On Weighted Shapley Values

8/13/2019 On Weighted Shapley Values

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Internat ional Journal of Game Theory, Vol. 16, Issue 3, pag.e 205 -2 22

n W eighted Shapley ValuesB y E K a la i I a n d D S a m e t 2

Ab s t r a c t Nonwmmetric Shapley values for coalitional form games with transferable utilityare studied. The nonsymmetries are modeled through nonsymmetric weight systems defined onthe players of the games. It is shown axiomatically that two families of solutions of this type arepossible. These families are strongly related to each other through the duality relationship ongames. While the first family lends itself to applications of nonsymmetr ic revenue sharing problemsthe second family is suitable for applications of cost allocation problems. The intersection ofthese two families consists essentially of the symmetric Shapley value. These families are also char-acterized by a prObabilistic arrival time to the game approach. It is also demonst rated that lackof symmetries may arise naturally when players in a game represent nonequal size constituencies.

1 I n t r o d u c t i o n

The Shap l ey 195 3b) va lue is cons i de r ed by m a ny gam e t heo r i s t s and econom i s t s a st he m a i n so l u t i on con cep t t o coope r a t i ve gam es w i t h t r ans f e r ab le u t i l i t y . These gam esand t h i s so l u t i on concep t have been app l i ed t o p r ob l em s o f r evenue sha r i ng and cos ta l loca t ions .

O n e o f t h e m a i n a x i o m s t h a t c h a r a c te r iz e t h e S h a p l e y v a lu e is o n e o f s y m m e t r y .The unde r l y i ng m o t i va t i on f o r u s i ng th i s ax i om i s t he a s sum pt i on t h a t exce p t f o r t hep a r a m e t e r s o f t h e g a m e s , t h e p l a y e r s ar e c o m p l e t e l y s y m m e t r ic . H o w e v e r , i n m a n yapp l i ca t i ons t h i s a s sum pt i on o f sym m et r y s eem s un r ea l i s t i c f o r t he s i t ua t i on t ha t i sbe i ng m ode l ed and t he u se o f nonsy m m e t r i c gene r a li z a ti ons o f t he Shap l ey va l ue w asproposed in such eases .

1 Ehud Kalai, Department of Managerial Economics and Decision Sciences, J. L. Kellogg GraduateSchool of Management, Northweste rn University, 2001 Sheridan Road, Evanston, Illinois 60201,USA.2 Dov Samet, Department of Economics , Bar-Ilan Universi ty, Ramat-Gan, Israel.Financial support for this research was granted by the National Science Foundation's EconomicsDivision, Grant No. SOC-7907542.

002 0-7 276 /87 /3/ 205 -22 2 $2.$0 9 1987 Physiea-Verlag, Heidelberg


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206 E. Kalai and D. Sam et

Consider , for exam ple , a s i tu a t ion involving tw o p layers . I f the tw o players co-ope ra te in a jo i n t p ro jec t t he y can gene ra te a un i t p ro f i t w h ich i s t o be d iv ided be -tween them . On the i r ow n the y can gene ra te no p ro f i t . The Sh ap ley va lue v iews th i ss i tua t ion a s be ing sym met r i c and wou ld a lloca te the p ro f i t f rom coop e ra tive equa llybe tween the two p laye rs . Howeve r , i n some app l ica t ions l a ck o f sym m et ry m ay bepre sen t in the unde r ly ing s i tua t ion . I t m ay be , fo r exam ple , t ha t a g rea te r e f fo r t i sneeded on the pa r t o f p laye r one than o n the pa r t o f p l aye r two in o rde r fo r theprojec t to succeed. Another example a r ises in s i tua t ions where p layer one representsa l arge cons t i tuency wi th m any ind iv idua ls and p laye r two ' s con s t i tuenc y i s comp osedof a sma l l num ber o f ind iv idual s . O the r exam ple s whe re la ck o f sy m m et ry i s p re sen tcan ea s i ly be cons t ruc ted fo r p rob lem s o f cos t a l loca t ions . A l so , l a ck o f sym m et r iym ay ar ise wh en d i f fe re nt barga in ing abi l it ies for d i f fe re nt p layers a re m odel led .The fami ly o f w e igh ted S hap ley values was in t rodu ced by Shap ley (1953a ) . E achweighted Shapley va lue assoc ia tes a posi t ive w eight wi th each p layer . These weightsa re the p ropor t ions in wh ich the p laye rs sha re in unan imi ty games . The symmet r i cShapley value is the special case where al l the weights are the same. In this paper weex ten d the no t ion o f we ig h t s to we igh t sys t em s enab l ing a we igh t o f z e ro fo rsom e p laye rs . We then de f ine in S ec t ion 2 the n o t ion o f the we igh ted Sha rp ley va luewi th a g iven we igh t sys t em and re la t e i t t o a p rocedure o f d iv idend a l loca tion tha twas proposed by Harsanyi (1959) (see a lso Owen 1982) for games wi thout s idepay-men t s . In Sec t ion 3 we g ive an equ iva len t de f in i t ion o f the w e igh ted Shap ley va lueby random orde rs wh ich gene ra l i z e the random orde r approach to the symmet r i cShap ley value . In Sec t ion 4 we g ive an a x iom a t i c cha rac te r iza t ion o f the fam i ly o fweighted Sh apley values - tha t i s , we provide a li s t of proper t ies o f a so lu t ion whichis sa t i sf ied by and on ly by weigh ted Shapley va lues.

Shap ley (1981) p roposed a lso a f am i ly o f w e igh ted cos t a l loca t ions schemes andaxiomat ica l ly charac te r ized , for exogenously g iven weights , the schemes assoc ia tedwit h these w eights . This fam ily of so lu t ions i s re la ted to the weigh ted Shap ley va luesby dua l i ty . We exp lo re fu r the r the re l a tionsh ip be twee n the se two fami li e s, p rov idean ax iom a t i za t ion o f the l a t t e r f ami ly (wh ich does no t u se the w e igh ts exp l i c i t ly inthe ax ioms a s Shap ley ' s ax ioms do ) an d ge t a s a r e su lt an ax iom a t i za t ion o f the sym-me t r i c Shap ley value wh ich does no t u se the sy m m et ry ax iom.

Owen (1968 and 1972) sho wed th a t we igh ted Shap ley values can be com pute dby a d i agona l fo rm ula p rov id ing ano the r in t e rp re t a t ion o f the we igh t s a ssoc ia t edwi th the p l aye rs . In Sec t ion 6 we ex tend the d iagona l fo rm ula fo r we igh t sys t emsand a l loca tion schemes .

F ina l ly , we no te tha t i f o ne accept s the ax iom s in Sec t ion 4 , one is ob liged to usea w e igh ted Shap ley va lue bu t n o recom m enda t ion o f the we igh t s is imp l i ed by theax ioms . The we igh t s shou ld be de te rmined b y cons ide r ing such fac to r s a s ba rga in ingabi l i ty , pa t ience ra tes , or past exp er ience . In S ec t ion 7 w e exam ine cases in w hichthe s ize o f the p layers (where the p layers themselves a re groups of indiv idua ls) a reappro pr ia te weights for the p layers .


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On Weighted Shapley Values 2 7

2 W e i g h t e d S h a p l e y V a l u e s

Le t N be a f i n it e s e t, the m em b er s o f wh i ch wi ll be ca l led players . Subs e t s o f N a r eca l l ed coal i t ions and N i s ca l l ed the grand coal i t ion . Se t INI = n . Fo r each co a l i t ionS w e d e n o t e b y E s t he t S I -d i m ens iona l Euc l i d i an space i ndexed b y t he p l aye r s o f S .A g a m e v i s a f u nc t i on wh i ch a s signs t o each coa l i t i on a rea l nu m b er and i n pa r t icu l a rv ~ ) = 0 . T h e s e t o f a ll ga m e s i s d e n o t e d b y F . A d d i t io n o f t w o g a m e s v a n d w i n Fi s de f i ned by v + w ) S ) = v S ) + w S ) f o r e ach S and m u l t i p l ic a t i on o f the gam e v bya sca la r a i s def ined by a v ) S ) = ~ v S ) f o r e ach coa l i t i on S . Th us F i s a vec t o r space .Fo r each coa l i t i on S t he u n a n i m i t y g a m e o f t h e c o a l it io n S , u s , is de f i ned by u s T ) =1 i f T _ 3 S a n d u s T ) = 0 o t he r wi s e . I t i s we l l kno w n t h a t t he f am i l y o f gam es{U s)s c_ N is a basis for F.

Th e Shap l ey va l ue r is t he l inea r f unc t i on r F -->EN , w h i c h f o r e a c h u n a n i m i t y1g a m e U s i s d e f i n e d b y ~i U s ) = ~ i f i E S and ~i U s ) = 0 o therwise . In tu i t ive ly , int he gam e U s any coa l i t i on wh i ch con t a i n s S can s p l i t one un i t be t ween i t s m em ber sa n d t h e r e f o r e p l a y e rs o u t s i d e S d o n o t c o n t r i b u t e a n y t h i n g t o t h e c o a l it io n th e yjo in . H ence , O~i u~) = 0 for i q~S. T h e m e m b e r s o f S o n t h e o t h e r h a n d s p li t e q u a l lyt he one un i t be t ween t hem s e l ves . S i nce {U s)s c_ N i s a basis to F and ~ is l inear , r i sde f i ned f o r a l l t he gam es . A w e i gh t ed Sh ap l ey va l ue gene r a li z e s t he Shap l ey va lue bya l lowi ng d i f f e r en t w ays t o s p l it o ne un i t be t w een t he m em ber s o f S i n u s . We p r e -s c r ibe a vec t o r o f pos i ti ve we i gh t s X = Ca i )t ~ n and i n each U s p l aye r s s p li t p r op or -t i ona l l y t o t he i r w e i gh ts . We wan t t o a l low s om e p l aye r s t o have we i gh t z e r o . Th i sm ean s t ha t i f t he y s p l i t one u n i t w i t h p l aye r s w ho have pos i t ive we i gh ts , t hey ge tze r o . B u t t he n we have t o s pec i f y how t he s e ze r o - we i gh t p laye r s stili t a un i t wh en nopos i ti ve - we igh t p l aye r i s wi t h t hem . Th i s b r i ngs u s t o t he f o l lowi ng l ex i cog r aph i ede f i n i t i on o f a we i gh t s ys t em .

A w e i g h t s y s t e m co i s a pa i r 0~, E ) wh ere X EEN++ and Z = Sz , . . . , S in ) i s anor de r e d p a r t i t i on o f N . A we i gh t s ys t em co = ~ , Y ,) is c a l led s imple i f Z = N ) . T h ewei gh t ed Shap l ey value w i t h w e i gh t s y s t em co is t he l i nea r m ap ~ o : P -* E N wh i chis d e f i n e d f o r e a c h u n a n i m i t y g a m e U s as fo l lows .

Le t k = m ax ] I S n S =/= 0) an d d en ot e S = S n Stc. T he n

r i u s ) = F , x i

f o r i E S a n d O o ~ ) i u s ) = 0 o therwise .I n o t h e r w or ds , t he w e i gh ts o f p l aye r s i n S i are 0 wi t h r e s pec t t o p l aye r s i n S

wi t h ] > i . T he po s i ti ve we i gh t s o f p l aye r s i n Si are u s e d o n l y f o r g a m e s u s s u c h t h a tno p l aye r f r o m S j w i t h ] > i is i n S . Obs e rve t ha t ~ is t he s ym m e t r i c ) Shap l ey va l ue


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2 0 8 E K a l a i a n d D S a m e t

i f and on ly i f co = (k , (N) ) and ~ i s p ropor t iona l to the vec to r (1 , 1 , . . . , 1 ). Ano the rcom puta t ion p rocedure o f r is a long the l ine s p roposed by Harsany i (1959) . Inth is proced ure each co a l i t ion S a l loca tes d iv idends to i t s m em bers a f te r a l l the propersubcoal lt io ns of S have done i t . T he d iv idend a l loca t ion proceeds as fo l lows . We f i rs ta l loca te to each p laye r i h i s wor th v ( { i ) ) . Suppose tha t a l l the coa l i t ions of s ize k orless have a l ready a l loca ted d iv idends and le t S be a coa l i t ion of s ize k + 1 . Denoteb y z ( S ) t he sum o f the d iv idends tha t mem bers o f S we re pa id by p rope r subcoa l i tionso f S . T h e n v (S ) - z ( S ) (which is poss ib ly 0) i s the amount tha t S wi l l a l loca te to i t smem bers . To de te rm ine how the am ou n t i s d iv ided , we de f ine the coa l i t ion S (wh ichis a subse t o f S) as above . The m emb ers o f S wi ll d iv ide v ( S ) - z ( S ) i n p r o p o r t i o n t othe i r we igh t s wh i l e the re s t o f the p laye rs in S ge t no th ing . The to ta l amoun t tha teach p layer accum ula ted a t the en d of the proce dure ( i .e ., a f te r N a l loca ted i t s d iv i-dends) i s exac t ly ( r To see th i s one can ea s ily p rove by induc t ion tha t i f v =

a s U s t hen fo r e ach coa l i t ion S , v ( S ) - z ( S ) = a s and the d iv idend a l loca t ion iss_clvthe re fo re the a l loca t ion o f the coe f f i c ien t s a s in acco rdance wi th the de f in i t ion o f~bto. A gene ra l i z at ion o f th i s p rocedure fo r the com puta t ion o f the Shap ley value waspropo sed by Maschler (1982 ) . The sam e genera l iza t ion applies a lso for Cto . We s ta r tby choos ing any coa l i t ion S wi th v ( S ) ~ 0 and a l loca t ing v ( S ) accord ing to ~ . In l a t e rs te p s o f t h e c o m p u t a t i o n w e c h o o s e f o r d i v id e n d a l lo c a t io n a n y S f o r w h i c h v ( S ) -z ( S ) r 0 where z ( S ) i s the sum of th e d iv idends pa id for the p layers in S by subcoal i-t ions o f S wh ich a l ready a l loca ted d iv idends (no t i ce tha t a coa l i t ion may be chosensevera l t imes in th is procedure) . The procedure ends when v (S ) - z ( S ) = 0 for a l l thecoa l i t ions . The pr oo f tha t such p rocedure a lways te rmina tes and g ives indee d ~oJ isthe same as in Maschler (1 982) . Harsanyi (1959 ) d ef ined a lso a proced ure o f weigh tedd iv idend a l loca t ion fo r games w i thou t s idepaymen ts . A fam i ly o f so lu tions ob ta inedby the se p rocedure s was ax ioma t i zed by Ka la i and Sam e t (1985) . They re fe r to the sesolu t ions as ega l i ta r ian , and i t i s shown there th a t the re s t r ic t ion of each ega l i ta r iansolu t ion to games wi th s ide paym ents i s a weigh ted Shapley va lue .In the n ex t s ec t ion we p rov ide a p robab i l is t ic approach to the we igh ted Shap leyvalues , one which genera l izes the probabi l i s t ic formula of the (symmetr ic ) Shapleyv a l u e

3 P r o b a b i li s t i c D e f i n i t i o n o f W e i g h t e d S h a p l e y V a l u e s

Le t ~ (S ) deno te the s e t o f a l l o rde rs R o f p laye rs in the co a l i t ion S . Fo r an o rde r Ri n R ( N ) w e d e n o t e b y BR i the s e t o f p laye rs p reced ing i i n the o rde r R . Fo r anorde red pa r t i t ion Z = ( 1 . . . . . S i n ) of N , ~ i s the s e t o f o rde rs fo r N in wh ich a ll


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On W eighted Shap ley Values 209

t he p l aye r s o f S i p r eced e t hos e o f S i+l f o r i = 1 , . . . , m - 1 . Each R i n ~ can bedes c r i bed a s R = ( R t . . . . , R m ) wh ere R i E IR(Si) , i = 1 , . . . , m.

Le t I S I = s and l e t k E Es + . We a s s oc ia t e w i t h ) , a p r obab i l i t y d i s t r i bu t i on Pxo v e r ~ ( S ) . F o r R = ( i l . . . . . i s) i n ~ ( S ) , w e d e fi n e

e x R ) = 11i=1 iXik

r

One wa y t o ob t a i n t h i s p r obab i l i t y d i s tr i bu t i on i s by a r rang i ng t he p l aye r s o f S in ano r de r , s t a r t i ng f r om t he end , s u c h t h a t t h e p r o b a b i l i t y o f a d d in g a p l a y e r t o t h ebeg i nn i ng o f a pa r t i a l l y c r ea t ed l i ne i s t he r a t i o be t ween h i s we i gh t and t he t o t a lwe i gh t o f t he p l aye r s o f S t h a t a r e no t ye t in t he l i ne .

W i th each we i gh t s ys t em co = ( ~ , Z ) w he r e ~ = ( S l . . . . . S i n ) we a s s oc i a t e a p r ob -ab i l i ty d i s t r i bu t i on P ~ ove r ~ ( N ) a s fo l lows . The d i s t r i bu t i on P r o van is hes ou t s i de

m~ z , a n d f o r R = ( R I , . . . , R m ) in ~ , Pco(R) = H whe r e k s i is t he p r o j ec -t io n o f X o n E s i . i= 1 P x s i ( R i ) 'F o r a g iv e n g a m e v a n d o r d e r R i n ~ ( N ) t h e c o n t r i b u t i o n o f p l ay e r i is G ( v , R ) =

v ( B R , i u { i ) ) - v ( B R ' i ) . We pr ove now :

Theorem 1: F o r e a c h p l a y e r i E N , wei gh t s ys t em 6o, and gam e v ,

~ , ~ ) ~ v ) = e ~ , ~ ( G ( v , ) )w h e r e t h e r ig h t h a n d s id e i s th e e x p e c t e d c o n t r i b u t i o n o f p l a y e r i w i t h r e s p ec t t o t h ep r ob ab i l i t y d i s t r i bu t ion P r o .

Proof . We say tha t i is last fo r S in th e ord er R i f i E S and S _C B g , i U {i} . Fo r a g ivenor de r R and p l aye r i t he co a l i t ion N~( BR ' i W {i}) i s ca l l ed t h e t a il o f i i n R. A coa l i -t i on T i s s a id t o be a t a i l f o r R i f f o r s o m e i , T i s a t ai l o f i i n R .

Le t r = (~, ( 1 , . . . , S in) ) be a w e i gh t s ys t em an d l e t S be a coa l i t ion . Deno t e k =m ax {1 IS n S 4= 0} _and S = S n S k. We sh ow tha t fo r ea ch i E S \ S , e ~ ( i i s las t fo rS ) = 0 , f o r e a c h i E S , P ~ ( i is l a st f o r S ) > 0 , a n d f o r e a c h ] , i E S :

P ~ ( i i s l as t for S) Xeto (J is last fo r S) ~,/ * )


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210 E. Kalai and D. Samet

I n d e e d , i f i E S \ S t hen i n o r de r t o be l a s t f o r S , i m us t be p r eceded by p l aye r s f r omSk wh i ch occu r s wi t h p r oba b i l i t y 0 . Now s uppos e i , j E S . L e t A = ( U S t ) \ S t h e nw e ha ve t t> k

P c o ( i i s l a s t f o r S ) = E P ~ ( T i s a t a i l o f i )T C A)2 P ~ ( T i s a t a i l o f i l T i s a t a i l ) P ~ ( T i s a t a i l )TC_A

= ~ , X i ( I / Y-, X r ) P t o ( T i s a t a i l )TC_A rE N\ T) nS k= X i h wh ere h i s pos i t ive .

Simi la r ly , P ~ ( ] i s l as t for S) = Xlh and (*) fo l lows .N o w c o n s i d e r t h e g a m e u s . The con t r i bu t i on o f i ~ S is 0 i n each o r de r and t hus

E p ~ o ( C i ( u s , ) ) = 0 = ( r T h e c o n t r ib u t i o n o f i E S in th e ord er R is 1 i f i i sl as t f o r S i n R , a n d i s 0 o th e r w i s e. I f i E S \ S t h e n

E p c o ( C i ( u s , )) = P t o ( i i s las t fo r S) = 0 = ( r

I f i , j E S , t hen

E p ~ ( C i ( u s , .) ) P ~ ( i i s l as t for S) Xe e ~ C i u s , )) e ~ ( j is l a st f o r s ) x /

B u tY_, E e , o q u s , . ) ) = E ~ , r ,i E N i E N C i ( u s , ) ) = Epr = 1.

O n t h e o t h e r h a n d , a s w e h a v e sh o w n :

Y~ l rp ~ (q (u s , ) )= ~_ F e ~ ( G ( u s , ) )i E N i ~ 8and t h e r e f o r e f o r e ach i E

g p ~ ( G ( u s , ) ) = iY.,_ k jiE

= O ~ ) i u s ) .


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On Weighted Shapley Values 2

Clear ly E p ~ C i v , - ) ) is a l i nea r m ap f rom r t o E and so i s r 162 and t he r e fo re ,s in c e t h e y c o i n c id e o n t h e b a s is c o n s is t in g o f t h e u n a n i m i t y g a m e s , t h e y c o i n ci d eon P . Q .E .D .

4 A n A x i o m a t i c C h a r a c t e r i z a t i o n o f t h e W e i g h t e d S h a p l e y V a lu e s

A so lu t i on fo r I~ i s a f unc t i o n ~ f ro m F to E ~v. Fo r a coa l i t i on S we de no te b y rt he sum ~ ~ i v ) . A coa l i t i on S i s s a id t o be a coali t ion of partners or a p - t y p eiEScoa l i t i on , i n t he gam e v , i f f o r e ach T C S and eac h R C_N \ S , v R U T ) = v R ) .

C o n s i d e r n o w t h e f o l l o w i n g a x i o m s i m p o s e d o n ~ . F o r a l l g a m e s v , w E P :1 . E f f i c i ency . ~ v ) N) = v N)2 . Add i t i v i t y . O v + w ) = r + r3. Posi t iv i ty . I f v i s m o n o t o n i c i .e . , v T ) >>-v S ) f o r e a c h T a n d S s u c h t h a tT _3 S) th en ~ v) i> 0 .4 . D u m m y P la y er . I f i i s a d u m m y p l a y e r i n th e g a m e v i .e . , f o r e a c h S, v S tJ

i}) = v S ) ) t h e n r = 0 .5. Partnership. I f S i s a p - t y p e c o a l i t io n i n v t h e n r = r f o r

each i E S .A x i o m s 1 - 4 a r e s t a n d a r d i n v a r i o u s a x i o m a t i z a t i o n s o f t h e S h a p l e y v a lu e . I n o r d e r t oe x a m i n e a x i o m 5 c o n s i d e r f i r st t h e c h a r a c t e r o f a p - t y p e c o a l it i o n . A p - t y p e c o a li t io nS in t he gam e v behav es i n a c e r t a in s ense l ike one i nd iv idua l i n t he g am e v s i nce a lli t s subcoa l i t i ons a r e c om ple t e ly pow er l e s s . I n t h i s s ense S behav es i n t e rna l l y t he s amein V as in Us. One can exp ec t t he r e fo re t ha t S w i ll t ake i t s sha re i n t he gam e v a s onei n d iv i d u a l a n d t h e n b a r g a in o v e r t h is s h a r e. T h i s is t h e c o n t e n t o f a x i o m 5 . ~ v ) S ) u si s a u n a n i m i t y g a m e i n w h i c h t h e m e m b e r s o f S b a rg a i n o v e r ~ v ) S ) w h i c h i s w h a tthe y r ece ived t oge the r i n ~ v ) . ~ i ~ v ) S ) u s ) i s wha t i r e ce ive s a s a r e su l t o f t h i sba rga in ing . Th i s shou ld be exa c t l y w ha t he r ece ived i n v .

T h e o r e m 2 : A s o l u t i o n ~ s a ti sf ie s a x i o m s 1 - 5 i f a n d o n l y i f t h e r e e x i s t s a w e i g h ts y s t e m c o s u c h t h a t ~ i s t h e w e i g h te d S h a p l e y v a l ue ~ .


8/18


Proof W e f ir s t s h o w t h a t f o r w = (X , ( S l , . . ., S i n ) ) , r s a ti sf ie s a x i o m s 1 - 5 . T o p r o v ee f f i c i e n c y w e o b s e r v e t h a t f o r e a c h v a n d R , ~ C i ( v, R ) = v ( N ) a n d t h e r e f o r ei ~ N

r = ~ ( ~ ) i ( v ) = ~ E p ~ G v , )) =E p~ ( ~ N C ~ v , ) ) = v N ) .i E N i ~ NT h e a d d i t iv i t y o f O w f o l lo w s f r o m t h e a d d i ti v i ty o f E p w a n d C / . T h e p o s i t i v i t y a n dt h e d u m m y p l a y e r a x i o m s fo l l o w a ls o i m m e d i a t e ly . T o c h e c k th e p a r t n e rs h i p a x i o ma s s u m e t h a t S i s a p - t y p e c o a l i t i o n i n a g a m e v . O b s e rv e f i rs t t h a t s i n ce S is o f p - t y p ea p l a y e r i i n S m a k e s a n o n z e r o c o n t r i b u t i o n i n a n o r d e r R o n l y i f i i s l a s t f o r S in R .N o w l e t k = m a x ( ] IS N S =~ ~b} and le t S = S c~ S k . Fo r i E S \ S the o rd ers in w hic hi i s l a s t f o r S h a v e p r o b a b i l i t y z e r o a n d t h e r e f o r e (r = O . F o r i E S we h a v e :

( r = Ee ~ ( G ( v , - ))= E E p w ( C / ( v , ) )l T is a t a i l o f i )P to (T i s a t a i l o f i )T C_ N \ S

B u t

E p t o ( C / ( v , ) l T i s a t ai l o f i )

i s t h e s a m e f o r e v e r y i E S s i n c e S i s o f p - t y p e . M o r e o v e r P ~ ( T i s a ta i l o f / ) i s o f t h ef o r m X i h ( T ) w h e r e h ( T ) i s t h e s a m e f o r e a c h i E S .

T h u s , t h e r e e x is ts a c o n s t a n t K s u c h t h a t f o r e v e r y i E S , ( ~ ) i ( v ) = X i K w h i c hs h o w s t h a t O w s a t i s f i e s t h e p a r t n e r s h i p p r o p e r t y .

N o w l e t ~ b e a s o l u t i o n w h i c h s a t is f ie s a x i o m s 1 - 5 a n d w e w i ll s h o w t h a t f o rs o m e we ig h t s y s t e m c o, ~ = r W e d e f in e f i r s t a we ig h t s y s t e m o3 = ( X , ( $ 1 . . . . . S r n ) )a s f o l lo w s . T h e c o a l i t i o n $ 1 c o n t a i n s a l l p l a y e r s i f o r w h i c h 4~i(UN ) 4= O, S l :/= r b e -c a u s e o f t h e e f f i c i e n c y a x i o m ) . 3 W e d e f i n e X i = r f o r e ac h i E S 1 A s s u m i n g th a tt h e c o a l i t i o n s S 1 , . . - , S k a r e a l re a d y d e f i n e d t h e n d e n o t e T = / V ' k (S 1 u . . . u S k ) a n dl e t S k + l c o n t a i n a ll t h e p l a y e r s i f o r w h i c h (~i(Ur) ~e 0 an d de f in e Xi = ~ i ( u r ) f o r a l li E S k 1 - ( S k + 1 is n o t e m p t y b e c a u se o f t h e e f f i c ie n c y a n d d u m m y p l a y e r a x i o m s . )B y t h e p o s i t iv i t y a x i o m , X > 0 . N o w f o r i = 1 . . . . . m w e d e f in e S i = S m _ i + l a n d w =Q k , S 1 , 8 2 , . . . , a m ) ) .

3 Th is is the o nly plac e where we use the efficiency axiom. Therefore we could use a muchweaker axiom, namely that for each S, r 8) ~ O. It is easy to see that such an axiom plus axiom 5imply efficiency.


9/18

On W eighted Sha pley Values 2 3

N ext w e show t ha t ~b is hom oge neou s , i . e . ~ t v ) = t ~ v ) for each gam e v and sca la rt . S i n ce e v e r y g a m e i s t h e d i f f e re n c e o f t w o m o n o t o n i c g a m e s i t i s e n o u g h , b y t h e a d -d i t iv i ty a x i o m , t o c o n s id e r o n l y m o n o t o n i c g am e s. A g a in b y a d d i t i v it y , h o m o g e n e i t yf o l l ow s f o r r a t i ona l s ca la rs . Le t v be a m o no t on i c gam e . Ch oose s equences o f r a ti ona l s{ rk} and { sk} w h i ch conve r ge t o t f r om above and be l ow , co r r e spond i ng l y . By t headd i t i v i t y and pos i t i v i t y ax i om s , r ) - ~ ( t v ) = r k - t ) v ) > ~ 0 and s i m i la r l y r -r > 1 0 . Bu t r ) - r ) = ( r k - s g ) C ( v ) - ~ 0 as k -+ ~ and t he r e f o r e r - +r a n d r ) = r k r ) - ~ t r w h i ch p r oves t he ho m og ene i t y o f ~b. S i nce bo t h rand ~w a r e l inea r m aps on F , i t su f f ic e s t o show a s w e do nex t t ha t ~ and ~ r co i nc i deo n e a c h u n a n i m i t y g a m e .

F o r a u n a n i m i t y g a m e U s de f i ne k = m a x ( ] I S ~ S ~ r and le t ,~ = S n S k . Le tkT = U S . Th e coa l i t i on S i s o f p - t yp e i n u r ( a s e ach subse t o f T i s) and b y t hej = lpa r t ne r sh i p ax i o m f o r e ach i E S

~ i u r ) = ~ i ~ u r ) S ) u s ) = ~ u T - ) S ) ~ i u s ) .

B y t h e d e f i n it i o n o f T t h e o n l y m e m b e r s o f T w h o h a v e n o n z e r o p a y o f f s i n U T aret h o s e o f S k , t h u s r = Z _ h > 0 a n d t h e r e f o r e

j ~ s~i u r )~ u s ) = Z_ h

I t f o l l ow s t ha t f o r i E S ,

h i~ i U s ) = F , _ h j

a n d f o r i q S S , ~ i ( U s ) = 0, i .e . , r = (r Q.E.D.

The f am i l y o f a l l w e i gh t ed Sh ap l ey va lue s r f o r si m p l e w e i gh t sys t em s co , c an a lsobe cha r ac t e r i zed b y s l i gh tl y chang ing t he pos i t i v i t y ax i om . We r ep l ace no w ax i o m 3b y t h e f o ll o w i n g on e .

( 3 ) P o s i t i v i t y . I f v i s m o n o t o n i c a n d t h e re a r e n o d u m m y p l a y e r s i n v t h e n ~ ( v ) > 0 .

T h e o r e m 3 : A so l u t i on ~ s a tis fie s ax i om s 1 , 2 , 3 , 4 , and 5 i f and on l y i f t he r e ex i st sa si m p l e w e i gh t sys t em ~ = ( h , ( N ) ) such t ha t r = r


10/18


Proo f. " I f co is a s imp le we igh t sy s t em then fo r e ach o rde r R in N (N ) , P r o ( R ) > 0 . I f vs a ti sf ie s th e c o n d i t i o n o f a x i o m 3 ' t h e n f o r e a c h p l a y e r i , C i (v; ) >1 0 a n d f o r s o m eR , C i ( v , R ) > 0 wh ich show s tha t 4ko(v) > 0 .T h e p r o o f o f t h e o t h e r d i r e c ti o n is a l o n g th e s a m e l in e o f th e p r o o f o f T h e o r e m 2 .T h e o n l y d i f fe r e n c e i s t h a t b e c a m e o f a x i o m 3 ' , r > 0 a n d t h e r e f o r e t h e p a r t i t io nb u i lt i n t h e p r o o f o f T h e o r e m 2 c o n t a in s o n l y N .

I n th e n e x t t h e o r e m w e s h o w t h a t w e i g h t e d S h a p l e y v a lu e s c a n b e a p p r o x i m a t e dby s imp le we igh t ed Shap l ey va lue s .

T h e o r e m 4 . Fo r each we igh t sy s t e m co = Q ,, (S l . . . . , S i n ) ) t he r e ex i s t s a s equence o fs imp le we igh t sy s t em s co = CA , (N) ) such t ha t f o r e ach g ame v , C t o t ( v ) - * ( % o ( v )when t ~ oo.

P r o o f L e t O < e < 1 a n d d ef in e f o r e a c h t , 1 < ~ l< ~ m a n d i E S t, X ~ = e t ( m - t + l ) X i ,and de f ine co t = CAt , (N) ) . I t is e a sy t o s ee t h a t f o r e ach S , ~ w t ( U s ) - * 4 )( U s) and s ince4ko and O w t are l inear , O j ( v ) ~ O ~ o(v ) fo r e ach v . Q .E .D .

5 D u a l i t y

T h e d u a l g a m e o f a g a m e v is d e n o t e d b y v * a n d i s d e f i n e d b y

v * ( S ) = v ( N ) - v ( N ~ S ) for e ach S C_ N.

T h e t r a n s f o r m a t i o n v ~ v * i s a o n e - t o - o n e l i n e a r m a p f r o m F o n t o i t se l f. In p a rt i c u la ruthe se t ( S ) S C _ N is a ba si s f o r F . O b se rv e th a t u ~ ( T ) = 0 f o r e a c h T w i t h T n s =

r a n d u ] ( T ) = 1 i f T n S : /: r We ca l l the gam e u~ th e r e p r e s e n t a ti o n g a m e f o r t h ec o a l i t i o n S . T h e g a m e u ~ h a s a n a tu r a l i n t e r p r e t a t i o n a s a c o s t- g a m e w h e r e u ~ ( T ) i st h e c o s t i n c u r r e d b y T . T h e p r e s e n c e o f a n y n u m b e r o f m e m b e r s o f S in T in c u r s au n i t c o s t ( c o m p a r e S h a p l e y 1 9 8 1 ) . F o r a w e i g h t s y s t e m c o = ( X , (S 1 . . . . . S i n ) ) w e

{ u s ) s c _ N asef ine a l inear m ap * 9 *t o. F ~ R N by def in ing ~b* on the bas is fo l lows .Fo r a g iven S d eno te k = m ax { ] IS n S : /: O} and l e t S = S n Sk . T hen fo r i E S ,

O ~ ) t u s ) = : ~ . x j

a nd r = 0 i f / ~ S .


11/18

On Weighted Shapley Values 2 5

An equ i va l en t r and om o r d e r app r o ach is de f i ned f o r *t o . F o r a n o r d e r R w edeno t e by R* t he r eve r se o r de r . Fo r a g i ven p r obab i l i t y d i s t r i bu t i on P ove r R( N)w e d e f i n e P * b y P * ( R ) = P ( R * ) . We now have t he f o l lowi ng equ i va lence .Theorem 1 * : For each p l aye r i , we i gh t sys t em co and gam e v ,

r = E ~ G v , .)) .

T h e p r o o f is a n a lo g o u s to t h e p r o o f o f T h e o r e m 1 , w h e r e t h e n o t i o n i is l as t fo r Si n R is r ep l aced by i i s f ir s t f o r S i n R wh i ch m eans S n B R , i = ~. T he so l u t i ons~ and r c an be r e l a ted i n a s i m p le way .

Theorem 5: Fo r each gam e v and w e i gh t sys t em co,

, L v ) = , ~ v * ) .

Proo f : Cons i de r t he gam e v = u~ . T hen v* = ( u~) * = u s , and by the def in i t ion o f @o~a n d $ * , r = r N o w l e t v = Z asU~. T h e nS N

~ , * v ) = : ~ * *Sr = ~ aS t o (Us ) = ~o( ~ , a sUs ) = $ ~ ( v * ) . Q . E . D .S _N S N S _NAn ax i om a t i c cha r ac t e r i z a t i on o f t he f am i l y { r is ob t a i ned by chang i ng ax i o m 5.We say t h a t a coa l i ti on S is o f p* - t yp e i n t he gam e v i f f o r e ach R D S and T C S ,v ( R \ T ) = v (R ) . Her e aga in , a s i n t he ca se o f p - t ype coa l i t ions , a p* - t yp e coa l i t ion canbe cons i de r ed a s one i nd iv i dua l r ep r e sen t ed by s eve ra l agan t s. Bu t in t he p* - t yp e casea n y n o n e m p t y s u b c o a l it io n o f a g e n ts h a s t h e s a m e e f f e c t o n t h e c o s t a s t h e c o a l it io no f a l l agen t s, wh i l e i n t he p - t ype ca se a ll the p r ope r subcoa l i ti ons o f agen t s a r e pow er -less . We ca l l a coa l i t ion of a p*- type a c oal i t ion o f representat ives. C o m m o n t o b o t hp - t y pe and p * - t y pe coa l i t i ons is t he f ac t t ha t t h e i nne r coa l i ti ona l s t r uc t u r e o f suchcoa l i t ions is t ri v ia l . A x i om 5* i s ana l ogous t o Ax i om 5 ; i t r equ i re s t ha t i f S i s a p* - t yp ecoa l i t i on i n t he gam e v , t hen t he cos t sha r ed by each one o f it s m em ber s can becom pu t ed by l e t t ing t he p l aye r s i n S ba r gai n ove r t he sp l i tt ing o f the t o t a l cos t sha r edby S i n r C l ea r l y by t he na t u r e o f t he p* - t yp e coa l i t i on t h i s ba r ga in i ng is r ep -r e s e n te d b y t h e g a m e u ~ .A x i o m 5 *: I f S is o f p * - t y p e i n v , t h e n r = r f o r e a c h i E S .


12/18

216 E Kalai and D Same t

T h e o r e m 2 * : A s o l u t i on s a ti sf ie s ax i om s 1 , 2 , 3 , 4 , and 5* i f and on l y i f t he r e ex is t sa weigh t sy stem ~ such th at r - - Cw.*

T h e p r o o f is a na l og o u s t o t h a t o f T h e o r e m 2 .O n e m i g h t e x p e c t t h a t r c a n b e o b t a i n e d f r o m r b y a n a p p r o p r ia t e t ra n s f o r-m a t i on o f t he we i gh t s ys t em . T o s ee t ha t t h i s is no t t he ca s e we exam i ne fi r st s im p l ewe i gh t s ys t em s .

T h e o r e m 6. Le t iNI >~ 3 . I f ~ = (p , (N) ) and 6o '= (h , (N) ) a re tw o s imple we ights y s te m s a n d O * ( v ) = r f o r e a c h g a m e v t h e n b o t h h a n d / a a re m u l t ip l e s o f t h evec t o r ( 1 , 1 . . . . . 1 ) and t hus bo t h r and ~ t o ' a r e t he Shap l ey va l ue.

P r o o f. ' A s s um e r 1 6 2 * . T h e n f o r a n y c o a li ti o n { i , ] } C _ N , ( r =( r Bu t

h ie ~ , ~ , ) i u { u } ) - ; , t + h ;

a n du 1( O ~ ) K u { u } ) = ( r } ) = ( r + u ( ; } - u { , j } ) = ~ i- - - ~ j

Th er e f o r e , fo r e ach i and ] i n N

h i ~ jh i + i i + P i

f r o m w h i c h w e c o n c l u d e t h a t k i i = h~/. tj for eac h i , ] e N . I t f o l l ows t ha t t he r e ex i s t sa pos i ti ve num ber C f o r w h i ch h t = - - f o r e ach i e N . Cons i de r no w a co a l i t ion { i,] , k}. We f ind tha t P~

( ~ , ) ~ ( u { ~ , j , k } ) = ? , i + ; v + x k C/~ C/~ j C/~k ( 1 )laiP k

P i k + P i g i + I~f~ t~


13/18


Using the p robab il is t ic de f in i t ion o f 4 we can com pute

_ u k ~ i~ ) i u i , i , k ) ) - 2 )ta i+ taj + la~ /ai +/~

/ .t i / a i / a k / a i / a k

Equa t ing the two expre ss ions (1 ) and (2 ) , d iv id ing by / l i / h : , and mu l t ip ly ing by ~ i +/aj +/ ak we f ind tha t :

1 i l a i l a i l a k- - + - - = 9 ( 3 )1 1 12 I d i 1 2 l d / l . l k I d i l 2 b l i l d k

We can ob ta in an equa t ion s imi la r to (3 ) fo r (4 r and (4 ) i app l i ed to the gameu ( i , i , k ) . By sym m et ry the r igh t hand s ide o f th i s equa t ion w i ll be the same a s in (3 )and the re fo re equa t ing the l e f t han d s ides we ge t :

- - - - = - -/~i +/~i /ai +/~k /~i +/ ai /a +/ ak

Fro m tha t we co nc lud e /a = /~ i and the re fo re k i = X . S ince th i s true fo r an y i , ] EN ,the p ro of i s com ple ted . Q .E.D .

C o r o l l a r y 1 : F o r I N ] /> 3 , t h e o n l y d i s t r ib u t i o n w h i c h is c o m m o n t o t h e f a m i l y o fdis t r ibut ions (Pr and the fam ily (P*~} where co ranges over a l l s imple weight system sis Pr o w here coo -- ((1 , . . . , 1), (N) ).

We can al so ob ta in a cha rac te r i za tion o f the ( sym met r i c ) Shap ley va lue , one wh ichd o e s n o t u s e th e s y m m e t r y a x i o m .

T h e o r e m 7 . Fo r IN[ i> 3 , a so lu t ion 4 sa ti s fi e s ax ioms 1 ,2 , 3 ' , 4 , 5 , and 5* i f and on lyi f i t is the Shap ley va lue .

Fo r N = 2 the re ex i st s a t r ans fo rm a t ion co ~ co* o f s imp le we igh t sys t ems suchtha t r = 4w* . I nde ed, i t i s easy to see th a t i f for co = (k , ( iV)) we se t co* = (k* , (N))w h e r e ~,* = ( ~ 2 , ~ 1 ) t h e n 4 = 4 t o . - W e s t a te n o w t h e e x te n s i o n o f T h e o r e m 6 togene ral we igh t sys t ems and om i t t he p roof .


14/18


15/18

On W eighted Shap ley Values 219

j u s t b y c h e c k i n g t h e e q u a l i ty f o r v = u s , s ince the r igh t hand s ide is linear in v (obse rvet h a t F u s x ) = I I x i ) I t i s e a s y t o s ee t ha t i f t he p l ay e r s a r ri val t im e i s d i s t r ibu t ed

t ~ sacco r d i ng t o t he ~ i s , t hen t he p r ob ab i l i t y t ha t t hey a r ri ve i n a c e r t a in o r d e r R isP c ~ R ) .No w de f i ne F~* by

F * x l . . . . . X n ) = v N ) - F v 1 - x l . . . . . 1 - x n )

I t i s e a s y t o chee k t ha t F v * x ) = F * x ) T h e r e f o r e

l a F * d g t t )r = ( r = o ax i ~ t ) d t d t

aF~ l aF~ d ~ d ~D en ot e r~t(t = 1 - ~t( t) an d ob ser ve th at __ax-T (~(t)) = __~x,x, (n(t)) an d d-7 = - d 7I t f o l l ows t ha t

o d ? i t )q~ *), v ) = f - - - - d t1 ~ x i I n t ) ) d t

r l t t ) can be i n t e r p r e t e d a s t he p r o bab i l i t y t ha t p l aye r i ar ri ve s a f t e r t i m e t

7 R e d u c t i o n o f p - t y p e a n d p * - t y p e C o a l i t io n sPa r t o f t he r ea s on i ng o f t he pa r t ne r s h i p ax i o m is t ha t a coa l i ti on o f pa r t ne r s c an bet r ea t ed i n a c e r t a i n s ens e a s one i nd i v i dua l . I n t h i s s ec t i on we s how how a p - t ypecoa l i t i on can be p r ac t i c a l ly de f i ned a s one p l aye r , t he r eb y reduc i ng t he s iz e o f t hegam e . L e t u s f i x a coa l i t i on S o w i t h m or e t h an one p l aye r . Cons i de r t he s e t N w h i chcons i s t s o f a l l the p layers of N ex cep t tha t a l l the p layers in So a re rep laced by a s inglep l aye r den o t ed by s , i. e ., N = ( Ar kSo) t3 {s ) . Fo r a ny gam e v on N we de f i ne a gam e z~o n - N b y ~ S ) = v S ) i f s ~ - S a nd ~ S ) = v S \ { s ) ) t d S o ) i f s E S . L e t o 3 = ( ; L ( S 1 ,. . .. S i n ) ) be a w e i gh t s ys t em f o r N , and l e t k b e t he h i ghes t i ndex f o r wh i ch Sk C~So =/= 0 . Th e we ight sy s tem ff~ = (X, (Sl . . . . . S in) fo r .~ i s d ef ined as fo l lows . F or eachi=/=s, )~t=)~i an d Xs = Z )~t. F o r e a c h ] = /: k , S j = S j \ S o a nd S k = S k \ S o ) t3 {S }.We can s t a t e no w t ~ o


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220 E Kalai and D Samet

T h e o r e m 9 : I f S O i s a p - t y p e c o a l i t i o n i n v t h e n f o r e a c h i q : s , ( r = ( w ) i ( v )a n d ( $ ~ ) s ( f ) = ~ ( r

iES oS i m i l a rl y , i f S o i s a p * - t y p e c o a l i t i o n t h e n f o r e a c h i ~ s , ( ~ * ) i ( ~ ) = ( ra n d ( r ~ ( r T o p r o v e t h i s t h e o r e m w e u se t h e f o l l o w i n g l e m m a s .iE oL e m m a l : I f i i s a d u m m y p la y er i n r a n d v = ~ a s U s t he n a s = 0 f o r e a c h Sw h i c h c o n t a i n s i . s _ N

P r o o f . ' By i n d u c t i o n o n t h e s i z e o f S . F o r S = { i} , 0 = v ( { i} ) = a ( i } . S u p p o s e w e p r o v e df o r a l l c o a l i t i o n s o f s i ze k w h ic h c o n t a in i a n d l e t S b e a c o a l i t i o n o f si z e k + 1 a n ds u c h t h a t i E S . T h e n 0 = v ( S ) - v ( S \ { i } ) = Y , a T - ~ a T = ~ a T . B u tTCS TC_S\i iETCSf o r i E T C S , a T 0 a n d t h e r e f o r e a s = 0 . Q . E . D .r

L e m m a 2 : L e t S O b e a p - t y p e c o a l i t i o n i n v a n d l e t v = S C _ Neac h S w hic h sa t i s f ie s S n So 4= 0 an d S n So 4= So .

a s U s t h e n a s = 0 f o r

P r oo f. F o r a c o a l i t i o n T a n d a g a m e v d e n o t e b y V T t h e r e s t r i c t i o n o f t h e g a m e u t oth e c o a l i t i o n T . S in v e v ~ v T i s a li n e a r m a p f r o m t h e s p a c e o f ga m e s o n N t o t h e s p a c eo f g a m e s o n T a n d s in c e u~ = 0 i f S r T i t fo l l o w s t h a t

v r = X a s u r = Y , a rs u s . ( * )SCN SC_TN o w i f T s a t is f i es T r3 S O 4= 0 a n d T r S o 4= S o t h e n a ll t h e p l a y e r s o f T N S o a r ed u m m i e s i n t h e g a m e v r . I n p a r ti c u la r , w e c o n c l u d e b y L e m m a 1 a n d ( * ) t h a t a r = 0 .

Q . E . D .

P r o o f o f T h e o r em 9 : I t c a n b e e a s il y s h o w n b y L e m m a 2 t h a t i f v = ~ a s U s t h e nf~ = ~_, a s a s + Z , a s a s . T h e r e f o r e f o r i : / : ssq~S s~

r = X - - a s + X a s~ e s Z ~ ,j ~ e s x 7 ,;i ~ S j ~ S

ZS e N \ S o l ~ s x i- - a S Y , ~ s = (r ) A v ) 9S 2 S o I ~ s X i


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Fo r i = s

~s i~S Or = ~ - - a S - - ~ ~ a S = ~ , r~ s z ~ i S~So Z ~ i ~Soj ~ s j

N o w i f S o is o f p * - t y p e i n v , t h e n S o is o f p - ty p e i n v * . T o p r o v e t h e s e co n d h a l f o ft he t h eo r em o ne ha s t o obs e r ve on l y t ha t v--)* = v* and us e t he equa l i t y o f Th eor e m 5 .

Q. E . D.

The f o l lowi ng co r o l l a r y f o l l ows f r om Th eor e m 9 . I t is i m po r t an t f o r app l ica t i ons i nwh i ch t he p l aye r s t hem s e l ves a r e , o r a r e r ep r e s en t i ng , g r oups o f i nd iv i dua ls . Such i st he cas e f o r exam pl e wh en t he p l aye r s a re pa r t i e s , c i ti e s , o r m anag em e n t boa r ds .The us e o f t he s ym m e t r i c Shap l ey val ue s eem s t o be u n j us t i f ied i n ce r t a in ca se s o f t h i st ype becaus e t he p l aye r s r ep r e s en t cons t i t uenc i e s o f d i f f e r en t s iz es . A na t u r a l c and i -da t e f o r a s o l u t i on is t he we i gh t ed Sh ap l ey va l ue whe r e t he p l aye r s a r e we i gh t ed byt he s ize o f t he cons t i t uenc i e s t hey s t and f o r . The f o l l owi ng co r o l l a r y s hows t ha t s ucha pro ced ure i s jus t i f i ed in the two spec ia l cases descr ibed be low .

Co r o l la r y 2 : L e t v be a gam e on N I NI = n ) i n w h i ch each p l aye r i i s a s e t o f i nd iv i d -uals M i w i t h rn i m em ber s . Cons i de r t he s e t o f i nd iv i dua ls N = LI M i and t he gam esv t and v2 de f i ned on / V a s f o l lows . Fo r each S C 3 I , i ~ n

v t S ) = v i I M i c S ) )v2 S ) = v { iI M i n S =/=0}) .

Le t w be t he s i m p l e we i gh t s ys t em r n 1 . . . . m n ) , N ) ) . T h e n f o r e a c h i

* ~ ) t O = , v ~ ) M i )

a n d

~ * ) i v ) = ~ v D M i )wh er e ~ is t he s ym m e t r i c Shap l ey va lue .


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222 E . Ka l a i a nd D. S a m e t : On W e i gh t ed S ha p l e y Va l ue s

e f e r e n c e s

Ha rsa ny i JC . (1959 ) A ba rga in i ng mod e l fo r c oo pe ra t i ve n -pe r son game s. In : Tuc ke r AW , Luc e R D(eds) C on t r i bu t i ons t o t he t he o ry o f ga me s IV (Anna l s o f Ma t he ma t i c s S t ud i e s 40 , pp 3 25 -355) . P r ince ton Un ivers i ty Press, Pr ince ton

Ka l a i E , S a me t D (1985) Mo no t on i c so l u t i ons t o ge ne ral c oope ra t i ve ga me s . Ec onom e t r i c a 53 / 2 :3 07 - 3 2 7

Ma sc hl er M (1982) The w or t h o f a c oope ra t i ve e n t e rp r is e t o e a c h me m be r . In : Ga me s , e c on om i cdyn amics and t ime se ries ana lys i s

O w e n G ( 1 9 6 8 ) A n o t e o n t h e S h a p le y v al u e. M a n a g e m e nt S c ie n c e 1 4 / 1 1 : 7 3 1 - 7 3 2Ow e n G (1972) Mul t i li ne a r e x t e ns i ons o f game s. Ma na ge me n t S c i e nc e 18 / 5 : P 6 4-P 79Owe n G (1982) Ga me t he o ry , 2nd e d . Ac a de mi c P re s s , Ne w YorkS ha p l e y LS (1953a ) Add i t i ve a nd non -a dd i t i ve s e t func t i ons . P hD The s i s , De pa r t me n t o f Ma t he -

mat ics , Pr ince ton Univers i tyS ha p l e y LS (1953b) A va l ue fo r n -pe r son ga mes . In : K uhn HW , Tuc ke r AW (e ds) C on t r i bu t i ons

t o t he t he o ry o f ga me s I I (Anna l s o f Ma t he ma t i c s S t ud i e s 28 , pp 307 -317) . P r i nc e t on Un i ve r -s i ty Press , Pr ince to n

S ha p l e y LS (1982) Di sc us san t s c o mm e n t . In : Mor i a r i t y S ( e d ) Jo i n t c os t a l l oc a t ion . Un i ve r s it y o fOklahoma Press , Tulsa

R e c e i ve d Nov e mb e r 1985

On Weighted Shapley Values

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