social choice and welfare volume 6 issue 2 1989 [doi 10.1007%2fbf00303166] j. snyder -- political...

8/12/2019 Social Choice and Welfare Volume 6 Issue 2 1989 [Doi 10.1007%2Fbf00303166] J. Snyder -- Political Geography a

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Soc Cho ice W elfare (1989) 6:103-12 5 S o c i a l h o i e eelfare Spr inger Ver lag 1989

P o l it i c a l G e o g r a p h y a n d I n te r e st -G r o u p P o w e rJ . S n y d e rDep artment of Econom ics, University of Chicago, 1126 Ea st 59th S treet, Chicago, IL 60637, USAReceived Ap ril 28, 19 87/A cceptedA pril 12, 1988

A b s t r a c t T h e r e l a t i o n s h i p b e t w e e n t h e d i s t r i b u t i o n o f a n i n t e r e s t g r o u p ' sm e m b e r s ( i . e . , v o t e r s ) a c r o s s l e g i s l a t i v e d i s t r i c t s a n d t h e g r o u p ' s a b i l i t y t oi n f lu e n c e g o v e r n m e n t p o l i c y is n o t w e ll u n d e r s t o o d . W e p r o p o s e a s i m p le m o d e lo f in t e r e s t g r o u p s a n d l e g is l a ti v e d e c i s i o n m a k i n g t o c h a r a c t e r i z e th i s re l a t i o n -s h ip f o r m a l l y , a n d a n a l y z e t h e m o d e l f o r t h e c a s e o f th r e e c o m p e t i n g i n t e r e s tg r o u p s . W e f i n d th a t a n i n t e r e s t g r o u p ' s e x p e c t e d p o w e r is m a x i m i z e d w h e n i tsm e m b e r s a r e e v e n l y d i v i d e d a c r o s s ( in + 1 ) /2 o f t h e m d i s t r i ct s , e x c e p t w h e n t h eg r o u p is s o m e w h a t s m a ll , in w h i c h c a s e it s e x p e c t e d p o w e r is m a x i m i z e d w h e n i tis e v e n l y d i v i d e d a c r o s s a l l m d i s tr i c ts . E x p e c t e d p o w e r d e c r e a s e s s h a r p l y a s ag r o u p ' s m e m b e r s b e c o m e c o n c e n t r a t e d i n f e w e r t h a n ( m + 1 )/2 d i st ri c ts .

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

D e s p i t e t h e w i d e ly h e ld v i e w t h a t s p e c i a l i n te r e s t g r o u p s h a v e a p r o f o u n d i n f lu e n c eo n p u b l i c d e c i s i o n s a t a l l l e ve l s o f g o v e r n m e n t i n a ll ty p e s o f p o l i t i c a l sy s t e m s , w ek n o w r e l a ti v e l y l it tl e a b o u t t h e s o u r c e s o f i n t e r e s t- g r o u p p o w e r . S y s t e m a t i c s t u d y o fi n t e r e s t - g r o u p i n f l u e n c e h a s p r o v e d t o b e e x c e e d i n g l y d i ff ic u l t, a n d t h u s m u c h o fw h a t h a s b e e n w r i t t e n is a n e c d o t a l o r s p e c u la t iv e . T h e r e a r e f e w g e n e r a l th e o r e t i c a lp r o p o s i t i o n s a b o u t t h e l ik e l y o u t c o m e o f i n t e r e s t -g r o u p p o l i ti c s, a n d f e w d e ta i le d ,w e l l - d o c u m e n t e d r e g u l a r i ti e s . I n p a r t i c u l a r , t h e r e is n o f u l l y s a t i s f a c t o r y e x p l a n a -t i o n w h y s o m e g r o u p s t e n d t o r ec e iv e m o r e b e n e f it s f r o m g o v e r n m e n t t h a n o t h e r s .

I n th i s p a p e r w e f o c u s o n a p a r t i c u l a r s o u r c e o f i n t e r e s t - g r o u p p o w e r , a g r o u p ' sm e m b e r s h i p a s a p o t e n t i a l b l o c k o f v o t e r s . W h i l e p o l i t ic a l s c ie n t is t s h a v e l o n gr e c o g n i z e d t h e p o t e n t i a l im p o r t a n c e o f s u c h v o t e - p o w e r , i t h a s re c e i v ed r e l a ti v e l yl it tl e a t te n t i o n i n m u c h o f t h e f o r m a l t h e o r e t i c a l li t e ra t u r e . F o r e x a m p l e , t h e s o -c a ll e d e c o n o m i c t h e o r y d e v e l o p e d b y S t ig l er (1 9 79 ), P e l t z m a n ( 19 7 6 ), a n d B e c k e r* The author would like to thank the participants of the Worksho p in Applications of Economicsatthe Universityof C hicago fpr the ir helpfulcomments, and the C enter for the S tudy of the Econom yandthe State for its financial support.


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(1983) adopts the view that voting and majorities are not the fund ment ldeterminants o f political influence even in democracies .1 These authors emphasizethe importance of lobbying and campaign contributions, and the free-riderproblems a group faces in trying to finance these activities. However, one of themost important rewards a group can offer a politician is the votes of its members.This is true even for fairly small groups, because the margin of victory for politicaloffice if often small - a block of several thousand votes can turn defeat into victory.Also, since the costs of voting are small (or perhaps even negative, after accountingfor psychic benefits ), groups probably do not face severe free-rider problems ingetting their members to vote. Thus, even a group that is too big to generate largerper-member campaign contributions (due to free-riding) may have a large amountof influence per member if its members vote as a block.

The available empirical evidence seems to confirm this view. Studies by Pincus(1977), Ray (1981), Lavergne (1983), Gardner (1985), and Miller (1986) find mixedsupport for the importance of free-riding. For example, contrary to the predictionbased on free-rider arguments, Miller and Pincus find a significant positiverelationship between group size and the per-member benefit to the group. Also,studies attempting to link campaign contributions by interest groups to Con-gressional voting on issues important to the groups have had mixed success.Silberman and Durden (1976), Chappell (1982), and Kau and Rubin (1982) foundevidence that interest-group contributions influenced Congressmen's votes on suchissues as the minimum wage, the Bl-bomber, the debt limit, the windfall profits tax,and wage and price controls. On the other hand, Chappell (1981), (1982), Welch(1982), and Wright (1985) found a weak relationship between contributions andvotes on legislation regarding cargo preferences, crude oil tax rebates, milk pricesupports, highway construction and regulation of physicians? There is strongerevidence that a group's vote-power matters, at least for the few groups studied.Welch (1982) found a strong positive relationship between the probability that aCongressman would vote in favor of higher dairy price supports and the amount ofdairy production (highly correlated with the number of daily farmers) in his district.Similarly, Kau and Rubin (1982) found that the number of union members, and thenumber of farmers, in a Congressman's district generally (although not always) hada significant impact on his vote over a wide range of issues impor tant to the groups.

If the votes of an interest group's members are an important source of itspolitical influence, then in a representative democracy with geographically definedconstituencies, the geographic distribution of the group's members is potentially acritical variable. Two groups of roughly the same size may have greatly differentamounts of political power because the members of one group are well-dist ributed in some sense, while those of the other group are not. Political scientists

1 Be ck e r (1 9 8 3 , p . 3 8 5 ) , h i s i t a l i cs .2 O n t h e o t h e r h a n d , t h e s e s t u d ie s p r o v i d e s t r o n g e v i d e nc e f o r a n o t h e r p r o p o s i t i o n o f t h e e c o n o m i ct h e o r y , t h a t h i g h e r d e a d w e i g h t c o s t s o f s u b s i d iz i n g a g r o u p l e a d t o l o w e r s u b s i d i e s f o r t h e g r o u p .3 O f c o u r s e , t h e s e r e s u l ts a re d i ff i cu l t t o i n te r p r e t b e c a u s e a C o n g r e s s m a n s v o t e o n a n y s i n g le bi ll m a yb e t h e o u t c o m e o f v o t e - t r a d i n g o r s o m e o t h e r s t r a t e g ic c o n s i d e r a t io n s , o r i t m a y b e p u r e l y s y m b o l i c . I t isd i f f ic u l t t o e s t i m a t e t h e t r u e e x t e n t o f a n i n t e r e s t g r o u p s i n f lu e n c e o v e r p o l i c y f r o m t h e a n a l y s i s o fi n d i v i d u a l b i l l s .


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Political Geo graphy and Interest-Group Power 105o f t e n n o t e t h a t s u c h g e o g r a p h i c a l c o n s i d e r a t i o n s m a y m a t t e r , 4 b u t t h e re a r e f e wg e n e r a l c o n c l u s i o n s a b o u t e x a c t ly h o w t h e y m a y m a t t e r . M o s t e x is t in g s tu d i e s h a v ed i s c u s s e d p a r t i c u l a r e m p i r i c a l c a s e s o r t h e o r e t i c a l e x a m p l e s ( s e e , f o r e x a m p l e ,S p i l e r m a n a n d D i c k e n s 1 9 7 4 ; J o h n s t o n 1 9 77 , 1 98 2 ; T a y l o r a n d J o h n s t o n 1 9 79 ). I nt h i s p a p e r w e a t t e m p t t o c h a r a c t e r i z e m o r e g e n e r a l l y t h e r e l a t i o n s h i p b e t w e e n t h ed i s t r i b u t i o n o f a g r o u p ' s m e m b e r s a n d t h e g r o u p ' s p o w e r .

W e a n a l y z e a s i m p l e f o r m a l m o d e l o f l eg i sl at iv e d e c i s io n m a k i n g a n d i n t e re s tg r o u p s i n w h i c h t h e s o l e s o u r c e o f a g r o u p ' s p o l i t i c a l i n f l u e n c e i s t h e v o t i n g p o w e ro f i ts m e m b e r s . T h e m o d e l a l l o w s u s t o d e s c r ib e t h e r e l a t io n s h i p b e t w e e n t h eg e o g r a p h i c d i s t r ib u t i o n o f a g r o u p ' s m e m b e r s , in p a r t i c u l a r t h e d e g r e e o fd i s p e rs i o n ', , a n d t h e g r o u p ' s p o l i ti c a l p o w e r , a s m e a s u r e d b y v a r i o u s p o w e r i n d e x esu s e d i n o t h e r s t u d i e s. W e c o n s i d e r o n l y t h e c a s e o f t h r e e g r o u p s d i v i d i n g a d o l l a r ,t h e n o w c l a s s ic p a r a d i g m o f r e d i s tr i b u t i v e p o l it ic s . O u r m a i n r e s u l t is t h a t e x c e p t f o rf a i r l y s m a l l g r o u p s , a g r o u p ' s a v e r a g e p o w e r is g r e a t e s t w h e n i t s m e m b e r s a r e e v e n l yd i v i d e d a c r o s s s l i g h t l y m o r e t h a n h a l f o f t h e l e g is l a ti v e d i s tr i c ts . A v e r a g e p o w e rd e c re a s es s h a r p l y a s th e g r o u p ' s m e m b e r s h i p b e c o m e s c o n c e n t r a t e d i n f e w e r t h a nh a l f o f t h e d i s tr i c ts . A v e r a g e p o w e r a l s o f a l ls a s t h e g r o u p b e c o m e s m o r e d i s p e r s e d ,b u t n o t a s s i g n if ic a n tl y . F o r s m a l l g r o u p s , a v e r a g e p o w e r i s h i g h e s t w h e n t h e g r o u pis v e r y w i d e l y d i s p e rs e d , w i t h m e m b e r s e v e n l y d i v i d e d a c r o s s a l l o f t h e d i s t r ic t s .

B e f o r e p r o c e e d in g , w e s h o u l d m e n t i o n s e ve ra l s h o r t c o m i n g s o f th e m o d e lp r e s e n t e d b e l o w , w h i c h a f f e c t t h e g e n e r a l i t y o f o u r r e s u l t s. F i r s t l y , w e t a k e t h e s e t o fi n t e r e s t g r o u p s , a n d t h e i r s i z e s a n d m e m b e r s h i p d i s t r i b u t i o n s , a s g i v e n . T h u s , w ea b s t r a c t f r o m t h e i m p o r t a n t p r o b l e m s o f r e d i st r ic t in g a n d g r o u p f o r m a t i o n a n dc h a n g e . ( O u r r e s u l t s d o t o u c h o n t h e s e is s u es h o w e v e r , s u g g e s t i n g a c h a r a c t e r i s t i c o fg r o u p s t h a t a r e l ik e l y t o t h r i v e i n t h e p o l i ti c a l a r e n a , a n d s u g g e s t i n g a ls o a f a c t o ri n f lu e n c i n g g r o u p s ' p r e f e r e n c es a b o u t h o w d i s tr i ct l in e s s h o u l d b e d r a w n . )S e c o n d l y , w e a s s u m e t h a t a l l l e g i s la t o r s a r e t h e s a m e , w h i l e in f a c t t h e y g e n e r a l l y a r en o t . I n t h e U . S . f o r e x a m p l e , s o m e l e g i s l a t o r s a r e m e m b e r s o r c h a i r m e n o fp a r t i c u l a r l y i m p o r t a n t c o m m i t t e e s o r s u b c o m m i t t e e s , o r h o l d p r i v i le d g e d p o s i t io n sw i t h i n t h e i r p a r t y . T h i r d l y , t h e r e a r e n o p o l i t i c a l p a r t i e s . P a r t i e s a r e c l e a r l y a ni m p o r t a n t c o n s i d e r a t i o n , b u t w e h a v e n o t y e t m a n a g e d t o i n c o r p o r a t e t h e m i n t o t h em o d e l . T h u s , w e a p p e a l t o t h e w i d e l y h e l d b e l i e f t h a t , a t l e a s t i n t h e U . S . C o n g r e s s ,p a r t y c o n t r o l is o f t e n r a t h e r w e a k , p a r ti c u l a r l y w i t h r e g a r d s t o s o m e i m p o r t a n tr e g u l a t o r y / r e d i s t r i b u t i o n a l is s ue s . L a s t l y , as n o t e d a b o v e , w e t r e a t t h e v o t i n g p o w e ro f a g r o u p ' s m e m b e r s h i p a s t h e o n l y s o u r c e o f th e g r o u p ' s i n f lu e n c e , i g n o r i n g o t h e rr e s o u r c e s s u c h a s c a m p a i g n c o n t r i b u t i o n s a n d d i r e c t l o b b y i n g e f f o r t s . F u t u r er e s e a rc h s h o u l d w o r k t o c o m b i n e t h e s e d i f fe r e n t f a c t o rs i n a s in g le f r a m e w o r k .

2 A Sim ple Model of Legislatures and Interest Group Pow erT h e r e is a s e t N = {1 . . . . . n } o f i n t e r e s t g r o u p s a n d a s e t M - - {1 . . . . . m } o f p o l i t ic a ld i s tr i ct s , m o d d . B o t h s e t s p a r t i t i o n t h e p o p u l a t i o n o f v o t e rs , s o e a c h v o t e r i s am e m b e r o f e x a c t l y o n e g r o u p a n d o n e d i s t r ic t . E a c h d i s t r ic t e le c ts o n e r e p r e s e n t a -t iv e t o t h e l e g i s la t u r e , b y m a j o r i t y - r u l e . T h e l e g i s l a t u r e i s a u n i c a m e r a l b o d y t h a t4 See, for exam ple, Hre ben ar and S cott (1982, pp. 31-32).


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1 0 6 J . S n y d e r

p a s se s l e g is l a ti o n b y m a j o r i t y - r u l e ; t h u s , a p r o p o s a l x d e f e a ts a n o t h e r p r o p o s a l y i fa n d o n l y i f t h e n u m b e r o f l e g is l at o r s w h o v o t e f o r x i s g r e a t e r t h a n t h e n u m b e r o fl e g i s la t o r s w h o v o t e f o r y .

I n d e c i d i n g h o w t o v o t e o n a n y p a i r o f p r o p o s a l s , a l e g is l a to r c o n s i d e rs o n l y h o wh i s c h o i c e a f f e c t s hi s c h a n c e s f o r r e e l e c t i o n . T h u s , a l e g i s l a to r v o t e s f o r x o v e r y i fa n d o n l y i f t h e n u m b e r o f v o t e r s in h i s d i s tr i ct w h o p r e f e r x t o y is g r e a te r t h a n t h en u m b e r w h o p r e f e r y t o x . 5 N o t e t h a t t h is s t r a t e g y is m o s t r e a s o n a b l e i f w e t h i n k o fe a c h p r o p o s a l a s s o m e t h i n g li ke t h e a l l o c a t i o n o f u t il it ie s t o t h e v a r i o u s g r o u p sp r o d u c e d b y a ll o f th e b i l ls p a s s e d d u r i n g t h e l eg i s la t iv e s es s io n . E x c e p t in a w o r l d o fr a t h e r n a i v e v o t e r s a n d / o r l e g i s la t o r s , a le g i s l a t o r ' s s t r a t e g y o n a n y s in g l e b i ll is n o ts o s t r a i g h t f o r w a r d ( e v e n i f h e i s c o n c e r n e d o n l y a b o u t t h e n e x t e l e c ti o n ), a s h e m u s tt a k e i n t o a c c o u n t t h e e f f ec t s o n h is c o n s t i t u e n t s o f o t h e r b i ll s t o b e c o n s i d e r e ddu r i ng t he l eg i s l a ti ve s e s s ion , an d t he po s s i b i l i t y f o r l og r o l l i ng ac r o s s b i l ls . 6

G i v e n t h is s t r u c tu r e , w e c a n d e f i n e a c o o p e r a t i v e g a m e a m o n g t h e g r o u p s t h a t i sp l a y e d t h r o u g h t h e l e g i sl a tu r e . L e t sij d e n o t e th e n u m b e r o f v o te r s fr o m g r o u p j t h a tl iv e i n d i s t r ic t i. W e s a y a c o a l i t i o n o f g r o u p s C c o n t r o l s d i s t r ic t i i f a n d o n l y i f i tsm e m b e r s c o l l e c ti v e ly a c c o u n t f o r m o r e t h a n h a l f o f t h e v o t e r s i n d i st r ic t i . W e c a l l Cwinning i f a n d o n l y i f it c o n t r o l s m o r e t h a n h a l f o f t h e d i s tr ic t s, 7 i .e . , i f a n d o n l y i f

) m + lW ( C ) = i Z s i j E sij~>= 2C jM C J( = ~S d e n o t e s th e c a r d i n a l i t y o f s e t S . ) T h i s is a n a t u r a l d e f i n i t i o n - a n y w i n n i n gc o a l i t i o n c o u l d e f f ec t iv e l y f o r c e t h e l e g is l at u re t o p a s s a p r o p o s a l b y p r o m i s i n g t os u p p o r t ( v i a it s m e m b e r s ' v o t e s in t h e n e x t e l ec t io n ) a n y l e g is l a to r w h o v o t e s f o r t h ep r o p o s a l , a n d t o o p p o s e a n y l e g i sl a to r w h o v o t e s a g a i n s t th e p r o p o s a l . I f th ec o a l i t i o n ' s t h r e a t s a n d p r o m i s e s a r e c r e d i b l e , t h e n a l l t h o s e r e p r e s e n t a t i v e s f r o md i s t r i c t s i n w h i c h t h e c o a l i t i o n ' s m e m b e r s c o n s t i t u t e a m a j o r i t y w i l l f a v o r t h ep r o p o s a l . S i n c e t h i s is a m a j o r i t y o f t h e d i s t r ic t s , t h e p r o p o s a l w i ll p a s s. A c o a l i t i o nt h a t i s n o t w i n n i n g i s losing.

A s s i g n i n g a v a l u e o f 1 t o a l l w i n n i n g c o a l i t io n s , a n d a v a l u e o f 0 to a ll lo s i n gc o a l i t i o n s , w e h a v e a s t r o n g , s i m p l e , p r o p e r g a m e . A s i n m o s t v o t i n g g a m e s , i ng e n e r a l th i s g a m e h a s n o c o r e . H o w e v e r , w e c a n u s e o n e o f se v e ra l v a l u e - t y p es o l u t i o n c o n c e p t s , s u c h a s v o t i n g w e i g h t s ( w h e n s u c h w e i g h t s e x i s t ) , t h e S h a p l e y -S h u b i k v a l u e , o r t h e B a n z h a f i n d e x , to d e t e r m i n e t h e r e l a ti v e p o w e r o f t h e d i f fe r e n tg r o u p s .s T o a p p l y t h e m o d e l t o a s p e ci fi c p o l ic y a r e a w e m i g h t w i s h t o m o d i f y th e m o d e l a s f o l lo w s : F i x t h ep o l ic y a r e a , a n d s u p p o s e t h a t e a c h d is t ri c t c o n t a i n s i n t e r e s t e d v o t e rs , w h o s u p p o r t o r o p p o s e th e i rr e p r e s e n t a ti v e s o l el y o n t h e b a s i s o f h is p e r f o r m a n c e w i t h r e s p e c t t o t h e g i v e n p o li c y a r e a , a n d

u n i n t e r e s t e d v o t e rs . A s s u m e t h e n t h a t a r e p r e s e n t a ti v e v o te s fo r a p r o p o s a l i f a n d o n l y if m o r ei n t e r e s t e d v o t e r s i n h i s d is t r i c t p r e f e r t h e p r o p o s a l t h a n o p p o s e i t . T h i s s t r a t e g y i s e s p e c i a l l y c o m p e l l i n g i ft h e l e g i s l a t o r e x p e c t s t h e u n i n t e r e s t e d v o t e r s i n h i s d is t r i c t t o d i v i d e t h e i r v o t e s e v e n l y b e t w e e n h i m a n dh i s o p p o n e n t , o r i f h e i s v e r y u n c e r t a i n a b o u t h o w t h e y w i ll v o te . I n a n y c a s e , i t i s t h e l o g i c a l s t r a t e g yw h e n e v e r m o r e v o t e s a r e p r e f e r r e d t o le s s .6 A s i m i la r c a v e a t a p p li e s t o t h e v a r i a t i o n p r o p o s e d i n F o o t n o t e 5 . I n p a r ti c u l a r , w e m i g h t h a v e t ow o r r y a b o u t l o g r o l l i n g a c r o s s p o l i c y a r e a s .7 T h i s d e f i n it i o n i s s a t i s f a c t o ry as lo n g a s th e n u m b e r o f d is t ri c ts a n d t h e t o t a l n u m b e r o f v o t e rs in e a c hd i s tr ic t a r e b o t h o d d , b u t m u s t b e m o d i f i e d i f e i th e r o r b o t h o f t h e s e a r e e v e n , to a c c o u n t f o r t h ep o s s i b i l it y o f ti e s. F o r t h e p u r p o s e s o f t h is p a p e r , w e i g n o r e t i e s.


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Political Geog raphy and Interest-Group Pow er 107A s i m p l e e x a m p l e w i l l i l l u s tr a t e t h e s e i de a s . C o n s i d e r f i v e i n t e r e s t g r o u p s , w h o s e

m e m b e r s a r e d i s t r ib u t e d a c r o s s f iv e e q u a l - s iz e d d i st ri c ts i n t h e f o l l o w i n g f a s h i o n :

GroupsDistrict 1 2 3 4 51 3 1 12 0 3 0 0 23 0 0 3 0 24 1 2 0 2 05 1 0 2 2 0

W e c a n e a s i l y v e r i f y t h a t t h e w i n n i n g c o a l i t i o n s a r e { 1 , 2} , { 1 , 3 }, { 1 , 4 } , { 2 , 3 , 4 } ,a n d a ll s u p e r s e t s o f t h e s e ( e .g . , { 1 , 2 , 3 } i s a l s o w i n n i n g ) . F o r e x a m p l e , g r o u p s 1a n d 4 t o g e t h e r a c c o u n t f o r t h r e e v o t e r s , h e n c e a m a j o r i t y , i n e a c h o f d i s t ri c t s 1 , 4 ,a n d 5 ; th u s { 1 ,4 } is a w i n n i n g c o a l i t i o n . T h i s g a m e c a n b e r e p r e s e n t e d a s aw e i g h t e d m a j o r i t y v o t i n g g a m e i f w e a s s i g n th e f o l l o w i n g w e i g h t s t o t h e g r o u p s :w t = 2 /5 , w 2 = w 3 = w4 = 1 /5 , a n d w s = 0 . T h e n a c o a l i t i o n C i s w i n n i n g i f a n d o n l y i f

w j > 1 /2 , t h a t i s, i f a n d o n l y i f i t h a s m o r e t h a n h a l f o f th e t o t a l w e i g h t . T h ej s cS h a p l e y - S h u b i k v a l u es ar e V1 = 1 /2 , V = V3 = V = 1 /6 , a n d V5 = 0 . T h i s e x a m p l ep o i n t s o u t s ev e r a l i n t e r es t i n g p h e n o m e n a : 1 ) S o m e w i n n i n g c o a li t io n s m a y h a v el es s t h a n h a l f o f t h e v o t e r s - e . g . , { 1 , 2} i s w i n n i n g , b u t i t h a s o n l y 2 / 5 o f t h e v o t e r s ;

2 ) g r o u p s iz e a n d p o w e r n e e d n o t b e h i g h l y c o r r e l a t e d - e .g . , g r o u p 1 is sm a l l e rt h a n g r o u p 2 , a n d t h e s a m e s iz e a s g r o u p s 3 a n d 5 , y e t i t h a s a h i g h e r v o t i n g w e i g h ta n d S h a p l e y v al u e ; a n d 3 ) e v e n a r e la t iv e l y la r g e g r o u p m a y h a v e n o p o w e r a t a ll -g r o u p 5 , w h i c h i s t h e s a m e s i ze as g r o u p s 1 a n d 3 , a n d l a r g e r t h a n g r o u p 4 , isp o w e r l e s s - t h a t is , i t m a y n e v e r a d d v a l u e t o a c o a l i t i o n b y j o i n i n g i t . T h e s ep h e n o m e n a a r e p e r h a p s m o r e n o t e w o r t h y c o n s i d e r i n g th a t a ll d i s tr i ct s a r e t h e sa m esize . 8

I n s u b s e q u e n t s e c t i o n s , w e w i ll c o n s i d e r a g r o u p o f s o m e f i x e d t o t a l s iz e ( i. e. ,s o m e f i xe d n u m b e r o f m e m b e r s ) , a n d c o m p a r e t h e g r o u p ' s r el a t iv e p o w e r u n d e rd i f f e r e n t d i s t r i b u t i o n s o f it s m e m b e r s a c r o s s t h e d i s tr i c ts . T h u s , i f t h e s iz e o f g r o u pj is t j , w e c a l l a d i s t r i b u t i o n s j = s i j . . . . , s m j ) f e a s i b l e i f a n d o n l y i f s ij > 0 f o r a l l i,a n d ~ s i j = t j . A l s o , t o a v o i d u n n e c e s s a r y t e c h n i c a l d e t a i ls , w e a l l o w t h e s u t o b e

i =n o n i n t e g e r s . T h a t is , w e t r e a t g r o u p s a s i f t h e y a r e i n f i n i t e l y l a r g e r e l a t iv e t o t h en u m b e r o f d is t ri c ts .

T h r o u g h o u t m o s t o f t h is p a p e r w e r es t ri c t a t t e n t i o n t o t h e sp e c ia l c a s e o f t h r e ei n te r e s t g r o u p s . I t is s t r a i g h t f o r w a r d t o s h o w t h a t w i t h o n l y th r e e g r o u p s t h e r e a r ej u s t f o u r p o s s i b l e s t r u c t u r e s (i. e ., c h a r a c t e r i s t i c f u n c t i o n s ) f o r t h e c o o p e r a t i v e g a m e :

8 In fact, even when ai1 groups are the sa m e size and all districts are the sam e size, we can finddistributions of the groups' m embe rs such that so me coalitions with ust slightly more than one-fourth o fthe voters are winning (this is no t suprising, since to obtain ha lf o f the voters in half of the districts oneneeds only one-fourth of all the vo ters), and distributions such tha t slightly less than ha lf of the groupsare pow erless.


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108 J . Snyd ere i t h e r o n e o f t h e g r o u p s i s a w i n n i n g c o a l i t i o n b y i t s e l f ( t h e re a r e t h r e e s t r u c t u r e s o ft h is t y p e ) , o r el se e v e r y c o a l i t i o n o f t w o o r m o r e g r o u p s i s a w i n n i n g c o a l i t i o n . I n a n yo f th e f i r s t t h r e e c a s e s , t h e g r o u p t h a t i s w i n n i n g c o n t r o l s t h e l e g i s l a t u r e - i . e. , is a

d i c t a t o r - a n d a n y o f t h e u s u a l p o w e r m e a s u r e s , i n c l u d i n g v o t i n g w e i g h ts , g i v e av a l u e o f 1 t o t h e w i n n i n g g r o u p a n d a v a l u e o f 0 t o t h e o t h e r g r o u p s ( a l s o , t h isa l l o c a t i o n i s t h e c o r e o f t h e g a m e ) . I n t h e f o u r t h c a s e, t h e s i t u a t i o n i s s y m m e t r i c f o ra ll g r o u p s , a n d a n y o f t h e u s u a l p o w e r m e a s u r e s g iv e e a c h g r o u p a v a l u e o f 1 /3 . W el et W j d e n o t e t h e p o w e r o f g r o u p j .

3 . D o m i n a ti o n D o e s n t T e l l U s V e r y M u c hI t s h o u l d b e c l e a r f r o m t h e e x a m p l e in t h e p r e v i o u s s e c t i o n t h a t a g r o u p ' s r e l a t iv ep o w e r d e p e n d s n o t o n l y o n t h e d i st r ib u t i o n o f i ts o w n m e m b e r s , b u t o n t h ed i s tr i b u t io n s o f t h e o th e r g r o u p s ' m e m b e r s a s w e ll . T h u s , t h e d e fi n i ti o n o f a g o o do r b a d d i s t ri b u t i o n o f v o t e r s m u s t t a k e i n t o a c c o u n t t h e d i s t ri b u t io n s o f a llg r o u p s .

A n o b v i o u s w a y t o r a n k t h e d i f f e re n t m e m b e r s h i p d i s t r i b u t io n s is b y ad o m i n a n c e r e l a ti o n . W e s a y t h a t a d i s tr i b u t io n s j o f a g r o u p j ' s m e m b e r s d o m i n a t e sa n o t h e r d i s t ri b u t i o n s j i f a n d o n l y i fs j g iv e s g r o u p j a t le a s t a s m u c h p o w e r a s d o e s s jr e g a rd l e s s o f th e d i s t r i b u t io n o f o t h e r g r o u p s ' v o t e r s , a n d s t ri ct ly m o r e p o w e r t h a n s jf o r s o m e d i s t r i b u t i o n s o f th e o t h e r g r o u p s . T h a t i s , i f w e p i c k , s a y , j = 1 , a n d l e tW 1 ( sl , . . . , s , ) d e n o t e t h e p o w e r o f g r o u p 1 g i v e n v o t e r d i s t r i b u t i o n s ( s l , . . . , s , ) , t h e nS l dom in a t e s s [ i f an d on ly i f W1 ( s l , s2 , . . . , s , ) > WI ( s [ , s z , . . . , s , ) f o r a l l ( s2 . . . . . , s , ) ,w i t h s t r i c t i n e q u a l i t y f o r s o m e ( s 2 . . . . , s , ) .

U n f o r t u n a t e l y , f o r m o s t c a s e s w e a r e c o n c e r n e d w i th , i t i s r a r e t h a t o n em e m b e r s h i p d i s t r i b u t io n d o m i n a t e s a n o t h e r , s o d o m i n a n c e i s n o t v e r y i n f o r m a t i v e .I n p a r t i c u l a r , i f t h e r e a r e t h r e e g r o u p s , w e h a v eP ro p o s i t io n 1. F i x t h e n u m b e r o f v o t e r s i n ea c h 9 r o u p a n d c o n s i d e r o n l y f e a s i b l eme mb ersh ip d i s t r ibu t i ons . I f g roup 1 is no t t he l a rges t g roup ( i . e . , t 1 = < m ax( t 2 , t3 ) )t h e n n o d i s tr i b u t io n o f 9 r o u p 1 s m e m b e r s d o m i n a t e s a n y o t h e r .P r o o f S e e A p p e n d i x .

T h u s , w h e n s t u d y i n g s m a l l o r a v e r a g e - s i z e d g r o u p s , t h e d o m i n a n c er e l a ti o n is c o m p l e t e l y u n i n f o r m a t i v e .

W h e n c o n s i d e ri n g t h e la r g e s t g r o u p , w e c a n fi n d s o m e m e m b e r s h i pd i s tr i b u ti o n s t h a t a r e d o m i n a t e d . F o r e x a m p l e i f g r o u p 1 is so la r g e t h a tq > ( 2 m / ( m + l ) ) ( t z + t 3 ) , t h e n i f i ts m e m b e r s a r e e v e n l y d i v id e d a m o n g a ll mdis t r ic t s - i . e . , i f s a = t l / m f o r a l l i - t h e n i t i s c e r t a i n t o c o n t r o l t h e l e g i s l a t u r e ,r e g a rd l e s s o f t h e d i s t ri b u t i o n o f th e o t h e r g r o u p s ' m e m b e r s . T h u s , a n y d i s t r i b u ti o ns~ s u c h t h a t W 1 ( s~ , s 2 , s3 ) < 1 fo r so m e s 2 and s 3 - f o r e xa m ple , an y s~ w i th s~t = 0 i nm o r e t h a n h a l f t h e d is t ri c ts - is d o m i n a t e d b y s l = ( q / m . . . . . t ~ /m ) . O r , i f g r o u p 1 isn o t q u i t e s o l a r g e , w e h a v e t h e f o l l o w i n gP ro p o s i t io n 2 . S u p p o s e m a x t 2 , t3) < tl < ( 2 m / ( m + 1)) (t2 + t3). I f s l i s a n y d i s tr i b u -t i on o f g roup 1 s vo t e r s such t ha t t he re ex i s t s k ~ M such t ha t ~ s i~ > t2 + t3 fo r a l l S

i e


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Political Geo graphy and Interest-Group Pow er ~09with k s S and 4~ S> (m + 1) / 2 , then there ex i s t s a f eas ib le di st ribut ion s [ tha tdominates s l .Proo f . S e e A p p e n d i x .

T h e d o m i n a t e d d i s t r ib u t i o n s 1 o f P r o p o s i t i o n 2 i s t o o c o n c e n t r a t e d , t h a t i s, i tp u t s t o o m a n y o f g r o u p l ' s m e m b e r s i n d i st ri c t k . T h e g r o u p w o u l d a l w a y s b e b e t t er -o f f ( h a v e m o r e r e l a ti v e p o w e r ) i f it s v o t e r s w e r e m o r e s p r e a d o u t , i n a n a p p r o p r i a t ef a s h i o n . W e w i l l f i n d r e s u l t s s i m i l a r t o t h i s i n s p ir i t i n th e n e x t s e c t i o n o f t h e p a p e r .

T h e m a i n p o i n t o f t h e p r e s e n t s e c ti o n h o w e v e r , e m b o d i e d i n P r o p o s i t i o n 1, ist h a t w e c a n n o t u s e th e d o m i n a n c e r e l a ti o n t o r a n k d i f fe r e n t m e m b e r s h i pd i s tr i b u ti o n s a n d m u s t t h u s t u r n t o s o m e w e a k e r n o t io n .

4 M embership Distribution and Average Power: Tw o SpecificationsG i v e n o u r i n a b i l it y t o e f f ec t iv e l y c o m p a r e d i s tr i b u ti o n s b y t h e d o m i n a n c e r e la t io n ,w e t u r n i n t h i s s e c t i o n t o a s o m e w h a t w e a k e r c r i t e r i o n . S p e c i f i c a l l y , w e r a n k ag r o u p ' s m e m b e r s h i p d i st r i b u ti o n s b y h o w m u c h p o w e r t h ey yi el d o n a v e r a g e ,w h e r e w e a v e r a g e o v e r t he p o s s i b le m e m b e r s h i p d i s t r ib u t i o n s o f t h e o t h e r g r o u p s .T h i s c r i t e r i o n i s l es s t h a n f u l ly s a t i s f a c t o r y s i n c e t h e c h o i c e o f a p r o b a b i l i t y m e a s u r eo v e r th e o t h e r g r o u p s ' m e m b e r s h i p d i s t r ib u t i o n s i s n e c e s sa r il y a rb i t r a r y . O n t h eo t h e r h a n d , o n c e a c h o i c e is m a d e , w e g e t s o m e c le a r c u t p r e d i c t io n s .

L a c k i n g m u c h p r i o r i n f o r m a t i o n t o g u i d e u s, e it h e r f r o m t h e o r y o r f a c ts a b o u tt h e a c t u a l d is t r ib u t i o n o f g r o u p m e m b e r s h i p d i s t ri b u t io n s i n t h e w o r l d , w e e x a m i n et w o a l t e r n a t i v e s w h i c h h a v e s o m e a p p e a l . A s i n th e p r e v i o u s s e c ti o n , w e t r e a t o n l yt h e ca s e o f t h r e e i n te r e st g r o u p s , a n d r a n k t h e d i ff e r en t m e m b e r s h i p d i s t r ib u t i o n s o fg r o u p 1 - t h u s, w e t a k e g r o u p l ' s d i s t r ib u t i o n a s k n o w n , a n d s p e c i fy a p r o b a b i l i t ym e a s u r e o v e r t he m e m b e r s h i p d i s tr i b u ti o n s o f g r o u p s 2 a n d 3.

T h e f i rs t f o r m u l a t i o n is m o t i v a t e d b y th e a r g u m e n t t h a t , h a v i n g n o s t r o n gr e a s o n s t o b e l ie v e t h a t s o m e d i s t r i b u t i o n s a r e m o r e o r le ss l i k e ly t o o c c u r t h a n o t h e r s( p e r h a p s b e c a u s e w e h a v e n o t t h o u g h t e n o u g h a b o u t t h e p r o b l e m ) , w e m i g h t a s w el la s s u m e t h a t a ll d i s t r ib u t i o n s a r e e q u a l l y l ik e ly . T h u s , w e f i x t h e t o t a l s iz e s o f g r o u p s2 a n d 3 a t t 2 = t 3 = 1 a n d a s s u m e t h a t th e m e m b e r s h i p d i s t ri b u t io n s S 1 2 . . . , S m 2 a n d(s13 .. .. , sin3 a r e i n d e p e n d e n t r a n d o m v e c t o rs , e a c h j o i n t ly d i s t ri b u t e d u n i f o r m l y v e r t h e m - s i m p l e x ( i ' e v e r { s j l ~ s ~ J = l } )

I n t h e s e c o n d f o r m u l a t i o n , w e a s s u m e s i m p l y t h a t f o r a ll d i s tr i c ts i ~ M= { 1 , . .. , m } a n d a l l g r o u p s j e { 2 , 3} , t h e s ij a r e i n d e p e n d e n t r a n d o m v a r i a b l e s ,d i s t r i b u t e d u n i f o r m l y o n [ 0 ,2 / m ] . N o t e f i r s t t h a t u n d e r t h i s s p e c i f ic a t i o n th e t o t a ls iz es o f g r o u p s 2 a n d 3 a re n o t f ix e d b u t r a n d o m , w i t h m e a n s E(t2) = E ( t 3 ) = 1 . M o r ei m p o r t a n t l y , n o t a ll m e m b e r s h i p d i s t r ib u t i o n s o f g r o u p s 2 a n d 3 ar e e q u a ll yp r o b a b l e ; r a t h e r , t h e g r o u p s ' m e m b e r s a r e m o r e l ik e l y t o b e s p r e a d o u t t h a nc o n c e n t r a t e d i n a f e w d i s tr i ct s . A l s o , t h e g r o u p s w i ll t e n d t o b e s m a l l e r i n t o t a l s i z e i ft h e ir m e m b e r s h i p s a r e c o n c e n t r a t e d . T h e s e m a y i n f ac t b e de s i ra b l e p r o p e r ti e s , f o r9 Alternatively,we might also assum e hat the groups' sizes, t2 and ta, are random variables, or at leastexam ine cases wh ere t2 + t3.


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110 J. Snyde rt w o r e a s o n s . F i r s t , i n s o m e c a s e s t h e r e m a y b e e m p i r i c a l r e a s o n s t o s u s p e c t t h a td i s t r ib u t i o n s i n w h i c h a g r o u p s m e m b e r s a r e f a i rl y e v e n ly s p r e ad o u t a r e m o r el ik e ly t o o c c u r t h a n o t h e rs . F o r e x a m p l e , i f g r o u p s a r e d e f i n e d b y o c c u p a t i o n a li n d u s t r y ( e . g . , s t e e l w o r k e r s , f a r m e r s , d o c t o r s , e t c . ) , t h e n s e r v i c e i n d u s t r y g r o u p s( e . g . , r e a l t o r s a n d i n s u r a n c e a g e n t s ) w i l l n e c e s s a r i l y t e n d t o b e s p r e a d o u tg e o g r a p h i c a l l y , r a t h e r t h a n c o n c e n t r a t e d i n o n e r e g i o n . I f s e rv i ce i n d u s t r i e s e m p l o ya l a rg e f r a c t i o n o f t h e p o p u l a t i o n a n d a r e p o l i t ic a l l y a c t iv e , t h e n w e m a y e x p e c t t os e e w i d e l y d i s t r i b u t e d g r o u p s i n v o l v e d i n p o l i t ic a l b a r g a i n i n g g a m e s r e l a ti v e m o r eo f t e n t h a n c o n c e n t r a t e d g r o u p s . A s e c o n d , p e r h a p s m o r e i m p o r t a n t , a r g u m e n t ist h e f o l l o w i n g . W e s h o w l a t e r in t h e p a p e r t h a t , u n d e r e i t h e r o f th e t w o s p e c i f i c a t io n so f t h e p r o b a b i l i t y m e a s u r e o v e r m e m b e r s h i p d i s t r i b u t io n s o f g r o u p s 2 a n d 3 , i fg r o u p l s m e m b e r s a r e g e o g r a p h i c a l ly v e r y c o n c e n t r a t e d , t h e n t h e g r o u p w i ll o na v e r a g e h a v e v e r y li tt l e p o w e r . T h u s , i f t h e b e n e f i t s a g r o u p r e c ei v e s f r o mg o v e r n m e n t a r e c l o s el y l i n k e d t o i ts p o w e r , w e m i g h t e x p e c t t h a t , o v e r t im e ,g e o g r a p h i c a l l y c o n c e n t r a t e d g r o u p s w o u l d d r o p o u t o f t h e p o li t ic a l g am e , g i v e nt h a t i t i s c o s t l y t o p l a y . T h i s w o u l d b e t r u e e s p e c i al ly f o r g r o u p s t h a t a r e b o t h l a r g ea n d c o n c e n t r a t e d , s in ce i n a d d i t i o n t o h a v i n g l it tl e p o w e r b a s e d o n t h e v o t i n gs t r e n g t h o f t h e i r m e m b e r s ( d u e t o t o o m u c h c o n c e n t r a ti o n ) , s u c h g r o u p s m a y f a ces e v er e fr e e - r id e r p r o b l e m s i n t r y i n g t o c o l l e c t o t h e r r e s o u r c e s ( s u c h a s m o n e y f o rc a m p a i g n c o n t r i b u t i o n s ) t h a t w o u l d i n cr e as e th e i r p o w e r . T h e l a t t e r p r o b l e m m a yn o t b e a s i m p o r t a n t f o r s m a l l , c o n c e n t r a t e d g r o u p s . T h i s a t t r i ti o n w o u l d s k e w t h ed i s t r i b u t i o n o f p o l i t i c a ll y a c t iv e g r o u p s i n a d i r e c t i o n c o n s i s t e n t w i t h t h e s e c o n df o r m u l a t i o n . T h u s , w h i l e n e i t h e r o f o u r s p e c i f i c a t i o n s i s , a s f a r a s w e k n o w , t h eo u t c o m e o f a m o r e g e n e r a l e q u il i b r iu m m o d e l i n w h i c h g r o u p s m a y e n t e r o r e x it , t h ef i r st s p e c i fi c a t io n , a s s i g n i n g e q u a l p r o b a b i l i t y t o a l l d i s t r i b u t i o n s , i s m o r e o b v i o u s l yo p e n t o c r i t i c is m a l o n g t h e s e l in e s .

5 Avera ge Pow er under the Secon d Spec i f i ca t ionT o c h a r a c t e r iz e t h e r e l a t io n b e t w e e n g r o u p l s m e m b e r s h i p d i s t r i b u t i o n a n d i tsa v e r a g e p o w e r , w e m u s t c a l c u l a t e t h e p r o b a b i l i t i e s o f th e f o u r p o s s i b le o u t c o m e s t ot h e p o l i ti c a l g a m e d e s c r i b e d i n S e ct . 2 a b o v e . G r o u p j c o n t r o l s t h e l e g i s l a tu r e i f a n do n l y i f i t c o n t r o l s ( m + 1 ) /2 o f t h e m d i s tr i c ts . T h u s , t h e p r o b a b i l i t y t h a t g r o u p jcon t ro l s the l eg i s l a tu re i s

k= m ~l {SI S=k} (5.1)

w h e r e P i j i s t h e p r o b a b i l i t y t h a t g r o u p j c o n t r o l s d i s t r ic t i. F o r e x a m p l e , i f m = 3 t h e nP j = P l j P 2 j ( 1 - P 3 j ) + P l j ( 1 - P 2 ~ ) P 3 j + (1 - P I j) P 2 ~ P 3 ~ + P l j P z j P 3 j .R e c a l l t h a t , u s i n g v o t i n g w e i g h t s ( o r a n y o f t h e u s u a l p o w e r i n d i c es ) to e v a l u a t et h e d i f f e r e n t o u t c o m e s , W 1 = 1 i f g r o u p I c o n t r o l s t h e l e g i s la t u r e , W x = 0 i fe i t h e r g r o u p 2 o r g r o u p 3 c o n t r o l s t h e l e g i s l a tu r e , a n d /4 11 =1 /3 o t h e r w i s e .T h us , p r o b ( W l = l / 3 ) = l - P 1 - P 2 - P 3 , a n d th e e xp ec te d p o w er o f g ro u p 1 isE (W t) = P1 + (1 /3) (1 - P~ - P z - P3) = (1/3) + (2/3) Pa - (1/3) P z + P3).


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Polit ical Ge ogra phy and Interes t-Gro up Power 111N o w f ix a d i s t r i b u t i o n S I = S l l . . . . . S ml ) f o r g r o u p 1. U n d e r t h e s e c o n d

s p e c i f i c a t i o n o f t h e m e a s u r e o v e r th e m e m b e r s h i p d i s t r i b u t i o n s o f g r o u p s 2 a n d 3 ,t h e s~j a r e a ll i n d e p e n d e n t , s o w e c a n t r e a t t h e d i s t r ic t s o n e a t a t im e . G r o u p 1 c o n -t r o l s d i s t r i c t i i f a n d o n l y i f s a > s f2 + s ~3 . T h u s , s i n c e si2 a n d s~3 a r e b o t h d i s t r i b u t e du n i f o r m l y o n [ 0 , 2/ m ], t h is o c c u r s w i t h p r o b a b i l i t y

P i l S i l ) =

m2 2Y S~l 0 _-_s. = m~ - T m - s ' m

41 < s nm

(5 .2)

G r o u p 2 c o n t r o l s d i s t r i c t i i f a n d o n l y i f si2 > S i l + S i 3 , o r s i2 - - s i3 > s , . T h u s , t h ep r o b a b i l i t y t h a t g r o u p 2 c o n t r o l s d i s t r ic t i i s

I s . 0 < s . < - - mP i 2 s , ) =0 - - < S i lm

(5 .3)

A l s o , b y s y m m e t r y P i 3 (S i l ) = P i 2 ( si l) . U s i n g ( 5 . 1 ) - ( 5 .3 ) w e c a n w r i te p r o b ( W 1 = 1),p r o b ( W 1 = 0 ) a n d E W 1 ) a s f u n c t io n s o f g r o u p l ' s m e m b e r s h i p d i s t r ib u t i o n s l .T h e s e f u n c t i o n s a re , u n f o r t u n a t e l y , q u i t e n o n c o n v e x , a n d t h u s s o m e w h a t d i f f i c u ltt o c h a r a c t e r i z e i n d e t a i l . A l s o , t h e r i g h t - h a n d s i d e o f (5 .1 ) b e c o m e s v e r y l o n g a s th en u m b e r o f g r o u p s g e t s l a rg e .

T o s i m p l i f y m a t t e r s , w e f o c u s o n d i s t r ib u t i o n s in w h i c h g r o u p l ' s m e m b e r s a r ee v e n l y d i v i d e d a c r o s s s o m e s u b s e t o f th e d i s t r i c ts , t h a t i s

~ t l / k , i e S ( 5 . 4 )s i l = [ O , i e M - S ,

w h e r e S is a s u b s e t o f M w i t h # S = k , f o r s o m e k < m , a n d t 1 i s t h e t o t a l s iz e o f g r o u p1. W h i l e t h i s i s a s m a l l s u b s e t o f a ll p o s s i b l e d i s t r i b u t i o n s , i t a l l o w s u s t o a d d r e s sc l e a n l y t h e i s s u e o f c o n c e n t r a t i o n v s. d i s p e r s i o n . U s i n g a n y o f t h e u s u a l m e a s u r e so f d i s p e r s io n , w e w o u l d f i n d t h a t a m o n g d i s t r i b u t i o n s s a t i s f y in g (5 .4 ), t h e d e g r e e o fd i s p e r s i o n i s i n c r e a s i n g i n k )

lo Fo r example, we might use the following definition, which has been studied by Atk inson and othe rsin the context of incom e inequality. Le t s 1 an d s~ be two mem bership distributions with s 11 > .. . --> ,,l,S;a >__... > s~a , an d ~ sll = ~ s;t . Then s 1 s m o r e d i s p e r se d (exhibits more dispers ion)th an s; if and only

i= ~=if there exists i* such th at s l --< ~i for all i N i*, an d sil >= ~x for a ll i > i*, w ith strict equ ality for some i.No te that this gives only a partia l ordering of the sets of all possible distributions. Thus, fo r example,(1,1,1,1,0) is m ore dispersed than (2,2,0,0,0), bu t it is neither more no r less dispersed than(2,1/2,1/2,1/2,1/2).


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112 J. Sny derW e c o n s i d e r d i f f e r e n t v a l u e s f o r t h e t o t a l s i z e t a o f g r o u p 1 , b e c a u s e t h e r e s u l t s

v a r y w i t h t h i s si ze . T a b l e s 1 - 3 a n d F i g s . 1 - 9 t e ll t h e s t o r y . I n t h e f i g u r e s , w e p l o tE W 1 ) , p r o b W 1 = 1 ) a n d p r o b W 1 = 0 ) a s f u n c t i o n s o f t 1 f o r t h e c a s e o f 3, 5 , a n d 7d i s t r ic t s . A l o n g e a c h c u r v e i n e a c h f i g u re , k is f i x e d ; t h u s , t h e r e a r e m c u r v e s i n e a c hf i gu r e w h e r e m i s t h e n u m b e r o f d i s tr i c t s ) , o n e fo r e a c h p o s s i b l e v a l u e o f k . T h e m o s ti n t e r e s ti n g o b s e r v a t i o n , r e c o r d e d i n T a b l e s 1 - 3 a n d F i g s . 1 - 3 , i sT a b l e

Entries give E W 1) Three districtsk fi 1/4 1/2 3/4 1 5/4 3/2 7/4 21 0 .1 01 6 0 . 1 5 6 3 0 . 1 6 6 7 0 . 1 6 6 7 0 . 1 6 6 7 0 . 1 6 6 7 0 . 1 6 6 7 0 .1 66 72 0 .1 1 3 5 0 . 2 0 6 4 0.2862 0.3652 0.4608 0.5873 0.7221 0.84383 0 .1152 0 .2067 0 . 2 7 6 4 0 . 3 3 3 3 0 . 3 9 0 3 0 . 4 6 0 0 0 . 5 5 1 5 0 .6 66 7

T a b l e 2

Entries give E W x) Five districtsk t a 1/4 1/2 3/4 1 5/4 3/2 7/4 21 0 .1 07 4 0 . 1 2 5 0 0 . 1 2 5 0 0 . 1 2 5 0 0 . 1 2 5 0 0 . 1 2 5 0 0 . 1 2 5 0 0 .1 25 02 0 .1 31 8 0 . 2 1 4 8 0 . 2 4 9 0 0 . 2 5 0 0 0 . 2 5 0 0 0 . 2 5 0 0 0 . 2 5 0 0 0 .2 50 03 0 .1 3 78 0 . 2 3 5 9 0.3008 0.3542 0.4388 0.5809 0.7475 0.89494 0 .1 40 1 0 . 2 3 9 5 0 . 3 0 0 5 0 . 3 4 0 9 0 . 3 9 0 7 0 . 4 8 5 0 0 . 6 3 7 9 0 .7 89 75 0 .1412 0 .2407 0 . 2 9 9 2 0 . 3 3 3 3 0 . 3 6 7 5 0 . 4 2 5 9 0 . 5 2 5 5 0 .6 66 7

Table 3

Entries give E W 1) Seven districtsk t 1 1/4 1/2 3/4 1 5/4 3/2 7/4 21 0 .1 02 5 0 . 1 0 4 2 0 . 1 0 4 2 0 . 1 0 4 2 0 . 1 0 4 2 0 . 1 0 4 2 0 . 1 0 4 2 0 .1 04 22 0 .1 42 4 0 . 2 0 5 1 0 . 2 0 8 3 0 . 2 0 8 3 0 . 2 0 8 3 0 . 2 0 8 3 0 . 2 0 8 3 0 .2 08 33 0 .1 53 1 0 . 2 4 6 5 0 . 2 8 7 7 0 . 2 9 1 7 0 . 2 9 1 7 0 . 2 9 7 1 0 . 1 9 2 7 0 .2 91 74 0 .1 57 4 0 . 2 5 8 3 0 . 3 1 1 7 0.3450 0.4138 0.5601 0.7519 0.92055 0 .1 59 6 0 . 2 6 2 0 0 . 3 1 3 4 0 . 3 3 9 6 0 . 3 8 2 9 0 . 5 0 2 9 0 . 6 8 5 1 0 .8 51 86 0 .1 6 09 0 . 2 6 3 7 0.3135 0 . 3 3 5 8 0 . 3 6 4 1 0 . 4 3 7 5 0 . 5 8 8 8 0 .7 68 57 0 .1618 0 .2647 0 . 3 1 3 3 0 . 3 3 3 3 0 . 3 5 3 3 0 . 4 0 2 0 0 . 5 0 4 9 0 .6 66 7

O b s e r v a t i o n 1 . F i x m a n d c o n s i d e r a l l d i s t r i b u t i o n s o f th e f o r m g i v e n i n 5 .4 ) . T h e r ee x i s t t * a n d ** t * *


11/23

Political Geography and Interest Group Power 113

1 . 0 -

9 -

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0.z~-

0 . 3 -

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0.1

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23

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o 5 '1 ~ ' 5 ~ 2 ' s ~ ~E (W ~ ) 3 D i s t r i c t s

Fig 1

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1 1 4 J . S n y d e r

1.0

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0 ~o 1 5 , 1 5 2 5 3E W I ) 7 D i s t ri c t s

Fig 3

R e c a l l t h a t E ( t 2 ) = E ( t 3 ) = 1. T h u s , O b s e r v a t i o n 1 i m p li e s t h a t i f g r o u p 1 isc o n s i d e r a b l y s m a l l e r t h a n t h e a v e r a g e s iz e o f t h e o t h e r g r o u p s t h e n i ts a v e r a g ep o w e r i s g r e a t e s t w h e n i t s m e m b e r s a r e e v e n l y d i v i d e d a c r o s s a l l m d i s t r ic t s , a n d i fg r o u p 1 is la r g e r t h a n t h e a v e r a g e s iz e o f t h e o t h e r g r o u p s t h e n i ts a v e r a g e p o w e r i sg r e a t e s t w h e n i t s m e m b e r s a r e e v e n l y d i v i d e d a c r o s s ( m + 1 ) /2 o f t h e d i s t ri c t s. W h i l ew e h a v e n o t p r o v e d t h i s f o r a r b i t r a r y m , th e p a t t e r n s e em s c le a r . L o o k i n g a g a i n a tT a b l e s 1 - 3 a n d F i g s. 1 - 3 , w e n o t e a l s o th e f o l l o w i n g i n t e r e s t i n g f e a t u r e :O b s e r v a t i o n 2 . F o r a n y t l , g r o u p 1 s e x p e c t e d p o w e r d e c li n es s h a r p l y a s k d r o p sb e l o w ( m + 1 ) /2 , a n d c o n t i n u e s t o d e c l i n e q u i c k l y a s k f a l ls t o 1 .

T h u s i f g r o u p 1 s m e m b e r s a r e c o n c e n t r a t e d i n le ss t h a n h a l f o f t h e d i s tr ic t s, t h eg r o u p w i ll h a v e l i t tl e e x p e c t e d p o w e r .

T h e r e a s o n s f o r t h e se r e s u lt s b e c o m e c l e a r e r i f w e t u r n t o F i gs . 4 - 9 d e s c r ib i n gp r o b (W 1 = 1 ) a n d p r o b ( W1 = 0 ) . 11 I f g r o u p 1 is c o n c e n t r a t e d i n f e w e r t h a n h a l f o ft h e d i s t ri c ts , t h e n i t h a s n o c h a n c e o f c o n t r o l l i n g t h e l e g is l a tu r e . A l s o , t h e m o r ec o n c e n t r a t e d g r o u p 1 b e c o m e s , t h e m o r e l i k e l y i t is t o b e p o w e r l e s s ; t h a t i s, th e l e ssl i k e ly i t i s t o p r e v e n t o n e o f t h e o t h e r g r o u p s f r o m c o n t r o l l i n g t h e l e g i s l a tu r e . T h i s is11 W e m i g h t a l so c o n s i d e r a n o n c o o p e r a t i v e g a m e i n w h i c h e a c h i n t e re s t g r o u p a c t s in i so l a t io n ,l o b b y i n g f o r s o m e p r o p o s a l i t f a v o r s . I f t h e p r o b a b i l i ty t h a t l e g i s la t o r s v o t e f o r g r o u p l s p r o p o s a l a r eg i v e n b y ( 5 .2 ) , a n d t h e g r o u p n e e d s t h e s u p p o r t o f a m a j o r i t y o f t h e l e g i s l a tu r e f o r it s p r o p o s a l t o p a s s ,t h e n t h e g r o u p s i n f l u e n c e i s m e a s u r e s i m p l y b y p r o b ( W 1 = 1 ) , o r P 1 . T h e s p e c i f i c a t i o n i n ( 5 . 2 ) i s n o tu n r e a s o n a b l e , a l t h o u g h i t i s o b v i o u s l y a s p e c ia l c a se , s o t h e r e l a t io n s h i p b e t w e e n P 1 a n d g r o u p l sm e m b e r s h i p d i s t r i b u t i o n i s o f s o m e i n t e r e s t i n i t s e l f .


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P o l it i ca l Ge ogra ph y a nd In t e re s t Group P owe r 115b e c a u s e i f g r o u p 1 h a s n o m e m b e r s i n a g i v e n d i s tr i ct t h e n i t h a s n o c h a n c e o fp r e v e n t i n g o n e o f th e o t h e r g r o u p s f r o m c o n t r o l l i n g t h a t d i s tr ic t . E x a m i n i n g ( 5 .3 ),w e se e t h a t i n e a c h d is t ri c t t h e r e a re c o n s i d e r a b l e d e c r e a s i n g r e t u r n s , i n t e r m s o fr e d u c i n g t h e p r o b a b i l i t y t h a t o n e o f th e o t h e r g r o u p s c o n t r o l s t h e d i st ri c t, t o h a v i n ga d d i t i o n a l g r o u p m e m b e r s i n th e d i s tr ic t . I n o r d e r f o r o n e o f t h e o t h e r g r o u p s t oc o n t r o l t h e l e g is la t u re , i t m u s t c o n t r o l m o r e t h a n h a l f o f th e d i s t r ic t s ; t h u s t h ep r o b a b i l i t y th a t g r o u p I is p o w e r l e s s is sm a l l e r w h e n t h e g r o u p h a s a fe w m e m b e r s i nm a n y d i s t ri c ts t h a n w h e n i t h a s m a n y m e m b e r s i n a f e w di st ri ct s, a t l e a st u p t o t h ep o i n t a t w h i c h i t h a s m e m b e r s i n h a l f o f t h e d i s tr i c ts .

A s g r o u p 1 b e c o m e s e v e n m o r e w i d e ly d i s tr i b u te d , t h e s i t u a t i o n is c o m p l i c a t e di n tw o w a y s . F i r s t l y , t h e g r o u p c a n s o m e t i m e s c o n t r o l t h e l e g i sl a t u re i ts e lf . W e s e ef r o m F i gs . 4 - 6 t h a t ( r eg a r d le s s o f t h e si ze o f g r o u p 1) t h e p r o b a b i l i t y t h a t t h is o c c u r sis g r e a t e s t w h e n g r o u p 1 s m e m b e r s a r e e v e n l y d i v i d e d a c r o s s ( m + 1 ) /2 d i st r ic t s , a n di t d e c r e a s e s m o n o t o n i c a l l y a s t h e g r o u p b e c o m e s m o r e a n d m o r e d i s p e rs e d . T h er e a s o n f o r t h is is n o t o b v i o u s , a s it in v o l v e s a t r a d e o f f : a s th e g r o u p b e c o m e s m o r ew i d e l y d i s p e r se d , i t h a s a h i g h e r p r o b a b i l i t y o f c o n t r o l l i n g s o m e d i s t ri c ts , a n d al o w e r p r o b a b i l i t y o f c o n t r o l l i n g o t h e r s . E v i d e n t l y , th e l a t t e r e f f e c t is m o r ei m p o r t a n t t h a n t h e f o r m e r , g i v e n th e s p e c i fi c a ti o n o f t h e p r o b a b i l i t y m e a s u r e o n s 2a n d s 3 . S e c o n d l y , t h e r e a r e o p p o s i n g e f fe c t s o n t h e p r o b a b i l i t y t h a t g r o u p 1 isp o w e r l e ss . A s t h e g r o u p b e c o m e s m o r e w i d e ly d is p e r se d , i t h a s a h i g h e r p r o b a b i l i t yo f p r e v e n t i n g th e o t h e r g r o u p s f r o m c o n t r o l l in g s o m e d i s t ri c ts , a n d a lo w e rp r o b a b i l i t y o f p r e v e n t i n g s u c h c o n t r o l i n o t h e r d i st ri ct s. W h i c h e f f e c t d o m i n a t e sd e p e n d s o n t h e s iz e o f g r o u p 1 t h e r e e x i s ts a t * * * < 1 s u c h t h a t i f t 1 < t * * *

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15/23

P o l it i ca l Ge og ra phy a nd In t e re s t Group P owe r 117

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118 J Snyd er

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Fig 9

t h e n p r o b W a = 0 ) d e c r e a s e s a s th e g r o u p b e c o m e s m o r e w i d e l y d i s p e rs e d , a n d i ft I > t * * * t h e n t h e o p p o s i t e h o l d s . T h u s , f o r t 1 > t * * * t h e r e is n o c o n f l i c t b e t w e e nin c re as in g p r o b W l = l ) a n d d e c re a si ng p r o b W i = 0 ) , a n d E W 1 ) i s c l ea r l ym a x i m i z e d w h e n k = m + l ) / 1 . F o r t i < t * * * h o w e v e r , t h e r e is a c o n f li c t, p r o -d u c i n g t h e t * a n d t * * o f O b s e r v a t i o n 1 .

T o s ee w h e t h e r o r n o t w e m i s s a n y t h i n g i n t e r e s ti n g b y r e s tr i c ti n g a t t e n t i o n t om e m b e r s h i p d i s t ri b u t i o n s o f t h e f o r m g i v e n b y 5 .4 ), w e in v e s t ig a t e t h e c a s e o f t h r e ed i s t r i c t s i n m o r e d e t a i l . T h e m o s t i m p o r t a n t r e s u l t i sroposition 3 . There ex i s t s t * such t ha t f o r any t i < t * , E W 1 ) /s u n i q u e ly m a x i m i z e d ,

over a l l f eas i b l e d i s t r ibu t i ons o f g roup 1 , when t he m e m b ers o f g roup 1 a re even l yd i v i ded over a l l t h ree d i s t r i c ts ; an d or any t 1 > t * , E W i ) i s un i que l y m ax i m i ze d wh ent he m em bers o f g roup 1 a re even l y d i v i ded over t wo o f t he t h ree d i st r ic t s .P r o o f . S e e A p p e n d i x .

T h u s , a t l e a s t f o r t h e c a s e o f t h r e e d i s t ri c t s, i n c l u d i n g d i s t r i b u t i o n s t h a t d o n o ts a t is f y 5 . 4 ) d o e s n o t c h a n g e o u r b a s i c f i n d i n g . W e s t r o n g l y s u s p e c t t h a t t h e s a m eh o l d s f o r la r g e r n u m b e r s o f d i s tr i c ts a s w e ll .I t w o u l d o f c o u r s e b e s t ill m o r e s a ti s fy i n g t o h a v e a m o r e g e n e r a l p r o p o s i t i o na b o u t t h e r e l a t io n b e t w e e n d i s p e r s io n a n d e x p e c t e d p o w e r , b u t w e l e a v e t h a t f o rf u t u r e w o r k .


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Political Ge ography and Interest-Group Pow er 1196 Avera ge Pow er under the F i r s t Spec i f i ca t ionT h e f i r s t s p e c i f ic a t i o n o f t h e p r o b a b i l i t y m o d e l o n s 2 a n d s3 in w h i c h s z a n d s3 a r ee a c h d r a w n u n i f o r m l y f r o m t h e m - s im p l e x ( w h e r e m is t h e n u m b e r o f d is tr ic t s) , ism o r e d i f f i c u l t t o w o r k w i t h t h a n t h e s e c o n d , b e c a u s e w e m u s t t r e a t a l l d i s t ri c t ss i m u l t a n e o u s l y . T h e r e s u l t s a r e s i m i l a r , h o w e v e r , t o t h o s e f o u n d f o r t h e s e c o n ds p e c i f i c a t io n i n S e c t . 5 , a t l e a s t f o r t h e c a s e w e a n a l y z e o f t h r e e d i s t r ic t s . I np a r t i c u l a r , w e h a v eP r o p o s i t i o n 4 . There exists t * < l such that for any t 1< t * , E W1) is uniquelymaximized, over all easible distributions of group 1, when the members of group 1 areevenly divided across all three districts; and for tl ~ t*, l] ,E W1 ) is uniquelymaximized when the members of group 1 are evenly divided across two of the threedistricts.Proof S e e A p p e n d i x .

N o t e t h a t w e h a v e n o t i n v e s t i g a t e d E W1) f o r t 1 > 1 . W e a l so f i nd t ha tp r o b (W ~ = 1 ) b e h a v e s e x a c t l y a s i n t h e s e c o n d s p e c i f ic a t i o n . T h a t i s , f o r a n y t~ < 1 ,p r o b ( W 1 = 1 ) is u n i q u e l y m a x i m i z e d ( o v e r a ll p o s s ib l e d i s t r i b u t io n s ) w h e n t h em e m b e r s o f g r o u p 1 a r e e v e n l y d i v i d e d o v e r t w o o f th e t h r e e d i s tr ic t s. T h e o n ed i f f e r e n c e o c c u r s i n t h e f u n c t i o n p r o b ( W ~ = 0 ) . W e f i n d t h a t f o r t h e f i r s t s p e c i f i -c a ti o n , p r o b ( W1 = 0 ) is u n i q u e l y m i n i m i z e d w h e n t h e m e m b e r s o f g r o u p I a r e e v e n l yd i v i d e d a c r o s s a ll t h r e e d i st r ic t s , f o r a n y t l < 1 . T h a t is , t h e s w i t c h p o i n t , t * * * < ] ,w h i c h e x i st s f o r t h e s e c o n d s p e c i f i c a t i o n ( re c a l l O b s e r v a t i o n 1 ), d o e s n o t e x i s t i n t h ef i r s t spec i f i c a t i on .

7 Conc lus ionsO n c e w e r e c o g n i z e t h e c o n f l i c t i n g e f f e c t s o f v a r i a b l e s s u c h a s g r o u p s iz e a n dg e o g r a p h i c c o n c e n t r a t i o n , i t is n o t s u r p r i s in g t h a t s t u d ie s a t t e m p t i n g t o d e t e r m i n et h e e f f e c t s o f f r e e - r id i n g o n p o l i t ic a l i n f lu e n c e , w h i c h h a v e u s e d t h e s e v a r i a b l e s t om e a s u r e t h e e x t e n t o f fr e e -r id i n g , h a v e m e t w h i c h o n l y m i x e d s u c ce ss . T h e e f f e ct s o fg r o u p s i z e a r e s t r a i g h t f o r w a r d : w h i l e l a r g e r g r o u p s m a y f a c e g r e a t e r f r e e - r i d e rp r o b l e m s , s u c h g r o u p s a l so h a v e m o r e v o t e r s a n d h e n c e p o t e n t i a l ly m o r e p o l i ti c a lc l o u t , s i n c e , a f t e r a ll , it is v o t e s t h a t p o l i ti c i a n s n e e d f o r r e e l e c t i o n . T h e e f f e c t s o fg e o g r a p h i c d i s p e r si o n a r e le ss o b v i o u s . M o r e d i s p e rs e d g r o u p s m a y f a c e m o r e f r e e -r i d in g a s d is t a n c e m a k e s o r g a n i z a t i o n a n d e n f o r c e m e n t m o r e d i ff ic u l t. O n t h e o t h e rh a n d , a s s h o w n i n t h is p a p e r , g r o u p s t h a t a r e c o n c e n t r a t e d i n o n l y a f ew l eg i sl at iv ed i s t ri c t s w i ll h a v e v e r y li tt l e p o w e r b a s e d o n t h e v o t i n g s t r e n g t h o f t h e i r m e m b e r s .A b s t r a c t i n g f r o m t h e f r e e -r i d e p r o b l e m , w e f i n d a n o p t i m a l d e g r e e o f d i s p e rs i o n ,b e y o n d w h i c h p o w e r d e c r e a s e s w i t h i n c r e a s ed d i s p e r si o n . T h u s , a f t e r a p o i n t, t h ee f fe c t s o f i n c r e a s e d d i s p e r s io n o n f r e e- r id i n g a n d v o t e p o w e r o p e r a t e i n t h e s a m ed i r e c t i o n .


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1 2 0 J . S n y d e rppendix

P r o o f o f P r o p o s i t i o n 1 . L e t s j a n d s [ b e t w o f e a s i b l e d i s t r i b u t io n s o f g r o u p 1 .W i t h o u t l o ss o f g e n e r a li t y , s u p p o s e t 2 > t l . A l s o w i t h o u t l o ss o f g e n e r a li t y ,s u p p o s e s[1 < s i t a n d s ~ l < s z l . W e f i n d f e a s i b l e d i s t r i b u t i o n s s z a n d s a s u c h t h a tW 1 ( s t , s 2 , s 3) < W 1 ( s [ , s z , s 3) , p r o v i n g t h a t s t d o e s n o t d o m i n a t e s ; . P i c k Sz a s f o l -l o w s 812 = 0 , SZt < 22 < r a i n ( s ~ l , 11 21 ) , Si2 -~- Si l A d f o r i t { 3 . . . . , ( m + 3 ) / 2 } ,/ .+3>/2w h e r e d = ( 2 / ( r n - 1 ) ) ' ~ t 2 - s 2 2 - _ i~=~_ s a ) , a n d s i 2 = 0 f o r i e { ( m + 5 ) / 2 . . . . , m } .P i c k s 3 s u c h t h a t si3 = 0 f o r i t { 2 . . . . . ( m + 3 ) / 2 }, a n d sia = 2 t 3 / ( m - 1 ) > 0 f o r a l li e { 1 , ( m + 5 ) / 2 . . . . , m } . T h e n s 2 a n d s 3 a r e b o t h f e a si b le . N o w d > ( 2 / ( m - 1 ) )( m + 3 ) / 2

t z - S l l - s 2 1 i~= 3 s a ) > = ( 2 / (m - 1 ) ) ( tz - t l ) > = O ; s o s i 2 > s i t = s i l + s i 3 , f o r a l li t { 2 . . . . . ( m + 3 ) /2 } . A n d ~ { 2 . . . ( m + 3 ) / 2 ) = ( m + 1 ) / 2 , s o W2 ( s l , s 2 , s 3 ) = 1 a n dt h u s W t ( s t , S z , S 3 ) = O ; b u t Wl ( s~ , s2 , s3 ) >=l / 3 , s i n c e s i 2 < s . + s i 3 f o r a l li t { 1 , 2 , ( m + 3 ) / 2 . . . . , m ) ( t h u s W 2 s t ,s 2 , s 3 ) ~ I ) a n d s ~ 3 < s i l + s i z f o r a l li t { 2 . . . . , ( m + 3 ) /2 } ( t h u s W 3( st , s2, s3) = 1 ). S o s 1 d o e s n o t d o m i n a t e s t . Q E DP r o o f o f P r o p o s i t i o n 2 . W i t h o u t l o s s o f g e n e r a l i t y , l e t k = 1 a n d l e t s 1 s a t i sf y( m + 1 ) / 2s 2 t < s a l < . . . < s m t . L e t d = ~ s i l - t 2 - t 3 > O . D e f i n e s ~ b y s ' l l = s , t - d a n di = 1s ;l = s i t + d / ( m - 1 ) f o r i t { 2, . . . , m } . N o t e t h a t s t 1 > 0 ( s i n c e s t t > d b y t h e a s s u m p -t i o n t h a t t t < ( 2 m / ( m + 1 ) ) ( t z ta ) ) , a n d ~ s ~l = ~ s a , s o s [ i s f e a s i b l e . A l s o , f o r/ = 1 i = 1 (m + 1)/2a ll S w i t h l t S a n d S > ( m + 1 ) / 2 , ~ S ~ z> t z+ t3 s i n c e ~ , s ~2 > ~ S ; z - d / 2

\ i e S i e S i = 1t2 + t 3 + d / 2 ) . W e n o w s h o w t h a t ( i) W 1 ( s l , S2, S3) ~---1 i m p l i e s W 1 ( s~ , s2 , s3) = 1 ,

a n d ( i i ) W (s'l , s2, s 3) = 0 i m p l i e s W 1 ( s t , s 2 , s 3) = 0 , w h i c h p r o v e s t h a t s~ d o m i -n a t e s s , .

( i ) I f W t ( s t , S z , S 3 ) = l t h e n t h e r e e x i s t s S w i t h ~ S = ( m + 1 ) / 2 s u c h t h a ts a > s ,2 > s ~a f o r a l l i e s. I f 1 S , o r i f I e S b u t s [ t > s t 2 + s 1 3 , t h e n s ~l > si2 si3 f o ra l l i ~ S , s o W 1(s'l, s2, s a ) = 1 . O t h e r w i s e , i f 1 e S a n d s~ 1 ---- 1 2 + s 1 3 t h e n t h e r e e x i s t sj E ( M - S ) s u c h t h a t s j l > S j z + S j 3 ( s i n c e ( ( M - S ) u { 1 } ) = ( m + l ) / 2 a n d t h u s

\ \2 s ~ l > t E + t 3 ) , i n w h i c h c a s e S ; l > s , a + S i a f o r a l l i e ( S - { 1 } ) w { j } , a n di~ (M--S)u{1} /a g a i n W l ( s ; , s z , s 3) = 1 .

( i i ) I f W t ( s [ , s 2 , s 3 ) = 0 t h e n e i t h e r W z ( s ; , s 2 , s 3 ) = 1 o r W 3 ( s [ , s 2 , s 3 ) = 1 .W i t h o u t l o ss o f g e n e r a l i t y , s u p p o s e t h e f o r m e r h o l d s . T h e n t h e r e e x i s t S w i t h# S = ( m + 1 ) / 2 s u c h t h a t s i2 > s ; t + si3 f o r a l l i ~ S . A l s o 1 S s i n c e i f 1 ~ S t h e n

s~2 > 2 s ; t + 2 s , ~ , w h i c h w o u l d i m p l y t h a t t z + t 3 ~ 2 s l2 - ~ s i3 > ~ Si lie i~ ie i~ I~S iea contradiction. Thus saS~x S~3 for all ieS,so Wt Sx, s2, s3) = 0. QED


19/23

P o l it ic a l G e o g r a p h y a n d I n t e r e s t - G r o u p P o w e r 12 1

P r o o f o f P r o p o s it i o n 3 . N o t e f i rs t t h a t i f t l > 8 / 3, t h e n p u t t i n g S l = ( t l / 2 , q / 2 , 0 ) ,g r o u p I c o n t r o l s t h e l e g is l at u re w i t h p r o b a b i l i t y 1 , w h i c h c a n n o t b e i m p r o v e d u p o n .T h u s , w e o n l y c o n s i d e r t l < 8 / 3. A l s o , s in c e p , = 1 f o r a l l s~l > 4 / 3 , w e c a n r e s t r i c ta t t e n t i o n t o s 1 s u c h t h a t s~l < 4 / 3 f o r a l l i. F i n a l l y , e x p l o i t i n g s y m m e t r y , w e c a ns up po s e t ha t s 31 < s 21 < s 11 . Su bs t i t u t i ng ( 5 .2 ) an d ( 5 . 3 ) i n t o ( 5. I ) an d pu t t i ng n = 3 ,w e h a v e

8 1 ( e 2 ~ 2 2 2 2 2 9 2~ - k 1 1 3 2 1 + S l l $ 31 t- $ 2 1 $ 3 1 - - ~ - 4 1 S ~ 1 $ 3 1 ) '9 9 4 2 2 _ _ g g _ _ S l l ) ) S 3 t . _ ~ _ g ~ 2 2 1 S 3 1- ( ( l - - g ( ~ ' - - S l l ) ) $ 2 1 - ~ ( 1 9 4 2 9 2

_ _ 9 ( 1 _ _ 9 4 2 2 2( ~ - - - $ 1 1 ) ) $ 2 1 S 3 1 ,( ~ _ _ 9 ( 3 4 _ _ _ $ 1 1 ) 2 ) ( ] 9 4 2 9 9 4 2- g x - s ~ O ) + ~ 0 - ~ - k - s 1 1 ) ) ~ 1

p l ( s l ) = _ 9 ( 1 9 4- 2 9 3 2 2 2- - g ( ~ - - - S l l ) ) ( 1 - - ~ - ( ~ - - $ 2 1 ) )$ 2 1 S 31 ,9 4 2 9 4 - 2( 1 - - ~ - ( ~ - - $ 1 1 ) ) ( ] - - g ( ~ - - - $ 2 1 ) )

a n d

P1 (s l ) = /

q _ ( ~ 9 4 2 _ _ ~- ~ ~ - s 1 1 ) ) 0 k -s ~ l) )~ - ( 1 9 ( 4 ~ ~ 2 ~ ( 1 9 4 2- - ~ ~ - - o Z l ] ] - - ~ ( ~ - - $ 3 1 ) )

- - 2 ( 1 ~ ( ~ S ) 2 ~ ( 1 9 4 2- ~ ( ~ - - $ 2 1 ) ) ( 1 9 4 2~ x - s ~ 0 )- 8 3 - - 1 1 )

i f s 1 1 < 2

i f s 2 1 = < ~ < s l l

i f s31 < 2 < Se

i f 2 < s 3 1

8 1 2 2 2 2~ ~ - s1 1 ) ~ - s2 1 ) + 2 - s1 0 2 ~ - s3 0 2_ [ ._ ( 2 $ 2 1 ) 2 / 2 - ,2 9 1 2 2 2 2 2 2 2~ - ~ a x ) - ~ - t ~ - s 1 1 ) ( ~ - s 2 0 ( ~ - s 3 z ) ) l f s a l < x

8 1 2 2 2 2~ ( ( g - s z l ) ( ~ - s 3 1 ) i f s2a


20/23

122 J. Sny der( t l , 0 , 0 ) ( h . 0 , 0 )

& ~ . /4 2 t l 2 ~ D. 4 f3J l t

h T ' - f - T / A ; ~ 7 ' '1 - 5 - ' ~ '5 - , -2/ I' '

// t ~ , - ~ - , y I / / ~ \/ / \( 0 , 0 . t 11 ( 0 , t l , 0 } ( 0 . 0 , t 1 ) ( 0 , t l , 0 )

( t~,0.0)AA t l 2 t l 2 I / ~( ~ . ~ - ~ , - ~ ~ , / ~ . .... \

/. 2 2 \1, t l l l l J [-~','~',U]\' 3 3 ' 3 ' , 11 1 1 ' , L Fig. lO

{0 .0 . t 1 {0 , t l , 0 )2 4 / 3 f o r a l l (Sz i , S3x) e ( 2 / 3 , 4 / 3 ) x ( 0 , 2 / 3 ) , so w e m us t ha ve s ix = S z i . Bu tt h e n0 L 3 L

+ ~ f o r a l l ( s2 i , s3x) ~ (2 /3 , 4 /3) x (0 , 2 /3) .3sxiN e x t w e s h o w t h a t f o r r eg i o n i v ) t h e o n l y i n t e r i o r p o i n t s a t i s f y i n g t h e f i r s t -

o r d e r c o n d i t i o n s is s 1 = ( t l / 3 , t a / 3 , i l l 3 ) .


21/23

P o l it ic a l G e o g r a p h y a n d I n t e r e s t - G r o u p P o w e r 1 23T u r n i n g t o r e g i o n i ) , w e s h o w t h a t t h e o n l y i n t e r i o r p o i n t s t h a t s a t i sf y t h e

f i r s t - o r d e r c o n d i t i o n s a r e s l = t l / 3 , t l/ 3 , q / 3 ) , a n d a p o i n t w i t h s ia=s2a a n dska = ( 27s3~ - 54 ~ a + 48 s i1 - 8)/27si1 (2 - 3siO, w h e r e e i t h e r i = 1 a n d k = 3 o r i = 3a n d k - - 1. (T h e p r o p e r c h o i c e , w h i c h d e p e n d s o n t h e v a l u e o f t a , s i m p l y i n s u r e st h a t Saa __ _saa. T h e p r o o f i s a s f o l l o w s :

0 L a L i f f e i t h e r s aa = s 2 a(~S l a - - 8 S 2 1o r

sa t = s ~ a 2 7 s 2 a - 1 8 ) - F S3 ( 4 8 - 3 6 s 2 0 - 8 ) / - 2 7 s ~ 1 + 3 6 - 5 4 s 2 1 ) s 3 1 + 1 8 s2 a )- f s z~ , s 3 1 .

S i m i l a r l y ,0 L 0 L~Sia ~s31

a n di f f s l l = s a a o r s l l = f ( s a a , s2 1) ,

0 L a L(~s2a - (~$3 i f f s 2a = s 3 a o r s2 1 = f ( s 3 1 , s H ) .

F o r a l l ( s 2 t , 3 1 ) @ ( 0 , 2 / 3 ) x ( 0 , 2 / 3 ) s u c h t h a t S za + s 3 a , f s2 1 , s3a ) =#f (s3a , s2a ) . T h u sw e m u s t h a v e s ix = s ~ l f o r s o m e i a n d j . S o , e i t h e r sa a = s 2 2 = s 3 ~ , o r sl l = s j a a n dS ki = f s l a , S , ) = (2 7S 31 - -5 4S ~1 + 4 8 S ~ 1 - - 8 ) / 2 7 S i a ( 2 - -3 S ~ 1 ) . B y t h e c o n s t r a i n t ss3 a < s 2 a < S a a , w e c a n g e t s e t j = 2 i n t h e l a t t e r c a s e .

F i n a l l y , w e a n a l y z e t h e b o u n d a r i e s . O n t h e b o u n d a r y w i th s31 = 0 , E W1 ]sa ) i sm a x i m i z e d ( f o r a n y t l) a t f i / 2 , f i / 2 , 0 ). N o t e t h a t t o s h o w t h i s f o r t a E ( 2 /3 , 4 /3 ) w em u s t c o n s i d e r r e g i o n s i ) a n d i i ) , a n d f o r t 1 e ( 4 /3 , 2 ) w e m u s t c o n s i d e r r e g i o n si i ) a n d i i i ) . A l o n g t h e b o u n d a r y w i t h s l l = 2 /3 , E ( W a I sl) i s a l w a y s i n c r e a s i n g i ns z l , s o E Wa l sa ) is m a x i m i z e d w h e n S ea i s a s l a r g e a s p o s s i b l e ( e i t h e r t I - 2 / 3 o r 2 / 3d e p e n d i n g o n t i ) . S i m i l a r l y , E WalsO s m a x i m i z e d a l o n g t h e b o u n d a r y s 3t = 2 /3 a tt h e p o i n t ( t 1 - 4 / 3 , 2 / 3 , 2 / 3) . A l o n g t h e b o u n d a r y s l l = 4 / 3 , E ( W ~ ]sa) is m a x i m i z e de i t h e r a t ( 4 / 3 , t l /2 - 2 / 3 , q / 2 - 2 / 3 ) o r ( 4 / 3 , t I - 4 / 3 , 0 ) , d e p e n d i n g o n t I .

W i t h a ll t h e c a n d i d a t e s f o r t h e g l o b a l m a x i m u m o f E W a l s l) a t h a n d , w e n o ws i m p l y c o m p a r e t h e m . T h e c o m p a r i s o n s a r e s t r a i g h t f o r w a r d e x c e p t f o r t h e p o i n tw i t h S~l = s 2 a , Ski = f ( s i a , S ~I) b e c a u s e f o r f i x e d ta i t is d i f f i c u l t t o c a l c u l a t e s a e x a c t l y .I t is m u c h e a s i e r in t h is c a s e t o f ix Sea a n d c o m p a r e E W a ( S l) ) a t t h e r e s u l t i n g p o i n ts~ w i t h m a x ( E ( W l l (T a s20 /2 , 71 s21)/2 , 0 ) ) , E W al ( t'a ( s2a ) /3 , 7a ( s 20 /3 , 71 ( s2 a) /3 ) ) ) ,w h e r e t'l sz l ) = 2 S zl + f ( s 2 t , S zl ) . O f c o u r s e , o n l y v a l u e s o f s2 1 s u c h t h a t 71 ( s 21 ) /2 < 2n e e d t o b e c o n s i d e r e d . T h e c o m p a r i s o n s r e v e a l t h a t f o r a l l t a < t * ( w h e r et * ~ 0 . 5 1 3 ) , E W l l S l ) is m a x i m i z e d a t t x /3 , ta /3 , f i /3 ) , w h i l e f o r a l l q > t ~ ,E W I [ S l ) is m a x i m i z e d a t q / 2 , q / 2 , 0 ) . Q E DP r o o f o f P r o p o si ti on 4 . R e c a l l t h a t u n d e r s p e c i f i c a t io n o n e , f o r e a c h j E { 2, 3 },s j= sa j , Sz j , Saj h a s a j o i n t d i s t r i b u t i o n t h a t i s u n i f o r m o n t h e u n i t s i m p l e x


22/23

124 J . Sny derf _3 ){sj[ ~ si~ = 12 . T h u s, i f Sl l __


23/23

Pol i t ica l Geo graph y and In te res t -G roup Pow er 125Ray E J (1981) The de te rminants of t a r i f fand non ta r i f f t rade res t r ic t ions in the Uni ted S ta tes . J Pol Econ

89 : 105-121Si lbe rman J I , Durd en G C (19 76) De te rm ining legis la tive pre fe rences on the min imu m wage : ane c onomi c app roa c h . J P o l Ec on 8 4 : 317-329Spilerm an S, Dickens D (1974) Who wil l gain and wh o wil l lose influence u nde r differe nt e lect ion rules.A m J S o c 8 0 : 4 4 3 -4 7 7St ig ler G (1971) The econom ic theory of regula tion . Be l l J E con M anag Sc i 2 :3 -2 1Ta y l o r P J , Johns t on R J (1979) Ge og ra phy o f el ec ti ons. Ho l me s a nd Me i e r , Ne w Yo rkWeinberger M , G reevey D U (eds) (1982) The PA C di rec tory . Bal linger, Cam bridgeWelch W P (1982) Camp aign con t r ibut ions an d legis la tive vot ing: m i lk mo ney and da i ry pr ice support s .W e s t P o l Q 35 : 478-495W right JR (1985) PAC s , cont r ibut ions , and ro l l cal l s : an organiza t iona l perspect ive . Am Pol Sc i Rev79 : 400 -414

social choice and welfare volume 6 issue 2 1989 [doi 10.1007%2fbf00303166] j. snyder -- political...

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