i uppt i uppt Axiom 2: Let {, 2,...,q(U)} i, j, [ U ], if we have v( Ui) v( Uj) with i, j, then: fi([ U], v) = f j([ U] h, v) Axiom 3: For any = [ U] v, w IG( U) ( v w )( ) v ( ) w ( ) = U,, if we have then: 339

7 Res. J. App. ci. Eng. Techno., 4(): , 22 f ( v w, U) = f ( v, U) f ( w, U) Theorem 6: Let φ = [ φ, φ ]: IG ( U ) R v IG ( U), define the function as foows: v([ T ] h ) v([ U ] h ) = φi ( v,([ U ] h ) i [ U ] = φi ( v,([ U ] h ) i [ T ] h φ ( v, U ) = φ ( v,[ U ] )( h h ) i i i uppu (9) v([ T ] ) = v U ([ ] ) = φi ( v,([ U ] h ) i [ U ] = φi ( v,([ U ] ) i [ T ] φ ( v, U ) = φ ( v,[ U ] )( h h ) i i From Eq. (9), we get: œ i upp U () where i h = [ U ] ( v i ( )) [ U ], i φ ( v,[ U ] ) β qt ( ) v( T ) = v([ T ] h )( h ) = v ([ T ] h )( h ) = v ([ U ] h )( h ) v( )) i [ U ] i U U [ ] [ ] i φ ( v,[ U ] ) = β ( v ( i) = φ ( v,[ U ] h )( h h ) i [ U ] = φ i ( v,[ U ] h )( h h ) i [ T ] -v ( )) [ ] i U!( [ ] )! β s U s [ U = ] [ U ] h! s!( [ U ] h s)! β [ = U ] h [ U ] h! Then φ is the unique interva hapey function for v in U. Proof. (Existence) Axiom : From Theorem 4 introduced by Tsurumi et a. (2), we know T is a carrier in U for v if ony if [ T ] is a carrier for v in [ U ], is a carrier for v [ T ] in U, where =, 2,..., q (U). [ ] Thus, we have: = φ i ( v,[ U ] ( h) i uppt = φ i ( v, U ) i uppt imiary, we have: φ i ( v, U ) = v( T ) i uppt From Eq. (9) (), we can easiy get Axioms 2, 3. (Uniqueness) ince can be uniquey expressed by: v v = ct ut T N v IG (N), where ut ( W ) = T W otherwise 34

8 Res. J. App. ci. Eng. Techno., 4(): , 22 φi ( v, ) φi ( v, T ) i uppt t s t s ct = v v ( ) ( ), ( ) ( ) T T From Axiom 3, we ony need to show the uniqueness of φ on unanimity games u u T where, [ ] [ ] i f U i T f U {, 2,...,q(U)}. From Axioms, 2, we get: i φ i ( u,[ U ] h ) = s otherwise i φ i ( u,[ U ] h ) = s otherwise Theorem 7: Let v IG( U), if the associated interva game v IG ( N) of v is convex, then φ ( v, U) ( v, U). Proof: From Theorem 6, we have: = φi ( v, U ), φi ( v, U ) [ v ( U ), v ( U )] From the convexity of v IG( N) Eq. (8), we get v is convex. Thus: v( U( i)) v( ) v( T U( i)) v( T) T, Uwith T i ó upp From Eq. (9) (), we have: φi ( v, ) φi ( v, T ) i uppt φi ( v, ) φi ( v, T ) i uppt where T, U with T. Hence, we have: φi ( v, U ) φi ( v, ) i upp where U. Namey, φi ( v, U ), φi ( v, U ) φ i ( v, ), φi ( v, ) = [ v( ), v( )] U. imiar to of v IG N ( ) v IG ( U ), when the associated interva game is convex, we get: {[ φi ( v, U ). φi ( v, U )] } is an IPMAF for v in U, an imputation for v in U. ince the type-2 interva games in IG ( U ) IG ( U ) are continuity monotone node creasing with respect to the payer participation eves, we know every payer interva hapey vaue, obtained by Eq. (6) (7) or Eq. (9) (), is an interva number, where their associated interva games are size monotonic. NUMERIAL EXAMPLE There are 3 companies cooperate to compete a project. Namey, the set of payers N = {, 2, 3}. ince there exist many uncertainty factors in the process of deveopment. The payers ony know the scope of the crisp coaition vaues, which are given in Tabe. Furthermore, the payers are ony sure the ower upper participation eves in this cooperation, which are given by: U() = [ U (), U ()] = [., 3 7.] Tabe : The crisp coaitions interva vaues v( ) v ( ) {} [2, 4] {, 3} [6, ] {2} [, 3] {2, 3} [5, 9] {3} [2, 3] {, 2, 3} [2, 2] {, 2} [5, ] 34

9 Res. J. App. ci. Eng. Techno., 4(): , 22 U( 2) = [ U ( 2), U ( 2)] = [ 6., 9. ] U() 3 = [ U (), 3 U ()] 3 = [., 8 ] Then this is a type-2 interva game. When the fuzzy coaition vaues their associated crisp coaition vaues have the reationship given in Eq. (3). Namey, this game beongs to IG ( U ). From Eq. (6) (7), we get the payer interva hapey vaues as are: ϕ ( v, U) = [ 23., 522. ] ϕ 2 ( v, U) = [ 63., 53. ] ϕ 3 ( v, U) = [ 263., 575. ] when the fuzzy coaition vaues their associated crisp coaition vaues have the reationship given in Eq. (8). Namey, this game beongs to IG( U ). From Eq. (9) (), we get the payer interva hapey vaues as are: φ ( v, U) = [ 68., 545. ] φ 2 ( v, U) = [ 38., ] φ 3 ( v, U) = [ , 5. 75] NLUIN We have researched two specia kinds of type-2 interva games, where the payer participation eves the fuzzy coaition vaues are both interva numbers. The research in his paper extension the earning scope of fuzzy games, can better appied in practica probems. But we ony discuss type-2 interva games, it wi be interesting to research the other fuzzy games with type-2 fuzzy payoffs, which combine the operations of fuzzy sets. AKNWLEDGMENT This study was supported by the Nationa Natura cience Foundation of hina (Nos 777, ). REFERENE Aubin, J.P., 974. oeur et vaeur des jeux fous à paiements atéraux. omptes Rendus Hebdomadaires D, 279-A: Aparsan Gök,.Z.,. Mique. Tijs, 29. ooperation under interva uncertainty. Math. Method. per. Res., 69: Aparsan G,.Z., R. Branzei. Tijs, 2. The interva hapey vaue: An axiomatization, ent. Eur. J. per. Res., 8: 3-4. Butnariu, D., 98. tabiity hapey vaue for an n- persons fuzzy game. Fuzzy et. yst., 4: Butnariu, D. T. Kroupa, 28. hapey mappings the cumuative vaue for n-person games with fuzzy coaitions. Eur. J. per. Res., 86: Branzei, R.,. Branzei,.Z. Aparsan Gök,. Tijs, 2. ooperative interva games: A survey. ent. Eur. J. per. Res., 8: Borkotokey,., 28. ooperative games with fuzzy coaitions fuzzy characteristic functions. Fuzz. et. yst., 59: Li,.J. Q. Zhang, 29. A simpified expression of the hapey function for fuzzy game. Eur. J. per. Res., 96: Meng, F.Y. Q. Zhang, 2. The hapey function for fuzzy cooperative games with mutiinear extension form. App. Math. Lett., 23: Meng, F.Y. Q. Zhang, 2.The hapey vaue on a kind of cooperative fuzzy games. J. omput. Inf., 7: Meng, F.Y. Q. Zhang, 2. The fuzzy core hapey function for dynamic fuzzy games on matroids. Fuzzy ptim. Decis. Ma., : Mares, M., 2. Fuzzy coaition structures. Fuzzy et. yst., 4: Mares, M. Vach, 2. Linear coaition games their fuzzy extensions. Int. J. Uncertain. Fuzz., 9: Maozzi, L., V. cazo. Tijs, 2. Fuzzy interva cooperative games. Fuzzy et. yst., 65: wen, G., 972. Mutiinear extensions of games. Manage. ci., 8: akawa, M. I. Nishizazi, 994. A exicographica soution concept in an n-person cooperative fuzzy game. Fuzzy et. yst., 6: prumont, Y., 99. Popuation monotonic aocation schemes for cooperative games with transferabe utiity. Game Econ. Behav., 2: Tsurumi, M., T. Tanino M. Inuiguchi, 2. A hapey function on a cass of cooperative fuzzy games. Eur. J. per. Res., 29: Tijs,., R. Branzei,. Ishihara, et a., 24. n cores stabe sets for fuzzy games. Fuzzy et. yst., 46: Yu, X.H. Q. Zhang, 29. The fuzzy core in games with fuzzy coaitions. J. omput. App. Math., 23: Yu, X.H. Q. Zhang, 2. An extension of cooperative fuzzy games. Fuzz. et. yst., 6: Zadeh, L.A., 973. The oncept of a Linguistic Variabe its Appication to Approximate Reasoning. American Esevier Pubishing ompany, UA. 342