The Shapley Value in Totally Convex Multichoice ... - Semantic Scholar
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Applied Mathematics Letters
Applied Mathematics Letters 13 (2000) 95 98
www.elsevier.n l/loeate/aml
The Shapley Value in Totally C o n v e x Multichoice G a m e s D.
A. AYOSHIN AND T.
TANAKA
Department of Information Science, Faculty of Science, Hirosaki University Department of Mathematical System Science, Faculty of Science and Technology Hirosaki Uifiversity Hirosaki 036-8.561, Japan <srdimaOsi><sltana~cc>. hirosaki-u, ac. jp
(Received March 1999; revised artd ac:cepted April I999)
Abstract--In this paper, we introduce a class of totally convex multichoice cooperative games a.nd prove that the Shapley value of such games is always in the core. @ 2000 Elsevier Science I~td. All rights reserved. K e y w o r d s - - M u l t i c h o i c e game, Shapley value, Total convexity, Core, Coalition.
1. I N T R O D U C T I O N Hsiao a n d R a g h a v a n [1] i n t r o d u c e d a class of multichoice c o o p e r a t i v e g a m e s and tbund its S h a p l e y value using an a x i o m a t i c a p p r o a c h . Later, Nouweland et al. [2] d e t e r m i n e d t h e S h a p l e y value ~br multichoice c o o p e r a t i v e g a m e s following its p r o b a b i l i s t i c i n t e r p r e t a t i o n . Howewu. t h e w~lues o b t a i n e d by t h e s e two m e t h o d s are q u i t e different. In our p a p e r , while avoiding the p r o b l e m of i n c o n s i s t e n c y of t h e S h a p l e y value between H s i a o - R a g h a v a n and Nouweland, we consider a n e c e s s a r y a n d sufficient c o n d i t i o n for t h e S h a p l e y vahle by Nouweland to be in t h e core of a m u l t i c h o i c e c o o p e r a t i v e game. It is well k n o w n t h a t in t h e class of c o o p e r a t i v e g a m e s in tile c h a r a c t e r i s t i c function form, the S h a p l e y value is in t h e core if t h e c h a r a c t e r i s t i c function is either convex [3], aw~rage convex [4], or t o t a l l y convex [5]. T h e l a t t e r p a p e r shows t h a t t h e class of t o t a l l y convex g a m e s inchnles tha.t of a v e r a g e convex games. W e discuss c o n d i t i o n s for t h e S h a p l e y value to be in t h e core tot t h e ('lass of multichoice c o o p e r a t i v e games.
2. M U L T I C H O I C E
COOPERATIVE
GAME
F i r s t of all, we d e s c r i b e t h e multichoice c o o p e r a t i v e g a m e ( M C G ) i n t r o d u c e d ill [1]. Let N -= {1, 2 , . . . , 'n} be t h e set of players, M~ = {0, 1, 2 . . . . ,m4} t h e set of a c t i v i t y levels of player i ¢ N . Vv'e a s s u m e t h a t m i = L, L ¢ R +, for all i c N as in [1]. A coalition ill M C G is d e n o t e d b y a v e c t o r s = ( S l , . . . , s n ) , w h e r e s, ¢ M i , i c N . shows a c t i v i t y level of p l a y e r i in t h e c o a l i t i o n s. If a p l a y e r does not c o o p e r a t e , his level of a c t i v i t y is set at zero. Hence,
0893-9659/00/$ - see fl'ont matter @ 2000 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(99)00216-5
Typeset 1)y ,4.~S-'I~F~\
96
D.A.
AYOSHIN
AND
T.
TANAKA
the coalition in which no player participates is specified by the zero vector 0 = ( 0 , . . . , 0). We denote the set of all coalitions by M = M1 x ..- x Mn. T h r o u g h o u t this paper, a coalition s A t = ( m i n { s l , tl}, min{s2, t 2 } , . . . , min{s~, t~}) is considered as the intersection of coalitions s and t, and a coalition s Vt = (max{s1, tl}, max{s2, t 2 } , . . . , max{sn, tn}) is a d m i t t e d as the union of s and t. A superadditive function v: M --* R 1 with v(O) = 0 is called a characteristic function of an M C G . It is easily seen t h a t v(m), m = ( m l , . . . , ran), is the m a x i m a l value of the characteristic function. We denote M C G by G(v, N). Consider an ( m + 1) x n-dimensional payoff m a t r i x ~ = (~ji) distributing v(m) a m o n g all players and their activity levels. A c o m p o n e n t ~ji shows the increase in payoff to player i when he changes his activity from level j - 1 to level j. It is said t h a t the payoff m a t r i x ~ is efficient if ~-~i=1 ~ j £m io ~ji = v(m) and it is level increase rational if ~ = 0 ~ji _> v ( ( 0 , . . . , 0, si, 0 , . . . , 0)), where i E N , s~ E M+ An efficient and level increase rational payoff m a t r i x is called imputation and considered as a solution of G(v, N). Let I(v, N) be the set of all i m p u t a t i o n s in G(v, N). We shall say t h a t the set C(v, N) = {f E I(v, N) { ~ s~#o ~'~jLO ~ ~ji >- V(S) for all s E M } is the core of G(v, N). 3.
TOTAL
CONVEXITY
In [2], the following procedure of construction was proposed for the Shapley value. Suppose t h a t a given coalition s E M is formed step-by-step, starting from the zero coalition 0 = ( 0 , . . . , 0). On each stage of the procedure, one of the players has to increase his activity level by 1. Thus, the coalition s E M will be created after k(s) = ~ i : ~ # 0 si steps, i.e., each player i E N will reach his level of activity s~ in s. Define M + = { ( i , j ) ] i E N , j E M~ \ {0}}. An admissible order is a bijection w : M + --~ { 1 , 2 , . . . ,~-~ieNmi} satisfying w((i,j)) < w((i,j + 1)) for all i c ~ N and j E {1, 2 , . . . , mi - 1}. T h e n u m b e r of the admissible orders for G(v, N) is
-
1-I
iEN
Take an a r b i t r a r y coalition s E M and then fix a player l E N , st ¢ 0. Suppose t h a t by an admissible order w, the given coalition s is created after the first k(s) steps, with the player l completing the formation of s. T h e n u m b e r of such orders is
al(s)
=
11 i:(sIs~- 1)~#0
(si!)
H ((L - si)!) i~N
where s I sl - 1 = ( S l , . . . , s t - l , s l - 1, sl+l . . . . ,sn). One can have t h a t f~l(s) = 0 if sl = 0. Nouweland et al. [2] showed t h a t ¢ = {¢j~}, where i = 1 , . . . , n, j = 0 , . . . , m,, and ¢¢~ =
~
ads)
a ( m ) Iv(s) - v(s I si - 1)]
(3.1)
8:Si=3
is the Shapley value of G(v, N). Let G s, s E M, be a s u b g a m e of the g a m e G(v, N). Suppose t h a t the characteristic function v s o f G s is the restriction of v to the set M s = {t E M I 0 < t i < si for e a c h i E N}. Denote the Shapley value of G s by Cs = {¢~}, i = 1 , . . . , N , j = 0 . . . . , si. We shall say t h a t according to the Shapley value, a coalition r E M s, s E M obtains ¢(r) = ~i:r~¢0 ~ 0 CJi in the g a m e G(v, N)
Convex Multichoice Games
97
and ~,s(~.) = E,:,-,#0 ~-:~jL0 ~' Cji in the s u b g a m e G ~, Now we will find a condition for 0 t,o be in
C(~', N). I n t r o d u c e functions di(s) = v(s) - v(s[si - 1), s E M , si - 1 _> O, i E N. Let t be a particular coalition in M . From (3.1), we have t, i
i:ti#O j = 0
f~(s)
t,
&(s) : i:tigiO E jE= 0 s:si=j E -9.(m)
=E
E
(3.2)
fh (s)
i:ti¢O s:(sAt)i
Applied Mathematics Letters
Applied Mathematics Letters 13 (2000) 95 98
www.elsevier.n l/loeate/aml
The Shapley Value in Totally C o n v e x Multichoice G a m e s D.
A. AYOSHIN AND T.
TANAKA
Department of Information Science, Faculty of Science, Hirosaki University Department of Mathematical System Science, Faculty of Science and Technology Hirosaki Uifiversity Hirosaki 036-8.561, Japan <srdimaOsi><sltana~cc>. hirosaki-u, ac. jp
(Received March 1999; revised artd ac:cepted April I999)
Abstract--In this paper, we introduce a class of totally convex multichoice cooperative games a.nd prove that the Shapley value of such games is always in the core. @ 2000 Elsevier Science I~td. All rights reserved. K e y w o r d s - - M u l t i c h o i c e game, Shapley value, Total convexity, Core, Coalition.
1. I N T R O D U C T I O N Hsiao a n d R a g h a v a n [1] i n t r o d u c e d a class of multichoice c o o p e r a t i v e g a m e s and tbund its S h a p l e y value using an a x i o m a t i c a p p r o a c h . Later, Nouweland et al. [2] d e t e r m i n e d t h e S h a p l e y value ~br multichoice c o o p e r a t i v e g a m e s following its p r o b a b i l i s t i c i n t e r p r e t a t i o n . Howewu. t h e w~lues o b t a i n e d by t h e s e two m e t h o d s are q u i t e different. In our p a p e r , while avoiding the p r o b l e m of i n c o n s i s t e n c y of t h e S h a p l e y value between H s i a o - R a g h a v a n and Nouweland, we consider a n e c e s s a r y a n d sufficient c o n d i t i o n for t h e S h a p l e y vahle by Nouweland to be in t h e core of a m u l t i c h o i c e c o o p e r a t i v e game. It is well k n o w n t h a t in t h e class of c o o p e r a t i v e g a m e s in tile c h a r a c t e r i s t i c function form, the S h a p l e y value is in t h e core if t h e c h a r a c t e r i s t i c function is either convex [3], aw~rage convex [4], or t o t a l l y convex [5]. T h e l a t t e r p a p e r shows t h a t t h e class of t o t a l l y convex g a m e s inchnles tha.t of a v e r a g e convex games. W e discuss c o n d i t i o n s for t h e S h a p l e y value to be in t h e core tot t h e ('lass of multichoice c o o p e r a t i v e games.
2. M U L T I C H O I C E
COOPERATIVE
GAME
F i r s t of all, we d e s c r i b e t h e multichoice c o o p e r a t i v e g a m e ( M C G ) i n t r o d u c e d ill [1]. Let N -= {1, 2 , . . . , 'n} be t h e set of players, M~ = {0, 1, 2 . . . . ,m4} t h e set of a c t i v i t y levels of player i ¢ N . Vv'e a s s u m e t h a t m i = L, L ¢ R +, for all i c N as in [1]. A coalition ill M C G is d e n o t e d b y a v e c t o r s = ( S l , . . . , s n ) , w h e r e s, ¢ M i , i c N . shows a c t i v i t y level of p l a y e r i in t h e c o a l i t i o n s. If a p l a y e r does not c o o p e r a t e , his level of a c t i v i t y is set at zero. Hence,
0893-9659/00/$ - see fl'ont matter @ 2000 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(99)00216-5
Typeset 1)y ,4.~S-'I~F~\
96
D.A.
AYOSHIN
AND
T.
TANAKA
the coalition in which no player participates is specified by the zero vector 0 = ( 0 , . . . , 0). We denote the set of all coalitions by M = M1 x ..- x Mn. T h r o u g h o u t this paper, a coalition s A t = ( m i n { s l , tl}, min{s2, t 2 } , . . . , min{s~, t~}) is considered as the intersection of coalitions s and t, and a coalition s Vt = (max{s1, tl}, max{s2, t 2 } , . . . , max{sn, tn}) is a d m i t t e d as the union of s and t. A superadditive function v: M --* R 1 with v(O) = 0 is called a characteristic function of an M C G . It is easily seen t h a t v(m), m = ( m l , . . . , ran), is the m a x i m a l value of the characteristic function. We denote M C G by G(v, N). Consider an ( m + 1) x n-dimensional payoff m a t r i x ~ = (~ji) distributing v(m) a m o n g all players and their activity levels. A c o m p o n e n t ~ji shows the increase in payoff to player i when he changes his activity from level j - 1 to level j. It is said t h a t the payoff m a t r i x ~ is efficient if ~-~i=1 ~ j £m io ~ji = v(m) and it is level increase rational if ~ = 0 ~ji _> v ( ( 0 , . . . , 0, si, 0 , . . . , 0)), where i E N , s~ E M+ An efficient and level increase rational payoff m a t r i x is called imputation and considered as a solution of G(v, N). Let I(v, N) be the set of all i m p u t a t i o n s in G(v, N). We shall say t h a t the set C(v, N) = {f E I(v, N) { ~ s~#o ~'~jLO ~ ~ji >- V(S) for all s E M } is the core of G(v, N). 3.
TOTAL
CONVEXITY
In [2], the following procedure of construction was proposed for the Shapley value. Suppose t h a t a given coalition s E M is formed step-by-step, starting from the zero coalition 0 = ( 0 , . . . , 0). On each stage of the procedure, one of the players has to increase his activity level by 1. Thus, the coalition s E M will be created after k(s) = ~ i : ~ # 0 si steps, i.e., each player i E N will reach his level of activity s~ in s. Define M + = { ( i , j ) ] i E N , j E M~ \ {0}}. An admissible order is a bijection w : M + --~ { 1 , 2 , . . . ,~-~ieNmi} satisfying w((i,j)) < w((i,j + 1)) for all i c ~ N and j E {1, 2 , . . . , mi - 1}. T h e n u m b e r of the admissible orders for G(v, N) is
-
1-I
iEN
Take an a r b i t r a r y coalition s E M and then fix a player l E N , st ¢ 0. Suppose t h a t by an admissible order w, the given coalition s is created after the first k(s) steps, with the player l completing the formation of s. T h e n u m b e r of such orders is
al(s)
=
11 i:(sIs~- 1)~#0
(si!)
H ((L - si)!) i~N
where s I sl - 1 = ( S l , . . . , s t - l , s l - 1, sl+l . . . . ,sn). One can have t h a t f~l(s) = 0 if sl = 0. Nouweland et al. [2] showed t h a t ¢ = {¢j~}, where i = 1 , . . . , n, j = 0 , . . . , m,, and ¢¢~ =
~
ads)
a ( m ) Iv(s) - v(s I si - 1)]
(3.1)
8:Si=3
is the Shapley value of G(v, N). Let G s, s E M, be a s u b g a m e of the g a m e G(v, N). Suppose t h a t the characteristic function v s o f G s is the restriction of v to the set M s = {t E M I 0 < t i < si for e a c h i E N}. Denote the Shapley value of G s by Cs = {¢~}, i = 1 , . . . , N , j = 0 . . . . , si. We shall say t h a t according to the Shapley value, a coalition r E M s, s E M obtains ¢(r) = ~i:r~¢0 ~ 0 CJi in the g a m e G(v, N)
Convex Multichoice Games
97
and ~,s(~.) = E,:,-,#0 ~-:~jL0 ~' Cji in the s u b g a m e G ~, Now we will find a condition for 0 t,o be in
C(~', N). I n t r o d u c e functions di(s) = v(s) - v(s[si - 1), s E M , si - 1 _> O, i E N. Let t be a particular coalition in M . From (3.1), we have t, i
i:ti#O j = 0
f~(s)
t,
&(s) : i:tigiO E jE= 0 s:si=j E -9.(m)
=E
E
(3.2)
fh (s)
i:ti¢O s:(sAt)i
