The Case Against Bullet,
Approval and Plurality Voting
Saari, D.G. and Newenhizen, J. van
IIASA Collaborative Paper
February 1985
Saari, D.G. and Newenhizen, J. van (1985) The Case Against Bullet, Approval and Plurality Voting. IIASA
Collaborative Paper. IIASA, Laxenburg, Austria, CP-85-003 Copyright © February 1985 by the author(s).
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THE CASE AGAINST BULLET. APPROVAL
AND PLURALITY VOTING
Donald G. S a a r i
Jill Van Newenhizen
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 the International Institute f o r Applied Systems Analysis
and which has received only limited review. Views o r opinions
expressed herein do not necessarily r e p r e s e n t those of t h e Institute, i t s National Member Organizations, o r o t h e r organizations
supporting t h e work.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
2361 Laxenburg, Austria
PREFACE
In this paper Don Saari pursues his inquiry into the paradoxes of voting systems and shows that bullet, approval and plurality voting can have
most unlooked-for consequences.
This research w a s carried out in collaboration with the System and
Decision Sciences Program.
ALEXANDER KURZHANSKI
Chairman
System and Decision
Sciences Program
THE CASE AGAINST BULLET. APPROVAL
AND PLURALtTY VOTING
Donald G. Saari
Jill Van Newenhizen
I t i s w e l l Known that p l u r a l i t y v o t i n g need not r e f l e c t the t r u e sentiments of
the uoters.
When there are more than two candidates running f o r the same o f f i c e ,
i s probable t h a t the candidate approved by most o f the voters won't win.
it
Examples
abound; perhaps the best Known one i s the Senatorial e l e c t i o n i n the s t a t e o f New
YorK d u r i n g 1970. Conservative James Buckler b e n e f i t e d fram a s p l i t uote f o r h i s two
l i b e r a l .opponents; he was e l e c t e d w i t h 3YL o f the vote even though 61% o f the
electorate preferred a l i b e r a l .
I n the 1983 Democratic Party primary campaign f o r
the mayor o f Chicago, the black candidate, Harold Washington, was e l e c t e d because o f
a s p l i t vote f o r h i s two w h i t e opponents, Jane Bryne and Richard Daley.
Indeed,
examples can be found i n many c l o s e l y contested e l e c t i o n s among three or more
candidates.
To i l l u s t r a t e the problem, consider the f o l l o w i n g example o f f i f t e e n v o t e r s and
three candidates A, B, and C.
Suppose the uoters' preferences are s p l i t i n the
f o l l o w i n g way: Six o f the v o t e r s have the r a n k i n g A)C)B;
B)C)A,
and f o u r haue the r a n k i n g C)B)A.
w i t h a t a l l y 6:5:4.
p r e f e r B t o A.
f i v e haue the r a n k i n g
The r e s u l t o f a p l u r a l i t y e l e c t i o n i s A)B)C
But, although A i s elected, a m a j o r i t y (60%) o f these v o t e r s
More s e r i o u s l y , a m a j o r i t y (60%) o f these v o t e r s p r e f e r C, the l a s t
place candidate, t o 4 , and 213 of them p r e f e r C t o B!
candidate, but t h i s f a c t isn't
So, C i s the p r e f e r r e d
r e f l e c t e d i n the e l e c t i o n r e s u l t s .
T h i s i s because
w i t h a p l u r a l i t y vote the v o t e r s can vote f o r o n l y t h e i r top ranked candidate.
Voters are aware of t h i s phenanena and r e a c t by using s t r a t e g i c v o t i n g
behavior.
T h i s i s manifested by the c m o n r e f r a i n near e l e c t i o n time o f 'Don't
waste your uote, mark your b a l l o t f o r
---.'
To remain viable, candidates must
devote valuable campaign time t o counter t h i s e f f e c t .
For instance, d u r i n g the 1984
p r e s i d e n t i a l primaries, Jesse Jackson urged h i s f o l l o w e r s t o vote f o r him r a t h e r
than s t r a t e g i c a l l y v o t i n g f o r Walter Mondale.
There i s l i t t l e question t h a t t h i s instrument o f democracy i s i n need o f
reform.
B u t , what s h o u l d be i t s replacement?
One r e f o r m measure, w h i c h i s i n t e n d e d
t o c a p t u r e how s t r o n g l y a v o t e r p r e f e r s h i s f a v o r i t e c a n d i d a t e ,
i s b u l l e t votinq.
Here a v o t e r h a s two v o t e s ; he can v o t e f o r h i s t o p r a n k e d c a n d i d a t e , h i s t o p two
c a n d i d a t e s , o r he can c a s t b o t h v o t e s f o r h i s t o p r a n k e d c a n d i d a t e .
For i n s t a n c e ,
i n t h e above example, i f a l l o f t h e v o t e r s had s u f f i c i e n t r e g a r d f o r c a n d i d a t e C,
then she w o u l d be e l e c t e d .
B u l l e t v o t i n g was used i n I l l i n o i s f o r c e r t a i n
l e g i s l a t i v e o f f ices.
A second proposed r e f o r m method i s a ~ ~ r o v av ol t i n q .
votes f o r
o f the c a n d i d a t e s he approves o f .
T h i s i s where a v o t e r
As such, when t h e r e a r e N
c a n d i d a t e s , t h e v o t e r has N c h o i c e s ; he can v o t e a p p r o v a l f o r h i s t o p i c a n d i d a t e s ,
i=l,..,N.
A g a i n f o r t h e above example, depending upon t h e degree w h i c h t h e v o t e r s
f a v o r c a n d i d a t e C, she may, o r mar n o t , emerge v i c t o r i o u s .
I t was
Approval v o t i n g e n j o y s the support o f several e x p e r t s i n t h i s f i e l d .
employed f o r a s t r a w b a l l o t d u r i n g the P e n n s y l v a n i a Democratic p a r t y c o n f e r e n c e i n
December, 1983, C11, and i t was used t o s e l e c t f a c u l t y members t o t h e N o r t h w e s t e r n
U n i v e r s i t y P r e s i d e n t i a l s e a r c h c m i t t e e i n November, 1983.
i t '..is
I n addition,
now used i n academic s o c i e t i e s such a s t h e E c o n a n e t r i c S o c i e t y , i n t h e s e l e c t i o n o f
members o f t h e N a t i o n a l Academy o f Science d u r i n g f i n a l b a l l o t i n g , and by t h e U n i t e d
N a t i o n s S e c u r i t y C o u n c i l i n t h e e l e c t i o n o f a S e c r e t a r y General.
B i l l s t o enact
t h i s r e f o r m a r e now b e f o r e the s t a t e l e g i s l a t u r e s o f New York and Vermont
.' t 2 1 .
Much o f t h i s s u p p o r t i s a consequence o f t h e a n a l y s i s o f i t s p r o p e r t i e s b y two
o f i t s foremost advocates, Steven Brams and P e t e r F i s h b u r n .
Most o f t h e i r
c o n c l u s i o n s , w h i c h h i g h 1 i g h t s e v e r a l of t h e d e s i r a b l e p r o p e r t i e s o f t h i s system, a r e
summarized i n t h e i r book 'Approval Voting'C31.
To demonstrate t h e s t r e n g t h o f
a p p r o v a l v o t i n g , o f t e n t h e y canpared i t w i t h those c m o n l y used systems w h i c h
d i s t i n g u i s h between two s e t s o f c a n d i d a t e s
v o t i n g i s t h e s p e c i a l case where k = l ;
c a n d i d a t e and a l l o t h e r s .
-
t h e t o p k and t h e r e s t . P l u r a l i t y
i t d i s t i n g u i s h e s between t h e top r a n k e d
Because o f the t e c h n i c a l d i f f i c u l t i e s i n v o l v e d , n o t a l l
of t h e p r o p e r t i e s o f 'approval
u o t i n g ' were f o u n d , and i n 131 i t wasn't
compared
w i t h a1 1 o t h e r v o t i n g systems.
A t a c o n f e r e n c e i n J u l y , 1984, one o f u s (DGS) p r e s e n t e d s e v e r a l n e g a t i v e
r e s u l t s c o n c e r n i n g t h e b e h a v i o r o f a p p r o v a l v o t i n g (see 14,531 a s p a r t o f a l e c t u r e
d i s c r i b i n g what can occur w i t h v o t i n g methods.
L a t e r , S. Brams p r i v a t e l y asked
whether t h e t e c h n i q u e s developed t o o b t a i n t h e s e r e s u l t s c o u l d expose o t h e r
p r o p e r t i e s o f approval uoting.
We s t a r t e d t h i s p r o j e c t w i t h t h e e x p e c t a t i o n t h a t
a p p r o v a l v o t i n g i s , i n sane sense, b e t t e r t h a n most systems.
B u t , we f o u n d t h a t i t
has s e v e r a l d i s t u r b i n g f e a t u r e s w h i c h makes i t worse than even t h e p l u r a l i t y v o t i n g
system.
I n d e e d , these p r o p e r t i e s appear t o be s u f f i c i e n t l y
bad
disaualif~
a ~ p r o v a lv o t i n a a s a v i a b l e r e f o r m a l t e r n a t i v e .
I n t h i s p a p e r , we r e p o r t on sane o f t h e s e n e g a t i v e f e a t u r e s w h i c h a r e s h a r e d b y
v o t i n g systems, such as a p p r o v a l u o t i n g , b u l l e t u o t i n g , c a r d i n a l u o t i n g , e t c . ,
t h e r e i s more t h a n one way t o t a l l y each v o t e r ' s r a n k i n g o f t h e c a n d i d a t e s .
f e a t u r e emphasized h e r e i s t h a t t h e e l e c t i o n outcome can be 'random"
r a t h e r than b e i n g decisiue.
More p r e c i s e l y ,
where
The
i n nature
i f there are N candidates, then t h e r e
a r e N! p o s s i b l e ways t o rank them w i t h o u t t i e s . The purpose o f an e l e c t i o n i s t o
d e t e r m i n e w h i c h one o f them i s t h e group's c h o i c e .
a given s e t o f voters'
I f a v o t i n g system i s d e c i s i v e ,
p r o f i l e s u n i q u e l y d e t e r m i n e s one o f these r a n k i n g s .
f o r a p p r o v a l and b u l l e t u o t i n g , t h e r e a r e a l a r g e number o f examples where
However,
N!
o u t c a n e s o c c u r f o r t h e same p r o f i l e ! T h a t i s , each v o t e r v o t e s h o n e s t l y a c c o r d i n g t o
h i s f i x e d r a n k i n g o f the candidates.
B u t , as t h e v o t e r s v a r y t h e i r c h o i c e o f how
t h e i r b a l l o t s w i l l be t a l l i e d , each o f t h e p o s s i b l e N! r a n k i n g s emerge!
T h i s phenomenon can be i l l u s t r a t e d w i t h t h e above example.
L e t w,y,z
denote,
r e s p e c t i v e l y , t h e number o f v o t e r s f r o m t h e t h r e e t y p e s o f v o t e r s who v o t e a p p r o v a l
f o r t h e i r t o p two c a n d i d a t e s r a t h e r t h a n j u s t t h e i r t o p r a n k e d c a n d i d a t e .
O(w(6,
O(y(5,
O(z(4,
and t h e t a l l y f o r A:B:C
i s 6:5tz:4twty.
t h a t 9 e l e c t i o n outcome i s a t t a i n a b l e f r o m these u o t e r s .
Then,
I t f o l l o w s immediately
For i n s t a n c e t h e r e s u l t
B>A>C o c c u r s when 212 ( a t l e a s t two f r o m t h e l a s t s e t o f v o t e r s v o t e f o r t h e i r t o p
two r a n k e d c a n d i d a t e s ) and wty(1.
Even t i e s a r e p o s s i b l e .
A deadlocked e l e c t i o n o f
-
&B=C
r e s u l t s f r o m z=1 and w+y=2, w h i l e t h e r e s u l t B=C>A r e s u l t s f r o m w+y-l=z>2.
T h i s example and t h e g e n e r a l r e s u l t i n d i c a t e s t h a t systems such a s approval and
b u l l e t v o t i n g possess f e a t u r e s which a r e more u n d e s i r a b l e t h a n even those o f t h e
p l u r a l i t y v o t i n g system!
J u s t t h i s s t o c h a s t i c , random n a t u r e o f t h e e l e c t i o n
outcome r a i s e s s e r i o u s q u e s t i o n s whether a p p r o v a l v o t i n g , and r e l a t e d methods, t r u l y
o f f e r any r e f o r m .
The proposed c u r e seems t o be much worse than t h e disease.
T h i s r e s u l t doesn't
of t h e p l u r a l i t~ v o t i n g .
a voter's
first,
second,
mean we a r e doomed t o a c c e p t and t o l i v e w i t h t h e f a i l i n g s
There a r e o t h e r ways t o t a l l y a b a l l o t which would r e f l e c t
..,
l a s t ranked candidates.
For i n s t a n c e , a Borda Count i s
where when t h e r e a r e N c a n d i d a t e s , N 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
candidate,
(N-1)
f o r h i s second r a n k e d c a n d i d a t e ,
candidate,
.., and
..., (N-k)
1 p o i n t f o r h i s l a s t ranked candidate.
top ranked
f o r h i s kTH ranked
(The Borda r a n k i n g f o r
t h e above example i s t h e d e s i r e d one of C>B>A w i t h a t a l l y o f 34:29:27.)
So, o u t o f
a l l p o s s i b l e ways t h e r e a r e t o t a l l y b a l l o t s , t h e p r o b l e m i s t o i s o l a t e those ways
which best capture the wishes o f the e l e c t o r a t e .
I t t u r n s o u t t h a t t h e unique
s o l u t i o n f o r t h i s p r o b l e m i s t h e Borda Count.161
2. The m a i n r e s u l t s .
Assume t h e r e a r e N13 c a n d i d a t e s denoted by ( a ~ , . . , a N l .
(WI,
...,
WJ~WK
WN)
Let
W=
be a v o t i n ~v e c t o r where i t s components s a t i s f y t h e i n e q u a l i t i e s
i f and o n l y i f j < k , and
WI>WN.
w e i g h t s t o be r a t i o n a l numbers.
s i m p l i f y the proofs.
Furthermore, we r e q u i r e a l l o f t h e
(The o n l y purpose o f t h e l a s t r e q u i r e m e n t i s t o
C l e a r l y i t doesn't
impose any p r a c t i c a l l i m i t a t i o n s because
f o r a l l commonly used methods the w e i g h t s a r e f r a c t i o n s and/or
integers.)
Such a
v e c t o r d e f i n e s t h e t a l l y i n g p r o c e s s f o r an e l e c t i o n
voter's
jTH
ranked candidate.
determines her f i n a l ranking.
p l u r a l it y vote.
.
Count.)
Let
EN be
p o i n t s are t a l l i e d f o r a
For example, t h e v e c t o r (1,0,..,0)
corresponds t o the
d e f i n e s t h e u s u a l Borda Count p r o c e d u r e .
g e n e r a l l y , a Borda V e c t o r i s whenever t h e d i f f e r e n c e s
nonzero c o n s t a n t f o r j=i,. ,N-1.
WJ
Then, the sum o f t h e p o i n t s t a l l i e d f o r a c a n d i d a t e
.. , I )
BN=(N,N-1,.
--
WJWJ+I
(More
a r e t h e same
An e l e c t i o n u s i n g such a v o t i n g v e c t o r i s a Borda
the vector N-l(l,l,..,l).
This isn't
a l l o f t h e components a r e equal, so i t can't
a v o t i n g v e c t o r because
d i s t i n g u i s h how c a n d i d a t e s a r e ranked.
A s i m p l e v o t i n a system i s where a s p e c i f i e d v o t i n g v e c t o r i s used t o t a l l y t h e
voters'
r a n k i n g s o f the candidates.
A a e n e r a l v o t i n a system i s where t h e r e i s a
s p e c i f i e d s e t o f a t l e a s t two v o t i n g v e c t o r s ,
any two o f these v e c t o r s i s n ' t
same.)
(WJ),
a scalar m u l t i p l e o f
where t h e d i f f e r e n c e between
EN.
(Hence, no two a r e t h e
Then, each v o t e r s e l e c t s a v o t i n g v e c t o r t o t a l l y h i s b a l l o t .
Examples.
l,O,..,O,
is l,O,..,
F o r t h e b u l l e t method, t h e s e t o f t h r e e v o t i n g v e c t o r s a r e
l,l,O,..,O,
0 , , , O , ,
2,0,..,0.
O,..,
For a p p r o v a l v o t i n g , t h e s e t o f N v e c t o r s
l , , . 0,
1 , 1 , . , 1
the v o t e r i s f r e e t o s e l e c t t h e v a l u e s o f t h e w e i g h t s
constraints.
WJ
For c a r d i n a l v o t i n g ,
subject t o c e r t a i n
F o r i n s t a n c e , t o s t a n d a r i z e t h e c h o i c e s , t h e w e i g h t s m i g h t be r e q u i r e d
t o sum t o u n i t y , o r t o be bounded above and below by s p e c i f i e d c o n s t a n t s .
The extreme i n d e c i s i v e n e s s f o r e l e c t i o n s w h i c h was d e s c r i b e d above i s
c h a r a c t e r i z e d i n the f o l l o w i n g d e f i n i t i o n .
D e f i n i t i o n 1. A g e n e r a l v o t i n g system f o r N c a n d i d a t e s i s s a i d t o be s t o c h a s t i c i f
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 all p o s s i b l e r a n k i n g s o f t h e c a n d i d a t e s
( w i t h o u t t i e s ) can r e s u l t f r a n t h e same p r o f i l e a s t h e v o t e r s v a r y t h e i r c h o i c e how
t h e i r b a l l o t s a r e t o be t a l l ied.
I n o t h e r words, f o r these examples o f v o t e r s ' p r o f i l e s , the outcome i s random
and r e f l e c t s t h e v o t e r s '
f l u c t u a t i o n s i n t h e i r choice o f t a l l y i n g procedure r a t h e r
than t h e i r r a n k i n g s o f t h e c a n d i d a t e s . Thus, t h e e l e c t i o n outcome c o u l d be
.
.. , e t c .
a ~ > a z > ..>an, or a n > a n - I > . . > a l , o r a1 > a ~ > .
where t h e
determining f a c t o r i s the voters'
c h o i c e o f how t h e i r b a l l o t s a r e t o be t a l l i e d ,
t h e i r rankings o f the candidates.
As i n the i n t r o d u c t o r y example, a l l r a n k i n g s ,
even those w i t h t i e s can o c c u r .
not
T h i s random f e a t u r e i s a p r o p e r t y we want t o a v o i d ,
s o i t i s i m p o r t a n t t o c h a r a c t e r i z e those general v o t i n g systems w h i c h have i t .
A1 1 g e n e r a l v o t i n g systems a r e
Theorem 1. Assune t h e r e a r e N)3 candidates.
stochast ic.
As i l l u s t r a t e d b y t h e i n t r o d u c t o r y example, a m a j o r c r i t i c i s m o f t h e p l u r a l i t y
v o t e i s t h a t i t need n o t r e f l e c t t h e u o t e r s '
t r u e wishes.
a measure of the t r u e s e n t i m e n t o f t h e v o t e r s .
To q u a n t i f y t h i s , we need
The one most commonly used i s the
Condorcet w i n n e r .
Candidate
D e f i n i t i o n 2. Assume t h a t t h e N)3 c a n d i d a t e s a r e ( a ~ , a z , . . , a ~ ) .
a r i s c a l l e d a Condorcet w i n n e r i f , i n a11 p o s s i b l e p a i r w i s e c a n p a r i s i o n s , a r
a l w a y s w i n s b y a m a j o r i t y v o t e . A Condorcet l o s e r i s an c a n d i d a t e w h i c h always
l o s e s by a m a j o r i t y v o t e i n a11 p o s s i b l e p a i r w i s e c a n p a r i s i o n s w i t h t h e o t h e r
candidates.
A Condorcet w i n n e r appears t o c a p t u r e the t r u e c h o i c e o f t h e v o t e r s ; a f t e r a1 1 ,
she was t h e c h o i c e o f a m a j o r i t y o f t h e e l e c t o r a t e when she was compared w i t h any
other candidate.
B u t , t h e i n t r o d u c t o r y example shows t h a t t h e p l u r a l i t y v o t e can
r a n k a Condorcet w i n n e r i n l a s t p l a c e and a Condorcet l o s e r i n f i r s t p l a c e !
A c t u a l l y , as i t i s s h w n i n 161, t h i s type o f b e h a v i o r i s c h a r a c t e r i s t i c o f a l l
s i m p l e u o t i n g systems w i t h t h e s o l e e x c e p t i o n o f the Borda Count.
Theorem 2.t61.
Suppose t h e r e a r e N)3 candidates. Then, f o r any
system o t h e r than a Borda Count, t h e r e e x i s t examples o f v o t e r s '
Condorcet u i n n e r i s r a n k e d i n l a s t p l a c e and t h e Condorcet 1o s e r
p l a c e . The Borda Count i s t h e u n i q u e method which never r a n k s a
l a s t p l a c e , and never r a n k s a Condorcet l o s e r i n f i r s t p l a c e .
How does a g e n e r a l v o t i n g system f a r e ?
simple v o t i n g
p r e f e n c e s rrhere t h e
i s ranked i n f irst
Condorcet r r i n n e r i n
I t t u r n s o u t t h a t i t can be even worse.
D e f i n i t i o n 3. A general v o t i n g system (W_J)
i s ' p l u r a l i t y 1 ike' i f there are
non-negat i ue s c a l a r s ( b ~ such
l
t h a t when the d i f f e r e n c e s be tween successive
components o f Z ~ J ~are
J Jcomputed, a1 1 b u t one are zero.
The summation d e f i n e s a v o t i n g v e c t o r , and the c o n d i t i o n i s t h a t t h i s v e c t o r
d i s t i n g u i s h e s between o n l y two s e t s of candidates.
This condition i s s a t i s f i e d
a u t m a t i c a l l y when the general v o t i n g method i n c l u d e s a v o t i n g v e c t o r o f t h i s type,
e.g.,
a p l u r a l i t y v o t i n g vector.
p l u r a l i t y 1i k e .
Thus, b o t h approval and b u l l e t v o t i n g are
A l s o , the c o n d i t i o n i s t r i v i a l l y s a t i s f i e d should the general
method i n c l u d e N-1 v o t i n g v e c t o r s which, a l o n g w i t h
independent s e t .
T h i s i s because the v e c t o r s span
EN,
form a l i n e a r l y
RN.
Theorem 3.
Suppose there are N)3 candidates. Choose a r a n k i n g f o r each o f the
( T h i s may be done i n a randan
N(N-1112 p a i r s o f candidates i n any manner you wish.
f a s h i o n ; the r a n k i n g s need n o t be t r a n s i t i v e . )
Assume t h a t a l l subsets o f more than
two c a n d i d a t e s are t o be ranked w i t h a p l u r a l i t y l i k e , general v o t i n g method. Then,
t h e r e e x i s t examples o f v o t e r s ' p r o f i l e s so t h a t
1) f o r each of the p a i r s o f candidates, a m a j o r i t y o f the v o t e r s have the
i n d i c a t e d preference, and
2) f o r each subset of three o r more candidates, t h e o u t c m e i s s t o c h a s t i c .
(As i t w i l l becme c l e a r i n the p r o o f ,
method i s ' p l u r a l i t y 1 ike'
general v o t i n g systems.
isn't
the c o n s t r a i n t t h a t the geceral v o t i n g
necessary; the c o n c l u s i o n holds f o r almost a1 1
We impose t h i s assumption because i t s i g n i f i c a n t l y
s i m p l i f i e s the p r o o f w i t h o u t i n c u r r i n g much o f a s a c r i f i c e t o g e n e r a l i t y
--
the
general methods which have been s e r i o u s l y considered o r used, such as approval o r
b u l l e t voting, s a t i s f y t h i s condition.)
T h i s theorem means t h a t there e x i s t examples o f v o t e r s '
p r o f i l e s where f o r each
p a i r ( a ~ , a ~ )a, m a j o r i t y o f the v o t e r s p r e f e r the candidate w i t h the smaller
subscript.
Thus, the r a n k i n g s o f the p a i r s are h i g h l y t r a n s i t i v e and imp1 y a group
r a n k i n g o f a l > a z > . . > a ~ . I n s p i t e o f t h i s , f o r the same v o t e r s , the a p p r o v a l
v o t i n g rankings o f
all
subsets o f three o r more candidates are s t o c h a s t i c .
Thus the
approval v o t i n g outcome o v e r a s e t o f more than two c a n d i d a t e s c o u l d depend more on
the q u i r k s and f o r t u n e s o f how t h e v o t e r s s e l e c t t o have t h e i r b a l l o t s t a l l i e d than
on how they rank t h e c a n d i d a t e s .
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 i s f o r N=3.
Our goal was t o compare approval v o t i n g w i t h the Condorcet w i n n e r .
This i s a
c o r o l l a r y of t h e theorem when the r a n k i n g s o f t h e p a i r s d e f i n e a Condorcet w i n n e r .
F o r i n s t a n c e , when k 4 , t h e r e e x i s t examples o f v o t e r s s o t h a t whenever a , i s
compared w i t h any o t h e r c a n d i d a t e , she always w i n s w i t h a m a j o r i t y v o t e .
Yet, when
these same v o t e r s use a p p r o v a l v o t i n g t o rank t h e c a n d i d a t e s , t h e outcome i s
s t o c h a s t i c whether t h e s e t o f c a n d i d a t e s i s i a l , a z , a a , a + I ,
a
, a , a ,
a
,
,
a
or a , a , a .
I n o t h e r words,
ia~,az,asI,
it i s
p r o b a b l e t h a t a Condorcet w i n n e r c o u l d w i n an approval e l e c t i o n o n l y by a c c i d e n t .
Moreover, t h e approval v o t i n g outcome over any o f these s u b s e t s o f c a n d i d a t e s i s
random, so i t i s n ' t
c l e a r who i s t h e approval winner.
T h i s t y p e o f an example i s
i m p o r t a n t because i t i l l u s t r a t e s t h a t a general v o t i n g method can be random even
when t h e r e i s a Condorcet w i n n e r ; a general v o t i n g method need n o t r e f l e c t the
voters'
t r u e views.
The g e n e r a l i z a t i o n o f the example i s h i g h l i g h t e d i n t h e
f 01 1 owing f o r m a l s t a t e m e n t .
Because we a r e n ' t
considering
p o s s i b l e s u b s e t s of
c a n d i d a t e s , t h e c o n d i t i o n on t h e general v o t i n g system i s r e l a x e d .
Assume t h e r e a r e N)3 c a n d i d a t e s which a r e t o be r a n k e d w i t h a
C o r o l l o r y 3.1.
g e n e r a l v o t i n g method. Assume t h a t t h e g e n e r a l v o t i n g system has a t l e a s t one
v e c t o r w h i c h i s n ' t a Borda v e c t o r . There e x i s t examples o f v o t e r s ' p r o f i l e s s o t h a t
even though t h e r e i s a Condorcet w i n n e r , t h e r a n k i n g o f the N c a n d i d a t e s i s
stochastic.
Again, t h e 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 i s r e s u l t f o r N=3.
t h i s r e s u l t extends t o a l l s u b s e t s o f . t h e N c a n d i d a t e s .
Furthermore,
Thus, a g e n e r a l method,
such as approval o r b u l l e t v o t i n g , need n o t r e f l e c t the w i s h e s o f t h e v o t e r s over
any subset o f t h e c a n d i d a t e s .
These n e g a t i v e s t a t e m e n t s about approval v o t i n g can be r e c o n c i l e d w i t h some o f
t h e p o s i t i v e ones w h i c h appear i n t h e l i t e r a t u r e .
settings,
I n particular, in certain
i t has been p r o v e d t h a t approval v o t i n g can r e s u l t i n a Condorcet w i n n e r
b e i n g r a n k e d i n f i r s t p l a c e C31.
I t f o l l o w s f r o m t h e above s t a t e m e n t s t h a t such
f a v o r a b l e c o n c l u s i o n s can o c c u r , b u t , they a r e j u s t one o f t h e many p o s s i b l e
s t o c h a s t i c f l u x u a t i o n s of t h e e l e c t i o n !
The c o r o l l a r y and t h e theorem a s s e r t t h a t
i f t h e v o t e r s behave by c h o o s i n g t o t a l l y t h e i r c h o i c e s i n c e r t a i n , s p e c i f i e d wars,
then these d e s i r a b l e outcanes w i l l r e s u l t .
But i f they don't,
then a n y t h i n g can
o c c u r . I n o t h e r words, r e s u l t s as u n d e s i r a b l e a s one may f e a r a r e p r o b a b l e .
Another imp1 i c a t i o n of Theorem 3 concerns those p r o c e d u r e s used t o c o n s i d e r
v o t e r s ' p r e f e r e n c e s n o t o n l y over the t o t a l s e t o f c a n d i d a t e s , b u t a l s o o v e r
subsets.
F o r example, t h e f o l l o w i n g i s a s t a n d a r d approach: F i r s t r a n k t h e N
c a n d i d a t e s , and t h e n d r o p the c a n d i d a t e who i s i n l a s t p l a c e .
Rerank t h e r e m a i n i n g
s e t and c o n t i n u e t h i s e l i m i n a t i o n procedure u n t i l o n l y t h e r e q u i r e d number o f
candidates remain.
voting,
Now, suppose we employ a g e n e r a l method, such as a p p r o v a l
t o r a n k t h e c a n d i d a t e s a t each s t e p of t h i s e l i m i n a t i o n procedure.
Theorem
3 shows t h a t t h e s t o c h a s t i c n a t u r e o f t h e c o n c l u s i o n c o u l d f o r c e t h e f i n a l r e s u l t t o
have no r e l a t i o n s h i p whatsoever w i t h how t h e v o t e r s r e a l l y r a n k t h e c a n d i d a t e s .
Theorem 3 can be extended.
For i n s t a n c e , c o n s i d e r a s i t u a t i o n where t h e r e a r e
4 c a n d i d a t e s , b u t t h e e l e c t i o n i s o n l y c l o s e l y c o n t e s t e d among t h r e e o f them.
Then
we m i g h t e x p e c t t h a t t h e s t o c h a s t i c e f f e c t a f f e c t s o n l y these t h r e e c a n d i d a t e s , n o t
t h e l a s t one.
T h i s does happen; r e s u l t s about ' p a r t i a l
stochastic'
e f f e c t s can be
obtained.
So f a r we've
candidates.
compared general v o t i n g methods o n l y w i t h the r a n k i n g s o f p a i r s o f
A n o t h e r t e s t i s t o c m p a r e i t w i t h p l u r a l i t y v o t i n g and o t h e r s i n g l e
v o t i n g methods.
After all,
p l u r a l i t y v o t i n g aren't
Condorcet w i n n e r .
i n the i n t r o d u c t o r y example, the c o n c l u s i o n s o f
i n d i c a t i v e o f the v o t e r s '
p r e f e r e n c e s as measured by a
The f o l l o w i n g statement compares t h e r e s u l t s o f a g e n e r a l v o t i n g
method w i t h any s i m p l e v o t i n g system.
Theorem 4. Assume t h e r e a r e N)4 c a n d i d a t e s . L e t ~JNbe a v o t i n g v e c t o r d e f i n i n g
a s i m p l e v o t i n g method, and assume t h a t a g e n e r a l v o t i n g method i s g i v e n where a t
l e a s t two of t h e v e c t o r s and En a r e 1 i n e a r l y independent. Then t h e r e e x i s t
examples o f u o t e r s ' p r o f i l e s s o t h a t
a. There i s a Condorcet w i n n e r .
b. I f &
I
i s n ' t a Borda v e c t o r , then the s i m p l e v o t i n g method h a s any
p r e v i o u s l y s e l e c t e d r a n k i n g o f the c a n d i d a t e s .
c. The g e n e r a l method i s s t o c h a s t i c .
I f t h e g e n e r a l v o t i n g method i s e i t h e r a p p r o v a l o r b u l l e t v o t i n g , t h e n t h e
conclusion holds f o r
N23.
So, w i t h t h e e x c e p t i o n o f t h e Borda Count, Theorem 4
i l l u s t r a t e s t h a t examples e x i s t where a n y t h i n g can occur w i t h the s i m p l e v o t i n g
scheme w h i l e t h e g e n e r a l method i s s t o c h a s t i c .
I n particular,
t h i s i m p l i e s the
e x i s t e n c e o f examples o f v o t e r s ' p r o f i l e s where the p l u r a l i t y outcome
does
r a n k the
Condorcet w i n n e r i n f i r s t p l a c e w h i l e approval v o t i n g has a s t o c h a s t i c e f f e c t .
Consequently,
i t i s p r o b a b l e t h a t the p l u r a l i t y e l e c t i o n r e s u l t s
v o t e r s ' w i s h e s w h i l e a p p r o v a l v o t i n g does n o t . Again,
r e f l e c t the
t h i s i s because t h e random
f l u x u a t i o n s o f a p p r o v a l v o t i n g a l law any type o f e l e c t i o n r e s u l t t o emerge.
A
second consequence of t h i s theorem, a l o n g w i t h Theorem 2,
i s t h a t among a l l s i n g l e
v o t i n g systems, o n l y t h e Borda Count r e f l e c t s t h e v o t e r s '
intent.
A r e m a i n i n g i s s u e i s t h e r o b u s t n e s s o f these c o n c l u s i o n s . That i s , can we
d i s m i s s these s t a t e m e n t s because t h e c o n c l u s i o n s occur o n l y w i t h some h i g h l y
p a t h o l o g i c a l example w h i c h i s h i g h l y u n l i k e l y t o o c c u r ?
We show t h i s i s n ' t
so; w i t h
any f a i r l y general d i s t r i b u t i o n o f u o t e r s ' p r e f e r e n c e s , these r e s u l t s have a
positive p r o b a b i l i t y of occurring.
Moreover, as i t w i l l become apparent i n the
p r o o f s , these s t o c h a s t i c outcomes a r e more 1 i k e l y t o occur i n those t y p e s o f
s i t u a t i o n s w h i c h have been used t o d i s c r e d i t t h e p l u r a l i t y system.
I t turns out
t h a t these random e l e c t i o n outcomes tend t o occur when t h e r e i s a c l o s e l y c o n t e s t e d
e l e c t i o n among t h r e e o r more c a n d i d a t e s .
To show t h a t these examples a r e p r o b a b l e , we need t o i n t r o d u c e a measure f o r
the d i s t r i b u t i o n o f the voters' p r o f i l e s .
I f t h e r e a r e N c a n d i d a t e s , then t h e r e a r e
N! p o s s i b l e r a n k i n g s o f them.
For each p o s s i b l e r a n k i n g , A,
f r a c t i o n o f the v o t e r s w i t h t h i s r a n k i n g .
l e t na be t h e
The sum o f these numbers e q u a l s u n i t y , so
t h e r e a r e N ! - 1 degrees of freedom; these numbers d e f i n e a u n i t cube, C(N),
p o s i t i v e o r t h a n t o f a N!-1 d i m e n s i o n a l space.
w i t h t h e ( r a t i o n a l ) p o i n t s i n C(N).
Voters'
i n the
p r o f i l e s can be i d e n t i f i e d
Because a l l o f t h e u s u a l c o n t i n u o u s
d i s t r i b u t i o n s i d e n t i f y a p o s i t i v e p r o b a b i l i t y o f occurrance w i t h an open s e t i n
C(N),
outcomes a r e p r o b a b l e i f t h e y a r e i d e n t i f i e d w i t h open s e t s i n C(N).
f o l l o w i n g theorem a s s e r t s t h a t t h i s c h a r a c t e r i z e s our a s s e r t i o n s .
The
Also, statements
a s s e r t i n g t h e a s y m p t o t i c , p o s i t i v e l i k e l i h o o d o f these examples, a s t h e number o f
v o t e r s grows, f o l l o w i m m e d i a t e l y . The 1 i m i t s a r e r e l a t e d t o t h e measure o f the open
s e t s i n C(N).
(See 14,51.)
Theorem 5. F o r each o f the above theorems, the s e t o f examples d e f i n i n g t h e
d e s c r i b e d b e h a v i o r c o n t a i n s an open s e t i n C(N).
F i n a l l y , we a r e l e f t w i t h t h e i s s u e o f r e f o r m .
g e n e r a l method does
not c o n s t i t u t e
The above d e m o n s t r a t e s t h a t a
a reform alternative f o r p l u r a l it y voting.
But,
t h i s doesn't mean we a r e f o r c e d t o l i v e i n t h e i m p e r f e c t w o r l d o f p l u r a l i t y v o t i n g .
I n Theorems 2 and 4 and i n C o r o l l a r y 3.1,
we see t h a t the Borda Count i s t h e u n i q u e
s i m p l e v o t i n g scheme w h i c h a v o i d s many o f t h e p i t f a l l s a s s o c i a t e d w i t h p l u r a l i t y
voting.
T h i s i n i t s e l f d e m o n s t r a t e s t h a t t h e Borda Count i s b e s t c a n d i d a t e , of t h e
v o t i n g methods, f o r r e f o r m .
For a more d e t a i l e d a n a l y s i s o f i t s p r o p e r t i e s a l o n g
w i t h a comparison of i t w i t h o t h e r s i m p l e v o t i n g methods, see 161.
3. P r o o f s
Proof of Theorem 1.
Assume t h e r e a r e NL3 c a n d i d a t e s { a , ,
t h e g e n e r a l v o t i n g system c o n s i s t s o f t h e v o t i n g v e c t o r s
. ., a ~ land
{ W J ~ , j=l,..,s,
that
where
512.
Then, each
WJ
i s a v e c t o r i n t h e N dimensional space RN.
r a n k i n g a l > a z > . ..>an,
p e r m u t a t i o n s of A.
and l e t P(A) be a g e n e r i c r e p r e s e n t a t i o n f o r t h e N !
For v o t i n g vector
the b a l l o t w i l l be t a l l i e d .
p e r m u t a t i o n o f the v e c t o r
if p(3,2,1),
L e t A denote t h e
WJ.
WJ,
any such p e r m u t a t i o n P(A) d e t e r m i n e s how
I n f a c t , t h i s t a l l y can be viewed as b e i n g a
Denote t h i s permuation by
WJP~A).
For instance,
then t h e s t a n d a r d r a n k i n g a l > a n ) a a d e f i n e s t h e v e c t o r (3,2,1).
The r a n k i n g a3)al)an d e f i n e s t h e p e r m u t a t i o n o f W_,
(2,1,3),
t o r e f l e c t that
f o r t h i s r a n k i n g , two p o i n t s a r e t a l l i e d f o r a l , one f o r an, and t h r e e f o r a3.
L e t n p t a ) denote t h e f r a c t i o n o f the v o t e r s w i t h t h e r a n k i n g o f t h e
The t a l l y o f a s i m p l e e l e c t i o n u s i n g W_J
c a n d i d a t e s P(A).
4.1
is
npca ) W J P ( a 1
where t h e summation i s over a l l N! p e r m u t a t i o n s P(A).
The outcome o f t h e e l e c t i o n
i s d e t e r m i n e d by a l g e b r a i c a l l y r a n k i n g t h e components i n t h i s v e c t o r sum.
There i s a g e o m e t r i c r e p r e s e n t a t i o n f o r t h i s a l g e b r a i c r a n k i n g .
i n d i f f e r e n c e h y p e r p l a n e i n RN g i v e n b y XI=XK.
I f t h e v e c t o r sum 4.1
Consider t h e
i s on t h e
s i d e o f t h i s h y p e r p l a n e , then a ~ r a n k s h i g h e r t h a n ax, and v i c e v e r s a .
XK)XI
I n p a r t i c u l a r , t h e N(N-1)/2
'ranking regions',
p o s s i b l e ' i n d i f f e r e n c e h y p e r p l a n e s ' d i v i d e s RN i n t o
and the f i n a l r a n k i n g o f t h e c a n d i d a t e s i s d e t e r m i n e d b y which
r a n k i n g r e g i o n c o n t a i n s the v e c t o r sum.
F o r a g e n e r a l v o t i n g system, l e t m r r t a ) denote t h e f r a c t i o n of those v o t e r s
w i t h a P(A) r a n k i n g t h a t e l e c t t o have t h e i r b a l l o t s t a l l i e d w i t h t h e
vector.
Then,
JTH
voting
t h e f r a c t i o n o f the t o t a l number o f v o t e r s w i t h t h i s t a l l y i s
n p t a ) m ~ r ( a ) . Consequently, t h e t o t a l t a l l y i s g i v e n b y t h e double sum
)[Srn~r
4.2
Again,
( a )WJP ta 1 1 .
Z n r (a
4
Pr A l
t h e r a n k i n g o f t h e c a n d i d a t e s i s determined by t h e r a n k i n g r e g i o n o f RN
w h i c h c o n t a i n s t h i s v e c t o r sum.
We r e p r e s e n t Eq. 4.2 as a mapping.
Toward t h i s end, l e t
Because each term d e f i n e s a p e r c e n t a g e , t h e s e t C n p t a ) I i s a ( r a t i o n a l ) p o i n t i n
the s e t Si(N!).
For each P ( A ) ,
the s e t Crnrrta)l i s i n S i ( s ) .
t h e e n t r i e s d e f i n e non-negative f r a c t i o n s which sum t o u n i t y . )
domain p o i n t i s i n the ( N ! - l ) ( s - l ) ~ !
T
= S i (N! ) x ( S i ( s ) ) " !
( T h i s i s because
T h i s means t h a t a
d i m e n s i o n a l space
.
Any r a t i o n a l p o i n t i n T c o r r e s p o n d s t o an example o f v o t e r s J p r o f i l e s a l o n g w i t h
t h e i r i n d i v i d u a l s e l e c t i o n o f v o t i n g v e c t o r s t o t a l l y the b a l l o t s .
Thus, Eq. 4.2
can be viewed a s b e i n g a mapping fran T t o RN
F:T
4.3
-------- )
R",
where F i s the sumnation.
D e f i n e t h e "complete i n d i f f e r e n c e " r a n k i n g i n RN t o be the l i n e g i v e n b y a l l
scalar multiples o f
EN.
The name comes fran the f a c t t h a t t h i s l i n e c o r r e s p o n d s
t o where t h e r e i s a complete t i e i n t h e r a n k i n g s of a l l o f t h e c a n d i d a t e s .
Notice
that
a)
t h e c u n p l e t e i n d i f f e r e n c e r a n K i n g i s the i n t e r s e c t i o n o f a l l o f t h e i n d i f f e r e n c e
h y p e r p l a n e s , and
b)
t h i s 1 i n e i s on t h e boundary o f a1 1 o t h e r r a n k i n g r e g i o n s .
To p r o v e t h i s theorem, we must show t h e e x i s t e n c e o f a n* i n S i
(N!
(a choice
of v o t e r s J p r o f i l e s ) s o t h a t as t h e v a r i a b l e m=CrnJpta)I v a r i e s , t h e image o f
F((n*,m))
meets a l l p o s s i b l e r a n k i n g r e g i o n s .
L e t n * c o r r e s p o n d t o where t h e r e i s an equal number o f v o t e r s w i t h each
p o s s i b l e r a n k i n g o f the candidates;
i .e.,
n*=(N!)-i(l
,l,..,l).
I t follows
i m m e d i a t e l y t h a t i f m i p t a ) = l f o r a l l c h o i c e s o f P(A) ( a l l v o t e r s choose t h e f i r s t
voting vector),
then Eq, 4.2 reduces t o Eq. 4.1,
complete i n d i f f e r e n c e l i n e .
equal.
and the image o f F i s on t h e
The same c o n c l u s i o n h o l d s i f a l l o f the m r p t a ) a r e
T h i s i s because t h e double summation can be i n t e r c h a n g e d t o o b t a i n s e p a r a t e
summations o f the type g i v e n i n Eq. 4.1,
complete i n d i f f e r e n c e l i n e .
each o f w h i c h y i e l d s a p o i n t on the
Denote t h i s domain p o i n t by (n*,m*).
The i d e a i s t h e f o l l o w i n g .
equal t o N, where,
held fixed.
Assume t h a t the Jacobian o f F a t (n*,m*)
has ranK
i n the c o m p u t a t i o n o f the Jacobian, t h e n p t a ) v a r i a b l e s are
(We t r e a t them as parameters.)
T h i s means t h a t t h e r e i s an open s e t
about the i n t e r i o r p o i n t m* w h i c h i s mapped t o an open s e t about t h e image
F
n
,
m
T h i s open s e t y i e l d s outcomes which can be a t t a i n e d w i t h the same
p r o f i l e o f v o t e r s (n*),
b u t where m, w h i c h i n d i c a t e s t h e c h o i c e o f v o t i n g v e c t o r s t o
t a l l y the b a l l o t s , varies.
Because an open s e t about any p o i n t on t h e l i n e o f
complete i n d i f f e r e n c e meets a l l r a n k i n g r e g i o n s , t h e c o n c l u s i o n f o l l o w s .
easy t o show t h a t t h e r e a r e r a t i o n a l c h o i c e s o f m w i t h t h i s p r o p e r t y .
(It i s
For d e t a i l s ,
see 14,51.)
Thus, the p r o o f i s completed i f we can determine c e r t a i n p r o p e r t i e s about the
Jacobian o f F a t (n*,m*).
There a r e two cases t o c o n s i d e r , and t h e y a r e based upon
t h e sum o f the components of each v o t i n g v e c t o r .
E i t h e r a t l e a s t two o f these sums
d i f f e r , or t h e y a r e a l l t h e same.
Assume t h a t a t l e a s t two of t h e sums d i f f e r .
For each P(A), e l i m i n a t e t h e
dependency of t h e components C r n r r t a ) l b y s e t t i n g m ~ r t a ) = l - ~ m r r t n ) .Then, the
jrt.
r a n k o f t h e J a c o b i a n o f F i s d e t e r m i n e d by t h e maximum number of independent v e c t o r s
i n subsets from
where P(A> ranges over a1 1 N! permutat i o n s o f A and where j = 2 , .
c h o i c e o f j where t h e sum of t h e components o f
components o f
MI,
say j = 2 .
There i s a
WJ doesn't equal the sum o f t h e
Moreover, w i t h o u t l o s s o f g e n e r a l i t y , we can assume
t h a t t h e sum o f t h e components o f
WI.
.,s.
bJ2
i s l a r g e r than t h e sum o f the c m p o n e n t s o f
What we show i s t h a t t h e s e t o f v e c t o r s
4.5
spans
CW_z~(a)-W_~rta))
RN.
T h i s w i l l complete t h e p r o o f .
~t f o l i o w s immediately t h a t
where t h e s c a l a r i s ( N - I ) !
L ~ J r t h i) s
w\-
a nonzero s c a l a r mu1 t i p l e of
t i m e s t h e sum o f the components o f
~JJ.
Thus,
EN
~ ( W _ ~ ~ ~ ~ , - W _ I Pi s~ Aa , p) o s i t i v e s c a l a r m u l t i p l e o f
Pt61
EN,
so t h i s v e c t o r i s i n
The simplex Si(N) has
the space spanned by the v e c t o r s i n Eq. 4 . 5 .
EN a s
a normal
v e c t o r , so the theorem i s proved i f t h e s i m p l e x i s spanned b y the v e c t o r s i n Eq.
4.5.
Each v e c t o r i n Eq. 4 . 5 can be viewed a s b e i n g a p e r m u t a t i o n of the components
of W-2-UI. L e t v e c t o r
be t h e p e r m u t a t i o n o f
g which
has t h e l a r g e s t v a l u e i n
t h e f i r s t component, t h e second l a r g e s t i n t h e second canponent, e t c .
i f W_z=(5,4,2,1)
then e(0,3,1,1)
and g1=(5,1 ,I,0),
a1 l p o s s i b l e permutat i o n s o f
and p ( 3 , 1 , 1 , 0 )
(For example,
.) 'The
set of
9,
CVP t a l l ,
4.6
agrees w i t h the s e t i n Eq. 4 . 5 .
Because
"
.
.--
?
multiple of
EN, \-I
can be
viewed as b e i n g a v o t i n g v e c t o r and t h e v e c t o r s i n Eq. 4.6 can be viewed as b e i n g
the v a r i o u s ways t o t a l l y b a l l o t s .
T h a t t h i s s e t spans S i ( N ) f o l l o w s i m n e d i a t e l y
from [ 7 , 4 3 .
Suppose t h e sums o f the components f o r each o f the v o t i n g v e c t o r s a r e t h e same.
We show t h a t t h e Jacobian o f F has r a n k N-1 and i t s image spans a s i m p l e x S i (N).
T h i s r e q u i r e s t h e f o l l o w i n g adjustment i n t h e p r o o f .
mapped t o an open s e t about F((n*,m*))
must meet a l l r a n k i n g r e g i o n s .
i n the simplex.
F i r s t , an open s e t about m* i s
However, such an open s e t
(The s i m p l e x has codimension one, and i t s normal
d i r e c t i o n i s g i v e n by the l i n e of complete i n d i f f e r e n c e . ) .
Thus, a l l we need t o
show i s t h a t t h e v e c t o r s i n 4 . 6 span t h e subspace o r t h o g o n a l t o
same argument g i v e n aboue.
EN.
T h i s i s the
T h i s completes the proof.
The p r o o f s of Theorems 3 and 4 depend h e a v i l y upon the p r o o f s and r e s u l t s i n
163.
E s s e n t i a l l y , the i d e a o f the p r o o f s i s t o use s p e c i a l ways i n which t h e v o t e r s
choose t h e i r v o t i n g v e c t o r s t o o b t a i n a s i m p l e v o t i n g systems.
Then, m o d i f i c a t i o n s
of the t y p e used i n t h e p r o o f o f Theorem 1 and r e s u l t s from l 6 1 l e a d t o a condi t i o n
o f the type where F((n*,m*))
i s on the 1 i n e of complete i n d i f f e r e n c e .
The Jacobian
c o n d i t i o n f o l l o w s f r o m the a n a l y s i s g i v e n i n the p r o o f s o f 161.
The f o l l o w i n g i s a consequence o f Theorems 5 and 7 i n [dl.
Proof o f Theorem 4.
L e m a . Suppose t h e r e a r e two s i m p l e u o t i n g v e c t o r s , V_I and 92 which,
a) f o r m a l i n e a r l y independent s e t a l o n g w i t h
and
b) a B o r d a u e c t o r i s n ' t i n t h e span o f these t h r e e u e c t o r s .
Rank t h e p a i r s of c a n d i d a t e s i n any way and r a n k t h e N c a n d i d a t e s i n any two ways.
Then, t h e r e e x i s t p r o f i l e s o f u o t e r s f o r w h i c h when t h e same u o t e r s c o n s i d e r each
p a i r o f c a n d i d a t e s , a m a j o r i t y p r r f e r t h e d e s i g n a t e d one. When these same u o t e r s
r a n k t h e N c a n d i d a t e s by t h e s i m p l e v o t i n g system V _ j , the outcome i s t h e j T n
r a n k i n g o f t h e candidates, j=1,2.
EM
A c c o r d i n g t o the statement of t h e theorem, the range space c o n t a i n i n g t h e t a l l y
o f t h e v a r i o u s subsets o f c a n d i d a t e s i s g i v e n by S=(RH)x(RH)x(R*)C
p=N(N-1)/2.
where
The f i r s t component space i s t h e t a l l y o f the s i m p l e v o t i n g system, t h e
second i s t h e t a l l y o f the g e n e r a l v o t i n g system, and t h e l a s t p c o m p ~ n e n t sc o n t a i n
t h e t a l l y o f t h e b i n a r y comparisons.
The domain i s T.
Thus, the o b v i o u s summat i o n s
d e f i n e t h e mapping
F*: T
--------
)
S.
The p r o o f f o l l o w s much as i n t h a t o f Theorem 1.
We show t h e e x i s t e n c e o f a s e t
o f p r o f i l e s n' f o r which the r a n k i n g s o f t h e simple system and t h e r a n k i n g s o f t h e
p a i r s o f c a n d i d a t e s a r e as s p e c i f i e d .
Moreover, n'
i s such t h a t t h e r e i s an
i n t e r i o r p o i n t , ma, i n the p r o d u c t o f t h e s i m p l i c e s w h i c h d e s i g n a t e how t h e v o t e r s
s e l e c t t h e i r t a l l y i n g v e c t o r s so t h a t F((n8,m'))
g e n e r a l system.
repeated.
i s complete i n d i f f e r e n c e f o r t h e
Then, t h e above argument c o n c e r n i n g the Jacobian o f F*,
is
The main d i f f e r e n c e i s t h a t i t i s e v a l u a t e d a t (nm,m') r a t h e r than a t
(n*,m*).
F i r s t we f i n d m'.
on P(A).
To do t h i s , we choose the m r p t a ) ' ~ t o depend on j b u t n o t
T h i s d e f i n e s a convex c o m b i n a t i o n o f t h e v o t i n g v e c t o r s w h i c h a r e
a v a i l a b l e t o t a l l y the r a n k i n g s .
I n t u r n , t h i s d e f i n e s a new v o t i n g v e c t o r ;
indeed,
i t d e f i n e s a continuum o f them where t h e dimension o f the continuum depends upon t h e
number o f 1 i n e a r l y independent v e c t o r s i n t h e g e n e r a l v o t i n g system.
Since we have
a continuum o f them a v a i l a b l e , and s i n c e a t l e a s t two v e c t o r s i n t h i s system d e f i n e
a t h r e e dimensional space w i t h
vector
VN
which, along w i t h
WN,
EN, we
can choose t h e mJ's t o o b t a i n a v o t i n g
s a t i s f i e s the c o n d i t i o n o f the lemma.
t h e l e m a , choose t h e r a n k i n g c o r r e s p o n d i n g t o
r a n k i n g corresponding t o
EN t o
VN
To use
t o be complete i n d i f f e r e n c e , t h e
be as s p e c i f i e d i n t h e theorem, and t h e p a i r w i s e
r a n k i n g s a s s p e c i f i e d i n t h e theorem.
The c o n c l u s i o n then f o l l o w s from t h e above
and t h e lemma.
Proof of Theorem 3.
I n t h i s s e t t i n g , t h e domain and t h e image o f F changes
d r a s t i c a l l y f r o m t h a t g i v e n above.
a t l e a s t two c a n d i d a t e s .
Thus,
Here we have 2N-(Ntl)
d i f f e r e n t subsets w i t h
t h e range space i s t h e c a r t e s i a n p r o d u c t over a l l o f
these s e t s o f E u c l i d e a n spaces o f the same dimension as the number o f c a n d i d a t e s i n
the subset.
The danain a l s o i s i n c r e a s e d s i g n i f i c a n t l y .
a g e n e r a l v o t i n g method.
F o r each s u b s e t , t h e r e i s
Thus, f o r each r a n k i n g o f t h e c a n d i d a t e s i n each s u b s e t ,
t h e domain i s i n c r e a s e d by a n o t h e r p r o d u c t o f a s i m p l e x r e f l e c t i n g t h e v a r i o u s
c h o i c e s t h e v o t e r s have t o t a l l y t h e i r b a l l o t s .
L e t t h e new domain, which i s a much
l a r g e r p r o d u c t space of s i m p l i c e s , be g i v e n b y T',
g i v e n b y R'.
The t a l l y o f t h e b a l l o t s s t i l l i s g i v e n by summations o f t h e type
found i n Eq. 4.2.
F':T'
and l e t t h e l a r g e r image space be
They d e f i n e a mapping
--------)
R',
As i n t h e statement o f t h e theorem, d e s i g n a t e f o r each p a i r o f c a n d i d a t e s which
one i s t o be p r e f e r r e d by a m a j o r i t y o f t h e v o t e r s .
t61.
We now appeal t o Theorem 6 i n
A consequence o f t h i s r e s u l t i s t h a t f o r 'mostm s i m p l e v o t i n g systems, 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 f o r each o f the p a i r s o f t h e c a n d i d a t e s , a m a j o r i t y
o f them f a v o r t h e d e s i g n a t e d c a n d i d a t e .
Yet, t h e i r rankings o f a l l subsets w i t h
t h r e e o r more c a n d i d a t e s i s complete i n d i f f e r e n c e .
"Most' r e p l a c e s the l i n e a r
independence c o n d i t i o n i n t h e lemma, and i t means t h a t the v o t i n g v e c t o r s f o r
v a r i o u s s u b s e t s don't
make a c e r t a i n d e t e r m i n a n t v a n i s h .
For our p u r p o s e s i t
s u f f i c e s t o n o t e t h a t f o r t h i s c o n d i t i o n i s s a t i s f i e d f o r any v o t i n g v e c t o r w h i c h i s
'plurality
like'.
To use t h i s theorem, we f i n d a s p e c i a l case of t h e g e n e r a l v o t i n g
system w h i c h i s a s i m p l e v o t i n g system s a t i s f y i n g t h e above.
F o r each subset o f more t h a n t h r e e c a n d i d a t e s , choose m ~ r t a ~ m T~h i.s ,
t h e n , d e f i n e s a convex c o m b i n a t i o n o f t h e v o t i n g v e c t o r s w h i c h a r e a v a i l a b l e t o
t a l l y the rankings.
Now, choose t h e m ~ ' s s o t h a t CmJIAJ d e f i n e s a v e c t o r o f
t h e t y p e i n D e f i n i t i o n 3.
vanish.
Such a v e c t o r doesn't make t h e d e t e r m i n a n t c o n d i t i o n
I f n o t a l l of t h e
J'S a r e p o s i t i v e , t h e n t h e y can be p e r t u r b e d s o t h a t
a l l a r e p o s i t i v e and t h e sum i s s t i l l a v e c t o r which s a t i s f i e s t h e n o n - v a n i s h i n g of
the determinant.
( T h i s i s because t h e d e t e r m i n a n t c o n d i t i o n i s an open c o n d i t i o n . )
Thus, f o r each subset, a c h o i c e of t h e Cm~3 can be made so t h a t t h e r e s u l t i n g
v e c t o r s o v e r a l l s u b s e t s do n o t s a t i s f y t h e v a n i s h i n g d e t e r m i n a n t c o n d i t i o n .
L e t m' c o r r e s p o n d t o t h e s e c h o i c e s o f C m ~ p t a ) 3over a l l s u b s e t s .
have from Theorem 6 i n 161 t h a t 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 , n',
v a r i o u s components o f F((n',m'))
t h e range.
so t h a t the
a r e on t h e l i n e o f complete i n d i f f e r e n c e , y e t t h e
r a n k i n g o f the p a i r s i s as designated.
t h e J a c o b i a n o f F',
Then we
What remains t o be shown i s t h a t t h e r a n k of
where n i s h e l d f i x e d ,
i s of the r a n k o f the t h e dimension of
B u t , w i t h t h e m o d i f i c a t i o n s o f t h e type g i v e n i n t h e p r o o f o f Theorem 3,
t h i s f o l l o w s from t h e p r o o f o f Theorem 6 i n [&I.
Indeed, t h e p r o o f o f Theorem 6 i s
based upon t h i s independence c o n d i t i o n h o l d i n g .
Proof of C o r o l l a r y 3.1.
isn't
Theorem 4 i n [&I a s s e r t s t h a t i f t h e v o t i n g v e c t o r
a Borda V e c t o r , t h e n f o r any r a n k i n g s o f t h e p a i r s o f a l t e r n a t i v e s and f o r any
r a n k i n g of t h e N a l t e r n a t i v e s , t h e r e i s a p r o f i l e o f v o t e r s w h i c h w i l l r e a l i z e a l l
o f these outcomes s i m u l t a n e o u s l y .
I n p a r t i c u l a r , the r a n k i n g s o f t h e p a i r s can be
chosen t o d e f i n e a Condorcet w i n n e r , and t h e r a n k i n g o f the N c a n d i d a t e s can be
chosen t o d e f i n e t h e complete i n d i f f e r e n c e r a n k i n g .
The h y p o t h e s i s o f t h e c o r o l l a r y
a s s e r t s t h a t t h e r e i s a t l e a s t one v o t i n g v e c t o r i n t h e general methods w h i c h i s n ' t
a Borda V e c t o r .
Thus, m'
can be chosen i n a manner s i m i l a r t o t h e above so t h a t t h e
r e s u l t i n g v o t i n g vector isn't
Borda.
Then, t h e same t y p e o f p r o o f goes t h r o u g h .
P r o o f o f Theorem 5 f o r h y p o t h e s i s o f Theorem 1.
The b a s i c i d e a s a r e
demonstrated f o r Theorem 1; s i n c e t h e ideas e x t e n d immediately f o r t h e o t h e r
theorems, we o n l y p r o v e t h i s case.
To p r o v e t h e theorem, a1 1 t h a t i s necessary i s t o show t h a t t h e r e i s an open
s e t U about n* i n S i ( N ) ( t h e space o f v o t e r s '
t h e r e i s an i n t e r i o r p o i n t m'
complete i n d i f f e r e n c e .
i n (Si(s))N!such
p r o f i l e s ) so t h a t i f n'
t h a t F((n',m'))
i s i n U, t h e n
i s on t h e l i n e o f
To do t h i s , we g i v e a g e o m e t r i c i n t e r p r e t a t i o n o f Eq. 4 . 2 .
For each r a n k i n g o f t h e c a n d i d a t e s , P(A),
t h e b r a c k e t e d term i n t h e double
summation Eq. 4 . 2 d e f i n e s t h e convex h u l l o f t h e v e c t o r s ( l d ~ t t a 1 ) . Thus, t h i s
means t h a t t h e double summation y i e l d s t h e convex h u l l o f t h e N! convex h u l l s .
f a c t t h a t t h e image of F((n*,m))
The
c o n t a i n s an open s e t means t h a t t h i s p a r t i c u l a r
convex c o m b i n a t i o n o f t h e convex h u l l s c o n t a i n s a open s e t around t h e p o i n t
F((n*,m*))
on t h e l i n e o f complete i n d i f f e r e n c e .
I t now f o l l o w s f r o m c o n t i n u i t y
c o n s i d e r a t i o n s t h a t the conclusion holds.
We c o n c l u d e w i t h an o b s e r v a t i o n which i n d i c a t e s t h a t t h e r e i s a l a r g e
l i k e l i h o o d o f a s t o c h a s t i c e f f e c t f o r a g e n e r a l v o t i n g system.
According t o the
above p r o o f , a measure of t h i s i s the abundance o f t h e p o i n t s (n',m')
F((n',m'))
i s on t h e l i n e o f complete i n d i f f e r e n c e .
independence argument,
such t h a t
B u t , a c c o r d i n g t o t h e above
i t f o l l o w s f r o m the imp1 i c i t f u n c t i o n theorem t h a t t h e
i n v e r s e image o f t h e complete i n d i f f e r e n c e l i n e i n T i s an a f f i n e space o f
codimension N-1.
To o b t a i n some f e e l i n g f o r t h e s i z e o f t h i s space, c o n s i d e r t h e
s e t t i n g where N=4 and where we a r e u s i n g an a p p r o v a l v o t i n g system (s=3).
Then, t h e
base p o i n t s d e f i n e a 58 dimensional l i n e a r subspace i n a 61 d i m e n s i o n a l space.
I n c i d e n t l y , these l a r g e dimensions a l r e a d y f o r o n l y 4 c a n d i d a t e s i n d i c a t e s 1 ) why
s u f f i c e i n t h e a n a l y s i s o f such v o t i n g schemes, 2)
s t a n d a r d methods won't
why we
used a c o n v e x i t y argument t o prove Theorem 5 i n s t e a d o f an i m p l i c i t f u n c t i o n
argument ( w h i c h w o u l d have i n v o l v e d a massive l i n e a r independence argument), 3) why
t h e s t o c h a s t i c e f f e c t o c c u r s (F i s t r y i n g t o f o r c e t h e i n p u t f r o m a 61 dimensional
space i n t o a 3 dimensional space, so we must expect such r e s u l t s ) , 4) why s i m p l e
v o t i n g systems don't
have as adverse e f f e c t s ( t h e subspace i s 20 d i m e n s i o n a l i n a 23
d i m e n s i o n a l space), and 5) t h a t t h e r e a r e many examples o t h e r t h a n those suggested
by t h e p r o o f o f the theorems.
(Because t h e 58 dimensional space i s a f f i n e , i t must
i n t e r s e c t t h e b o u n d a r i e s o f T. The b o u n d a r i e s correspond t o examples of v o t e r s '
p r o f i l e s where t h e r e a r e no v o t e r s w h i c h have c e r t a i n r a n k i n g s o f the N c a n d i d a t e s . )
A c k n o w l edgemen t s
The work o f b o t h a u t h o r s was s u p p o r t e d , i n p a r t , b y NSF Grant IST 8415348. As
m e n t i o n e d i n t h e i n t r o d u c t i o n , t h i s work was p a r t i a l l y s t i m u l a t e d b y q u e s t i o n s and
comments made by S. Brams, C a r l Simon, and t h e o t h e r p a r t i c i p a n t s a t t h e J u l y , 1984,
'Workshop on Mathematical Models i n P o l i t i c a l Science' which was o r g a n i z e d b y John
A l d r i c h and C a r l Simon. T h i s conference was j o i n t l y sponsored b y the NSF I n s t i t u t e
f o r M a t h e m a t i c s and i t s A p p l i c a t i o n s and t h e Department o f P o l i t i c a l Science, b o t h
a t t h e U n i v e r s i t y of Minnesota, M i n n e a p o l i s , Minnesota.
R e fe r c n c c s
1.
N a g e l , J., A debut f o r approval v o t i n g , PS W i n t e r 1984, l7, 62-65.
Brams, S., and P. F i s h b u r n , h e r i c a ' s u n f a i r e l e c t i o n s , The Sciences, Now. 1983,
2.
r e p r i n t e d i n E a s t e r n A i r l i n e s ' Review Magazine, Feb. 1984.
3.
Brams, S.,
and P. F i s h b u r n , A ~ p r o v a lU o t i n q , B i r k h a u s e r , Boston, 1982.
4.
S a a r i , D. G., The u l t i m a t e o f chaos r e s u l t i n g from w e i g h t e d v o t i n g systems,
Advances i n A p p l i e d Mathematics 3 (19841, 286-308.
5. S a a r i , D. G., The source of some paradoxes from s o c i a l c h o i c e and p r o b a b i l i t y ,
NU Center f o r Math Econ D i s c u s s i o n paper M609, June, 1984.
6 . S a a r i , D. G., The o p t i m a l r a n k i n g method i s t h e Borda Count, NU D i s c u s s i o n
p a p e r , Jan. 1985.
7 . S a a r i , D . G., I n c o n s i s t e n c i e s o f w e i g h t e d v o t i n g systems, M a t h o f O p e r a t i o n s
Research z ( 1 9 8 2 1 , 479-490.