Sympathy and Social Choice

7/28/2019 Sympathy and Social Choice

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E X T E N D E D S Y M P A T H Y A N D T H E P O S S I B I L I T Y O FS O C I A L C H O I C EK E N N E T H J . A R R O W

1 . S o c i a l C h o i c e a n 0 t h e C o n c e p t o f J u s t ic eTh e d o m a i n o f t h e t h e o ry o f so c i al c h o ic e i s n o t i n p r i n c ip l e th esame as th a t o f a t heo ry o f j u s t i ce . In som e d i rec t ions , i t is c l ea r lywider , s ince i t i s supp osed to cov er a ll dec i s ion tha t m us t b e mad eco l l ec t ive ly . On the o the r hand , i t i s poss ib l e to ho ld tha t p ropos i -t io n s a b o u t d i s tr i b u ti v e j u s t ic e a r e n o t n e c e s sa ri ly p ro p o s i t i o n s a b o u t

co l l ec t ive dec i s ions . Indeed , t heor i es such as Noz ick ' s [10 : Chap te r7 ] a rgue tha t j u s t i ce i s the resu l t o f i nd iv idua l ra the r t han co l l ec t ivedecis ions .Neve r the less , I th ink i t sa fe to say tha t t he tw o the or i es a rei n t i m a t e l y l in k e d . S u p p o se w e a c c e p t t h e P a re t o c r i t e r i o n a s p a r t o ft h e d e f i n i ti o n o f a s a t i s f a c t o ry t h e o ry o f so c ia l c h o i c e. Th e n , i n a n yg iven feas ib le se t o f a l te rna t ives , a p r inc ip l e o f soc ia l cho ice se l ec t sa m o n g t h e m e m b e r s o f t he P a r e t o e f f i c ie n t s u b s e t. T h e s e c o n s t i t u t ethe adm iss ib l e se t o f red i s t r ibu t ions . F ur th e r , w i th in th i s subse t , an yt w o d i s t r ib u t i o n s t h a t y i e ld t h e s a m e s a t i sf a c t io n s t o e a c h in d i v id u a la re regarded as ind i f fe ren t . He nce , t he p r inc ip l e o f soc ia l cho icese l ec t s a m o n g a l te rn a t iv e d i s t r ib u t i o n s o f s a t is f a c t io n s . I t t h e r e fo rese rves the sam e func t ion as the p r inc ip l e o f d i s t r ibu t ive jus t i ce a ndmigh t be iden t i f i ed wi th i t .A s fo r th e o t h e r d i r e c t i o n , I c o n t i n u e t o u p h o l d t h e v i e w t h a tp r inc ip l es o f j u s t i ce a re indeed in t en ded as c r i t e r i a fo r soc ia ldec i s ions . The dec i s ion migh t i ndeed be to l e t t he ac tua l d i s t r ibu t ionb e g o v e rn e d b y so m e d e c e n t r a li z e d m e c h a n i sm . B u t t h is is i t s e l f ad e c i si o n . Th i s is t h e t r a d i t io n t h a t h a s b e e n p re d o m i n a n t i n w e l f a r ee c o n o m i c s , a s i n th e w o rk o f B e rg so n [4 ] o r S a m u e l so n [1 4 :C h a p t e r V I I I ] , w h ic h I a c c e p t e d in m y o w n e a r li er w o r k [ 1 ] a n d itis a l so c e r t a i n ly t h e v i ew o f R a w l s [1 2 ] a n d , fo r t h a t m a t t e r , o f t h e

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KENNETH J. ARROW

u t il it a ri a n s . A fu l le r d e fe n se o f t h e d o c t r i n e t h a t p ro p o s i t i o n s a b o u tj u s t ic e a r e h y p o t h e t i c a l so c ia l d e c is io n s m u s t a w a i t a n o t h e r o c c a s i o n .

2 . Th e S t ru c t u re o f S o c ia l D e c i s io n P ro b l e m s .I w il l s i m p l y m a i n t a i n w i t h o u t d i s cu s s io n t h e v i e w t h a t t h eo u t c o m e o f a so c ia l c h o i c e p ro c e d u re ( c o n s t i t u t i o n , so c ia l w e l f a r efu n c t i o n , so c ia l d e c i si o n fu n c t i o n , o r w h a t e v e r o t h e r t e rm is u se d ) isa n o rd e r i n g ( t r a n s i ti v e , c o m p l e t e , a n d r e f l e c ti v e r e l a t io n ) o f a l t e rn a -t i v e so c i a l s t a t e s . Th i s fo rm u l a t i o n d o e s n o t e x c l u d e t h a t some

a sp e c t s o f r e so u rc e a l l o c a t i o n a r e r e s e rv e d t o p r i v a t e d e c i s i o n (o r t odec i s ion b y sub-soc ie t ie s ) . I t is s imp ly assum ed as g iven tha t t he r e i sa r a n g e o f d ec i si o n s a b o u t t h e d i s t r i b u t i o n o f g o o d s w h i c h i s t o b em a d e b y s o c ie t y . T h e r e a r e p r o b l e m s w i t h t h e f u ll f o r m u l a t i o n o f ad i v i s i o n o f c h o i c e b e t w e e n p r i v a t e a n d so c i a l r e a l m s (b e c a u se t h ed e c i s i o n s m a y b e c o m p l e m e n t a ry o r su b s t i t u t e ) , b u t a g a i n t h e sep ro b l e m s w ill n o t b e d e a l t w i t h h e re .A s i n d i c a t e d , I re t a in t h e v i e w t h a t t h e o u t c o m e sh o u l d b e a no rd e r in g . P re su m a b l y , t h e w e a k e s t p o s s ib l e o u t c o m e o f a c o n s t i t u -t i o n w o u l d b e a c h o i c e fu n c t i o n , a m a p p i n g o f e a c h f e as ib l e s e t o fa l te rn a t i v e s i n t o a c h o se n su b se t . A g a i n , o n e c a n n o t a v o i d so m ec o n s i s t e n c y r e l a ti o n s a m o n g t h e c h o i c e s f ro m d i f f e r e n t f e a s ib l e s e ts .V a r i o u s c o n d i t i o n s w e a k e r t h a n a fu ll o rd e r in g h a v e b e e n p ro p o se d ,b u t , i n m y j u d g e m e n t , t h e y d o n o t l e a d t o i n t e r e s t i n g c o n c l u s i o n s .U n l e s s t h e y a r e i m p l a u s i b l y w e a k , t h e se c o n d i t i o n s d o n o t l e a d t on o t i c e a b l y g re a t e r f r e e d o m in t h e s e l e c t i o n o f so c ia l c h o i c e p ro c e -dures .

W h a t r e m a i n s is t h e d e t e rm i n a t i o n o f t h e so c i al o rd e r i n g . O n w h a tda ta i s i t based? In pa r t i cu la r , how does i t re l a t e to ind iv idua lp re f e r e n c e s o v e r so c i a l s t a t e s , w h a t m i g h t b e t e rm e d , " i n d i v i d u a lu t i li t ie s . " F o r p u rp o se o f t hi s p a p e r , I a m a c c e p t i n g t h e v i e w p o i n t o ft h e u t i l it a r ia n s a n d o f w e l f ar e e c o n o m i c s . I t is a s su m e d t h a t e a c hi n d iv i d u al h a s so m e m e a su re o f t h e s a t i s f a c t i o n h e d ra w s f ro m e a c hsoc ia l s t a t e and tha t t he soc ia l o rde r ing i s de t e rmined by thespec i f i ca t ion o f t hese u t i li t ie s fo r a l l poss ib l e soc ia l s t a t e s .I w i ll r e t u rn sh o r t l y t o t h e m e a su re a b i l i t y p ro p e r t i e s o f u ti li ti e s ;a t t h e m o m e n t , I u s e th e t e r m i n a n e u t r a l m a n n e r c o n s i s te n t w i t ht h e i r b e i n g e i t h e r o rd i n a l o r c a rd i n al a n d i n t e rp e r so n a l l y c o m p a ra b l eo r n o t . A t t h i s p o i n t , I w a n t t o r a i s e a n o t h e r i s su e , w h a t p ro p e r t i e so f the soc ia l s t a t e a re a rgument s in the ind iv idua l u t i l i t y func t ions .Tw o l e a d in g c a n d i d a t e s a r e (1 ) a m e a su re o f t h e in d i v i d u a l 'ss a t i s f a c t i o n d e r i v e d f ro m t h o se a sp e c t s o f t h e so c i a l s t a t e t h a t2 2 4


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EXTENDED SYMPATHY AND SOCIAL CHOICE

c o n c e rn h i m a n d (2 ) h i s e v a l u a t io n o f th e e x t e n t t o w h i c h t h e so c ia ls t a te i s a j u s t o r o t h e rw i se so c i a l ly s a t i s f a c t o ry a l l o c a t io n . (O fc o u r se , c o m b i n a t i o n s o f t h es e t w o v i e w p o i n t s a r e p o s s ib l e .) In m yw o r k o n t h e i m p o s s i b i l i t y t h e o r e m , I r e g a r d e d e i t h e r i n t e r p r e t a t i o na s p e rm i s s i b l e , i n t h e s e n se t h a t t h e i m p o ss i b i l i t y w a s v a l i d u n d e re i t h e r [1 : p p . 1 7 -1 8 ] . S e n [1 6 ] h a s a rg u e d t h a t m y su g g e s te dc o n d i t i o n s o n so c i a l c h o i c e a r e m o re a p p ro p r i a t e t o t h e i n t e rp re t a -t io n o f c h o i c e a m o n g p r i n c ip l es o f j u s ti c e t h a n t o t h e d e t e r m i n a t i o no f a j u s t a l l o c a t i o n o f s a t i s f a c t i o n s . I r e m a i n u n c o n v i n c e d t h a t t h esa m e i s su e s d o n o t a r i s e u n d e r b o t h i n t e rp re t a t i o n s , a n d I t h i n k t h ere su lt s t o b e p re se n t e d i n th i s p a p e r a re v a li d i n e i t h e r i n t e rp re t a t i o n .H o w e v e r , I h a v e t h e f i r st in t e rp re t a t i o n i n m i n d p r i m a r i l y h e re . A f t e ra ll, i f w e a r e s e e k i n g a c o n c e p t o f j u s ti c e , i t is n o t v e ry s a t i s f a c t o ryt o s t a r t w i t h t h e id e a t h a t e v e ry i n d iv i d u a l h a s a l r e a d y fo rm e d ac o n c e p t w h i c h e n te r s i n t o t h e d e t e r m i n a t i o n o f t h e s o c ia l o u t c o m e .In t h is l a s t s t a t e m e n t , I t h i n k I a m a t o n e w i t h R a w l s . B u t i t isn o w c l e a r t o m e t h a t i n a n o t h e r w a y t h e re i s sh a rp d i v e rg e n c e ; R a w l sc l ea r ly does n o t base the jus t aUc oa t ion on ind iv idua l u ti l it i e s bu tr a t h e r o n t h e d i s tr i b u ti o n o f th e " p r i m a r y g o o d s " w h i c h p e r m i ti n d iv i d u al s t o m a x i m i z e w h a t e v e r u t i li t y fu n c t i o n s t h e y c h o o se ( fo rR a w l s ' s m o s t e x t e n d e d s t a t e m e n t o f th is p o s i t io n , s ee [ 1 3 ] ,e spec ia l ly pp . 551-4 ) .Th e w o rk I a m r e p o r t i n g o n h e re h a s a n i ro n i c r e l a t i o n t o R a w l s ' sd i f f e r e n c e p r i n ci p le . U n d e r c e r t a in e p i s t e m o l o g i c a l a s su m p t i o n sab ou t ind iv idua l u t il i ti e s , a soc ia l cho ice a pp ro ac h l eads to Ra wls 'd i f f e r e n c e p r in c i p le - b u t i n t e rm s o f u t il i ti e s , n o t p r i m a ry g o o d s .

3 . T h e l n v a ri a n c e o f S o c i a l C h o i c e U n d e r T r a n s f o r m a t i o n so f U t i l it i e s .

Th e n e w r e su l t s i n so c i a l c h o i c e t h e o ry t h a t I w i sh t o sk e t c h h e ret o d a y w e r e d e v e l o p e d in d e p e n d e n t l y b y t w o y o u n g s c h o la r s , S t e v e nS t r a sn i c k , in a n u n p u b l i sh e d d o c t o ra l d i s s e r t a t i o n i n p h i l o so p h y a tH a r v a r d , [ 1 7 ] , a n d P e t e r J . H a m m o n d , a n E n gl is h e c o n o m i s t , i n ar e c e n t ly p u b l i s h e d p a p e r , [ 7 ] . T h e w o r k s h a v e a l r e a d y b e e n in f lu -e n t i a l i n m a n u sc r i p t a n d h a v e i n p a r t i c u l a r l e d t o a n e x c e l l e n ts y n t he s is b y t h e B e lg ia n e c o n o m i s t s , C l a u d e d ' A s p r e m o n t a n d L o u i sGevers , [6 ] , and i t i s t he i r expos i t i on which I sha l l l a rge ly f611ow.M o s t o f t h is p a p e r w i ll b e d e v o t e d t o a n e x p o s i t i o n o f t h e f o r m a lt h e o ry , t h o u g h I sh a l l o m i t p ro o f s ; a t t h e e n d , I w i l l m a k e so m ec o m m e n t s o n i n te r p r e ta t io n .In m y b o o k , I u se d t h e p re f e r e n c e o rd e r i n g s o f i n d iv i d u a ls o v e r

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KENNETH J. ARROW

s o c i al s t a t e s a s t h e v a r i a b le s w h i c h d e t e r m i n e d t h e s o c i a l o r d e r i n g s .A s is w e l l k n o w n , a n a l t e rn a t i v e e q u i v a l e n t r e p r e s e n t a t i o n o f i n di -v i d u al p r e f e r e n c e w o u l d h a v e b e e n a r e a l- v a lu e d u t i li t y f u n c t i o n o v e rs o ci al s ta t e s , w h i c h , h o w e v e r , w o u l d h a v e m e a n i n g o n l y u p to am o n o t o n e t r a n s f o r m a t i o n J S in c e n o i n te r p e rs o n a l c o m p a r i s o n sa p p e a r e d i n m y a p p r o a c h , t h e u t i l it y f u n c t i o n s o f t h e d i f f e r e n ti n d iv i d u al s c o u l d b e s u b j e c t to i n d e p e n d e n t m o n o t o n e t ra n s -f o r m a t i o n s .

T h e t e r m , " m e a n i n g , " w h i c h m a y s e e m t o ra is e f r i g h te n i n gs e m a n t i c i s su e s , h a s a p r e c i s e m e a n i n g . I f w e r e p r e s e n t in d i v i d u a lp r e f e r e n c e s b y u t i l it y f u n c t i o n s , t h e s o c i al c h o i c e w i ll h a v e t o b ei n v ar ia n t u n d e r i n d e p e n d e n t m o n o t o n e t r a n f o r m a t i o n s f o r d i f fe r e n ti n d i v i d u a l s .

L e t ,N = s e t o f i n d i v i d u a ls ,X = s e t o f p o s s i b l e s o c i a l a lt e r n a t i v e s .

Any given e a s i b l e s e t o f a l te r n a t i v e s is , t h e n , a s u b s e t o f X .A u t i l i ty f u n c t i o n , u ( x ) , i s a r e a l- v a l u e d f u n c t i o n o n X . I t d e f i n e sa n o r d e r i n g o v e r X i n t h e u s u a l w a y , x i s p r e f e r r e d t o y a n d o n l y i f

u ( x ) > u ( y ) . A s p e c if i c a ti o n o f u t i l i ty f u n c t i o n s , o n e f o r e ac h i nd i-v i d u a l , u i ( x ) ( i = l . . . . . n ) , c a n a l so b e r e g a r d e d a s a s in g l e f u n c t i o nu ( x , i) o v e r t h e C a r t e s ia n p r o d u c t X x N . L e t ,

U = s e t o f r e a l- v a l u e d f u n c t i o n s o n X X x .F r o m n o w o n , t h e t e r m , " u t i l i t y f u n c t i o n , " w i l l r e fe r t o a n y

m e m b e r o f U . I n th e s e t e rm s , a s o c ia l w e l f a r e o r c o n s t i t u t i o n ( t o u s em y n o w - f a v o r i te t e r m ) c a n b e d e f in e d :

Def in i t ion . A cons t i tu t ion is a f u n c t i o n , f , m a p p i n g U i n t oo r d e r in g s o f X .T h a t i s , w e a s s o c i a t e t o e a c h u t i l i t y f u n c t i o n ( i n t h e p r e s e n t s e n s e ,

i .e ., a u t i l it y f u n c t i o n f o r e a c h i n d i v i d u a l ) a s o c i a l o i d e r i n g o f s o c i a ls t a t e s .L e t u b e a u t i l i t y f u n c t i o n , a n d l e t g i ( i= l . . . . . n ) b e n ( s t r ic t l y )

m o n o t o n e f u n c t io n s f r o m r e al n u m b e r s t o r ea l n u m b e r s . L e t a u t i l i t yf u n c t i o n , u ' , b e d e f i n e d b y ,

u ' ( x , i ) = g i [ u ( x , i ) ] .I f w e m a i n t a i n t h e s tr i c tl y o r d in a l a p p r o a c h o f A r r o w [ 1 ] a n d a l sop r o h i b i t i n t e r p e rs o n a l c o m p a r i s o n s , t h e n t h e r e w o u l d b e n o o p e r a -t io n a l d i s t in c t i o n b e t w e e n u a n d u ' ; t h e o n l y e v i d e n c e w e h a v e is t h eo r d e r i n g o f s o c i a l s t a t e s f o r e a c h i n d i v i d u a l , a n d t h a t is t h e s a m e f o r2 2 6


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u a nd u ' . He nce , th is a sce t ic v iew poin t wo uld requ i re tha t th eo rd e rin g s d e f in e d b y u a n d u ' b e t h e s a me .Ordinal Invariance. I f t h e r e e x i s t s t ri c t l y m o n o to n i c f u n c t io n s gt(i= l . . . . . n ) f r o m r e al n u mb e r s t o r e al n u m b e r s s u c h t h a t ,

u ' ( x , i ) = g a [ u ( x , i )] f o r a l l x a n d i ,t h e n f ( u ) = f (u ') .As i s we l l known ( see Arrow [1 ] ) , Ord ina l Invar iance , toge :he rwi th t h e o th e r a s s u mp t io n s u s u a lly ma d e a b o u t t h e c o n s t i t u t i o n ( s een e x t s e c t i o n ) , imp l ie s t h a t t h e r e e x i s ts n o c o n s t i t u t i o n .Th e q ues t ion the n a r ises , a re the re less dem anding fo rm s o fin v ar ia n c e wh ic h p e rm i t t h e e x i s te n c e o f s a t i s f a c to ry c o n s t i t u t i o n sa n d wh ic h c a n b e j u s t if i e d i n t e rms o f a c tu a l o r a t l e a st h y p o th e t i c a lo b s e rv a t i o n ?As i s we l l k n o w n , i f c a rd in a l i n t e rp e r s o n a l c o mp a r i s o n s o f u t i l i t yd i f fe rence s a re regarded as mean ingfu l , then the sum o f u t i l it i e sd e f in e s a c o n s t i t u t i o n . Th e t r a n s fo rma t io n s u n d e r wh ic h t h e c o n s t i -t u t i o n is t o b e i n v a ri a n t mu s t b e s u c h t h a t t h e t ru th -v a lu e o fs t a te m e n t s o f th e f o r m ,u(x , i) - u (y , i) = k [u (x , j ) - u (y , j ) ] ,i s p rese rved . Th is obv ious ly res t r ic t s the t rans fo rmat ions to l inea rt rans fo rm at ions w i th the same (pos i t ive ) coe f f ic ien t fo r al l ind iv id .ualsCardinal-Difference lnvariance. I f t h e r e e x i s t n u m b e r s , a t, b ( ) O ) ,s u c h t h a t ,

u ' ( x , i ) = a t + b u ( x , i ) f o r a ll x a n d i ,t h e n f (u ) = f (u ' ) .T h e n t h e c o n d i t io n ,, z ( x , ) z u ( y , ),s a t is f i e s C a r d i n a l - D i f f e r e n c e I n v a r i a n c e . O r d e r i n g b y s u m o f u t il it ie sa l s o s a t i s f i e s h e o t h e r c o n d i t i o n s ( i n d e p e n d e n c e o f i r r e l e v a n t a l te r -n a t i v e s , P a r e t o p r i n c i p l e , n o n y m i t y ) o f t h e f o l l o w i n g s e c t i o n .

I w a n t t o c o n c e n t r a t e , h o w e v e r , o n a d i f f e r e n t i n v a r i a n c e p r i n -c i p l e . T h i s o n e p e r m i t s i n t e r p e r s o n a l o r d i n a l c o m p a r i s o n s . T h a t i s ,w e r e g a r d a s m e a n i n g f u l s t a t e m e n t s o f t h e f o r m ,

i n d i v i d u a l i i n s t a t e x i s b e t t e r o f f t h a n i n d i v i d u a l i n s t a t e y .W h a t e v e r o n e m a y t h i n k o f i n t e r p e r s o n a l c o m p a r i s o n s , a t l e a s t t h e s ea r e o r d i n a l a n d t h e r e f o r e m a y b e i n t e r p r e t e d a s h y p o t h e t i c a l c h o i c e .I d e f e r m o r e d e t a i l e d d e f e n s e a n d c r i t i q u e t o s e c t i o n 7 .

I n t h i s c a s e , u ( x , i ) m a y b e i n t e r p r e t e d a s t h e u t i l i t y d e r i v e d b yi n d i v i d u a l i i n s t a t e x . S t a t e m e n t s o f t h e f o r m ,

u ( x , i ) > u ( y , j ) ,2 2 7


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KENNETH J. ARROW

a r e t o b e p r e s e r v e d u n d e r t r a n s f o r m a t i o n s . T h u s , t h e p e r m i t t e dt r a n s f o r m a t i o n s a r e m o n o t o n e t r a n s f o r m a t i o n s o f t h e w h o l e u t il i tyf u n c t i o n , b u t n o t t r a n s f o r m a t i o n s w h i c h d i f f e r f r o m i n d iv i d u al t oi n d i v i d u a l .Co-ordinal Invariance. I f t h e r e e x i s ts a s t r ic t l y m o n o t o n e f u n c -t i o n , g f r o m r e al n u m b e r s t o r e al n u m b e r s s u c h t h a t ,u ' ( x , i ) = g [ u ( x , i ) ] , a l l x a n d i ,t h e n f ( u ) = f ( u ') .4 . O t h e r C o n d i t i o n s o n S o c i a l C h o i c e .

W e r e t a i n c o n d i t i o n s o n s o c ia l c h o i c e l ik e t h o s e i n [ 1 ] , t h o u g hr e s t a t e d t o b e c o m p a t i b l e w i t h th e p r e s e n t d e f 'm i t i o n o f a c o n s t i -t u t i o n a s a m a p p i n g f r o m u t i l i t y f u n c t i o n s r a th e r t h a n f r o m n - t u p le so f o r d e r in g s .S i n c e , f o r e a c h u , f ( u ) is a n o r d e r i n g , t h e n o t a t i o n ,

x f l u ) y ,m e a n s ,x i s a t l e a s t a s g o o d a s y i n t h e s o c i a l o r d e r i n g i n d u c e d b y t h eu t i l i t y f u n c t i o n u .T h e s t ri c t p r e f e r e n c e o r d e r in g d e t i n e d b y u w il l b e d e n o t e d b yf p ( u ) .T h e ( c o n t r o v e r s i a l) a s s u m p t i o n o f i n d e p e n d e n c e o f i rr e le v a n ta l t e r n a ti v e s w i ll b e s t a t e d h e r e o n l y f o r p r e f e r e n c e s , i. e. , f o r c h o i c e

f r o m t w o - m e m b e r s et s. O n l y t h a t p a r t o f t h e a s s u m p t i o n w a s u s e d inI l l .Binary Relevancy. I f u a n d u ' a r e s u c h t h a t ,u ( x , i ) = u ' ( x , i ) a n d u ( y , i ) = u ' ( y , i ) , a ll i,t h e n x f ( u ) y i f a n d o n l y i f x f l u ' ) y .T h e d e m o c r a t i c c o n d i t i o n t h a t a l l i n d i v i d u a l s a r e t o c o u n t e q u a l l yw i ll b e h e r e r e p r e s e n t e d in t h e s t r o n g f o r m o f s y m m e t r y a m o n gi n d iv i d u a ls , in s t e a d o f t h e v e r y w e a k n o n - d i c t a t o r s h i p c o n d i t i o n .A n o n y m i t y . L e t s b e a p e r m u t a t i o n o f N . I f ,u ( x , i ) = u ' [ ( x , s ( i) ] f o r a ll x a n d i ,t h e n f ( u ) = f ( u ' ) .T h e P a r e t o c o n d i t io n u s e d i n A r r o w [ 1 ] is th e w e a k c o n d i t i o n ,Weak P areto. I f u ( x , i ) > u ( y , i ) , a ll i , t h e n x f e ( u ) y .F o r c e r t a i n p u r p o s e s , w e w i ll w a n t t h e s t r o n g e r c o n d i t i o n t h a t i f a tl e a st o n e i n d iv i d u a l i s b e t t e r o f f b y a c h a n g e , w h i l e n o n e a r e h u r t ,t h e c h a n g e s h o u l d b e m a d e .Stron g P areto. I f u ( x , i ) > u ( y , i ) , a ll i , a n d , f o r s o m e j , u ( x , j ) >u ( y , j ) , t h e n x f p ( u ) y . -2 2 8


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E X T E N D E D S Y M P A T H Y A N D S O C I A L C H O I C E

Fina l ly , t he re is an in t e r es t ing gene ra l i za t ion o f the Pare toc o n d i t i o n , a n i m p l i c a t io n o f w h i c h h a s i n t e r e st i n g c o n se q u e n c e s . U pto th i s po in t , w e have t aken the r ange o f ind iv idua l s , N , a s g iven . Bu tsu p p o s e w e a s su m e t h a t w e h a v e a c o n s t i t u t i o n f o r e v e r y s e t o fi n di v id u a ls . W e w o u l d e x p e c t t o h a v e so m e c o n s i s t e n c y c o n d i t i o n sa m o n g t h ese c o n s t i tu t i o n s . I n d e e d , t h e P a r e t o p r i n c ip l e is o n e su c h .I f t he re is on ly o ne ind iv idua l in the wo r ld , t he soc ia l o rde r ing iss i m p l y h i s . Th e n t h e W e a k P a r e t o p r i n c i p l e c a n b e i n t e r p r e t e d a sasse r t ing tha t i f al l one- ind iv idua l sub se t s p re fe r x to y , t he n so doesthe wh o le so c ie ty ; wh i le the S t rong Pa re to p r inc ip les says tha t i f a llone- ind iv idua l subse t s w eak ly p re fe r x to y , w h i le a t l eas t one has as t r o n g p r e f e r e n c e , t h e n so c i e t y p r e f e r s x t o y . Th e o n e - i n d i v i d u a lsubse t s a r e a pa r t i t i on o f the en t i r e se t o f vo te r s . The n i t isr e a so n a b le t o e x t e n d t o p r i n c ip l e t o c o v e r a ll p a r t i t io n s o f t h e v o t e r s ;f o r e a c h su b se t, w e a r e a s su m i ng th a t t h e c o n s t i t u t i o n p r e sc r i b e s am ap p ing o f the u t i l i ty func t ion ( r es t r i c ted to the ind iv idua l s in tha tsubs e t ) i n to a soc ia l o rde r ing .S ince N i s now var i ab le , we d e f ine ,UN = se t o f r ea l-va lued func t ions on X X N .

A c o n s t i t u t i o n f o r a n y g i v e n s e ts o f v o t e r s N i s n o w su p p o se d t odef ine soc ia l o rde r ings fo r u t i l i ty func t ion s on e very subse t o f vo te r s .Definition. A c o n s t i t u t i o n i s a f a m i l y o f f u n c t i o n s , f N , d e f i n e df o r a ll se t s o f v o t e r s N , m a p p i n g U N i n t o o r d e ri n g s o f X .i f u e U N , so m e N , a n d M C N t h e n u M w ill b e t h e f u n c t i o n ures t r i c ted to ind iv idua ls in M and so be longs to UM. Hence , fo r anyu e UN , fM(UM) i s de f ined fo r every M CN .Generalized Strong Pareto. (a) I f N consis ts o f the s ingle indi -v idua l , i, fN(u) i s t he o rde r ing de f ined b y the u t i l i t y in d ica to r u (x , i ).( b ) I f Q i s a pa r t i t i on o f N , and , fo r all M e Q, x fM (UM) y , w h i le fo rs o m e M 'e Q, x fM,P(UM,Y, the n x fNP(U) y.Th i s p r inc ip le was s t a t ed by S t r asn ick . I t had been used ea r l i e r byY o u n g ( p . 4 4 ).Here we use the genera l i zed S t rong Pare to p r inc ip le on ly in thespec ia l fo rm of ,Elimination of Indifferent bTdividuals. Le t N b e p a r t i t i o n e d i n t oM ' a n d M " . S u p p o se u e U N U ' e U N a n d U M ' = U ' M ' w h i l e , for al l xand y , u (x , i ) = u (y , i ) and u ' (x , i ) = u ' (y , i ) f o r a l l i e M ". T h e n f N ( u )= fN(U') .

5 . T h e T h e o r e m s .W e k n o w o f c o u r se t h a t O r d in a l l n ,t a ri a n ce , B i n a ry R e l e v a n c y ,

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K E N N E T H J . A R R O W

A n o n y m i t y , a n d t h e We a k P a r et o c o n d i t io n a re i n c o m p a t i b le . I f ,how ever , w e replace Ordinal Invar iance by Co-ordinal Invar iance , thecond i t ions a re indeed satis fi ed and , in fac t , b y the m axim in p rin-ciple, i.e., x f ( u ) y i f a n d o n l y i f r a in u ( x , i ) = m i n u ( y , i ) .

This co nd i t ion a lso sa tisf ies El im inat ion o f Ind if fer en t Individualsand the Genera l ized W eak Pare to p r inc ip le (an obv ious ana logue o fthe General ized Strong Pare to pr inciple g iven in the precedingsec tion) . I t does no t sa t i s fy the S t rong Pare to p rinc ip le ; how ever , a sSen [15: p . 138, fn . 11] has note d, a s imple mo dif ic a t ion wil l. leadto sa t is fac t ion.Lex i ca l Max i mi n Pr i nc i p l e . For any a l t e rna t ive x and u t i l i ty"fun ct io n u , rank the individuals in increasing orde r o f u(x , i ) , and le ti (x , k ) be th e k th rank ing ind iv idua l ; t ie s can be b roken a rb i t r a r ily .For any pa i r o f a l t e rna t ives x , y , l e tk (x ,y ) = min (k lu [x ,k ] ~ u [y , i (y ,k ) ] ),i f ide fmed . Then x fP(u) y i f and on ly i f u [x , i (x , k (x ,y ) ) ] > u [y , i(y ,k (x ,y ) ) ] ] .Th e Lex ical Maximin Principle satisfies Co.ord inal Invarian ce, Bi-n a r y R e l e v a nc y , An o n y m i t y , a n d t h e G e n e ra li z ed S t ro n g P a r e t oc o n d i t i o n .I t i s no t , however , the on ly p r inc ip le sa t i sfy ing these cond i t ions .Indeed , the m axim ax princ ip le ,x f l u ) y i f a n d o n l y i f m a x u ( x , i ) = m a x u ( y , i ),i i

also sat is fies Co-ordinal Inva r iance , Bina ry , Relev ancy, A no ny m ity ,and the W eak Pare to c ond i t ion ; and c lea r ly , the Lex ica l M aximaxpr inciple , def ined in the obvious w ay , sa tisf ies a ll the co nd i t ionss ta ted above for L exical M inimax.W hat is surprising is tha t these are the on ly tw o such con di t ion s;a n d , b y m a k i n g a v e r y we a k e q u i t y a s s u m p t i o n , t h e L e x i c a l Ma x i -m ax p r inc ip le can be ru led ou t .We def 'me two more cond i t ions ; they wi l l no t be regarded aspr ima ry , bu t the i r r e la tions w i th the o the r cond i t ions wi ll be s t a ted .S t r ong Eq u i ty . Fo r a l l u e U N ' a ll x and y in X, and a ll i and j inN, i f u(x , g) = u(y , g) for g :~ i , j , and u(y , i) < u(x , i ) < u(x , j )


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E X T E N D E D S Y MP A T H Y A N D S O C I A L C H O I C E

The dua l a ssump t ion to S t rong Equ i ty is ,I n e q u i t y . For a l l ueUN, a l l x and y in X, and a l l i and j in N, i fu(x , g) = u(y , g) fo r g 4 : i , j , a nd u(y , i ) < u(x , i ) < u(x , j ) < u(y , j ) ,then y fP(u) x .The be t t e r - o f f ind iv idua l a l w a y s prevails.T h e o r e m 1 . I f the con s t i tu t ion satis fie s B inary Re levan cy , A non y-m i ty , Co-ord ina l ity Invar iance , and E l imina t ion o f Ind i f fe re n t Ind i -v idual s, then e i the r S t rong E qui ty o r In equ i ty ho lds .This resul t may seem surpr is ingly s t rong, and i t s proof takesnumerous s teps . However , an in tu i t ive ske tch can be g iven fo r thecase o f two ind iv idua ls . In the f i r st p lace , the assum pt ions o f B inaryR e l e v a n c y , An o n y m i t y , a n d E l i m i n a t i o n o f I n d i f f e r e n t I n d i v i d u a l scan easi ly be sh ow n to im ply ,N e u t r a l i t y . I f s is a pe rm uta t ion o f X, and u (x , i ) = u ' [ s (x ) , i ] fo ra ll x and i , then x f (u ) y i f and o n ly i f s (x ) f lu ' ) s (y ) .Chang ing the nam es o f the a l te rna t ives does no t ma t te r . Suppose ,t h e n , b o t h S t r o n g E q u i t y a n d I n e q u i t y f a i l. T h e f a il u re o f S t r o n gEq ui ty impl ies the ex i s tence o f u , x , y , i and j , suc h th a t ,

u ( y , i ) < u ( x , i) < u ( x , j ) < u ( y , j ) , y f P ( u ) x . ,T h e f a il u re o f I n e q u i t y im p l ie s t h e e x is t en c e o l u , x , y , i ', a n d i_'s u c h t h a t ,U t (y l . I t t1)


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KENNETH J. ARROW

I f w e a d d t h e S t ro n g P a re t o p r i n c ip l e t o t h e a s su m p t i o n s o f Th e o -r e m 1 , t h e n Th e o re m s I , 2 a n d 3 t o g e t h e r t e l l u s w e h a v e r e d u c e dt h e r a n g e o f p o s s i b l e c o n s t i t u t i o n s t o t w o , t h e Le x i c a l M i n i m a x a n dthe Lex ica l Max imax . To e l imina te the second , i t i s su f f i c i en t t od e n y t h e I n e q u i t y a s s u m p t i o n , in s t ea d o f i m p o s i n g t h e a p p a r e n t l ys t ro n g e r S tro n g Eq u i t y c o n d i t i o n . A n a s su m p t i o n w h i c h c o n t r a d i c t sIn e q u i t y i s,Minimal Equity. There ex i s t u e UN ' x e X , y e X , and j e N , suchtha t , fo r a ll i = j , u (y , i )


11/15


t h e a s s u m p t i o n s ; a t l e a s t w i t h s o - o r d in a l i n v a r i a n c e , t h e r e i s n oi n tr in s ic w a y o f d i s ti n g u is h i n g h i m f r o m h i s f o r t u n a t e c o m p e e r .

I n d e e d , t h e q u e s t i o n c a n b e ra i s ed w i t h re g a r d t o t h e o r ig i n a lp o s i t i o n o f R a w l s . W h y , i n t h a t s i t u a t i o n , s h o u l d n o t t h e i g n o r a n to n e s e a c h l o o k t o t h e b e s t r a t h e r t h a n t h e w o r s t p o s s i b i li t y ? W e h a v et o i n t ro d u c e s o m e a d d it io n a l c o n s i d e r a t i o n b e y o n d t h o s e o f f o r m a ls y m m e t r y . R a w l s in d e e d i n t r o d u c e d t h e f a m i l ia r n o t i o n o f r is k a v e r-s io n . B u t t h is i s g e t t in g c l o s e to t h e d o c t r i n e s o f u t i l i ta r i a n i s m . I f w ea s s u m e d i m i n i s h in g m a r g i n a l u t il i ty ( w h e t h e r i n t h e r is k - b e a t in g.s en se o r s o m e o t h e r ) , t h e n w e c a n i n d e e d e x p l a i n t h e a s y m m e t r i ct r e a t m e n t o f b e s t - o f f a n d w o r s t - o f f ar e t h o s e w i t h h i g h e r m a r g i n a lu t i l i t y .

B u t o b v i o u s l y w e h a v e n o w i n t r o d u c e d a w h o l e r a n g e o f c o n s i d e r -a t io n s b e y o n d t h e se p e r m i t t e d b y c o - o r d in a l i n v a ri a n c e a n d a ls ob e y o n d t h o s e p e r m i t t e d i n R a w l s i a n t h e o r y . T e c h n i c a l l y , t h e i n v a r -i a n c e r e q u i r e m e n t s a r e b e i n g w e a k e n e d , a n d m o r e p o s s i b i l i t i e s a r eb e in ~ i n t r o d u c e d .A s e c o n d a r g u m e n t f o r m o v i n g in t h is d i r e c t io n is t h e s e c o n d o ft h e t w o r e s e r v a ti o n s a b o u t t h e p r e s e n t l in e o f d e v e l o p m e n t t h a t Ii n d i c a te d a b o v e . I t c a n b e p u t a s a c o n t i n u i t y r e q u i r e m e n t . S u p p o s ea c h a n g e f r o m x t o y d im i n is h e s th e u t i l it y o f th e w o r s t - o f f b y s o m ev e r y s m a l l a m o u n t b u t i n c r e a se s t h e u t i li t y o f al l o t h e r s b y a g r e a td e a l. S u r e l y i t s e e m s r e a s o n a b l e t o a r g u e t h a t i f t h e lo s s t o t h ew o r s t - o f f is s m a l l e n o u g h a n d t h e g a in t o e v e r y o n e e ls e la r ge e n o u g h ,s o c i e t y s h o u l d p r e f e r y t o x . I n d e e d , i f t h e r e w e r e no l o s s t o t h ew o r s t - o f f a n d a g a i n t o a l l o t h e r s , t h e L e x i c a l M a x i m i n r u l e w o u l dc al l y a s t r i c t i m p r o v e m e n t ; h e n c e , b y a n y k i n d o f c o n t i n u i t y a rg u -m e n t , t h e p r e f e r e n c e f o r y o v e r x s h o u l d b e m a i n t a i n e d i f t h e u t i li t yl o ss t o t h e w o r s t - o f f is s u f f i c i e n t l y s m a l l.

I t m a y b e o b j e c t e d t h a t , u t i li ti e s b e i n g o r d i n a l ly m e a n i n g f u l , c o n -t i n u i t y w i t h r e s p e c t t o u t i li ti e s is n o t a ll t h a t n a t u r a l o r e v e n m e a -n i n g f u l. B u t c o n s i d e r i n s t e a d t h e d i s t r i b u t i o n o f g o o d s w h i c h d e t e r -m i n e s t h e u t i li ti e s o f th e d i f f e r e n t i n d i v id u a l s . T h i s m a y b e r e g a r d e da s a p o i n t in a E u c l i d ea n v e c t o r s p a ce o r s o m e o t h e r t o p o l o g i c a ls p a c e . I t is p e r f e c t l y m e a n i n g f u l t h e n t o p o s t u l a t e t h a t p r e f e r e n c e sa re c o n t i n u o u s w i t h r e s p e c t t o t h e u n d e r l y i n g t o p o l o g y . T h u s , i f ( x ,i ) i s p r e f e r r e d t o ( y , j ) , t h e n ( x ' , i) i s p r e f e r r e d t o ( y ' , j ) f o r al l x ' a n dy ' s u f f i c i e n t l y c l o s e t o x a n d y , r e s p e c t i v e l y . W i th s u c h a r e s t r i c t i o no n t h e u n d e r l y i n g p r e f e r e n c e s , w e c a n r e s t r ic t o u r s e l v e s to u t i l i t yf u n c t i o n s w h i c h a re c o n t i n u o u s i n t h e so c ia l s t a te . T h e n c o n t i n u i t yo f t h e c o n s t i t u t i o n w i t h r e s p e c t t o t h e u t i l it y f u n c t i o n i s a n o r d i n a l-l y in v a r i a n t a n d t h e r e f o r e c e r t a i n ly c o - o r d i n a l l y in v a r i a n t p r o p e r t y .

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KENNETH J. ARROW

T h u s , a d d i n g a c o n t i n u i t y r e q u i r e m e n t t o t h e h y p o t h e s e s o fTh e o re m 4 le ad s t o a n im p o s s ib i li t y t h e o re m .Th i s i s b y n o me a n s a " fo rma l " ma t t e r . C l e a r ly , t h e i n tu i t i o nb e h in d t h e c o n t in u i ty r e q u i r e m e n t is a s ma ll s t e p i n t h e d i r e c t i o n o fu t il it a r ia n e th ic s ; e v e n t h e w o r s t -o f f m e m b e r o f t h e s o c i e ty mig h t b ema d e to s u f f e r i f t h e r e is e n o u g h b e n e f i t t o o th e r s . Th e a s s u mp t io nof d imin ish ing marg ina l u t i l i ty impl ies wi th regard to usua l po l icya l te rn at iv e s t h a t t h e r e a r e b e t t e r w a y s o f imp ro v in g t h e l o t o f b e t t e r -o f f m e m b e r s t h a n b y h u r t i n g t h e w o r s t- o f f.Bu t the re is o ne s t r ik ing case , o f g rea t p rac tica l im po r tan ce ,where our in tu i t ion i s in favor o f u t i l i t a r ian ism i s some fo rm asaga inst any m in im ax ru le . I r e fe r to a l loca t ion over t ime . Ty p ica l ly ,we e x p e c t f u tu r e g e n e ra t io n s t o b e b e t t e r o f f t h a n w e ar e . S h o u ld wes av e fo r t h e m e i t h e r d i r e c t ly o r i n t h e fo rm o f p u b li c i n v e s tme n t s ? Ama x im in ru le wo u ld s u r e ly s a y n o . Bu t i f in v e s tme n t i s p ro d u c t iv e ,s o t h a t , i n t e rms o f g o o d s , t h e n e x t g e n e ra t i o n ga in s mo re t h a n welose , we usua lly fee l tha t som e inves tm en t is w or thw hi le even thou gh4the rec ip ien ts wil l be be t te r o f f than we a re .

7 . Th e O p e ra t i o n a l S ig n if ic a n c e o f I n t e rp e r s o n a l Ord in a lCo mp a r i s o n s .T he d i scuss ion o f the la s t sec t ion sugges ts tha t , i f any th ing , evenin t e rp e r s o n a l o rd in a l c o mp a r i so n s a r e n o t s u f f ic i e n t t o t a k e a c c o u n to f o u r i n tu i ti o n s o f j u s ti c e a s d e r iv e d f ro m a s o c ia l c h o i c e fr a me -wo rk . Th e r e q u i r e me n t o f Co -o rd in a l I n v a r i a n c e ma y s t i l l b e t o os t r ic t . In th i s sec t ion , I wi l l address the oppos i te ques t ion ; whe the r

such compar i sons have any mean ing , i . e . , whe the r the invar iancecr i te r ion shou ld no t pe rhap s be even s tr ic te r .Th e p o s s ib i li ty o f s u c h c o mp a r i s o n s h a s a l re a d y b e e n d e f e n d e d ind i f f e r e n t f o rms b y S u p p e s [1 8 ] , Ko lm [8 : P a r t C ] , a n d m y s e l f [ 1 :p p . 1 1 4 -1 1 5 1. I w i ll c o m m e n t o n ly b r i e f l y , t o a v o id r e p e a t in g m year l ier arguments excess ively .Th e c o n c e p t o f a p r e f e r e n c e o rd e r in g o r , b y e x t e n s io n , o f a u t i l i tyfunc t ion i s re la ted to hypo the t ica l cho ices . I t s usua l use i s in ac o m p le t e t h e o ry , s a y o f i n d iv id u al b e h a v io r , i n wh ic h t h e p r e f e r e n c eorder ing and the feas ib le se t jo in t ly de te rmine the chosen a l te rna -t ive . Th e p re fe renc e o rde r ing i s thoug h t o f a s g iven be f o re the feasi -b l e s e t i s k n o w n a n d th e r e fo r e d e t e rmin e s c h o i c e s a m o n g a l l p o s sib lepa i r s o f a l terna t ives . Th e feas ib le se t p resc r ibes w hich a l te rna t ives a re2 3 4


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EXTENDE D SYMPATHY AND SOCIAL CHOICE

i n f a c t a v a il a b le . I t m a k e s s e n se , th e r e f o r e , t o in c l u d e i n o u r i n f o r -m a t i o n c h o i c e s w h i c h a r e n o t in f a c t f e as ib l e t h o u g h t h e y a r e c o n -ce ivab l e .

N o w w e c a n s a y t h a t a m o n g t h e c h a r a c t e r i s t i c s w h i c h d e t e r m i n ea n i n d i v i d u a l ' s s a t i s f a c t i o n a r e s o m e w h i c h a r e n o t , a t l e a s t a t t h em o m e n t , a l te r a b le . A n i n d i v id u a l w h o is ill c a n m e a n i n g f u l l y b e sa idt o p r e f e r b e i n g w e ll . I f i n f a c t t h e r e w e r e s o m e m e d i c a l m e a n s o fc u r e , w e w o u l d t e s t t h i s p r e f e r e n c e b y a s k i n g i f h e w i ll p u r c h a s e t h es er vic e s. B u t c l e a rl y t h e p r e f e r s - . . . . . . . ~ b e t h e re w h e t h e r o r n o tm e d i c i n e w a s u se f u l.W e m a y s u p p o s e t h a t e v e r y t h i n g w h i c h d e te r m i n e s a n i n d i v id u a l 'ss a t is f a c t io n is in c l u d e d i n t h e l is t o f g o o d s . T h u s , n o t o n l y t h e w i n eb u t t h e a b i li ty t o e n j o y a n d d i s c ri m i n a te a r e i n c l u d e d a m o n g g o o d s .I t is , i n f a c t , t ru e t h a t o n l y s o m e o f th e g o o d s s o d e f i n e d ar e t r a n s -f e ra b l e a m o n g i n d iv i d u a ls w h i l e o t h e r s a r e n o t . B u t t h a t c o n s i d -e r a t i o n e n t e r s i n t o t h e d e f i n i t io n o f t h e f e a s ib l e s e t, n o t t h a t o f t h eo r d e r i n g . I f w e u s e t h i s c o m p l e t e l is t, t h e n e v e r y o n e s h o u l d h a v e t h es a m e u t i l i t y f u n c t i o n f o r w h a t h e g e t s o u t o f th e s o c i a l s ta te 9 T l ~ sd o e s n o t , o f c o u r s e , m e a n t h a t i n d iv i d u al s a gr e e o n t h e u t i l it y o f as o c i a l s t a t e , s i n c e w h a t t h e y r e c ei v e f r o m a g i v e n s t a t e d i f f e r s a m o n gind iv idua l s .

T h e l o c a t i o n o f i n t e r p e r s o n a l d i f f e re n c e s i n a li st o f q u a l it ie s h a sb e e n d e f e n d e d b y P a s c a l [ 1 1 : 3 2 3 ] :

W h a t is t h e E g o ? . . . D o e s h e w h o l ov e s s o m e o n e o n a c c o u n t o fb e a u t y r e a ll y lo v e t h a t p e r s o n ? N o ; f o r t h e s m a l l -p o x , w h i c h w il lk ill b e a u t y w i t h o u t k il li ng t h e p e r s o n , w i ll c a u se h i m t o lo v e h e rn o m o r e .

A n d i f o n e l ov es m e f o r j u d g m e n t , m e m o r y , h e do e s n o t n o tl ove me, f o r I c a n l o s e t h e s e q u a l i t ie s w i t h o u t l o s i n g m y s e l f .W h e r e , t h e n , is t h is E g o , if i t b e n e i t h e r i n t h e b o d y n o r t h e s o u l ? 9 F o r i t is i m p o s s i b le a n d w o u l d b e u n j u s t t o l o v e t h e s o u l o f ap e r s o n i n t h e a b s t r a c t . . . . W e n e v e r , t h e n , lo v e a p e r s o n , b u to n l y q u a l it ie s .

L e t u s , t h e n , j e e r n o m o r e a t t h o s e w h o a r e h o n o r e d o n a c c o u n to f r a n k a n d o f f ic e ; f o r w e lo v e a p e r so n o n l y o n a c c o u n t o fb o r r o w e d q u a l i t i e s .F o r m a l l y , w e s u p p o s e a s p a c e , Y , w h i c h d e f i n e s t h e r a n g e o f

p o s s i b le i m p l i c a t i o n s o f a s o c i a l s t a t e f o r a n i n d i v i d u a l . S i n c e t h es t a t e d e f i n e s , f o r e v e r y i n d i v i d u a l , e v e r y t h i n g t h a t c h a r a c t e r i z e s h iss a t i s f a c t i o n s , t h e s p a c e Y i s t h e s a m e f o r a l l i n d i v i d u a l s . I t i n c l u d e sg o o d s , ta s te s , a n d t h e r e a c t io n s o f o t h e r s t o t h e e x t e n t t h a t i n d i -v i d u a l s c a r e a b o u t e a c h o t h e r . A l l i n d i v i d u a l s h a v e t h e s a m e p r e f e r -

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K E N N E T H J . A R R O Wa n c e s o v e r Y . Le t u (y ) b e t h e o rd i n a l ly d e f i n e d u t i l it y i n d i c a t o r .Ea c h s t a t e i n x d e f i n e s i m p l i c a t i o n s fo r e v e ry i n d id i v id u a l . L e t G i (x )( i = l , . . . , n ) b e f o r e a c h in d iv id u a l a m a p p i n g f r o m X t o Y , e x p r e s si n gt h e se i m p l i c a ti o n s . Th e n w e c a n i d e n t i fy ,

U (X , i ) = U [ G i ( x ) ] ,and th i s possesses Co-ord ina l Invar i ance .Th i s i s a t l e a s t o n e w a y t o i n t e rp re t a n d d e fe n d i n t e rp e r so n a lo rd i n a l c o m p a r i so n s . (Th e re a r e o t h e r s . )I c a n n o t , h o w e v e r , c o n c l u d e w i t h o u t a d m i t t in g so m e d i f f ic u l t ie s .I c a n t h i n k o f tw o , t h o u g h p e r h a p s t h e y a r e t h e s a m e l o o k e d a ts o m e w h a t d i f f e re n t l y . F o r o n e t h i ng , if y o u r s a t is f a c ti o n d e p e n d s o nso m e i n n e r q u a l it ie s t h a t I d o n o t p o s se s s, t h e n I r e a l ly h a v e n o t h a dt h e e x p e r i e n c e w h i c h w i l l e n a b l e m e t o j u d g e t h e s a t i s f a c t i o n o n ew o u l d d e r i v e f ro m t h a t q u a l i t y i n a s so c i a t i o n w i t h so m e d i s t r i b u t i o no f g o o d s . H e n c e , m y j u d g m e n t h a s a p r o b a b i l i t y e l e m e n t i n i t a n dt h e re fo re w i ll n o t a g ree w i t h y o u r j u d g m e n t . B u t i t is e s s e n t ia l t o t h ep re se n t c o n s t ru c t i o n t h a t t h e c o m p a r i so n s o f i n d iv i d u a l i i n s t a t e xw i t h i n d iv i d u al j i n s t a t e y b e th e s a m e w h e t h e r t h e c o m p a r i s o n ism ade by i , j , o r a t h i rd ind iv idua l , k .Th e s e c o n d d i f f i c u l t y i s t h a t r e d u c i n g a n i n d iv i d u a l to a sp e c if i e dl is t o f qua l i ti e s i s den y ing h i s i nd iv idua l i ty in a d eep sense . Th e l a s tl in e o f t h e q u o t a t i o n f ro m P a sca l sh o u l d gi ve o n e p a u se . I n a w a yt h a t I c a n n o t a r t i c u l a t e w e l l an d a m n o n e t o o su re a b o u t d e fe n d i n g ,t h e a u t o n o m y o f in d iv i du a ls , a n e le m e n t o f m u t u a l i n c o m -m e n s u r a b i l i ty a m o n g p e o p l e s e e m s d e n i e d b y t h e p o s s ib i li ty o f in t e r -p e r so n a l c o m p a r i so n s . N o d o u b t i t i s so m e su c h f e e l i n g a s t h i s t h a th a s m a d e m e s o r e l u c t a n t t o s h i f t f r o m p u r e o rd i n a li s m , d e s p it e m yd e s ir e t o s e e k a b a s is fo r a t h e o ry o f ju s t i c e .

H A R V A R D U N I V E R S I T YC A M B R I D G E , M A S S A C H U S E T T S 0 2 1 3 8

U . S . A .

N O T E SS t r ic t ly s p e ak i n g , n o t e v e r y p r e f e r e n ce o r d e r i n g c a n b e r e p r e s e n t e d b y ar e a l - v a l u e d i n d i c a t o r , b u t t h i s r e s t r i c t i o n c a n b e n e g l e c t e d h e r e . I t i s n or e s t r i c ti o n i f t h e n u m b e r o f a l t e r n a t i v e s i s f i n i t e .R o b e r t N o z i ck h a s s tr e s se d t h i s p o i n t t o m e .

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E X T E N D E D S Y M P A TH Y A N D S O C I A L C H O I C ETh e spec ia l ro l e o f t he ex t rem es has appe a red in a pa ra l l e l con tex t , t ha t o fd e c i s io n - ma k i n g u n d e r u n c e r t a i n t y , w h e r e w e d o n o t w i s h t o a s c r ib e p r o b a -b i l i t ie s (d i r ec t ly o r i n d i rec t ly ) to t he poss ib l e s t a te s o f t he wo r ld . Thes t a t e s o f t he wor ld co r re sp ond to i nd iv idua l s , t he ac t ions t o be chosen toa l t e rna t ive soc i a l s t a te s . Ce r t a in approaches im ply co -o rd ina l i nva r i anca .S e e M fln or [ 9 : Th e o r e m 4 ] a n d A r r o w a n d H u r w i c z [ 3 ] .I do n o t f i nd R awls ' s t heo ry o f j us t sav ing [12 : sec t ion 44] a t a i l c l e a t ; i tseem s to avo id a r ig id m ax im in ru l e w i thou t p rov id ing a c l ea t subs t i t u t e .See Ar row [2 ] and Dasgup ta [5 ] .

R E F E R E N C E S[1 ] K.J . Arrow , Soc ial Choice and Individual Values, 2 n d E d i t io n , N e w Y o r k ,a n d N e w H a v e n , C o n n . 1 9 6 3 .[2 ] K.J . Ar row , "R aw ls ' s P r inc ip l e o f Jus t Sav ing ," Swedish J , Econ . , 1 9 7 3 ,75 , 323-335 .[3 ] K.J . Ar row a nd L. Hu rwicz , "A n O pt im a l i t y Cr i t e r ion fo r Dec is ion-Ma-k i n g U n d e r I g n o r a n c e , " in C .F . C a r te r a n d J .L . F o r d ( e d s. ), Uncer ta in tyand Exp ecta t ions in Econom ics , O x f o r d , 1 9 7 2 .[4 ] A. Be rgson , "A R eform ula t ion o f Ce r t a in Aspec t s o f We lfare Ec ono m ics ,"Quart. J. Econ., 1938 , 52 , 310-334 .[5 ] P . D asgup ta , "O n Som e Al t e rna t ive Cr i t e ri a fo r Jus t i ce Be tween Gen e ra -t i o n s , " J . Pub . Econ., 1974 , 3 , 405-423 .[ 6 ] C . d ' A s p r e m o n t a n d L . G ev ers , " E q u i t y a n d t h e i n f o r m a t i o n a l b a s is o fco l l ec t ive cho ice ," Rev. Econ. S tud. , f o r t h c o mi n g .[ 7 ] P .L H a m mo n d , " Eq u i t y , A r r o w ' s C o n d i t io n s , a n d R a w l s ' D i f f e re n c e P r i n -c ip l e , " Econometrica, 1 9 7 6 , 44, 793-804 .[8] S.C. K olm , Just ice e t Equi td , Paris, 1972.[9 ] J . M i lnor , "Gam es aga ins t Na tu re , " In R .M. Thra l l , C .H. Co om bs , andR.L . Davis (eds .) , Decision Processess, New York , 1954 .s ,[10] R . Nozick , Anarc hy , S ta te , and Utopia, N e w Y o r k , 1 9 7 4 .[ 11 ] B. Pascal, Pens~es, New York , 1965 .[ 12 ] J. Raw ls, A Th eo r y o f Ju s ti ce , Cambridge , Mass. , 1971.[13] J . Rawis , "Fa i rne ss to G ood ness , " Philosophical Review, 1 9 7 5 , 84 ,5 3 6 - 5 5 4 .[14] P . A . Samue l son , Foundat ions o f Econom ic Ana lysi s, Cam br idge , Mass .,1947 .[ 15 ] A.K . Se n, Collective C hoice and Social Welfare, San Franc i sco , 1970 .[ 1 6 ] A .K . S e n , " S o c i a l C h o i c e Th e o r y : A R e - e x a mi n a t i o n , " E co n o me t r i ca1 9 7 7 , 45, 53-89 .[ 17 ] S.L. St rasnick , Preference Pr ior i ty and the M aximiza t ion o f Soc ia l Wel -

fare, Doc to ra l d i sse r t a t i on , Ha rvard Un ive r s i ty , 1975 .[ 18 ] P . Suppes , "So m e Fo rm a l M odels o f Grad ing Pr inc ip l e s , " Syn these , Dec .1 9 6 6 , 16, 284-306 .[ 1 9 ] H .P . Y o u n g , " A n A x i o m a t i z a t i o n o f B o r d a ' s R u l e , " J . Econ. Theory.Sep t . 1974 , 9, 53 -52 .

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Sympathy and Social Choice

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Transcript of Sympathy and Social Choice