The Webster Method of Apportionment

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

THE WEBSTER METHOD OF APPORTIONMENT

M.L. B a l i n s k i H.P. Young

J u n e 1979 WP-79-49

W o r k i n g P a p e r s a r e i n t e r i m r e p o r t s o n work o f t h e I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s a n d h a v e r e c e i v e d o n l y l i m i t e d r e v i e w . V i e w s o r o p i n i o n s e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e - s e n t t h o s e o f t h e I n s t i t u t e o r o f i t s N a t i o n a l Member O r g a n i z a t i o n s .

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 L a x e n b u r g , A u s t r i a

ABSTRACT

Several results concerning the problem of U.S. 'Congressional apportionment are given which together indicate that a method

first proposed by Daniel Webster (also known as "Major Fractions") is fairest judged on the basis of common sense, Constitutional requirement, and precedent.

Key words: Congress/representation/fair division/U.S.

Constitution.

THE WEBSTER METHOD OF APPORTIONMENT M.L. B a l i n s k i a n d H . P . Young

Y a l e U n i v e r s i t y , N e w Haven, C o n n e c t i c u t and I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s

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

The C o n s t i t u t i o n of t h e U n i t e d S t a t e s r e q u i r e s t h a t t h e House o f R e p r e s e n t a t i v e s b e a p p o r t i o n e d among t h e s e v e r a l s t a t e s a c c o r d i n g t o t h e i r c e n s u s p o p u l a t i o n s . V a r i o u s m e t h o d s f o r s o d o i n g h a v e b e e n a d v a n c e d o v e r t h e y e a r s , b e g i n n i n g i n 1792 a f t e r t h e f i r s t c e n s u s . F o u r d i f f e r e n t methods h a v e b e e n u s e d . I n s t u d y i n g t h e d i f f e r e n c e s b e t w e e n t h e methods t h e r e emerge s e v e r a l c r i t e r i a w h i c h we b e l i e v e t o b e m o s t i m p o r t a n t by r e a s o n o f

common s e n s e , C o n s t i t u t i o n a l r e q u i r e m e n t , a n d p r e c e d e n t .

The aim o f t h i s n o t e i s t o s e t down, f o r t h e r e c o r d , s e v e r a l r e s u l t s d e s c r i b i n g t h e i n t e r p l a y b e t w e e n t h e s e c r i t e r i a which t o g e t h e r i n d i c a t e t h a t o n e method b e s t a n s w e r s t h e n e e d s . D e - t a i l e d p r o o f s w i l l a p p e a r e l s e w h e r e .

D e f i n i t i o n s a n d E l e m e n t a r y P r o p e r t i e s

An a p p o r t i o n m e n t p r o b l e m i s s p e c i f i e d by an s - v e c t o r ( s z 2 ) o f r a t i o n a l numbers p

-

⁼ ( p l

, . . .

^{, p s )}

^,

^{a l l pi >} 0 , a n d a n i n t e g e r h o u s e s i z e h 2 - 0. An a p p o r t i o n m e n t o f h among s i s a n i n t e g e r - s - v e c t o r a

-

= ( a l I . . . , a s ) 2 ^-

-

0 w i t h Eiai = h . An a p p o r t i o n m e n t method i s a m u l t i - v a l u e d f u n c t i o n M ( p ; h )

- -

so t h a t , f o r e a c h p > 0

-

a n d h - _- > 0 , M

-

i s a s e t o f a p p o r t i o n m e n t s a of h among s ( s o m e t i m e s

-

u n i q u e , sometimes n o t )

.

^{M(') -.,} d e n o t e s a method f o r f i x e d ^{S ;} M ( " ~ ) _-., f o r f i x e d s and h. A p a r t i c u l a r M - s o l u t i o n i s a s i n g l e -

..d

v a l u e d f u n c t i o n f , ^..d w i t h , ..f ( p ; h ) ^..d = a ~ M ( p , h ) .

-

⁵

The q u o t a o f s t a t e i f o r h , s i s qi = pih/l j p j . The l o w e r q u o t a i s L q i l : t h e u p p e r q u o t a r q i l .

*

The f o l l o w i n g e l e m e n t a r y p r o p e r t i e s d e f i n e more e x p l i c i t l y what i s meant by a method t h a t a p p o r t i o n s ' a c c o r d i n g t o numbers'.

Method M

-

^{i s} homogeneous when aEM(X?;h)

-

^-., i f and o n l y i f a ~ M ( ? ; h ) -.,

-

f o r a l l r a t i o n a l X > 0. I t i s p r o p o r t i o n a l i f a ^..d = q

-

^{i s} u n i q u e i n M(?;h)

-

whenever t h e q u o t a s qi a r e a l l i n t e g e r . These p r o p e r - t i e s a r e e s s e n t i a l t o t h e v e r y i d e a of p r o p o r t i o n a l i t y . A method i s symmetric i f f o r any p e r m u t a t i o n n o f 1 , .

. .

^{, s ,} _{( a T}

^,. .

^I

a n ( s ) ) E M ( ( ~ n ( ~ ) * * . ' P n ( s ) ) ; h ) i f and o n l y i f a .-- E ~ ( p ; h ) ^{- . . d}

.

^{~ h u s}

o n l y t h e numbers c o u n t , n o t t h e names of s t a t e s .

F i n a l l y , a method i s n o n - d e g e n e r a t e i f pn+

-

^p

-

^{and a}

- EM(^";^)

..,-

f o r a l l n i m p l i e s a

-

E M(p;h) ^{. . d -}

.

So, i f t h e pn a r e a s e q u e n c e o f

-

i n c r e a s i n g l y a c c u r a t e e s t i m a t e s of t h e t r u e p o p u l a t i o n p , -., a l l o f which a d m i t t h e a p p o r t i o n m e n t a by M,then s o d o e s p .

- - -

T h i s i s a t e c h n i c a l p r o p e r t y t h a t a l l o w s f o r a j u s t h a n d l i n g of t i e s .

These f o u r p r o p e r t i e s a r e m e t by a l l methods which h a v e , t o o u r knowledge, e v e r b e e n p r o p o s e d , and w e assume them i n t h e s e q u e l u n l e s s o t h e r w i s e n o t e d .

D i v i s o r Methods

A r a n k - i n d e x r ( p , a ) , a 2

-

0 i n t e g e r and p > 0 r a t i o n a l i s any r e a l v a l u e d f u n c t i o n s a t i s f y i n g r ( p , a ) > r ( p , a + l ) . The H u n t i n g t o n method b a s e d on r ( p , a ) [ 6 ] i s

.

~ ( p ; h )

- -

= {a

- - -

2 0 : a ⁱ integer, Lai = h ~ maxi '(pitai) Lmin - a;

>o

r(pjfaj-1)

I *

1

A r a n k - i n d e x d e t e r m i n e s a method by a s s i g n i n g p r i o r i t i e s i n t h e a l l o c a t i o n o f s e a t s by t h e f o l l o w i n g r e c u r s i v e r u l e on t h e s i z e of t h e h o u s e ( h ' l

-

h ) : a t h ' = 0 s e t a l l ai = 0; i f a appor-

-

t i o n s h ' < h , t h e n a n a p p o r t i o n m e n t o f h '

+

1 s e a t s i s f s u n d by g i v i n g o n e more s e a t t o some s t a t e maximizing r ( p i , a i ) .

*

L x l d e n o t e s t h e g r e a t e s t i n t e g e r ( - x , r x l t h e s m a l l e s t

i n t e g e r

2

- x.

A d i v i s o r c r i t e r i o n d ( a ) , a

2

0 i n t e g e r , i s any r e a l v a l u e d monotone i n c r e a s i n g f u n c t i o n . The d i v i s o r method b a s e d on d ( a )

i s t h e H u n t i n g t o n method b a s e d on r ( p , a ) = p/d ( a )

.

^{W e} a d o p t t h e c o n v e n t i o n t h a t p ^> q i m p l i e s p/O > q/O.

A d i v i s o r method i s r e g u l a r i f e i t h e r a < d ( a ) - a

+

1 f o r a l l a , o r a 5 ^- d ( a ) < a

+

1 f o r a l l a .

Lemma 1 . A d i v i s o r method i s p r o p o r t i o n a l i f and o n l y i f i t i s r e g u l a r .

I t i s o f i n t e r e s t t o know t h a t v i r t u a l l y a l l o f t h e m e t h o d s p r o p o s e d - - w i t h t h e n o t a b l e e x c e p t i o n o f H a m i l t o n ' s - - h a v e b e e n r e g u l a r d i v i s o r m e t h o d s . T h e s e have r e c e i v e d d i f f e r e n t names and d e s c r i p t i o n s i n v a r i o u s c o u n t r i e s and t i m e s . To t h e b e s t o f o u r knowledge t h e y s h o u l d b e c r e d i t e d i n t e r m s o f e a r l i e s t d i s c o v e r y a s f o l l o w s . J o h n Q u i n c y Adams' method [ I ] h a s d ( a ) ⁼ a ;

James D e a n ' s method [ I 7 1 ( h e was P r o f e s s o r o f Astronomy and Math- e m a t i c s a t Dartmouth and t h e U n i v e r s i t y o f Vermont) h a s d ( a ) = 2 a ( a + 1 ) / ( 2 a + l ) . E . V . H u n t i n g t o n ' s method of e q u a l p r o p o r t i o n s [ 1 2 , 1 3 ] ( h e was P r o f e s s o r o f M a t h e m a t i c s a t H a r v a r d ) h a s d ( a ) =

.

D a n i e l W e b s t e r ' s method [ I 71 h a s d ( a ) = a

+

1 / 2 .

Thomas J e f f e r s o n ' s method [ I 51 h a s d ( a ) = a

+

1

.

T h e s e a r e a l l r e g u l a r , h e n c e p r o p o r t i o n a l . H u n t i n g t o n u n i f i e d t h e s e " h i s t o r i c f i v e methods" t h r o u g h h i s t e s t o f i n e q u a l i t y a p p r o a c h [ 1 2 , 1 3 ] and showed how t h e y c o u l d b e computed r e c u r s i v e l y u s i n g d i v i s o r f u n c t i o n s . I n t h e e i g h t e e n t h and n i n e t e e n t h c e n t u r i e s t h e methods w e r e d e s c r i b e d i n d i f f e r e n t ( t h o u g h e q u i v a l e n t ) terms u s i n g t h e

i d e a o f a n i d e a l d i s t r i c t s i z e o r common d i v i s o r , A . F i r s t a

X

i s s p e c i f i e d , t h e n t h e numbers pi/X a r e u s e d t o d e t e r m i n e t h e a p p o r t i o n m e n t s a whose sum d e t e r m i n e s h. F o r e x a m p l e , Adams

'

i

method r o u n d s up a l l f r a c t i o n s , t h a t i s , s e t s a i = Tpi/XI;

J e f f e r s o n ' s d r o p s a l l f r a c t i o n s , t h a t i s , s e t s a i = Lpi/A-l; a n d W e b s t e r ' s method r o u n d s t o t h e n e a r e s t i n t e g e r , t h a t i s , s e t s

a = Lpi/A

+

1/21.

i

J e f f e r s o n ' s method was u s e d f o r t h e a p p o r t i o n m e n t s b a s e d on t h e c e n s u s e s o f 1790 t h r o u g h 1840. W e b s t e r ' s method was u s e d f o r 1910 and 1930. H u n t i n g t o n ' s method o f e q u a l p r o p o r t i o n s was u s e d f o r 1930 - - i t happened t o a g r e e w i t h W e b s t e r ' s - - a n d s i n c e 1940 it h a s b e e n t h e law o f t h e l a n d .

House M o n o t o n i c i t y

Another e a r l y method i s A l e x a n d e r H a m i l t o n ' s [ I l l , , r e - i n v e n t e d and u s e d f o r t h e c e n s u s e s o f 1850 t h r o u g h 1900 u n d e r t h e name

" V i n t o n ' s Method o f 1850". I t f i r s t g i v e s t o e a c h s t a t e i i t s lower q u o t a LqiJ; t h e n a s s i g n s o n e a d d i t i o n a l s e a t t o e a c h o f t h e

1

^(qi

-

LqiJ) s t a t e s h a v i n g t h e l a r g e s t r e m a i n d e r qi

-

LqiJ. ~ u t it a d m i t s t h e infamous Alabama p a r a d o x i n which an i n c r e a s e i n

t h e h o u s e c a n r e s u l t i n some s t a t e s l o s i n g s e a t s .

A method M

-

i s house monotone i f t h e r e e x i s t s f o r any p some

-

M-solution f f o r which f ( p ; h+l

- - - -

)

2

^- f ($;h) f o r a l l h .

-

C o n g r e s s i o n a l d e b a t e makes c l e a r t h a t o n l y house monotone methods c a n b e c o u n t e - nanced. A l l ~ u n t i n g t o n methods a r e house monotone; i n d e e d t h e q u e s t f o r house monotone methods i s what m o t i v a t e d H u n t i n g t o n ' s work (see a l s o W i l l c o x [ I 81 )

.

U n i f o r m i t y

An i n h e r e n t p r i n c i p l e o f f a i r d i v i s i o n i s : e v e r y s u b d i v i s i o n o f a f a i r d i v i s i o n must b e f a i r . I n t h e c o n t e x t o f a p p o r t i o n m e n t t h i s p r i n c i p l e c a n b e f o r m u l a t e d as f o l l o w s : M

-

i s u n i f o r m [ 8 ] i f ( a , b )

-

^-. E M ( p , q ; h ) i m p l i e s ( i ) _{- . - -} a

- - - E M ( ~ ; I ~ ~ ~ ) ^,

and ( i i ) i f a l s o a 1 € ~ ( p ; Z a i )

-

^-.

-

t h e n ( a ' , b ) € M ( p , q ; h ) .

- -

^{T h a t i s ,} an a p p o r t i o n m e n t

- - . -

a c c e p t a b l e f o r a l l s t a t e s i s a c c e p t a b l e i f r e s t r i c t e d t o any s u b s e t o f s t a t e s c o n s i d e r e d a l o n e ; moreover, i f t h e r e s t r i c t i o n a d m i t s a d i f f e r e n t a p p o r t i o n m e n t of t h e same number o f s e a t s t h e n u s i n g it i n s t e a d r e s u l t s i n a n a l t e r n a t e a c c e p t a b l e a p p o r t i o n m e n t f o r t h e whole.

Theorem 1 . I f a method i s u n i f o r m and p r o p o r t i o n a l , t h e n it i s h o u s e monotone.

I n f a c t t h e p r o o f r e q u i r e s , i n a d d i t i o n t o u n i f o r m i t y , o n l y t h a t two s t a t e s h a v i n g i d e n t i c a l p o p u l a t i o n s c a n n o t have appor- t i o n m e n t s d i f f e r i n g by more t h a n one s e a t . ( T h i s r e s u l t was l a t e r i n d e p e n d e n t l y n o t e d by Hyllarpd [ 1 4 ] . ) S i n c e t h e Hamilton method i s n o t h o u s e monotone it i s n o t u n i f o r m .

Theorem 2 . A method i s u n i f o r m and p r o p o r t i o n a l i f and o n l y i f i t i s a H u n t i n g t o n method.

T h i s f o l l o w s d i r e c t l y from an e a r l i e r c h a r a c t e r i z a t i o n of Huntington methods [ 6 1 and Theorem 1 .

P o p u l a t i o n M o n o t o n i c i t y

U n i f o r m i t y i n h e r e n t l y b e a r s t h e i d e a t h a t a method s h o u l d be a p p l i c a b l e t o a l l problems w i t h a l l p o s s i b l e house s i z e s and numbers o f s t a t e s . A c r i t i c might c o u n t e r t h a t i n many s i t u a t i o n s s and h a r e f i x e d : i n t h e U n i t e d S t a t e s h = 435 and s = 50. So, l e t u s f i x s and h.

A c e n s u s p r o v i d e s p o p u l a t i o n s p

-

= ^{( p l}

^{,.. .}

, p S ) . But t h e s e change o v e r t i m e , and e r r o r s i n c e n s u s numbers may y i e l d v a r i o u s p ' s .

-

A method must behave r e a s o n a b l y when a p p l i e d t o d i f f e r e n t

p ' s .

Many d e f i n i t i o n s f o r s u c h b e h a v i o r a r e c o n c e i v a b l e . The o b v i o u s m a t h e m a t i c a l c h o i c e i s t o compare two p ' s i d e n t i c a l i n

-

a l l s t a t e p o p u l a t i o n s s a v e o n e , and a s k t h a t a method n e v e r a s s i g n t o t h e one s t a t e h a v i n g g r e a t e r p o p u l a t i o n f e w e r s e a t s . A c t u a l p o p u l a t i o n c h a n g e s o v e r t h e y e a r s do n o t p r o d u c e s u c h s i t u a t i o n s .

A method M ( s , ~ )

-

= M*

-

( p ) ( h a v i n g f i x e d s and h )

-

i s p o p u l a t i o n monotone i f a EM* ( p )

- - - - - - ,

a ' €>I* ( p ' ) and p i / p ! 3 - = > p i / p j imply t h a t

a ! < a i and a ' > a o c c u r s o n l y i f p j / p ' = p i / p and ( a , ,

. . .

^{, a}

j,

1 j j j J

. . .

^{, a ; ,}

. . .

^{, a s ) E M *}

^{( p )} .

T h i s a v e r s t h a t i f p o p u l a t i o ' n s c h a n g e , a p p o r t i o n m e n t s s h o u l d n o t change by g i v i n g more s e a t s t o a s t a t e w i t h r e l a t i v e l y s m a l l e r p o p u l a t i o n and l e s s s e a t s t o a s t a t e w i t h r e l a t i v e l y g r e a t e r p o p u l a t i o n ( u n l e s s t h e r e i s a " t i e " ) .

Theorem 3. F i x s

f

3 and h. M ( " ~ )

-

i s p o p u l a t i o n monotone i f and o n l y i f it i s a d i v i s o r method.

The r e s u l t i s n o t t r u e f o r s = 3: a counter-example e x i s t s f o r h = 7 . And, of c o u r s e , t h e d i v i s o r i s a f u n c t i o n o f s and h .

C o r o l l a r y . M

-

i s u n i f o r m and p o p u l a t i o n monotone i f and o n l y i f it i s a r e g u l a r d i v i s o r method.

I n v o k i n g u n i f o r m i t y t o g e t h e r w i t h p o p u l a t i o n m o n o t o n i c i t y r e s u l t s i n a d i v i s o r i n d e p e n d e n t of s and h , which i s what one would n a t u r a l l y e x p e c t . I n f a c t , we have shown t h a t u n i f o r m i t y and p r o p o r t i o n a l i t y , t o g e t h e r w i t h t h e v e r y weak demand t h a t

P i ^> p j must i m p l y a i

2

a s u f f i c e s t o c h a r a c t e r i z e d i v i s o r j '

methods ( a c t u a l l y a somewhat more g e n e r a l r e s u l t o b t a i n s i f p r o - p o r t i o n a l i t y i s d r o p p e d ) [ 2 1 . H y l l a n d [ I 4 1 h a s r e c e n t l y f o u n d a s i m i l a r r e s u l t .

S a t i s f y i n g Quota

The most p r i m i t i v e r e q u e s t f o r a method o f a p p o r t i o n m e n t i s t h a t i t s h o u l d g u a r a n t e e t o e a c h s t a t e a t l e a s t i t s l o w e r q u o t a and a t most i t s u p p e r q u o t a , LqiA 5 - a . < r q i l , f o r a l l i . Methods

1 =

w i t h t h i s p r o p e r t y a r e s a i d t o s a t i s f y q u o t a . The H a m i l t o n

method i s p r e d i c a t e d o n i t , a s i s t h e Quota method [ 3 , 9 1 . I t i s a n u n f o r t u n a t e f a c t t h a t i t i s s i m p l y i m p o s s i b l e t o h a v e a method which s a t i s f i e s q u o t a t o g e t h e r w i t h o t h e r f u n d a m e n t a l c r i t e r i a .

Theorem 4 . T h e r e i s n o u n i f o r m method t h a t s a t i s f i e s q u o t a . Theorem 5 . F i x s

2

- 4 a n d h l a r g e ( h

-

s + 3 s u f f i c e s ) . T h e r e

i s n o ' p o p u l a t i o n monotone method M

-

( S f h ) t h a t s a t i s f i e s q u o t a .

So e v e n f o r f i x e d s t h e r e i s no method M _" which r e c o n c i l e s t h e p r i m i t i v e w i s h t o s a t i s f y q u o t a w i t h t h e n e c e s s i t y o f p o p u l a - t i o n m o n o t o n i c i t y . F o r s = 3 a s p e c i a l r e s u l t o b t a i n s .

Theorem 6 . The method o f Webster i s t h e u n i q u e d i v i s o r method which s a t i s f i e s q u o t a f o r s ⁼ 3 .

S a t i s f y i n g q u o t a - - a s d e s i r a b l e a s i t may b e - - i s incom- p a t i b l e w i t h u n i f o r m i t y a n d w i t h p o p u l a t i o n m o n o t o n i c i t y f o r f i x e d s a n d h. W e c o n c l u d e t h a t it must b e abandoned. And t h i s , w e w i l l see, c a n b e d o n e a t e s s e n t i a l l y no c o s t . I n p a r t i c u l a r , w e d i s c a r d t h e Quota method a s w e l l a s a l l q u o t a t o n e methods

[ 7 1 .

W e a r e l e f t w i t h t h e c l a s s o f r e g u l a r d i v i s o r methods.

B i a s

Why h a s H u n t i n g t o n ' s method o f e q u a l p r o p o r t i o n s b e e n re- t a i n e d f o r U . S . C o n g r e s s i o n a l a p p o r t i o n m e n t from among t h e f i v e h i s t o r i c d i v i s o r methods? I f o n e i n s p e c t s e x a m p l e s , it i s i m -

m e d i a t e l y e v i d e n t t h a t a s a p p l i c a t i o n o f Adams' method it s u c c e e d e d

by a p p l i c a t i o n o f D e a n ' s , t h e n H u n t i n g t o n ' s , W e b s t e r ' s a n d

J e f f e r s o n ' s , s o l u t i o n s t e n d more a n d more t o f a v o r l a r g e s t a t e s o v e r s m a l l . T h i s b e h a v i o r c a n b e proved ( [ 9

I ,

Theorem 1 )

.

Two r e a s o n s w e r e u s e d t o a d o p t H u n t i n g t o n ' s method: ( 1 ) it i s i n t h e m i d d l e o f t h e f i v e from t h e p o i n t o f view o f f a v o r i n g s m a l l a s v e r s u s l a r g e

* ,

( 2 ) i t i s b a s e d on a m e a s u r e o f p a i r w i s e i n e q u a l i t y o f r e p r e s e n t a t i o n between s t a t e s w h i c h ( w h i l e a r b i t r a r y ) seems p r e f e r a b l e t o t h o s e m e a s u r e s o f i n e q u a l i t y which c h a r a c t e r -

i z e t h e o t h e r f o u r methods ( [ I 0,161 )

.

I n t h e s e r e p o r t s no a b s o l u t e s t a n d a r d f o r d e t e r m i n i n g w h e t h e r a method f a v o r s s m a l l a s a g a i n s t l a r g e s t a t e s was s e t down. The d e s i r e t o c h o o s e a method w h i c h i s " u n b i a s e d " i n i t s award o f s e a t s t o s m a l l and l a r g e s t a t e s i s w e l l f o u n d e d , a n d i s r o o t e d i n t h e " h i s t o r i c compromise" i n which t h e S e n a t e was g i v e n re- p r e s e n t a t i o n i n d e p e n d e n t o f p o p u l a t i o n , and e a c h s t a t e was a s - s u r e d o f a t l e a s t o n e s e a t i n t h e House.

Suppose t h a t a p a i r o f s t a t e s w i t h p o p u l a t i o n s ( p , q ) , p > q r e c e i v e ( a , b ) s e a t s . I f a / p > b/q t h e n t h e l a r g e r s t a t e i s f a v o r e d w h e r e a s i f a / p < b/q t h e s m a l l e r s t a t e i s f a v o r e d . I n h e r e n t t o u n i f o r m i t y i s t h e t r u e - t o - l i f e f a c t t h a t a s t a t e j u d g e s how w e l l o r how b a d l y it h a s b e e n t r e a t e d by making com- p a r i s o n s w i t h i t s s i s t e r s t a t e s ' a l l o c a t i o n s . I n d e e d , t h i s ob- s e r v a t i o n was a t t h e o r i g i n o f H u n t i n g t o n ' s a p p r o a c h , a l t h o u g h h e t h e n d e v e l o p e d methods b a s e d on a d m i t t e d l y a r b i t r a r y m e a s u r e s o f i n e q u a l i t y between s t a t e s ' r e p r e s e n t a t i o n . B y d e f i n i t i o n a u n i f o r m method a p p o r t i o n s s e a t s among e v e r y two s t a t e s i n t h e same manner a s i t would w e r e t h e two c o n s i d e r e d a l o n e . There- f o r e , c o n s i d e r t h e s e t S ( a , b ) o f a l l two s t a t e p r o b l e m s ( p , q )

( n o r m a l i z e d , by h o m o g e n e i t y , t o p

+

q = 1 ) w h i c h y i , e l d t h e ap- p o r t i o n m e n t ( a , b )

,

a > b 2 - 1 ( i m p l y i n g , by p o p u l a t i o n monoton- i c i t y , p

2

q ) . A d i v i s o r method d ( * ) i s u n b i a s e d i f t h e m e a s u r e o f t h e s u b s e t o f S ( a , b ) o f t h o s e p o p u l a t i o n s f o r w h i c h t h e s m a l l s t a t e i s f a v o r e d i s t h e same a s t h e m e a s u r e o f t h e s u b s e t f o r which t h e l a r g e s t a t e i s f a v o r e d , f o r a l l p a i r s ( a , b ) , a > b 2 - 1 . So, i n d e p e n d e n t l y o f t h e m a g n i t u d e s o f a and b , a n u n b i a s e d method d ( * ) n e i t h e r f a v o r s s m a l l n o r l a r g e o v e r t h e s e t o f a l l p r o b l e m s .

*

I t was f o r t u n a t e , f o r t h i s l o g i c , t h a t t h e number o f methods c o n s i d e r e d was odd.

Theorem 7 . The u n i q u e u n i f o r m , p o p u l a t i o n monotone, and u n b i a s e d method i s t h a t o f Webster.

D u a l l y , one might approach t h e c o n c e p t of " b i a s " by f i x i n g ( p , q )

,

p

+

q = 1

,

^p

4

q , and c o n s i d e r i n g t h e a p p o r t i o n m e n t s of h = 1 , 2 , 3 ,

...,

h* s e a t s , where h* i s t h e s m a l l e s t i n t e g e r f o r which ph* and qh* a r e i n t e g e r . A method i s " u n b i a s e d " i f t h e number o f t i m e s t h e s m a l l s t a t e i s f a v o r e d i s t h e same a s t h e number o f t i m e s t h e l a r g e s t a t e i s f a v o r e d , f o r a l l p a i r s ( p , q ) , p + q = 1 , p q . By t h i s d e f i n i t i o n t h e method o f Webster i s a g a i n t h e u n i q u e u n i f o r m , p o p u l a t i o n monotonerand " u n b i a s e d "

method.

S p e c i f i c a p p o r t i o n m e n t s f o r a g i v e n problem c a n be a n a l y z e d f o r b i a s . I n s p e c t e a c h p a i r o f a l l o c a t i o n s t o s t a t e s ( a , b ) where

a > b 2 - 1 and d e f i n e t h e b i a s r a t i o t o be t h e number of t i m e s

t h e s m a l l e r s t a t e i s f a v o r e d d i v i d e d by t h e t o t a l number of com- p a r i s o n s . One c a n n o t e x p e c t any r e g u l a r d i v i s o r method t o y i e l d a p e r f e c t b i a s r a t i o o f .5: f o r some problems t h e r a t i o s t e n d t o be h i g h , f o r o t h e r s low. B i a s i s a c o n c e p t c o n c e r n i n g many

problems and s o must be a p p l i e d o v e r many problems. W e have t a k e n t h e 19 c e n s u s p o p u l a t i o n s o f t h e U n i t e d S t a t e s ( 1 790-1970 i n c l u s i v e ) and found a p p o r t i o n m e n t s by e a c h of t h e h i s t o r i c f i v e methods t o g e t h e r w i t h t h e i r r e s p e c t i v e b i a s r a t i o s f o r e v e r y c a s e

( s e e T a b l e 2 ) .

To compare t h e o v e r a l l t e n d e n c i e s o f t h e f i v e methods c o u n t f o r e a c h method t h e number o f t i m e s t h e s m a l l e r s t a t e i s f a v o r e d o v e r a l l 19 problems and d i v i d e by t h e number o f comparisons t o o b t a i n t h e b i a s r a t i o o v e r t h e c o u r s e o f U . S . C o n g r e s s i o n a l

h i s t o r y (see T a b l e 1 ) . H u n t i n g t o n ' s method o f e q u a l p r o p o r t i o n s , now i n u s e , h a s b i a s r a t i o .562 and d e c i d e d l y f a v o r s t h e s m a l l s t a t e s .

J.Q. Adams J. Dean E.V. Huntington D. Webster T. Jefferson

T a b l e 1 . B i a s r a t i o o v e r 1790-1970 U.S. Censuses

J . Q . Adms J. Dean E.V. Huntington D. Webster T. J e f f e r s o n

T a b l e 2 . B i a s r a t i o f o r e a c h U.S. Census p o p u l a t i o n

The more d e t a i l e d y e a r l y f i g u r e s o f T a b l e 2 show t h a t f o r some s p e c i f i c p r o b l e m s (1880 i s t h e o n e e x a m p l e ) H u n t i n g t o n ' s method i s l e s s b i a s e d t h a n W e b s t e r ' s , w h i l e f o r o t h e r s ( e . g . ,

1820, 1920) t h e r e v e r s e h o l d s . T h i s i s u n a v o i d a b l e . O v e r a l l t h e s t a t i s t i c s s u s t a i n t h e a n a l y s i s : t h e Webster method i s i n - d i c a t e d i f b i a s i s t o b e a v o i d e d .

Minimum R e q u i r e m e n t s

The U.S. C o n s t i t u t i o n r e q u i r e s t h a t e a c h s t a t e r e c e i v e a minimum o f 1 s e a t , F r a n c e a s s u r e s e a c h o f i t s d e p a r t e m e n t s a t l e a s t 2 s e a t s , t h e European P a r l i a m e n t h a s f i x e d minimum numbers of s e a t s a t t a c h e d t o e a c h o f t h e c o u n t r i e s a n d r a n g i n g between

6 a n d 36. None o f t h e above developments h a s e x p l i c i t l y a c c o u n t e d f o r a minimum r e q u i r e m e n t o t h e r t h a n z e r o . However, w i t h ap-

p r o p r i a t e m o d i f i c a t i o n s o f d e f i n i t i o n s , t h e t h e o r e m s c a n b e ex- t e n d e d and t h e f u n d a m e n t a l c o n c l u s i o n s a r e t h e same.

C o n c l u s i o n

Methods o f a p p o r t i o n m e n t m u s t b e a n a l y z e d by i d e n t i f y i n g t h e c r i t e r i a t h e y s a t i s f y ( o r d o n o t s a t i s f y ) and by o b s e r v i n g t h e i r b e h a v i o r when u s e d f o r a c t u a l problems.

The argument o f t h i s p a p e r may b e summarized a s f o l l o w s . P o p u l a t i o n m o n o t o n i c i t y f o r f i x e d s (=50) and h (=435) means

t h a t a d i v i s o r method must b e u s e d . A d j o i n i n g u n i f o r m i t y n a r r o w s t h e c h o i c e t o a r e g u l a r d i v i s o r method d e f i n e d i n d e p e n d e n t l y o f s and h , and g u a r a n t e e s h o u s e m o n o t o n i c i t y . The r e q u i r e m e n t i n a d d i t i o n t h a t a method n o t b e b i a s e d t o w a r d s s m a l l o r l a r g e s t a t e s l e a v e s b u t o n e method: t h a t o f Webster.

The m a j o r c a s u a l t y a p p e a r s t o b e t h e l a c k o f a g u a r a n t e e t h a t a p p o r t i o n m e n t s s a t i s f y q u o t a . I n s i s t i n g upon t h a t g u a r a n t e e would r u l e o u t a l l p o p u l a t i o n monotone methods and a l l u n i f o r m methods. T h a t i s t o o g r e a t a p r i c e . I n f a c t t h e method o f Webster d o e s " b e s t " among t h e r e g u l a r d i v i s o r methods i n s a t i s -

f y i n g q u o t a , and f o r t h r e e r e a s o n s .

F i r s t , a s w e h a v e s e e n , it s a t i s f i e s q u o t a f o r s = 3 , a n d i s t h e o n l y d i v i s o r method which d o e s . Second, w e s a y a method

M

-

i s r e l a t i v e l y w e l l r o u n d e d - - " a l m o s t " s a t i s f i e s q u o t a

--

i f f o r a E M t h e r e

- -

i s no p a i r o f s t a t e s w i t h a i < qi

-

1/2 and a j > q j

+

^1/2. The method of Webster i s c h a r a c t e r i z e d a s t h e u n i q u e d i v i s o r method which i s r e l a t i v e l y w e l l rounded [4]. T h i r d , e m p i r i c a l o b s e r v a t i o n makes c l e a r t h a t t h e e v e n t of a Webster a p p o r t i o n m e n t n o t s a t i s f y i n g q u o t a i s e x t r e m e l y u n l i k e l y . A Monte C a r l o e x p e r i m e n t c o n f i r m s t h i s : f o r s = 50, h = 435,

20,000 p o p u l a t i o n s w e r e c h o s e n u n i f o r m l y o v e r t h e s i m p l e x { p

-

; l p i = 1

,

435p. ¹ 2 ^- .5}. The method o f Webster v i o l a t e s q u o t a 37 t i m e s . T h i s e x t r a p o l a t e s t o l e s s t h a n one v i o l a t i o n o f q u o t a

i n 5000 y e a r s .

W e c o n c l u d e w i t h D a n i e l W e b s t e r , " . . . l e t t h e r u l e b e , t h a t t h e p o p u l a t i o n s h a l l b e d i v i d e d by a common d i v i s o r , a n d , i n a d d i t i o n t o t h e number o f members r e s u l t i n g from s u c h d i v i s i o n , a member s h a l l b e a l l o w e d t o e a c h s t a t e whose f r a c t i o n e x c e e d s a m o i e t y o f t h e d i v i s o r " ( [ I 71, p. 1 2 0 ) .

Acknowledgements

T h i s work was s u p p o r t e d by t h e N a t i o n a l S c i e n c e F o u n d a t i o n u n d e r G r a n t MPS 75-07414. W e a r e i n d e b t e d t o t h e I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d Systems A n a l y s i s f o r h a v i n g s u p p o r t e d o u r work on a p p o r t i o n r n e n t p r i o r t o 1978 and f o r h a v i n g made i t pos- s i b l e f o r u s t o c o n t i n u e t o work t o g e t h e r a t t h e same l o c a t i o n

i n 1978 and 1979.

References

[I] Adams, John Quincy, Letter to Daniel Webster, (Feb. 20, 1832) Microfilm Edition of the paper of Daniel Webster, Ann Arbor, Mich., U. Microfilms, Reel 8, Frames 0098862009894.

[2] Balinski, M.L. (1 978) , "How should Congress be apportioned?'", invited talk, ORSA National Meeting, New York,

1

May.

[3] Balinski, M.L. and H.P. Young (1 974), "A new method for Congressional apportio&ent," Proc.Nat.Acad.Sci.USA

71 4602-4606.

[4] Balinski, M.L. and H.P. Young (1977), "Apportionment schemes and the quota method," Am.Math.Month. 84 - 450-455.

[51 Balinski, M.L. and H.P. Young (1979), "Criteria for propor- tional representation," 0per.Res. - ^{27 80-95.}

[61 Balinski, M.L. and H.P. Young (1977), "On Huntington methods of apportionment," SIAM J.Appl.Math. C, - 33 607-618

[7] Balinski, M.L. and HOP. Young (1979), "Quotatone apportionment methods," Math.of O.R. - ^{4, 31-38.}

[8] Balinski, Y.L. and H.P. Young (1 978) , "Stability, coalitions and schisms in proportional representation systems,"

Amer.Pol.Sci.Rev. - 72 848-858.

[9] Balinski, M.L. and H.P. Young (1975), "The quota method of apportionment," Amer.Math.Month. - 82 701-730.

[lo] Bliss, G.A., E.W. Brown, L.P. ..Eisenhart and R. Pearl

(1

929), Report to the President of the National Academy of

Sciences, 9 February.

[I11 Hamilton, Alexander (1966), The Papers of Alexander Hamilton, Vol. XI (February 1792 - June 1792), Harold C. Syrett, editor, Columbia University Press, N.Y. 228-230.

[I21 Huntington, E.V. (1928), "The apportionment of Representatives in Congress," Amer.Math.Soc.Trans.

-

30 85-110.

[I 31 ~untington, E.V. (1 921

)

, "The mathematical theory of the apportionment of representatives," Proc.Nat.Acad.Sci.USA 7 123-127.

-

[I 41 Hylland, ~ a n u n d (1 97

8)

, "Allotment methods: procedures for proportional distribution of indivisible entities,"

mimeographed report.

[I 51 Jefferson, Thomas (1 904) , The Works of Thomas Jefferson,

Vol. VI, Paul Leicester Ford, editor, G.T. Putnam's Sons,

New York, N.Y. 460-471.

[I61 Morse, Marston, J. von Neumannand Luther Eisenhart (1948), Report to the President of the National Academy of Sciences, 28 May.

[I71 Webster, Daniel (1903), The Writinqs and Speeches of Daniel Webster, Vol. VI, National Edition, Little, Brown E Co., Boston, Mass. 102-123.

[I81 Willcox, W.F. (1916), "The apportionment of representatives,"

Amer.Econ.Rev. VI, - Supplement 1-16.