The Optimal Ranking Method in the Borda Count

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

THE OPTIMAL RANKING METHOD IS THE BOE2DA COUNT

Donald G. S a a r i

February 1985 CP-85-4

C o l l a b o r a t i v e P a p e r s r e p o r t work which has not been performed solely at t h e International Institute f o r Applied Systems Analysis and which h a s received only Limited review. Views o r opinions expressed herein do not necessarily r e p r e s e n t those of the Insti- tute, i t s National Member Organizations, o r o t h e r organizations supporting t h e work.

INmRNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

PREFACE

This i s a comprehensive investigation of the Borda count method, which i s shown to come out fairly well when compared with other ranking methods.

This research was carried out in collaboration with the System and Decision Sciences Program.

ALEXANDER

KURZHANSKI Chairman

System and Decision Sciences Program

THE OPTIMAL W i GMETHOD IS THE BORDA COUNT

C e n t r a l t o s o c i a l c h o i c e i s the development o f techniques t o aggregate

i n d i v i d u a l r a n k i n g s o f N a l t e r n a t i v e s i n t o a group r a n k i n g . Many approaches e x i s t , b u t i f N)3, none does what we r e a l l y what i t t o do. The d i f f i c u l t y i s t h a t a l t h o u g h

i t i s c m o n t o t r e a t t h e group's r a n k i n g as i f i t were t r a n s i t i v e , i t need n o t be.

I n t h i s paper, I'll analyze t h i s s o c i a l c h o i c e problem f o r v o t i n g methods t o show what can occur. ( T h i s i s where, f o r a g i v e n s e t o f w e i g h t s WI,

..,

^{wn, W J}

p o i n t s a r e t a l l i e d f o r a v o t e r ' s J T ~ p l a c e a l t e r n a t i v e . ) For instance, i t i s

s t a n d a r d t o c l a i m t h a t p l u r a l i t y v o t i n g i s among t h e worse methods t h a t can be used.

We s u p p o r t t h i s a s s e r t i o n by c h a r a c t e r i z i n g the i n c o n s i s t e n c i e s o f i t s e l e c t i o n r e s u l t s . ( I n a r e l a t e d paper C101, i t i s shown t h a t t h e proposed r e f o r m method o f 'approval v o t i n g ' 121 has f e a t u r e s even worse than p l u r a l i t y v o t i n g . ) Then, 1'11 propose a r e s o l u t i o n f o r t h i s s o c i a l c h o i c e problem by d e t e r m i n i n g what i s t h e 'best' v o t i n g method.

To see t h e problem, c o n s i d e r a h y p o t h e t i c a l s i t u a t i o n where n i n e people s e l e c t a camnon luncheon beverage. Four o f them have t h e r a n k i n g beer (b) over wine ( w i ) over w a t e r (wa) (b)wi)wa), t h r e e have t h e r a n k i n g wa>wi)b, and two have t h e r a n k i n g wi)b)wa. BY use o f t h e p l u r a l i t y v o t i n g scheme ( o n l y your f i r s t p l a c e a l t e r n a t i v e

i s t a l l i e d ) , t h e group's r a n k i n g i s b)wa)wi. I f t h i s r a n k i n g were t r a n s i t i v e , then, s h o u l d beer be u n a v a i l a b l e , water would be t h e group's second c h o i c e . But, 2/3 o f these people p r e f e r wine t o water; indeed, a m a j o r i t y of them p r e f e r wine t o b e e r ! The r a n k i n g s of t h e p a i r s can be r e v e r s e d when c o n s i d e r e d s e p a r a t e l y ; t h u s t h e outcanes of p l u r a l i t y v o t i n g need n o t be c o n s i s t e n t .

As i t i s w e l l known, Arrow's theorem 111 a s s e r t s f o r N)3 t h a t t h i s phenamenon o c c u r s f o r

anr

n o n - d i c t a t o r i a l a g g r e g a t i o n technique which s a t i s f i e s c e r t a i n

s t a n d a r d c o n d i t i o n s . I t always i s p o s s i b l e t o f i n d an example o f v o t e r s '

p r e f e r e n c e s where the group's r a n k i n g o f N)3 a l t e r n a t i v e s i s n o t c o n s i s t e n t w i t h how t h e same group, u s i n g t h e same ( o r any o t h e r s p e c i f i e d ) procedure, r a n k s same p a i r

of a l t e r n a t i v e s . Universal consistency o f the outcome i s an i m p o s s i b i l i t y .

Yet, d e c i s i o n s must be made, so i n d i v i d u a l rankings must be aggregated i n t o a group ranking. Consequently, even though a l l v o t i n g methods are flawed, we need t o determine the 'best' one. To do t h i s , the goal f o r the s e l e c t i o n o f an aggregation technique must be relaxed. Our u n r e a l i s t i c dream was t o f i n d a procedure which always y i e l d s a t r a n s i t i v e ordering; a more r e a l i s t i c o b j e c t i v e i s t o f i n d those v o t i n g techniques which minimize the damage t o consistency. We w i l l show t h a t

.For

v o t i n q methods, the Borda Count i s the uniaue answer. T h i s i s where n - j p o i n t s are t a l l i e d f o r a v o t e r ' s j T n place a l t e r n a t i v e . (A r e l a t e d issue a r i s e s f o r c e r t a i n r a n k i n g methods o f nonparametric s t a t i s t i c s . Again, the r e s o l u t i o n i s the Borda Count

.

The Borda Count i s optimal f o r several reasons; the f i r s t i s w i t h respect t o the rankings of p a i r s o f a1 t e r n a t i v e s . I f sane a1 t e r n a t i v e i s p r e f e r r e d t o a1 1 o t h e r s by m a j o r i t y votes (i.e., i t i s a Condorcet winner), then i t shouldn't be ranked l a s t i n the r a n k i n g o f the N a l t e r n a t i v e s . But, the i n t r o d u c t o r y example

i l l u s t r a t e s t h a t a Condorcet winner (wine) can be ranked l a s t by the p l u r a l i t y vote.

We show t h a t the Borda Count i s the uniaue method which never ranks a Condorcet winner i n l a s t lace. nor a Condorcet l o s e r i n f i r s t .

I f a v o t i n g method i s t o be judged s u p e r i o r , i t must be d e c i s i v e ; i t must a d n i t fewer r a n k i n g i n c o n s i s t e n c i e s than any other method. To i n v e s t i g a t e t h i s question, 1'11 introduce sane n a t u r a l measures o f the i n c o n s i s t e n c i e s p e r m i t t e d by a v o t i n g method. Again, i t w i l l t u r n out t h a t the Borda Count i s the unique, best s o l u t i o n .

To g a i n a f l a v o r o f the type o f measures which w i l l be used, consider the set o f f o u r a l t e r n a t i v e s ( a ~ , a t , a r , a + J . T h i s set has one subset of f o u r

a l t e r n a t i v e s , f o u r subsets o f three a l t e r n a t i v e s , and s i x subsets o f two

a l t e r n a t i v e s . For each subset, s p e c i f y a v o t i n g method; t h a t i s , s p e c i f y the number o f p o i n t s which are t o be t a l l i e d f o r a v o t e r ' s j T H ranked a l t e r n a t i v e where j

ranges over the number o f a1 t e r n a t ivcs. Let

g

denote the c o l l e c t ion o f these eleven

b a l l o t i n g methods. Then, given W_ and the uoters' p r o f i l e s , the group's o r d i n a l r a n k i n g s f o r each o f the eleven subsets i s determined. As the v o t e r s ' p r o f i l e s v a r y over a l l p o s s i b l e choices, we o b t a i n the s e t , Ru, o f a l l p o s s i b l e o r d i n a l r a n k i n g s o b t a i n e d f r o m

W.

So, an element o f Ru i s a l i s t i n g o f the eleven o r d i n a l r a n k i n g s r e s u l t i n g fran sane p r o f i l e o f voters. L e t

FB

when a l l o f the subsets are ranked by a Borda v e c t o r . So, RB i s a 1 i s t i n g o f a11 p o s s i b l e Borda rankings.

C l e a r l y , Ru c o n t a i n s a l l t r a n s i t i v e o r d e r i n g s ; such an outcane r e s u l t s when a l l of the v o t e r s have an i d e n t i c a l r a n k i n g o f the f o u r a l t e r n a t i v e s . So, i f lRul

i s the c a r d i n a l i t y o f Ru, then IRul-4! i s the number o f p o s s i b l e r e s u l t s which are n o t t r a n s i t i v e . (For N a l t e r n a t i v e s , I R u l - N ! i s the number o f n o n - t r a n s i t i v e outcanes.) I n t h i s way, lRul i s a measure o f the i n c o n s i s t e n c i e s a d m i t t e d by a s e t o f v o t i n g methods. We show t h a t

the

unique. minimum value f o r l R u l occur2 o n l y i f the v o t i n q methods f o r a l l subs_ets o f a l t e r n a t i v e s # r e Borda Counts, i.e.,

I R u l l l R e l f o r a l l choices o f

W.

One d i f f i c u l t y w i t h lRul i s t h a t i t doesn't i n d i c a t e what are the

i n c o n s i s t e n t rankings. For instance, i t doesn't e l i m i n a t e the p o s s i b i l i t y t h a t t h e r e a r e Borda r a n k i n g s which, i n sane sense, ' v i o l a t e t r a n s i t i v i t y ' more so than any r a n k i n g i n t r o d u c e d by sane other v o t i n g system. T h i s can't happen because Re

i s c o n t a i n e d i n Ru f o r a11 choices o f

Y.

&y n o n t r a n s i t i v e r a n k i n a a d m i t t e d

& 1

Borda Count a l s o i s admitted by any other s e t o f u o t i n q methods.

-

A consequence o f the above i s t h a t the Borda Count admits o n l y those

i n c o n s i s t e n c i e s which are unavoidable. Thus, we need t o k n w what they are; we need t o c h a r a c t e r i z e Re. I t t u r n s out t o be i n e f f i c i e n t t o catalogue t h i s s e t ; so, we

i n t r o d u c e some simple methods which p e r m i t s one t o e a s i l y answer questions about Re. To i l l u s t r a t e the types of r e s u l t s which now are possible, we d e r i v e

necessary and s u f f i c i e n t c o n d i t i o n s f o r an a l t e r n a t i v e t o be Borda ranked f i r s t , l a s t , e t c . I n keeping w i t h a dominant theme o f s o c i a l choice, these c o n d i t i o n s are based upon how t h e v o t e r s ranked the p a i r s o f a l t e r n a t i v e s .

2 . V o t i n g M e t h o d s

L e t t h e N13 a l t e r n a t i v e s be < a l , a r , . ..,an). Assume t h a t each v o t e r has an o r d i n a l , complete (a11 a l t e r n a t i v e s are i n c l u d e d i n the r a n k i n g ) , t r a n s i t i v e r a n k i n g o f t h e N a l t e r n a t i v e s . A l i s t i n g o f the r a n k i n g s f o r t h e v o t e r s i s c a l l e d ' a ' p r o f i l e ' , A b a l l o t i n g o r a v o t i n g method i s where t h e group r a n k i n g i s determined from v o t e r s ' p r o f i l e s i n t h e f o l l o w i n g way: Given p = ( w ~ , w r , . . , w ~ ) , W J

p o i n t s a r e t a l l i e d f o r a v o t e r ' s j t n p l a c e a l t e r n a t i v e . Then, the s e t o f

a l t e r n a t i v e s a r e o r d e r e d a c c o r d i n g t o the sum o f p o i n t s each a l t e r n a t i v e r e c e i v e s . T h i s f i n a l o r d e r i n g can be determined e i t h e r by a s s e r t i n g t h a t t h e s m a l l e r the

t o t a l , t h e h i g h e r the r a n k i n g ( a r e v e r s e d method), o r t h e l a r g e r the t o t a l , t h e h i g h e r t h e r a n k i n g ( a monotone method). I n t h e l a t t e r case, t h e w e i g h t s s a t i s f y t h e c o n d i t i o n s t h a t wK2wr i f and o n l y i f k < j , and t h a t wl)wn. F o r a r e v e r s e d

method, these i n e q u a l i t i e s a r e reversed. For example, a p l u r a l i t y v o t e i s a monotone method w i t h p = ( l , O ,

...,

^0). For s i m p l i c i t y o f e x p o s i t i o n , assume t h a t

the W K ' S a r e a1 1 r a t i o n a l numbers. ( T h i s doesn't impose any p r a c t i c a l

l i m i t a t i o n s . The o n l y t h e o r e t i c a l l i m i t a t i o n occurs s h o u l d < W J ) be a s e t o f

c a n p l e t e l y i r r a t i o n a l numbers; here, c e r t a i n statements a s s e r t i n g t h e p o s s i b i l i t y o f e l e c t i o n r e s u l t s w i t h i n d i f f e r e n c e among a l t e r n a t i v e s may n o t h o l d . See t7,81 f o r an e x p l a n a t i o n o f t h i s . )

A c c o r d i n g t o t h e above, v o t i n g methods d i f f e r by the c h o i c e o f the v o t i n g v e c t o r s used i n t h e t a l l y i n g process. However, two methods may be e q u i v a l e n t because t h e y always y i e l d t h e same group r a n k i n g . For i n s t a n c e , i t i s c l e a r t h a t t h e outcane of an e l e c t i o n i s t h e same whether the v o t e r s ' r a n k i n g s are t a l l i e d w i t h W_n o r w i t h aW_n where a i s a nonzero constant. ( I f a i s n e g a t i v e , then one

system i s a monotone method w h i l e the o t h e r one i s a r e v e r s e d method.) L i k e w i s e ,

the outcome remains i n v a r i a n t should the preferences be t a l l i e d by u s i n g WN+~EN.

Here b i s a nonzero scalar and the complete i n d i f f e r e n c e v e c t o r EN i s N 1 , . . , 1 . Consequently,

D e f i n i t i o n 1. Two u o t i n g u e c t o r s ~ N and I HM2 are r a i d t o be e q u i v a l e n t i f they and the u e c t o r d e f i n e a two dimensional l i n e a r subspace o f RM.

T h i s d e f i n e s an equiualence r e l a t i o n and the equiualence classes o f u o t i n g v e c t o r s and methods. I n what f o l l m u s , we e x p l o i t t h i s equiualence by n o r m a l i z i n g

the u o t i n g v e c t o r s . As a f i r s t n o r m a l i z a t i o n , we consider o n l y monotone u o t i n g methods. The f o l l o w i n g c h a r a c t e r i z e s an important equivalence c l a s s .

D e f i n i t i o n 2. A Borda Count over N)3 a l t e r n a t i u e s i s where the vote t a l l y v e c t o r ,

WN, has the p r o p e r t y t h a t wgwg+l i s the sure nonzero constant f o r

-

K=1,2,..,N-I. Denote both a Borda u e c t o r and the equiualence c l a s s o f Borda u e c t o r s

bY

p.

Vector y1=(1,2,..,N) i s a reversed Borda v e c t o r , w h i l e

92=(2N-2,2N-4,...,2,01 i s a monotone Borda v e c t o r . They b o t h belong t o the same equivalence c l a s s because ~(N~EN-V_I)*Z.

The NL3 a l t e r n a t i v e s d e f i n e a f a m i l y o f 2N-(N+1) subsets, each o f which has a t l e a s t t u o a l t e r n a t i v e s . For each subset, s e l e c t a v o t i n g v e c t o r . (For the subsets of two a l t e r n a t i v e s , we can assume t h a t the u o t i n g methods are the same.) L e t

g

denote these 2N-(N+1) v o t i n g vectors. Should these u e c t o r s a11 be Borda, denote t h e combined u e c t o r by

B.

A g i v e n and a choice o f uoters' p r o f i l e s u n i q u e l y determines the o r d i n a l r a n k i n g s f o r the 2n-(N+1) subsets. L e t Ru be the s e t obtained by v a r y i n g the voters' p r o f i l e s over a l l p o s s i b l e choices. One o f the main r e s u l t o f t h i s paper, which e s t a b l i s h e s the s u p e r i o r i t y o f the Borda Count, i s given i n the f o l l m u i n g

theorem. ( I n t h i s and several other statements, we a s s e r t t h a t c e r t a i n conclusions h o l d f o r 'most' u o t i n g systems. 'flost' means "almost a l l ' i n a measure t h e o r e t i c

sense, o r 'open-dense' i n a t o p o l o g i c a l sense. More p r e c i s e l y , i t wi 1 1 mean a1 1 v e c t o r s

W

except those where the vector components s a t i s f y a s p e c i f i e d , s t r i c t , a l g e b r a i c r e l a t i o n s h i p among each o t h e r

-

see Section 5. I t i s o f importance t o note t h a t i f

g

c o n s i s t s s o l e l y of p l u r a l i t y v o t i n a methods. then i t

&

i n t h i s general c 1 ass.

Theorem 1.

l e a s t two a1 t o rank the

L e t N13. Consider the f a m i l y of a l l P - ( N + l ) d i f f e r e n t subsets o f a t t e r n a t i ues, and 1 e t r e p r e s e n t the c o l l e c t ion o f v o t i n ~ v e c t o r s rdop t e d d i f f e r e n t subsets. Then

I f #

g,

then the f i r s t s e t s t r i c t l y c o n t a i n s the second, Moreover, f o r most c h o i c e s o f Id, Ru c o n t a i n s a11 p o s s i b l e rankings.

T h i s means t h a t any v o t i n g method o t h e r than the Borda Count admits more n o n t r a n s i t i ve rank i ngs than those o b t a i n e d b y the Borda Count. Furthermore, i t f o l l o w s f r a t h e s t r i c t containment t h a t the Borda Count i s the unique b a l l o t i n g r e s o l u t i o n f o r t h i s s o c i a l choice problem. The f o l l o w i n g statement extends t h i s c o n c l u s i o n t o s u b f a m i l i e s o f subsets o f a l t e r n a t i v e s .

r o l l a r y 1 . 1 Out o f the 2N-(N+l) subsets of a t l e a s t two a l t e r n a t i v e s , s e l e c t a fami 1 Y o f K of t helm. L e t

Y

be the c o l 1 e c t i o n o f the v o t i ng v e c t o r s used t o rank t h i s f a m i l y o f subsets, and l e t R'u be a11 p o s s i b l e e l e c t i o n outcanes from t h i s f a m i l y . Then R'u c o n t a i n s R'e.

We do n o t a s s e r t t h a t the f i r s t set s t r i c t l y c o n t a i n s the second one because t h e r e a r e fami 1 i e s o f subsets where R'u=R'e independent o f the choice o f

u.

Often t h i s i s c h a r a c t e r i z e d by R'g c o n t a i n i n g a l l p o s s i b l e r a n k i n g s o f the subsets o f a l t e r n a t i v e s .

A second important f e a t u r e o f Theorem 1 i s t h a t , f o r most choices of

Y,

Ru c o n t a i n s e v e r y t h i n g ! Since any type o f inconsistency can occur, these are the worse systems which can be used. As asserted, p l u r a l i t y v o t i n g i s i n t h i s general c l a s s . The f o l l o w i n g c o r o l l a r y f o l l o w s immediately.

Cor o 1 f ash i t h a t

l a 1.2. For each o f the 2n-(N+l) subsets, s e l e c t , i n an a r b i t r a r y

on, sane r a n k i n g o f t h e a l t e r n a t i v e s . Then, t h e r e e x i s t p r o f i l e s o f v o t e r s s o t h e i r r a n k i n g o f each subset i s t h e s p e c i f i e d r a n k i n g .

Example: There e x i s t p r o f i l e s o f v o t e r s so t h a t t h e i r plur;!

!.

^{:I- kings}

change w i t h t h e number o f a l t e r n a t i v e s ; e.g., t h e group's p l u r a l i t y r a n k i n g s a r e a1 )as)aa)aq, b u t a a > a s ) a ~

,

a q > a s ) a ~

,

a q ) a a ) a ~

,

^and

aq)aa)a2, b u t a ~ ) a a i f f J<k.

Suppose f o r a v o t i n g v e c t o r Hn t h e r e i s o n l y one choice o f J so t h a t W J - w ~ + 1 # 0 . Such a v o t i n g system can d i s t i n g u i s h between o n l y two s e t s of

a l t e r n a t i v e s ; e.g., j-1 c h a r a c t e r i z e s t h e p l u r a l i t y v o t i n g v e c t o r which

d i s t i n g u i s h e s o n l y between t h e t o p ranked a1 t e r n a t i v e and a1 1 o t h e r s . A common1 y used method f o r committee s e l e c t i o n i s t o i n d i c a t e your 'top k ranked a l t e r n a t i u e s ' . Again, t h i s system d i s t i n g u i s h e s between o n l y two subsets. I t t u r n s o u t t h a t i f a l l

the v o t i n g canponents o f

W

d i s t i n g u i s h o n l y between two subsets o f a l t e r n a t i v e s , then W_ i s i n t h e general c l a s s where any outcome can occur. Thus, t h e above c o r o l l a r y h o l d s f o r a l l of these systems.

Theorem 1 a s s e r t s t h a t f o r most systems t h e r e need n o t be any r e l a t i o n s h i p whatsoever between how t h e v o t e r s rank t h e v a r i o u s subsets of a l t e r n a t i v e s . But, f o r o t h e r systems, what type o f r e l a t i o n s h i p s can be e x t r a c t e d ? We s t a r t our answer o f t h i s q u e s t i o n by examining the p o s s i b l e r a n k i n g s o f p a i r s o f a l t e r n a t i u e s . The m o t i v a t i o n i s t h a t i f t h e group's outcome were t r a n s i t i v e , then the r a n k i n g o f N

a l t e r n a t i v e s would u n i q u e l y e s t a b l i s h the group's r a n k i n g s o f the p a i r s o f a l t e r n a t i v e s . For i n s t a n c e , i f t h e group's r a n k i n g i s a~)az)...)an, then, t o p r e s e r v e t r a n s i t i v i t y , a m a j o r i t y of t h e v o t e r s would p r e f e r a ~ t o aJ i f and o n l y i f k < j .

I t has been r e c o g n i z e d f o r a l o n g time t h a t t r a n s i t i v i t y among the r a n k i n g s o f p a i r s need n o t e x i s t ; c y c l e s can occur. One of t h e o l d e s t examples, known as the

Condorcet t r i p l e t , i s where t h e p r o f i l e s o f t h r e e v o t e r s a r e al)az)aa, az)a3)a1, and a 3 ) a l ) a z . A s i m p l e computation shows t h a t , by v o t e s o f 2 t o 1, a l l a z , a z l a a , b u t a r ) a l . The i n s i d i o u s e f f e c t s of such c y c l e s

have been i l l u s t r a t e d by t h e p r a c t i c a l c o n s i d e r a t i o n s o f agenda m a n i p u l a t i o n , t h e e f f e c t s o f 'seeding' on t h e c o n c l u s i o n s o f tournaments, e t c . ( W h i l e t h e r e i s a v e r y

l a r g e l i t e r a t u r e on t h i s s u b j e c t , I suggest t h e r e f e r e n c e s [4,5,91.) The f o l l o w i n g theorem a s s e r t s t h a t t h e r e need n o t be any r e l a t i o n s h i p whatsoever among t h e

r a n k i n g s o f p a i r s o f a l t e r n a t i v e s ;

rnr

outcome i s p o s s i b l e . ( T h i s r e s u l t i s a s l i g h t g e n e r a l i z a t i o n o f t h a t g i v e n i n 193.1

Theorem 2. Consider the (N;2)=N(N-1)/2 p a i r s of a l t e r n a t i v e s Car,ar>. For each p a i r

of

a l t e r n a t i v e s , t h e r e a r e t h r e e d i f f e r e n t r a n k i n g s r a r ) a r , a r < a r , o r ar-a* ( ' a ~ i s i n d i f f e r e n t t o a*'). T h i s d e f i n e s a s e t o f 3(Nf23

sequences; each

of

the (N;2) e n t r i e s o f a sequence d e s i g n a t e s the r a n k i n g f o r a s p e c i f i c p a i r

of

a l t e r n a t i v e s . Any such sequence can r e a l i z e d ; t h e r e e x i s t v o t e r s ' p r o f i l e s w c h t h a t f o r each p a i r o f a l t e r n a t i v e s t h e d e s i g n a t e d c h o i c e r e s u l t s b y a m J o r i t y v o t e f o r s t r i c t p r e f e r e n c e and a t i e v o t e f o r i n d i f f e r e n c e . That is, R'u can be e q u a t e d w i t h t h e s e t of a1 1 p o s s i b l e sequences o f r a n k i n g s o f p a i r s o f a1 t e r n a t i u e s .

F o r N-3, t h e Condorcet c y c l e i s an example o f t h i s theorem. For N=4, t h i s theorem means, f o r instance, t h a t a ~ ) a z , a z ) a r , a r ) a ~ , a + > a ~ ,

a2>a+, a l l a r e r e a l i z e d by m a j o r i t y v o t e s from t h e same s e t o f v o t e r s p r o f i l e s and t h e s e same v o t e r s a r e e v e n l y s p l i t between a r and a+. I n g e n e r a l , a11

p o s s i b l e c y c l e s , subcycles, o r a n y t h i n g e l s e can be c o n s t r u c t e d b y means o f m a j o r i t y v o t e .

T h i s r e s u l t imposes lower bounds on t h e c o n s i s t e n c y o f v o t i n g independent the c h o i c e o f t h e v o t i n a method used t o r a n k N a l t e r n a t i u e s . T h i s can be seen w i t h

-

a p r o f i l e o f v o t e r s f o r t h e above example. Independent o f how these v o t e r s r a n k t h e s e t o f f o u r a l t e r n a t i u e s , the outcome must be i n c o n s i s t e n t w i t h how t h e same v o t e r s r a n k a t l e a s t two p a i r s of a1 t e r n a t i v e s . ( T h i s i s because the p a i r w i s e r a n k i n g s o f the two s u b s e t s (a1 ,az,asl and (a1 ,az,a+l form cycles.) T h i s

i l l u s t r a t e s Arrow's theorem f o r v o t i n g methods, and i t imposes a lower bound on the degree o f consistency which can be achieved through. voting.

Because t h e r e need not be any r e l a t i o n s h i p among the rankings o f the p a i r s o f a l t e r n a t i v e s , i t might be argued t h a t the search f o r consistency should be

r e s t r i c t e d t o those p r o f i l e s where there i s order among the ranKings o f the p a i r s . The goal, then, would be t o determine whether t h i s r e l a t i o n s h i p i s r e f l e c t e d i n t h e r a n k i n g o f the N a l t e r n a t i v e s . For instance, c y c l e s need not always occur; t h e r e a r e s i t u a t i o n s where, by m a j o r i t y votes, c e r t a i n a l t e r n a t i v e s m e r g e as c l e a r f a v o r i t e s , or as c l e a r l o s e r s . Such a l t e r n a t i v e s , which were i d e n t i f i e d by Condorcet, o f t e n are used as the standard f o r comparison f o r the consistency of a v o t i n g method.

D e f i n i t i o n 3. A l t e r n a t i v e ag i s c a l l e d a Condorcet winner i f i n a11 p o s s i b l e p a i r w i s e comparisons u i t h the other a l t e r n a t i v e s , ag always u i n s by a m a j o r i t y vote. A l t e r n a t i v e a r i s c a l l e d a Condorcet l o s e r i f i n a11 p o s s i b l e p a i r w i s e comparisons u i t h the o t h e r a l t e r n a t i v e s , ag always l o s e s by a m a j o r i t y vote.

For consistency, a v o t i n g method should rank a Condorcet winner i n f i r s t place and a Condorcet l o s e r i n l a s t place. But, t h i s need not be the case; the

i n t r o d u c t o r y example demonstrates, and i t f o l l o w s i n general from C o r o l l a r y 1.2, t h a t p l u r a l i t y v o t i n g can rank the Condorcet winner i n l a s t place and the Condorcet l o s e r i n f i r s t place. The next theorem a s s e r t s t h a t , w i t h the exception o f the Borda Count, t h i s and much worse phenomena can occur f o r v o t i n g method. On1 y the Borda Count always r e f l e c t s the rankings o f the p a i r s o f a l t e r n a t i v e s .

Theorem 3. L e t N)3 a l t e r n a t i v e s k given, and l e t WN be a v o t i n g method t o rank the N a l t e r n a t i v e s . Consider the r e l a t i o n s h i p between the rankings of the N a l t e r n a t i v e s and the (N;2) p a i r s o f a l t e r n a t i v e s . I f p # p , then R'u

c o n t a i n s a1 1 p o s s i b l e cambinat ions

of

the rankings o f p a i r s o f a1 t e r n a t i u e s and the ranking. o f the N a l t e r n a t i v e s . The Borda uector, BN, never ranks a Condorcet winner i n l a s t place, nor a Condorcet l o s e r i n f i r s t place. There i s no v o t i n g

system which always ranks the Condorcet winner i n f i r s t place and the Condorcet l o s e r i n l a s t place.

Thus, w i t h the exception of the Borda Count, there need not be any r e l a t i o n s h i p

whatsoever between the r a n k i n g s o f p a i r s of a l t e r n a t i v e s and the r a n k i n g s o f the N a l t e r n a t i v e s . I n o t h e r words, even when we r e s t r i c t a t t e n t i o n t o those p r o f i l e s where t h e r e i s o r d e r i n the r a n k i n g s o f p a i r s , we don't f i n d added c o n s i s t e n c ~ w i t h

t h e r a n k i n g o f the N a l t e r n a t i v e s . Indeed, the f o l l o w i n g statement d i s p l a y s an extreme s i t u a t i o n where t h e p a i r s do possess o r d e r , b u t i t i s a t odds w i t h the r a n k i n g o f t h e N a l t e r n a t i u e s .

C o r o l l a r y 3.1. Suppose t h a t

N13

and t h a t t h e adopted v o t i n g v e c t o r ,

p ,

i s n o t a Borck v e c t o r . Then t h e r e e x i s t p r o f i l e s o f v o t e r s

so

t h a t , b y r l j o r i t y votes, t h e p a i r s o f a l t e r n a t i v e s a r e r a n k e d a ~ > a r i f and o n l y i f j < k . Yet, t h e i r

Wn

r a n k i n g o f t h e N a l t e r n a t i v e s i s t h e r e v e r s a l : an)an-~>...)a~.

3. A V e c t o r Space A p p r o a c h

A l t h o u g h t h e above statements demonstrate t h e s u p e r i o r i t y of the Borda Count over o t h e r v o t i n g methods, they do n o t adequately d e s c r i b e Re n o r Ru. To remedy

t h i s , we need a more complete d e s c r i p t i o n o f v o t i n g systems. We s t a r t by r e l a t i n g c a r d i n a l r a n k i n g s w i t h o r d i n a l r a n k i n g s .

I n t h e N dimensional space RM, i d e n t i f y the kTn component xu w i t h t h e kTn a l t e r n a t i v e aa. A v e c t o r ~ ( x I , . . , x M ) can be i n t e r p r e t e d as a c a r d i n a l r a n k i n g o f t h e N a l t e r n a t i u e s where l a r g e r v a l u e s o f xu denote ' s t r o n g e r ' p r e f e r e n c e f o r au. The hyperplane xa=xr d i v i d e s R M i n t o t h r e e r e g i o n s ; t h e

two open r e g i o n s denote s t r i c t o r d i n a l p r e f e r e n c e (e.g., xa)xr corresponds t o where a r i s p r e f e r r e d t o o r ) , and the hyperplane corresponds t o i n d i f f e r e n c e

between t h e two a1 t e r n a t iues. By a1 l o w i n g k and j t o v a r y over a l l p o s s i b l e (N;2)=N(N-1)/2 p a i r s of i n d i c e s , t h e (N;2) hyperplanes d i v i d e RM i n t o r e g i o n s

r e p r e s e n t i n g a l l p o s s i b l e o r d i n a l r a n k i n g s o f t h e N a l t e r n a t i v e s . A connected open s e t i s a ' r a n k i n g r e g i o n s ' w i t h s t r i c t preferences among a l t e r n a t i u e s ; those r e g i o n s c o n t a i n e d i n the hyperplanes are r a n k i n g r e g i o n s w i t h i n d i f f e r e n c e among o r between

sane o f the a l t e r n a t i u e s . The l i n e passing through

1

and

EN

corresponds t o

complete i n d i f f e r e n c e among the a l t e r n a t iues; t h i s 1 ine i s the i n t e r s e c t ion o f the (N;2) ' i n d i f f e r e n c e ' hyperplanes.

L e t A denote the r a n k i n g a ~ ) a ~ ) . . ) a ~ . I f WN i s a v o t i n g method, then, because i t i s a monotone method, i t i s i n the c l o s u r e o f the r a n k i n g r e g i o n of A.

( I f any two o f the canponents o f WN are the same, t h i s uector i s on the boundary;

otherwise, i t i s i n the i n t e r i o r . ) Vector W_n r e p r e s e n t s the t a l l y f o r a uoter w i t h preference giuen by A. Denote t h i s dependency by

pa.

Any other r a n k i n g o f

the N a1 t e r n a t i u e s i s a permutation o f A, P(A). The t a l l y f o r the r a n k i n g o f such a u o t e r i s a permutation o f WMa; denote i t by WNpta). (bJNpta) i s i n the c l o s u r e

of the r a n k i n g r e g i o n defined by P(AI.1 I f there are n'pta) v o t e r s w i t h the r a n k i n g P(A), then the f i n a l t a l l y i s

3.1 n'r (a )U"? (a

where the summation index, P(A), ranges ouer N! permutations o f A. The group r a n k i n g i s determined by the r a n k i n g r e g i o n which c o n t a i n s t h i s sum.

The r a n k i n g i s i n u a r i a n t should the sum be d i v i d e d by n, the t o t a l number o f uoters. I f nrca)=n'rca)/n i s the f r a c t i o n o f the v o t e r s w i t h r a n k i n g P(A), then the rum becomes

3.2

f

n ~ t n ) W _ n r c n ) ~

Because the u a r i a b l e s CNp(a)l are non-negatiue and sun t o u n i t y , Eq. 3.2 can be i n t e r p r e t e d as r e p r e s e n t i n g a convex c a n b i n a t i o n o f the uectors C p r t a ) l . T h i s s e t i s i n the a f f i n e plane c o n t a i n i n g these uectors. Our a n a l y s i s i s s i m p l i f i e d when t h i s plane i s a l i n e a r subspace o f RN. T h i s m o t i u a t e s the f o l l o u i n g .

Vector N o r m a l i z a t i o n Assumption: The rum of the camponentr of a v o t i n g uector equals zero.

Examples: a) The standard uector f o r p l u r a l i t y v o t i n g ouer N a l t e r n a t i u e s i s l , O , . , O . A normalized uector i s (N-1,-1,-I,..,-1).

b) F o r k 2 , we always use (1,-1).

T h i s assumption f o r c e s t h e v o t i n g v e c t o r s and the sum i n Eq. 3.2 t o be i n t h e l i n e a r subspace o f RN which i s o r t h o g o n a l t o

EN.

Denote t h i s N-1 dimensional subspace b y EN. For b 3 , t h e r a n k i n g r e g i o n s o f E3 a r e g i v e n i n F i g u r e 1.

F o r k=2,...,N-1, c o n s i d e r a subset o f k a l t e r n a t i v e s , and l e t WK be t h e

v o t i n g method adopted t o r a n k t h i s subset o f a l t e r n a t i v e s . For any v o t e r ' s r a n k i n g o f t h i s s u b s e t , t h e t a l l y o f t h e b a l l o t i s g i v e n b y the a p p r o p r i a t e p e r m u t a t i o n o f

the components o f WK. However, t h i s permutat i o n o f - W K - a l s o can -be indexed b y

how t h i s v o t e r r a n k s a l l N a l t e r n a t i v e s , n o t j u s t t h i s r e l e v a n t subset. So, f o r any p e r m u t a t i o n o f A, l e t WKrtn) be the unique p e r m u t a t i o n o f W K w h i c h corresponds

t o how

the

s p e c i f i e d k a l t e r n a t i v e s a r e r a n k e d i n P(A). For i n s t a n c e , suppose W4, k=3, and t h e s p e c i f i e d r a n k i n g i s al)an)a3. There a r e f o u r c h o i c e s o f P(A)

w h i c h p r e s e r v e t h i s r a n k i n g

--

t h e y a r e t h e f o u r ways i n which a4 can be p o s i t i o n e d w i t h i n t h i s r a n k i n g o f t h r e e a l t e r n a t i u e s . Thus, f o r e x a c t l y f o u r d i f f e r e n t c h o i c e s of the s u b s c r i p t P(A), t h e v e c t o r s bJ3rtn) agree and r e p r e s e n t t h e v o t e t a l l y f o r t h e sane r a n k i n g o f t h e t h r e e a1 t e r n a t i v e s .

L e t (N;k) r e p r e s e n t t h e usual c o m b i n a t o r i c s m b o l

(f)

Each r a n k i n g o f k

a l t e r n a t i v e s i s p r e s e r v e d i n p r e c i s e l y (N;N-k)=(N;k) d i f f e r e n t p e r m u t a t i o n s o f P(A), s o t h e v e c t o r WKrtn) i s g i v e n b y (N;k) d i f f e r e n t s u b s c r i p t s P(4). The group's

r a n k i n g of these a l t e r n a t i v e s i s g i v e n by t h e r a n k i n g r e g i o n o f EK which c o n t a i n s t h ~ v e c t o r sum

3.3

I:

^{n ~ ( A}

) W K t

^{( A ) a}

T o model how t h e same v o t e r s would r a n k the N a l t e r n a t i v e s w i t h t h e method WN

and a subset o f k a l t e r n a t i v e s w i t h t h e method ^{W K ,} we use t h e space ENxEK.

The r a n k i n g r e g i o n s i n t h i s p r o d u c t space a r e g i v e n by t h e p r o d u c t o f r a n k i n g r e g i o n s i n t h e canponent spaces. The outcome i s the r a n k i n g r e g i o n which c o n t a i n s

t h e v e c t o r sum

3.4 ~ n r c n ) ( W N r ~ a ) , W K r c a ) ) . I f E ( W _ N , W K ) , t h i s can be represented by the sum

3.5 nrca)Wrca)

T h i s equation has an i n t e r p r e t a t i o n s i m i l a r t o t h a t o f Eq. 3.2, and the sum i s i n the convex h u l l o f the v e c t o r s { W p t a ) ) . To understand what n o n t r a n s i t i v e outcanes can r e s u l t , we need t o k n a which r a n k i n g r e g i o n s meet t h i s convex set.

A unique l i n e a r subspace i s spanned by the convex set d e f i n e d i n Eq 3.5. What s i m p l i f i e s our a n a l y s i s i s t h a t both the l i n e a r subspace and the convex h u l l meet the same r a n k i n g r e g i o n s o f EnxEK. ( T h i s w i l l be sham i n Section 5 . )

Therefore, the task of determining the elements o f R'u i s equivalent t o

d e t e r m i n i n g which r a n k i n g r e g i o n s o f ENxEK meet the l i n e a r subspace spanned by the v e c t o r s C W r t a ~ )

=

{(Wnrta),WKcta))l. Denote t h i s subspace by V U .

Moreover, the dimension o f the convex set and V U are the same, so t h i s carmon dimension serves as another measure of the number o f n o n t r a n s i t i v e group r a n k i n g s which can occur.

To i l l u s t r a t e t h i s , Theorem 2 w i l l be expressed i n terms o f the vector space V u . For t h i s , and f u t u r e statements, we impose the f o l l o w i n g o r d e r i n g on the

l i s t i n g of the (N;2) p a i r s o f a l t e r n a t i v e s : A given p a i r (ar,ar) i s l i s t e d w i t h index j < k . The p a i r s are l i s t e d i n the order k=j+I,..,N,j=l,..,N-1; i.e.,

a , a ,

. . ^,

^{a a a a}

^,

^{a - a} Each p a i r o f

a l t e r n a t i v e s ( a r , a r ) , j < k , i s represented by a space Ez where, because o f the ordering, the vector (1,-1) i n d i c a t e s t h a t ar i s p r e f e r r e d t o a r . Thus, the space o f a l l p a i r s i s represented by ( E z ) ( n ) z ) , and the above imposes an o r d e r i n g on t h i s space.

Theorem 4. Consider a l l (N;2) p a i r s

of

a l t e r n a t i u e s , and l e t

k

be the v e c t o r

of

v o t i n ~ methods. Then UP i s the t o t a l space (Ez)tN;z), and i t has dimension

<N;2).

Since Vp agrees w i t h the space ( E z ) ( N f z ' , i t meets each o f the r a n k i n g

r e g i o n s . Thus, Theorem 2 f o l l o w s . The importance o f the dimension o f Vp i s t h a t i t imposes a lower bound on the dimension of Vu when r e s u l t s are compared over a l l 2N-(N+1) subsets o f a l t e r n a t i v e s . T h i s i s because when Vu i s computed, i t must r e f l e c t t h i s freedom from consistency among the r a n k i n g s o f p a i r s . Thus, a lower bound t h e dimension qf Vu fpy

a

aeneral problem i s (N;2).

When d i f f e r e n t s e t s o f a l t e r n a t i v e s are ranked, the subspace Vu can vary

depending on the s c a l a r n o r m a l i z a t i o n s adopted f o r the v o t i n g components o f W_. For instance, t h e space spanned by the permutations o f the v o t i n g v e c t o r s

(3,1,-1,-3;1,0,-1) and (3,1,-1,-3;5,0,-5) d i f f e r even though i n b o t h cases the s e t o f 4 and the s e t o f 3 a l t e r n a t i v e s are ranked by Borda Counts. (As we have shown, the r a n k i n g r e g i o n s which meet these two subspaces are the same.) So, t o compare v e c t o r spaces, we need t o impose a s c a l a r n o r m a l i z a t i o n . Because other v o t i n g v e c t o r s w i l l be compared w i t h the Borda Count, the o n l y standards we impose are f o r

*2 and f o r the Borda Count; the n o r m a l i z a t i o n f o r the other v e c t o r s w i l l be determined as needed.

S c a l a r N o r m a l i z a t i o n Assumption: For N)3 a l t e r n a t i v e s , the normalized Borda v e c t o r i s N - l . , N + l - 2 i , l The v o t i n g v e c t o r used f o r N=2 i s (1,-1).

Example: For b 4 , the Borda v e c t o r i s (3,1,-1,-3).

BY t a k i n g a vector approach and by s t a n d a r i z i n g the Borda Count, sharper

c o n c l u s i o n s are p o s s i b l e . To i l l u s t r a t e t h i s , an improvement o f Theorem 3 f o l l ~ s . Here we a r e comparing the r a n k i n g o f the N a l t e r n a t i v e s w i t h the r a n k i n g s o f the p a i r s of a1 t e r n a t i u e s , so the space i s EHx(Ez)tHjz). The f i r s t v o t i n g

component o f

w(W_N,F)

r a n k s the N a l t e r n a t i v e s . The remaining v o t i n g components rank the p a i r s o f a l t e r n a t i v e s where

11

i s the vector i n Theorem 4.

D e f i n e the v e c t o r s < z n w l , K=l,..,n, i n Enx(EZ)(Nit) i n the f o l l o w i n g

way. The E N component o f has the v a l u e -(N-1)/N i n the kTH component, and l / N i n a11 o t h e r s . The

Ez

component o f ~ N Ki s z e r o i f t h i s component space

Ez

does n o t r e p r e s e n t a p a i r which i n c l u d e s ax. I f E2 r e p r e s e n t s ( a r , a ~ ) , t h e

component i s (1,-1) i f k < j , o t h e r w i s e i t i s (-1,1). T h i s c h o i c e r e f l e c t s t h a t a r i s t h e p r e f e r r e d a l t e r n a t i v e .

These k c t o r s can be i n t e r p r e t e d i n t h e f 01 1 owi n g manner. The camponen t s i n ( E z ) t N i z ) d e s i g n a t e t h a t a r i s a Condorcet w i n n e r . The EN component

d e s i g n a t e s t h a t a r i s r a n k e d i n l a s t p l a c e w h i l e a11 o t h e r a l t e r n a t i v e s a r e t i e d f o r f i r s t . F o r W 3 , these v e c t o r s a r e Z~,I=(-2/3,1/3,1/3;1,-l;lj-i;O,O),

&,z=(1/3,-2/3,1/3;-1,1;0,0;1 _,-I), and &,3=(1/3,1/3,-2/3;-1,1;0,0;-1 _,I),

Theorem 5. Assume t h e h y p o t h e s i s o f Theorem 3. I f g N # p , then Vu i s t h e t o t a l space ENx(EzI(MiZ). The space VB i s a (N;2) dimensional space c h a r a c t e r i z e d by t h e normal u e c t o r s < ~ N K ) . .

I t i s remarkable t h a t t h e dimension o f UB e q u a l s t h e t h e o r e c t i c lower bound of (N;2)! I t i s i m p o s s i b l e t o do b e t t e r w i t h o u t e l i m i n a t i n g t h e r a n k i n g s o f p a i r s , s o t h i s i s a n o t h e r argument s u p p o r t i n g t h e s u p e r i o r i t y of t h e Borda Count.

Because we can't do b e t t e r than t h e Borda Count, we need t o know these r a n k i n g s w h i c h cannot be avoided. Any such r a n k i n g d e f i n e s a s e t of v e c t o r s f r o m a r a n k i n g r e g i o n . T h i s r a n k i n g i s Borda a d n i s s i b l e i f and o n l y i f a t l e a s t one v e c t o r f r a n t h i s s e t i s o r t h o g o n a l t o o f the

ZNK

v e c t o r s . The proof of the f o l l o w i n g

s t a t e m e n t i l l u s t r a t e s t h i s . The f i r s t a s s e r t i o n improves upon Theorem 3 because i t r e l a x e s t h e c o n d i t i o n t h a t an a l t e r n a t i v e must be a Condorcet winner t o a v o i d b e i n g Borda r a n k e d l a s t . The second statement i l l u s t r a t e s how Theorem 5 can be used t o f i n d s u f f i c i e n t c o n d i t i o n s f o r an a l t e r n a t i v e t o be Borda ranked f i r s t . R e l a t e d r e s u l t s a r e e a s i l y d e r i v e d .

C o r o l l a r y 5.1. a) L e t f(k,,j) be the d i f f e r e n c e between the f r a c t i o n s of the v o t e r s r e f e r r i n g ar t o ar and those p r e f e r r i n g ar t o a&. I f

F ( k ) = f f ( k , j l i s p o s i t i v e , ar rill n o t be Borda ranked i n l a s t place, nor t i e d f o r l a d t place. I f FM(k) i s negative, ar w i l l n o t be Borda ranked i n f i r s t place, nor t i e d f o r f i r s t place.

b) I f FM(l))N-2, then a1 i s Borda ranked i n f i r s t place. If FM(l)<2+4, then a1 i s Borda ranked i n l a s t place.

I f a r i s a Condorcet winner, then f(k,j))O f o r a11 choices o f j. T r i v i a l l y , FN(k)>O, s o a r cannot be Borda ranked l a s t . Hawever, i t i s easy t o c o n s t r u c t examples where an a l t e r n a t i v e a r has F(k)>O even though i t i s n ' t a Condorcet

winner. Thus, t h i s r e s u l t s i g n i f i c a n t l y improves upon Theorem 3. These i n e q u a l i t i e s are r e v e r s e d i f ar i s a Condorcet l o s e r .

What we r e a l l y want are necessary and s u f f i c i e n t c o n d i t i o n s f o r an a l t e r n a t i v e t o be Borda ranked i n k T n place, k=l,..,N, based upon how the v o t e r s rank the

p a i r s o f a l t e r n a t i v e s . T h i s can't be done based s o l e l y upon the o r d i n a l rankings of the p a i r s , b u t i t can w i t h the added i n f o r m a t i o n o f how d e c i s i v e l y each a l t e r n a t i v e won or l o s t i n the p a i r w i s e comparisions. The f o l l o w i n g statement describes the close 1 ink between the Borda Count and the rankings o f the p a i r s .

C o r o l l a r y 5.2. Given a p r o f i l e o f v o t e r s , compute Fn(k), k=l,..,N. The Borda t a l l y i s (Fn(l),F(2),..,Fn<N)). Thus, the a l g e b r a i c r a n k i n g o f <FM(k)l

d e t e r n i n e s the Borda r a n k i n g o f <at).

The Borda r a n k i n g r e f l e c t s how d e c i s i v e l y an a l t e r n a t i v e f a r e s i n the p a i r w i s e comparisons w i t h the other a l t e r n a t i v e s . From t h i s , a case can be made t h a t

Borda winner i s p r e f e r a b l e t o a Condorcet winner. One s u p p o r t i n g argument i s t h a t a

-

Borda outcane i s robust w h i l e a Condorcet winner need not be. For instance, i t i s easy t o c o n s t r u c t examples where a1 i s the Condorcet winner by v i r t u e o f b a r e l y w i n n i n g m a j o r i t y votes over a2 and ag, y e t a2 wins d e c i s i v e l y over a3.

Here, a2 emerges as the Borda winner. Now, a s l i g h t change i n the voters' r a n k i n g s o f a1 and as would change the Condorcet winner t o a2, but i t wouldn't a f f e c t the Borda ranking. The reason, o f course, i s t h a t a Condorcet winner i s

determined by o r d i n a l r a n k i n g s w h i l e a Borda outcome r e f l e c t s the s t r e n g t h o f a p a i r w i s e v i c t o r y . A s i m i l a r type o f r o b u s t n e s s argument c h a r a c t e r i z e s t h e s i t u a t i o n s when a Condorcet winner i s n ' t Borda r a n k e d f i r s t .

P r o o f of t h e C o r o l l a r i e s . Proof o f C o r o l l a r y 5.2. For a g i v e n p r o f i l e o f v o t e r s , t h e outcame over the s e t o f N a l t e r n a t i v e s and the (N;2) p a i r s o f

a l t e r n a t i v e s i s g i v e n b y t h e r a n k i n g r e g i o n w h i c h c o n t a i n s t h e sum

3.6 n r ( a ( a )

where

e(BN,P)

i s t h e normal i z e d Borda v e c t o r i n ENx(EZ)(Nlfl. L e t t h e EN component o f t h i s v e c t o r ( t h e Borda outcane) be g i v e n by (XI,..,XN). Because

Eq. 3.6 i s an a d n i s s i b l e outcome, t h i s v e c t o r sum i s orthogonal t o & w l k=l,..,N.

Take t h i s s c a l a r p r o d u c t . The v a l u e o f t h a t p a r t o f t h e s c a l a r p r o d u c t r e s u l t i n g f r o m the EN components i s ( - X K ( N - ~ ) / N ) + ( ~ ( X J / N ) . B u t , because o f t h e v e c t o r

i

@lr

n o r m a l i z a t i o n , Z x r = 0 , so ~xJ/N=-xK/N. Thus, t h e c o n t r i b u t i o n t o t h e s c a l a r

;&R

p r o d u c t f r o m t h e EN components i s - X K .

Because o f t h e f o r m o f & K , the p a r t o f the s c a l a r product c o r r e s p o n d i n g t o the space ( E 2 ) t N j z ) i s FN(k). Thus, the o r t h o g o n a l i t y c o n d i t i o n l e a d s t o t h e d e s i r e d conc 1 u s i on

3.7 xw

=

FN(k),

T h i s campletes t h e p r o o f o f t h e C o r o l l a r y 5.2.

P r o o f o f C o r o l l a r y 5.1. P a r t a. Because f ( j , k ) = - f ( k , j ) ,

ZFN(K)=O.

^So,

e i t h e r a11 o f the FM(k)'s a r e z e r o ( t h e group's r a n k i n g i s complete i n d i f f e r e n c e ) , o r t h e r e a r e some which a r e p o s i t i v e and sane which a r e n e g a t i v e . I n t h e l a t t e r case, i f FN(k)lO, then i t f o l l o w s from C o r o l l a r y 5.2 t h a t aw can't be Borda ranked l a s t , nor t i e d f o r l a s t . S i m i l a r l y , i f FN(k)iO, can't be Borda ranked

f i r s t , nor t i e d f o r f i r s t .

P a r t b. T h i s proof i n v o l v e s n o t h i n g more than showing t h a t FNtl))N-2 i m p l i e s t h a t FNt1))FNtk). To do t h i s , we need to.determine the maximum values f o r the xr's. To i l l u s t r a t e the ideas, we r e s t r i c t a t t e n t i o n t o M 3 ; the proof f o r N13 i s s i m i l a r .

Assume t h a t FB(1))l. The vector outcome i n Eq. 3.8 must be i n the convex h u l l o f the 6 permutations o f the Borda v e c t o r (2,0,-2). T r i v i a l l y , the maximum values f o r the x r ' s occur on the boundary of t h i s set. Because F3(1))0, a1 must be ranked e i t h e r f i r s t o r second, and e i t h e r a2 o r a3 occupies the other

top two p o s i t i o n s . Assume w i t h o u t l o s s o f g e n e r a l i t y t h a t a1 and a2 are the top two r a t e d a l t e r n a t i v e s . T h i s assumption determines an edge o f the convex h u l l : t(2,0,-2)+(1-t)(0,2,-2)

=

(2t,2-2t,-21, where O(tL1. The assumption F3(1))1 f o r c e s t)1/2, which i n t u r n f o r c e s the r a n k i n g t o have at i n f i r s t place. T h i s completes the proof. For N)3, other r e s u l t s f o l l o w by usinq the surfaces o f the convex h u l l r a t h e r than j u s t the edges.

We end t h i s s e c t i o n w i t h our main r e s u l t .

Theorem 6. L e t N13 a1 t e r n a t i u e s be giuen. Consider the f a m i l y

of

a1 1 2n-(Ntl) subsets o f a t l e a s t two a1 t e r n a t iues. L e t T be the space

En~..x(EK)tMj~)x..~(Ez)~Mjz).

For each subset

of

a l t e r n a t i u e s , s e l e c t a

v o t i n g method, and l e t i n T be the v e c t o r c o n s i s t i n g o f a11

of

the u o t i n g methods.

a) Any

&

has a n o r m a l i z a t i o n

so

t h a t VB i s a l i n e a r rubspace o f Vu. I f

W l ,

then VB i s a proper subspace.

b) Vm i s a (Nj2) dimensional l i n e a r subspace o f T. The normal v e c t o r s for

VI

are found i n the f o l l o w i n g manner. For each subset

of

k a1 t e r n a t iues, the v e c t o r s

<ZKJ)

are defined. T h e w v e c t o r s can be extended t o T by a l l w i n g the new u e c t o r camponents t o be the z e r o uectors. The s e t o f a l l such v e c t o r s span the

1 inear space which i s normal t o VB.

C ) For most c h o i c e r of Id, VU=T.

Theorem 1 and Corol l a r y 1.1 are special cases o f Statement a. Theorems 4 and 5 have imposed a lower bound on the dimension o f VB; the remarkable f a c t i s t h a t

even though we are c o n s i d e r i n g a11 p o s s i b l e subsets o f a l t e r n a t i v e s , the dimension

o f VE has n o t changed; i t s t i l l equals (N;2). T h i s again demonstrates the

e f f i c i e n c y o f t h e Borda v e c t o r s . Because the normal v e c t o r s t o We, a r e s p e c i f i e d , the elements i n Re can be computed i n manner s i m i l a r t o t h a t given above. T h i s , then, c o n s t i t u t e s a simple t o o l t o determine p o s s i b l e Borda ranKings.

If f o r same subset o f k a l t e r n a t i v e s the a p p r o p r i a t e camponent o f

B

i s r e p l a c e d w i t h a n o t h e r v o t i n g v e c t o r , then the dimension o f t h e new v e c t o r space i n c r e a s e s b y - 1 E s s e n t i a l l ~ , the new v e c t o r space i s VB augmented by EK i n t h e

a p p r o p r i a t e space. T h i s i s one way i n w h i c h t h e VU spaces come about. A second way, w h i c h w i l l be discussed i n the s e c t i o n on p r o o f s , i s where t h e r e i s a a l i n e a r

c a m b i n a t i o n between t h e v o t i n g methods a t d i f f e r e n t l e v e l s which a r e o f a v e r y s p e c i f i c t y p e .

I n S e c t i o n 2, we s t a t e d t h a t p l u r a l i t y v o t i n g scheme i s i n t h i s general c l a s s o f 'most' v o t i n g systems. Thus

C o r o l l o r y 6.1. Assume the h y p o t h e s i s o f Theorem 6. I f a1 1

of

the v o t i n g cunponents of

W

c o r r e s p o n d t o t h e p l u r a l i t y u o t i n g scheme, t h e n VU

=

T.

From t h i s c o r o l l a r y , i t f o l l o w s t h a t t h e r e e x i s t v o t e r s ' p r o f i l e s l e a d i n g t o , say, t h e p l u r a l i t y r a n K i n g s a1 )az)a3)a4, a4)ar)az, a1 ) a i ) a 4 ,

PI )aa=a4, a q ) a ~ ) a j , w h i l e the p a i r s o f a l t e r n a t i v e s a r e ranked as g i v e n

i n t h e example f o l l o w i n g Theorem 2. O f course, t h i s same c o n c l u s i o n h o l d s f o r any

g

where t h e canponent v o t i n g v e c t o r s d i s t i n g u i s h o n l y between two subsets o f a1 t e r n a t i v e s .

I f N=3, a1 1 of t h e Borda r a n k i n g s can be o b t a i n e d by use o f Theorem 5. _{I f W4,} then t h e normal space t o VB i s n i n e dimensional. T h i s increased dimension means t h a t t h e r e a r e a l a r g e number of i n c o n s i s t e n t r a n k i n g s which are n o t Borda adnritted.

Indeed, t h e numbers a r e so l a r g e , t h a t a simple l i s t i n g would n o t be reasonable.

But, q u a l i t a t i v e r e s u l t s of the n a t u r e g i v e n i n C o r o l l a r i e s 5.1 and 5.2 are p o s s i b l e by u s i n g t h e same type o f methods.

C o r o l l a r y 6.2. For a given p r o f i l e of v o t e r s , the Borda rankings o f a subset of k a l t e r n a t i v e s C i s given by the a l g e b r a i c rankings o f ~ ~ c ( j ) = t f ( j , i ) where the w m a t io n i s over i # j

,

ⁱ

,

^jrC.

Example. To i l l u s t r a t e t h i s , we f i r s t show how a Condorcet winner over a

s p e c i f i c subset o f a l t e r n a t i v e s f a r e s over other subsets. Suppose t h a t N=4 and th.at a1 f o r j-2,3 by m a j o r i t y votes. BY the above r e s u l t s , a1 can't be Borda

ranked l a s t i n subset Ca~,az,aol. But, j u s t f r a n the knowledge t h a t a ~ ) a a , i t f o 1 lows t h a t independent of how the group ranks a1 and a,, a1

can't be Borda ranked i n f i r s t place i n Cal,az,a4> w h i l e Borda ranked l a s t i n the t o t a l s e t . T h i s i s because the f i r s t c o n d i t i o n i m p l i e s t h a t

F 3 ( l ) = f ( 1 , 2 ) + f ( 1 , 4 ) M , w h i l e the second c o n d i t i o n i m p l i e s t h a t F ~ ( l ) = F 3 ( 1 ) + f ( 1 , 3 ) < 0 . Because f(1,3))0, t h i s i s a c o n t r a d i c t i o n .

We conclude t h i s s e c t i o n w i t h a c m e n t concerning the p r o b a b i l i t y t h a t an i n c o n s i s t e n t r a n k i n g occurs. The p r o f i l e o f v o t e r s are represented by the s e t s Cnrta11. Thus, because they sum t o u n i t y , they can be i d e n t i f i e d w i t h the

r a t i o n a l p o i n t s i n the p o s i t i v e o r t h a n t o f a N!-1 dimensional space. Assume t h a t the p r o f i l e s o f v o t e r s are d i s t r i b u t e d i n such a manner t h a t the r a t i o n a l p o i n t s i n any open s e t i n t h i s space has p o s i t i v e p r o b a b i l i t y o f occurring. Then, i t t u r n s o u t t h a t f o r any choice o f

W,

pnr a c h i s s i b l e aroup rankinas w i t h s t r i c t preference be tween

the

a1 t e r n a t i ves has a pos i t i ve ~ r o b a b i 1 i t v

fi

occurr i nq.

T h i s can be proved by a simple vector a n a l y s i s argument s i m i l a r t o t h a t given above b u t w i t h W_ instead o f

a.

An a l t e r n a t i v e , geanetric approach i s t o note t h a t the outcomes are given by convex combinations o f the v e c t o r s Cgrta,> where the ( r a t i o n a l ) c o e f f i c i e n t s i n d i c a t e the number o f v o t e r s w i t h each r a n k i n g o f the a l t e r n a t i v e s . As we w i l l see, i f t h i s convex h u l l on VU meets a r a n k i n g r e g i o n w i t h s t r i c t preferences, then t h i s i n t e r s e c t i o n forms an open subset o f Vu. From

t h i s i t f o l l o w s immediately t h a t 1) there are an i n f i n i t e number o f examples,

indeed, t h e examples c o r r e s p o n d t o t h e r a t i o n a l p o i n t s i n an open s e t o f N!-1 space, 2) t h e examples need o n l y s a t i s f y i n e q u a l i t y c o n s t r a i n t s , and 3) as t h e number o f v o t e r s , n, approaches i n f i n i t y , then the p r o b a b i l i t y t h a t such a r a n K i n g o c c u r s approachs a p o s i t i v e l i m i t (which i s determined by t h i s open s e t i n N!-1 space).

4. Sane E x t e n s i o n s

The purpose o f t h i s s e c t i o n i s t o e x t e n d the above r e s u l t s i n two d i f f e r e n t d i r e c t i o n s . The f i r s t i s t o a d n i t a d d i t i o n a l v o t i n g methods over a f i x e d s e t o f a l t e r n a t i v e s t o determine how adverse of an e f f e c t t h i s has on c o n s i s t e n c y . The second i s t o determine whether t h e r e a r e f a m i l i e s o f s u b s e t s which a d n i t more c o n s i s t e n c y among the r a n k i n g s than suggested above.

I n S e c t i o n 2, a s t a n d a r d e q u i v a l e n c e r e l a t i o n f o r v o t i n g methods was g i v e n . The b a s i c i d e a was t h a t two methods a r e e q u i u a l e n t s h o u l d t h e y y i e l d t h e same group r a n k i n g f o r any c h o i c e o f v o t e r s ' p r o f i l e s . But, i s t h i s the b e s t one can do; does t h i s d e f i n i t i o n c a p t u r e a11 o f t h e r e l a t i o n s h i p s which p r e s e r v e t h i s i n v a r i a n c e o f group outcome? We show t h a t i f two v o t i n g methods a r e n o t e q u i v a l e n t a c c o r d i n g t o t h i s d e f i n i t i o n , then t h e r e e x i s t p r o f i l e s o f v o t e r s where the outcomes d i f f e r . Indeed, much more can occur; s h o u l d t h e r e be s e v e r a l v o t i n g methods which cannot be cannot be expressed i n terms o f each o t h e r , then the same p r o f i l e of v o t e r s can l e a d t o t o t a l l y u n r e l a t e d group r a n k i n g s .

D e f i n i t i o n 4 L81. L e t <WNJ), j=l,..,k, be a s e t o f k v o t i n g v e c t o r s used t o rank N a l t e r n a t i v e s . They a r e s a i d t o be ' c o m p l e t e l y d i f f e r e n t ' i f t h e y and t h e v e c t o r

EN

a r e l i n e a r l y independent.

I f t h e v o t i n g v e c t o r s a r e v e c t o r normalized, then we don't need t o use

EN.

When k=2, t h e a s s e r t i o n t h a t two v o t i n 9 methods a r e c o m p l e t e l y d i f f e r e n t means t h a t

they are not equivalent i n the sense of D e f i n i t i o n 1. The next theorem asserts t h a t i f there are k completely d i f f e r e n t v o t i n g vectors, t h e r e need not be any

r e l a t i o n s h i p msrlg the same group's rankings o f the same a l t e r n a t i u e s . Our

a s s e r t i o n t h a t D e f i n i t i o n 1 captures a11 o f the r e l a t i o n s h i p s l e a d i n g t o invariance of outcomes f o l l o w s f o r k=2.

Theorem 7 t81. L e t < F a ) , J=l,P,..,k<N be a w t o f k c a p l e t e l y d i f f e r e n t u o t i n g methods t o rank a s e t of N13 a l t e r n a t i u e s . L e t T' be t h e space ( E M ) # and l e t @

haue

PJ

as i t s j r @ vector component, j=l,..,k. Then VurT'. That i s , s e l e c t any k o r d i n a l r a n k i n g s o f the N a l t e r n a t i u e s . Then t h e r e e x i s t p r o f i l e s of v o t e r s so t h a t when the same u o t e r s rank the N a l t e r n a t i u e s b y u s i n g the jTu v o t i n g

method, the outcome i s the J T ~ s e l e c t e d ranking, J=l, ..,I(.

Even the Borda vector doesn't provide any & . . . T h i s i s b e c c - - t h e Borda v e c t o r d e r i v e s i t s pawer from i t s s e n s i t i v i t y t o i n t e r a c t i o n e f f e c t s over subsets o f a l t e r n a t i u e s ; here we are considering o n l y one subset o f a l t e r n a t i u e s . As an example, t h e r e e x i s t p r o f i l e s o f v o t e r s so t h a t t h e i r Borda r a n k i n g i s

a ~ ) a z ) a a ) a + , t h e i r p l u r a l i t y r a n k i n g i s a+)aa)az)al, and t h e i r

r a n k i n g by d e s i g n a t i n g the top two a l t e r n a t i v e s (weight vector (1,1,0,0) w i t h a normal i z a t i o n o f (1,1,-1 ,-I)) i s as)a+)a~)az.

I n the p r e v i o u s s e c t i o n , the value (N;2) arose b o t h as the dimension o f the subspace o f p a i r s o f a l t e r n a t i v e s and as the dimension o f UB. From t h i s and Theorem 5, i t may appear t h a t the s o c i a l choice problems are caused by the

i n c o n s i s t e n c i e s i n the rankings o f p a i r s o f a l t e r n a t i u e s . Should t h i s be so, then i t would be n a t u r a l t o ignore the b i n a r i e s ; namely, i n the i n t e r e s t o f f i n d i n g added consistency, perhaps the usual b i n a r y relevancy condi t ion should be replaced w i t h a k - f o l d releuancy c o n d i t i o n . However, i t t u r n s out t h a t t h e r e i s no advantage i n doing t h i s . For instance, i f a t t e n t i o n i s r e s t r i c t e d o n l y t o the subsets o f k a1 t e r n a t i v e s , the minimal dimension value o f (N;2) s t i 1 1 i s obtained by a Borda Count, and i t i s l a r g e r f o r any other v o t i n g method. However, there a r e . f a m i l i e s where the dimension o f Us i s smaller than (N;2). They are c h a r a c t e r i z e d a t the