legros cost allocation ijgt 1986

8/14/2019 Legros Cost Allocation IJGT 1986

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International Journal o f Game Theory, Vo l. 15, Issue 2 , page 109 -1 19Al locat ing Jo in t Cos ts by M eans o f the N uc leo lus 1B y P . L e g r o s2

Abstract: This paper presents a sufficient condition for the nucleolus to coincide with th e SC RBmethod vector and for nonemptiness of the core. It also studies the reasonableness and the mono-tonicity of the nucleolus under this condition. Finally it analyses the class of games satisfying thecond ition and compares it with the classes of convex games, subconvex game s and th e class Q ofDriessen and T ijs.

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

R e g u l a t o r y a u t h o r it i e s s u c h a s th e F e d e r a l C o m m u n i c a t i o n s C o m m i s i o n o r t h e T en -n e s se e V a l l e y A u t h o r i t y h i s t o r i c a l l y h a v e d e t e r m i n e d t a r if f s b a se d o n t h e p r in c i p l e o ff u l l y d i s t r i b u t e d c o s t s. T h i s t y p e o f p r i c in g h a s b e e n a n a l y s e d i n R a n s m e i e r ( 1 9 4 2 ) ,H e a n e y a n d D i c k i n s o n ( 1 9 8 2 ) , H e a n e y ( 1 9 7 9 ) , J a m e s a n d Le e ( 1 9 7 1 ) , Y o u n g e t a l.( 1 9 8 0 ) a n d B r a e u t i g a m ( 1 9 8 0 ) .

S h u b i k ( 1 9 6 2 ) m a d e o n e o f t h e f i r st a p p l ic a t io n s o f g a m e t h e o r y t o t h e s t u d y o ft h e j o i n t c o s ts a l lo c a t io n p r o b l e m . S u b s e q u e n t l y m a n y a u t h o r s ( L i tf l e ch i l d 1 9 7 5 ;L o e h m a n a n d W h i n s t o n 1 9 7 4 ; C h a m p s a u r 1 9 7 5 ; S h a r k e y 1 9 8 2 a ) h a v e e s ta b li s h ed t h em e t h o d o l o g i c a l s i m il a ri ty b e t w e e n t hi s p r o b l e m a n d t h e p r o b l e m o f fi n d in g a s o lu t i o ni n a n n - p e r s o n c o o p e r a t i v e g a m e . S p e c i f i c a l l y , c o n s i d e r a n e n t e r p r i s e p r o d u c i n g a c e r -t a i n s e r v ic e f o r a s e t o f N m a r k e t s . A s s o c i a t e w i t h e a c h s u b s et S o f N t h e c o s t C ( S ) o fp r o d u c i n g t h e s e rv i ce f o r t h e I S I m a r k e t s a l o n e . F a u l h a b e r ( 1 9 7 5 ) s h o w s t h a t i f t h er e g u l a t o r i m p o s e s a z e r o p r o f i t c o n s t r a i n t , t h e e x i s t e n c e o f s u b s i d y - fr e e p r ic e s i s e q u i v -a l e nt t o t h e n o n e m p t i n e s s o f t h e c o r e o f th e g a m e ( N , 6 ) . S u f f ic i e n t c o n d i ti o n s f o r t h ee x i st e n ce o f t h e c o r e c a n b e f o u n d i n S h a r k e y a n d T e l se r ( 1 9 7 8 ) .

T h e s u f f ic i e n t c o n d i t i o n p r e s e n t e d b e l o w d e f in e s a cl as s o f g a m e s w i t h n o n e m p t yc o r e s f o r w h i c h t h e S e p a ra b l e C o s t R e m a i n i n g B e ne f it (S C R B ) m e t h o d c o i n c i d es w i t ht h e n u c l e o l u s o f t h e c o s t g a m e . T h e c o i n c id e n c e o f th e S C R B m e t h o d a n d t h e n u c l e o lu s

1 T hi s is an enlarged version of a paper presented at the Ec ono me tric Society European M eeting,Pisa, August 1983. Part of th e results w ere obtained w hile I was visiting the M EDS Department ofNorthwestern University. I would lik e to thank Theo Driessen, Ehud Ka lai, Steph T ijs, MitsuoSuzuki and tw o anon ym ous referees of this journa l for helpful com ments. I gratefully acknowledgethe financial suppo rt of the Minist~re des Relations Ext6rieures (France).2 Patrick Legros: University of P aris XII, L a Varenne St Hilaire, France. Present address: Cali-fornia Institute o f Technology, Pasadena, California 91125, USA .0 0 2 0 - 7 2 7 6 /8 6 /0 2 0 1 0 9 - 1 1 9 $ 2 .5 0 9 1986 Phy sica-Verlag, Heidelberg, W ien


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110 P. Legroshas a l r eady been obse rved fo r a 3 -pe r son game in S t r a f f in and H eaney (198 1) and fo r a5 - p e rs o n g a m e i n S u z u k i a n d N a k a y a m a ( 1 9 7 6 ) . S u z u k i a n d N a k a y a m a ( 1 9 7 4 ) a ls o o b -ta in th i s t echnica l resul t in the g enera l case for a weake r con di t io n than i s s ta ted here .3O ne o f t he pu rpose o f t he p r e sen t pape r is t o show t ha t t he s t r onge r cond i t i on p rov i desan i n t u it i ve mo t i va t i on fo r behav i o r w h i ch l eads t o t h i s re su l t . I nde pen den t l y , D r i es senand Ti js ( 1983 a ) p rove t ha t t he r -va lue o f T ij s ( 198 1) co i nc i des w i t h t he nu c l eo l us andw i th the S CRB v ecto r for al l games belonging to a cer ta in c lass . We show th at th i s c lassis d i s j o in t f r om our s , excep t f o r t he case o f h i gh ly sym met r i ca l games .

The pape r i s o rgan i zed as f o l l ow s . Sec t i on 2 p r e sen t s a cond i t i on w hi ch de f i nes anew c la ss o f games and fo r w h i ch t he co re is no ne m pt y an d t he SC R B me t ho d l eads t ot he nuc l eo l us. Sec t i on 3 ana l yses t he r easonab l eness and t he m on o t o n i c i t y o f t henuc leolus for th i s class. Sect ion 4 com pares th i s c lass wi th tho se d ef ined by Sha pley(1971 ) , Sha rke y (198 2b) and D r i es sen and Tij s ( 198 3a ) . Sec t i on 5 p re sen t s some f i na lc o m m e n t s .

2 T h e S C R B M e t h o d a n d t h e N u c l e o l u sLe t (N , v ) be t he s av ing game as soc i at ed w i t h t he cos t a l l oca t ion p rob l em , The fu nc -t i on v is de f i ned by ,

v ( S ) = ~ C ( i ) - C ( S ) , fo r al l S c Ni E s

N o t e t h a t v is a l w ays ze ro -norma l i zed ( v ( i ) = 0 fo r a ll i in N ) . We requ i r e on l y t ha t v bem o n o t o n i c ,

v ( S ) > i v ( T ) , for a ll T C S C N.Th e se t o f i mpu t a t i on s i s de f i ned a s,

X = ( x @ ~ n s . t. x ( N ) = v ( N ) a n d x i >1 O , i = 1 , . . . , n )w h e r e x ( S ) d e n o t e s N x i .i E s

Clear ly , eve ry individual ly ra t ional fu l l cos t a l locat ion schem e y ( i .e . Yi


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A l l o c a t i n g J o i n t C o s t s b y M e a n s o f t h e N u c l e o l u s 1 1 1

w h e r e~ ( N ) = 1 , o ~ > 0 , i = 1 , . . . ,nS C i = v ( N ) - v ( N - i ) , i = 1 , . . . , nN S C = v ( N ) - S C ( N )

I t i s imm e d ia te tha t x i s a lwa ys Pa r e to op t ima l , bu t tha t x i m a y be pos i tive o r ne ga tivea n d t h a t x m a y n o t b e a n e l e m e n t o f t h e c o r e . W e n o w c o n s i d e r t h e n a iv e c as e w h e r ea i = 1 I n f o r a l l i in N a nd w e de no te by Y the ve c to r ~ = S C + N S C / n .

Th e e xc e s s o f a c oa l i t ion S w i th r e s pe c t to ( w . r . t . ) x i s de f ine d by e ( S , x ) = v ( S )- x ( S ) . I f the e xc e s se s a r e a rr a nge d in de c r e a s ing o r de r , l e t O ( x ) b e th e r e s u l t ing ve c to ri n ~ 2 n - a . A n i m p u t a t i o n x is p r e f e r r e d t o a n i m p u t a t i o n y w h e n e v e r 0 ( x ) i s s m a l le rtha n | w . r . t , t he l e x ic og r a ph ic a l o r de r ing on IR2 n - 1 . T h e n u c l e o l u s i s t h e s e t o fpo in t s in X wh ic h a r e m os t p r e f e r r e d in th i s qua s i- o r de ring on I Rn . Sc hm e id te r ( 1969 )p r ove s tha t the n uc le o lu s a lwa ys e x i s ts , i s un ique , a nd i s a n e le m e n t o f e ve r y e - c o r e .An in tu i t ive in te r p r e ta t io n is tha t the nuc le o lu s min imiz e s d i s s a t is f a c t ion , w i th p r io r i tyto c oa l i t ions wh ic h a r e m os t d i s sa t is f ie d .

W e g ive the f o l lowing de f in i t ion be f o r e s t a t ing a f i r s t r e s u l t .D e f i n i t i o n . A g a m e i s p s e u d o - c o n v e x i f S C i >1 v ( S ) - v ( S - i ) , for a ll i in N and a l l Si n N .

P s e u d o - c o n v e x i t y re q u i re s t h a t t h e m a x i m u m o f t h e s e p ar a bl e c o n t r i b u t i o n o f ap la y e r to a ll c oa l i t ions is re a c he d a t th e h ighe s t le ve l o f c oo pe r a t ion , i . e .N . Co n t r a r yt o S h a p l e y ' s n o t i o n o f c o n v e x i t y , t h e r e i s n o s n o w b a ll e f f e c t h e r e , b u t o n l y a n i n c en -t ive f o r the p la ye r s to r e a c h the h ighe s t l e ve l o f c oo pe r a t io n .T h e o r e m 1 : I f v i s p s e udo - c on ve x a nd s a ti sf ie s the c on d i t ion ,( P I) n - S C / I > - ( n - 1) " N S C

= S C + N S C / n i s th e n u c l e o lu s o f t h e g a m e a n d t h e c o r e is n o n e m p t y .P roo f. " L e t ( PC G) a n d ( PCG I ) de no te r e s pe c t ive ly the c la ss o f p s e udo - c onve x ga me sa nd the c la ss o f p s e udo - c onv e x ga mes w h ic h s a t i s fy ( P I ) . L e t S = { i 1 . . . . , i s } b e a n yc o a l i t i o n w i t h s m e m b e r s a n d d e n o t e S k t h e c o a l i ti o n o b t a i n e d f r o m S b y d e l e ti n g t h ek f i r s t p la ye r s , S k = S - ( i l . . . . . i k } w i th k = 1 , . . . , s , a nd l e t SO = S . I f v is in (PC G ) ,

S C i k > ~ v ( S x _ I ) - v ( S k ) , k = 1, . . . ,s .S u m m i n g o v e r t h e e l e m e n t s o f S - i s a n d n o t i n g v ( i ) = 0 for a ll i in N ,

s - -1 s - - IS C i k >1 2 ( v ( S k _ l ) - V ( S g ) ) = v ( S ) . (1 1k = l k = l


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112 P. Legro sI f w e c o n s id e r S = N - j , i t is i m m e d i a t e t h a t ( 1 ) i m p l i e s ,

N S C < O . (2 )Fr om (PI ) and (2 ) , we hav e ~ / /> 0 fo r a ll i i n N . " S ince ~ ( N ) = v ( N ) w e h a v e Y E X .

I f s ~< n - 2 , by (1 ) , ( 2 ) and (P I ) ,

e ( N - i, x ) - e ( S , ~ ) = N S C / n - ( v ( S ) - S C ( S ) - s " N S C / n )S--1

: ~ , S C i k - v ( S ) + S C i s + ( s + 1 ) " N S C / n > 1 0 .k= l

(3 )

W e no te t ha t :~ /> 0 fo r a ll i in N and (3 ) a r e t he cond i t i ons s t a t ed by Suzuk i andN a k a y a m a ( 1 9 7 4 ).Fo r eve ry v ec to r y 4= ~ in X , t he re wi l l ex i s t an i i n N such tha t e ( N - i, ~ ) 1 ~ ( s c j - x j ) / n] E Nt h is c o n d i t i o n s t a te s t h a t w h a t a p l a y e r i n N - i c a n e x p e c t t o r e ce i v e f r o m i is a t le a s ta s g rea t a s t he ave rage u to p ia l o s s w . r . t , any impu ta t io n . W e no te t ha t a su f f ic i en t con -

4 T hi s esult has been later rediscovered by Ow en (1968).


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Allocating Joint Costs by Means of the Nucleolus 113d i t i on fo r t h i s i nequ a l i t y t o ho l d i s ,

S C i / ( n - 1) > ~ S C - x i , f o r a l l ] i n N ,i .e . t he u t o p i a l o s s o f eve ry p l aye r ] i n N - i i s le ss t han w h a t i can be a sked t o pay t o ].

F i na l l y i t i s i n t e r e s ti ng t o no t e t ha t f o r games in (PC G I ) , ~ is j u s t t he vec t o r w h i chmi n i mi zes the d i s t ance be t w een t he u t op i a vec t o r and t he i mp u t a t i on s e t , i . e . ~so l vesr ain ]~ ( S C i - Y i ) 2 . 5y E X iE N

3 S o m e P r o p e r t i e s o f t h e N u c l e o l u sM i l n or ( 1 9 5 2 ) w as t h e f ir s t t o i n t r o d u c e t h e c o n c e p t o f re a s o n ab l e o u t c o m e . A p a y o f fvec t o r is r easonab l e i f x i 0 ),and l e t ~ be a so l u t i on c onc ep t w h i ch a ss igns t o each n -pe r son game v a un i que vec t o r

5 Necessaryand sufficient conditions for th is result are given in L egros.


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114 P. Le go s~v i n I1 n . Th en q~ i s mo no ton ic , b y M egiddo 's (19 74 ) def in i t ion i f q~w >1 ~sv for a ll i inN . The nuc l eo l us i s no t i n gene ra l mono t on i c (Meg i ddo 1974) . For games i n (PC G I ) ,how ever , t he nuc l eo l us i s mono t on i c : i f n u v an d n u TM a re t he nuc l eo l us o f t he games va n d w d e f i n e d a b o v e , t h e n n u TM = n u ~ . + A / n fo r al l i in N.R e c e n t l y , Y o u n g ( 1 9 8 2 ) h a s i n t r o d u c e d t h e c o n c e p t o f s t ro n g m o n o t o n i c i t y o f aso l u t i on . Le t v i ( S ) be t he con t r i bu t i on o f p l aye r i t o coa l i t i on S ,

I v ( S ) - v ( S - i ) i f i E Sv i ( S ) = [ v ( S U i) - v ( S ) i f i ~ S.

C ons i de r t w o games v and w su ch t ha t v i (S ) vS(S) fo ra ll S C N , and t he nuc l eo l us is no t s t r ong l y m on o t on i c .

In t h i s examp l e , the c on t r i bu t i on s o f p l aye r 5 t o a l l coa l i ti ons a re con s t an t - upto e - in bo th games v and w, whi le the re la tive va lues of the coal i t ions are qui te d i f -f e r en t i n bo t h games. Th i s r emark he l ps t o unde r s t and w h y t he n uc l eo l us i s no t s t r ong-l y m o n o t o n i c a n d w h y i m p o s i n g s t r o n g m o n o t o n i c i t y m a y n o t b e n a t u r a l . A q u i t es impl e exam pl e m ay c l a r if y th i s po i n t .

L e t IN I = 3 and t w o games v and w such t ha t 6 :V q ) = O , i = l , 2 , 3 ; v ( S ) = 1 i f lS I = 2 ; v ( N ) = 1w ( i )= 0 , i = 1 , 2 , 3 ; w ( 1 2 ) = w ( 1 3 ) = 2 ; w ( 2 3 )= 2 0 0 ; w ( N ) = 2 0 0 .

H ere v i ( S )


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Allocating Joint Costs by Means of the Nucleolus 1154 S o m e C o m p a r i s o n sDef i n i t i on 2 (Sh aple y 1971 ) : A gam e is co nv ex i f i t s charac ter i s t i c fun ct io n sa ti sf iesv ( S U T ) + v ( S 0 T ) >~ v ( S ) + v ( T ) fo r a ll S , T C N.Def i n i t i on 3 ( S h a r k e y 1 9 8 2 b ) : L e t Bo = ~b and B = (B 1 , . . . ,B k } b e an a rb i t r a ry pa r t i-i - -1t i o n o f N . L e t Q = ( Q 1 , - . - , Q k ) b e a c o l l e c ti o n o f c o a li ti o n s s u c h t h a t Qi c U B]]= oan d B i u Qi =~N, i = 1 . . . . k . Th e g ame ( iV, v) i s subc onv ex i f i t i s t rue tha t fo r al lsuch co l l ec t i ons B and Q ,

k~, (v (B i U Qi ) - v( Q i))


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116 P. Legros

k l B i )o , ( 4 ) i s t r u e f o r a l l t = 1 . . . . , k , a n d b e c a u s e B i s a p a r t i t i o n o f N , v ? = v ( N ) ,w h i c h p r o v e s t h a t v E ( S C G ) . iC o n s i d e r n o w t h e ( 2 n - 1 ) p a r t i t i o n s B s = { S , N - S ) w h e r e S C N . L e t Q s =

{ r S - i ) w h e r e i E S. B s and Q s c l e a r ly s a t i s f y D e f i n i t i o n 3 . S o , i f v E ( S C G ) ,v ( S ) + v ( N - i ) - v ( S - i ) < . v ( N ) . ( 6 )

( 6 ) i s t r u e f o r a l l S C N , a n d s o v E ( P C G ) . Q . E . D .

Corollary: ( C G ) = ( S C G ) = ( P C G ) i f J N I = 3 .

P r o of ." N o t e t h a t ( C G ) = ( P C G ) if I N I = 3 a n d u s e T h e o r e m 2 . Q . E . D .

W e s h o w n o w t h a t ( C G ) a n d ( S C G ) a r e d i s ti n c t f r o m ( P C G I ) w i t h a s e q u e n ce o f e x -a m p l e s .

Claim 1: ( C G ) - ( P C G I ) :~ ~ ,C o n s i d e r t h e 4 - p e r s o n g a m e , v (/ ) = 0 , i = 1 , 2 , 3 , 4 ; v ( S ) = 5 i f [S [ = 2 ; v ( 1 2 3 ) = 1 6 ;

v ( 1 2 4 ) = 1 2 ; v ( 1 3 4 ) = 1 0 ; v ( 2 3 4 ) = 1 4 ; v ( N ) = 2 6 .T h i s g a m e i s c o n v e x b u t SC 4 = 1 0 < 3 9 / 2 = - 3 / 4 - N S C a n d s o v q~ ( P C G I ) .

Claim 2 : ( C G ) n ( P C G I ) 4= ~ .L e t I N I = 4 a n d v ( / ) = 0 , i = 1 , 2 , 3 , 4 ; v ( S ) = 2 / 5 i f l S l = 2 ; v ( S ) = 1 i f l S I = 3 a n d

v ( N ) = 8 / 5 .T h i s g a m e is c o n v e x a n d ( P I ) is s a ti s fi e d : f o r all i, SC i = 3 / 5 = - 3 / 4 "NSC.

Claim 3: ( P C G I ) - ( S C G ) :~ ~ .L e t I 'N [ = 5 a n d v ( i ) = 0 f o r a l l i i n N ; v ( S ) = 2 5 i f iS ] = 2 ; v ( 1 2 3 ) = v ( 1 2 4 ) = 2 5 ;

v ( S ) = 5 0 i f I S I = 3 a nd S q~ { 1 2 3 , 1 2 4 } ; v ( N - 5 ) = 7 5 ; v ( N - / ) = 7 4 i f i E N - 5 ;v ( N ) = 1 0 0 .SC s = 2 5 a n d S C / = 2 6 i f i v~ 5 . S o , - 4 / 5 . N S C = 2 3 . 2 < S Q f o r a l l i i n N , a n d vs a t is f ie s ( P I ) . I t i s r o u t i n e t o c h e c k t h a t v i s p s e u d o - c o n v e x , s o v E (P C G I ) . C o n s i d e rn o w t h e f a m i l ie s B = { 1 , 2 3 , 4 , 5 ) a n d Q = { r 1 , 1 2, 1 } ; B a n d Q s a t i s f y t h e c o n d i -t i o n s o f D e f i n it i o n 3 , b u t w e h a v e ,

4 (v (B U Qi ) - v (Qi ) ) = v ( 1 2 3 ) + v ( 1 2 4 ) + v ( 1 5 ) - v ( 1 2 ) = 1 0 2 > v ( N )i=1a n d v ~ ( S C G ) w h i c h p r o v e s t h e c l a i m .

W e c o n s i d e r n o w t h e c la s s Q p r o p o s e d b y D r i e s s e n a n d T i js ( 1 9 8 3 a ) . T h i s c l as s i sd e f i n e d a s f o l l o w s ,

= ( v E Q s . t. SC (N ) - v (N )


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A l l o c a t i n g J o i n t C o s t s b y M e a n s o f t h e N u c l e o l u s 1 1 7

w h e r eQ = { v s . t . a < . S C , a ( N ) v ( N ) / n = - - . N S C an d v satisfies (PI). Q.E.D.nR e m a r k . " I t i s easy to show tha t as n becom es la rge enoug h, (PCG I) i s equa l to E G . In-deed , i f v E (PC GI) , by (7 ) and (P I ) ,

n - 1n

( S C ( N ) - v ( N ) )


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118 P. Legros5 C o n c l u d i n g R e m a r k sW e s h o w i n t hi s p a p e r t h a t t h e S C R B m e t h o d c a n b e s u p p o r t e d b y a n o r m a t i v e c o n c e p tw h e n e v e r t h e c o s t f u n c t i o n d e f in e s a s a vi n g g a m e w h i c h is i n ( P C G I ) . M o r e o v e r , w ep r o p o s e i n S e c t i o n 2 a n i n t e r p r e t a t i o n o f t h e r e s u lt i n t e r m s o f u t o p i a a n d e n t r a n c ef e es . I t is i n t e r e s ti n g t o m a k e t h e a n a l o g y b e t w e e n t h is i n t e r p r e t a t i o n a n d M o u l i n ' sA L D B ( A u c t i o n i n g t h e l e a d er s h ip w i t h d i f f e r e n t ia t e d b i d s ) m e c h a n i s m .

I n t h e A L D B p r o c e d u r e , t h e p l a y e r s a c t n o n c o o p e r a t i v e l y a n d f ir st c h o o s e as t h el e a de r t h e p l a y e r w h o h a s t h e l o w e s t b i d ( i .e . t h e a m o u n t o f m o n e y a p l a y e r w a n t s t og e t f r o m e a c h o t h e r p l a y e r t o b e c o m e t h e l ea d e r ). O n c e t h e le a d e r p r o p o s e s a d e c is i on( i. e . c h o o s e s a g i ve n s t a te a m o n g t h e s e t o f p o s s ib l e a l te r n a t iv e s t o g e t h e r w i t h a v e c t o ro f t r a n s f e rs ) , t h e p l a y e r s v o t e u n a n i m o u s l y t o a c c e p t t h e d e c i s i o n . I f t h e d e c i s i o n i sr e j e c t e d , t h e l e a d e r i s e l i m i n a t e d f r o m t h e d e c i s io n p r o c e s s a n d a n o t h e r l e a d e r is c h o s e n .T h i s p r o c e d u r e i s r e p e a t e d u n t i l a de c i s io n i s u n a n i m o u s l y a p p r o v e d . M o u l i n ( 1 9 8 1 )s h o w s t h a t i f t h e p l a ye r s b e h a v e i n a s o p h i st i ca t e d w a y , t h is m e c h a n i s m i m p l e m e n t st h e S C R B v e c t o r a n d t h e o p t i m a l b i d f o r e a c h p la y e r is e x a c t l y - N S C / n , w h e r e t h ef u n c t i o n v i s d e f i n e d b y v ( S ) = m a x E u i (a ) , a l l S CN, w i t h u i b e i n g t h e u t i l i t ya ~ A i ~ Sf u n c t i o n a n d A b e i n g t h e s e t o f p o s si b le s t a t e s .

R e f e r e n c e sBraeutigam RR (1980) An analysis of fully distributed cost pricing in regulated industries. BellJournal of Economics 11 (1) :1 82- 19 6Champsaur P (1975) How t o share the cost of a public good. International Journal of GameT h e o r y 4 ( 3 ) : 1 1 3 - 1 2 9Davis M, M aschler M (1965) The kernel of a cooperative gam e. Naval Research Logistic Quarterly1 2 : 2 2 3 - 2 5 9Driessen T, Tijs S (1983a) The r-value, the nucleolus and the core for a subclass of games. Methodsof Opera tions Research 46:395--40 6Driessen T, Tijs S (1983b) The r-value as a feasible compromise between utopia a nd disagreement.Re port 8 312, Department of Mathematics, Catholic University, Nijmegen

Faulhaber GR (1975) Cross subsidization: pricing in public enterprises. American Eco nom ic Review6 5 : 9 6 6 - 9 7 7Heaney JP (1979) Efficiency/equity analysis of environmental problems - a game theore ticperspective. In: Brain s SJ, Schotter A, Schw6diauer G (eds ) Applied game theory . Physica-Verlag , 352 -36 9Heaney JP, D ickinson RE (1982) M ethods for apportioning the c ost of a water resource project.Water Resources Research 18(3) :47 6- 48 2James LD, Lee RR (1971) E conomics o f water resource planning. Series in Wate r Resources andEnvironmental Engineering. M cGraw-Hill, New Yo rkKikuta O (1976) On the contribu tion of a player to a game. International Journal of Game Theo ry5 (4) :199 -20 8Legros P (198 4) Allocation des cot~ts et form atio n des coalitions: une approache par la th6orie desjeux. Doctoral dissertation, University o f Paris XIILittlechild SC (1975) Com mon costs, fixed charges, clubs and games. Review of Eco nom ic Studies42 (1 ) :117 -1 24Loehman E, Whinston A (1974) An axiomatic approach to cost allocation for public investment.Public Finance Q uarterly 2(2) :2 36 -2 51


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A l l o ca t i n g J o i n t C o s t s b y M ean s o f t h e N u c l eo l u s 1 1 9

M as ch l e r M (1 9 6 3 ) n -Pe r s o n g am es w i t h o n l y 1 , n - 1 , an d n -p e rm i s s i b le co a l i ti o n s . J o u rn a l o fM a t h e m a t i c a l A n a ly s is a n d A p p l i c a ti o n s 6 : 2 3 0 - 2 5 6

M e g i d do N ( 1 9 7 4 ) O n t h e n o n m o n o t o n i c i t y o f t h e b a r g a in i n g s e t , t h e k e r n e l a n d t h e n u c l e o l u s o f ag a m e . S IA M J o u r n a l o f A p p l ie d M a t h e m a t ic s 2 7 ( 2 ) : 3 5 5 - 3 5 8

M i l n or JW ( 1 9 5 2 ) R e a s o n a b l e O u t c o m e s f o r n - P e r so n G a m e s . R a n d C o r p o r a t i o n R M 9 1 6M o u l i n H (1 9 8 1 ) Im p l em e n t i n g j u s t an d e f f i c i en t d ec i s i o n m ak i n g . J o u rn a l o f Pu b l i c E co n o m i cs1 6 : 1 9 3 - 2 1 3

O w en G (1 9 6 8 ) n -Pe r s o n g am es w i t h o n l y 1 , n - 1 , an d n -p e r s o n co a l it i o n s . A m er i can M at h em a t i -c a l S o c i e t y , P r o c e ed i n gs 1 9 , p p 1 2 5 8 - 1 2 6 2

R a n s m e i e r J S ( 1 9 4 2 ) T h e T e n n e s s e e V a l l e y A u t h o r i t y : a c a s e s tu d y i n t h e e c o n o m i c s o f m u l t ip l ep u rp o s e s t r eam p l an n i n g . T h e V an d e rb i l t U n i v e r s i t y P res s, N as h v i l l e

Sch m ei d l e r D (1 9 6 9 ) T h e n u c l eo l u s o f a ch a rac t e r i s t ic fu n c t i o n g am e . S IA M J o u rn a l o f A p p l i edM a t h e m a ti cs 1 7 ( 3 ) : 1 1 6 3 - 1 1 6 9

S h a p l e y L S ( 1 9 7 1 ) C o r e o f c o n v e x g a m e s. I n t e r n a ti o n a l J o u r n a l o f G a m e T h e o r y 1 : 1 1 - 2 6Sh ark e y WW (1 9 8 2 a ) Su g g es t io n s fo r a g am e- t h eo re t i c ap p ro a ch t o p u b l i c u t i l i t y p r i c in g an d co s t

a l l o c a ti o n . B e l l J o u r n a l o f E c o n o m i c s 1 3 ( 1 ) : 5 7 - 6 8S h a r k e y W W (1 9 8 2 b ) C o o p e r a t i v e g a m e s w i t h l a rg e c o r es . I n te r n a t i o n a l J o u r n a l o f G a m e T h e o r y1 1 ( 3 / 4 ) : 1 7 5 - 1 8 2

Sh ark e y WW, T e l s e r L G (1 9 7 8 ) Su p p o r t ab l e co s t fu n c t i o n s fo r t h e m u l t i p ro d u c t f i rm . J o u rn a l o fE c o n o m i c T h eo r y 1 8 : 2 3 - 3 7

Sh u b i k M (1 9 6 2 ) In cen t i v es , d ecen t r a l ized co n t ro l , t h e a s s i g n m en t o f j o i n t co s t s an d i n t e rn a lp r i c in g . M an ag em en t S c i en ce , A p r i l : 3 2 5 -3 4 3

S t r a ff i n g P D , H e a n e y J P ( 1 9 8 1 ) G a m e t h e o r y a n d t h e T e n n e s s e e V a l l e y A u t h o r i t y . I n t e r n a t io n a lJ o u r n al o f G a m e T h e o r y 1 0 : 3 5 - 4 3

S u z u k i M , N a k a y a m a M ( 1 9 74 ) T h e c o s t a s s ig n m e n t o f th e c o o p e r a t i v e w a t e r r e so u r c e d e v e lo p -m e n t : a g am e t h eo re t i ca l ap p ro ach . Pap e r p re s en t ed a t t h e In t e rn a t i o n a l Wo rk s h o p o n B as icPro b l em s o f G a m e T h eo ry , Sep t em b er 2 t o 1 7 , 1 9 7 4 , B ad Sa l zu f l en . T h i s p ap e r has b een p u b -l is h ed i n J ap an es eS u z u k i M , N a k a y a m a M ( 19 7 6 ) S a m e t it l e. M a n a g e m e n t S c ie n c e 2 2 ( 1 0 ) : 1 0 8 1 - 1 0 8 6

T i j s S (1 9 8 1 ) B o u n d s fo r t h e co re an d t h e z -v a l u e . I n : M o es ch l i n O , Pa l l a s ch k e D ( ed s ) G am et h e o r y a n d m a t h e m a t i c a l e c o n o m i c s . N o r t h - H o l la n d , p p 1 2 3 - 1 3 2

Wes l ey E (1 9 7 1 ) A n ap p l i ca t i o n o f n o n s t an d a rd an a l y s is t o g am e t h eo ry . T h e J o u rn a l o f Sy m b o l i cL o g ic 3 6 ( 3 ) : 3 8 5 - 3 9 4

Y o u n g H P ( 1 9 8 2 ) A x i o m a t i z i n g t h e S h a p l e y v a lu e w i t h o u t l i n e a r it y . W P 8 2 - 64 , I I A S AY o u n g H P, O k ad a N , H as h i m o t o T (1 9 8 0 ) C o s t a l l o ca t i o n i n w a t e r r e s o u rces d ev e l o p m e n t - a ca se

s t u d y o f S w e d e n . R R 8 0 - 3 2 , I I A S A

R ece i v ed J an u a ry 1 9 8 5R ev i s ed v e r s i o n A p r i l 1 9 8 5

legros cost allocation ijgt 1986

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