The Source of Some Paradoxes
from Social Choice and
Probability
Saari, D.G.
IIASA Collaborative Paper
August 1986
Saari, D.G. (1986) The Source of Some Paradoxes from Social Choice and Probability. IIASA Collaborative Paper.
IIASA, Laxenburg, Austria, CP-86-025 Copyright © August 1986 by the author(s). http://pure.iiasa.ac.at/2860/
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THESOURCEOFSOMEPARAD0XE;SFROM
SOCIAL CHOICE AND PROBABILITY
Donald G. S a a r i
August 1986
CP-86-25
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 h a s not been p e r f o r m e d solely
at t h e International Institute f o r Applied Systems Analysis and which
h a s r e c e i v e d only limited review. Views o r opinions e x p r e s s e d h e r e i n
d o not necessarily r e p r e s e n t t h o s e 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 APPLIET SYSTEMS ANALYSIS
2361 Laxenburg, Austria
FOREWORD
This p a p e r o f f e r s a powerful, simple method f o r understanding many "paradoxes" in social choice and probability t h e o r y . The a p p r o a c h i s a geometrical one;
t h e underlying principle e m e r g e s from a wide v a r i e t y of examples ranging from
elections a n d agenda manipulation to gambling a n d conditional probabilities.
Alexander B. Kurzhanski
Chairman
System a n d Decision S c i e n c e s P r o g r a m
THE; SOURCE OF S O m P A W O E S FROM
SOCIAL CHOICE AND PROBABTLITT
DonaLd G. Saari
Department of Mathematics, Northwestern University, Evanston, Illinois
The social choice l i t e r a t u r e and t h e probability l i t e r a t u r e are filled with
descriptions of "paradoxes". A s w e show h e r e , many of them c a n b e explained and
extended by using t h e same, simple, geometric argument. Our extensions include
s e v e r a l new r e s u l t s about t h e intransitivities of election results o v e r subsets of alternatives, t h e cycles of agenda manipulation, gambling, and Simpson's paradox
from conditional probability. Furthermore, we prove t h a t t h e s e paradoxes must
accompany t h e modeling in a robust fashion. Our a p p r o a c h a p p e a r s t o be new, i t i s
elementary (based on t h e open mapping principal), and i t uncovers new examples.
Indeed, one point t h a t emerges i s t h e e a s e with which paradoxes (i.e., a p p a r e n t
contradictions in a relationship) can a r i s e .
Our argument extends beyond social choice and probability, but we emphasize
these two a r e a s because of t h e i r familiarity and t h e i r importance as standard
modeling tools f o r economics and decision analysis. Examples from probability a r e
discussed in Sections 2 and 3; examples from social choice a r e discussed in Section
3. To simplify t h e exposition, we use d i s c r e t e random variables. However, all this
work easily generalizes t o more general models.
The flavor of o u r r e s u l t s is indicated by the following two prototype examples.
In t h e following sections, w e show how they are r e l a t e d and how they c a n b e extended.
1.1 Conditional Probability and Simpson's Paradox
Suppose a c e r t a i n d r u g is t e s t e d in Chicago ( C ) a d in Los Angeles (C'). A test
g r o u p (T) r e c e i v e s t h e new d r u g , and a c o n t r o l g r o u p (T') r e c e i v e d t h e s t a n d a r d
t r e a t m e n t . Some people are r e t u r n e d t o h e a l t h (H), while o t h e r s are n o t (H').
Suppose t h a t in both communities t h e new d r u g i s judged t o b e successful b e c a u s e
i t c u r e s t h e sick with a h i g h e r r a t i o t h a n t h e s t a n d a r d t r e a t m e n t
Is i t possible f o r t h e a g g r e g a t e d t e s t r e s u l t s to h a v e t h e r e v e r s e conclusion
P(H:T)
< P(H:T')?
I t is, a n d t h i s i s known as Simpson's p a r a d o x . An explanation
(which d i f f e r s from t h a t given h e r e ) a n d how i t r e l a t e s to t h e "sure thing" principle i s given by Blyth (1972a). Examples with r e a l d a t a are given b y Wagner (1982).
This inconsistency phenomenon t u r n s o u t to b e a c h a r a c t e r i s t i c of models
based on conditional probability o r t h e combination of random v a r i a b l e s . When additional conditions are introduced, almost any imaginable extension c a n o c c u r . F o r
instance, s u p p o s e t h a t t h e tests are conducted using facilities provided b y univers i t i e s (U)a n d p r i v a t e l a b o r a t o r i e s (U'). T h e r e e x i s t examples whereby t h e new
d r u g i s unsuccessful a t e a c h university and at e a c h l a b o r a t o r y a s w e l l as in e a c h
community, b u t it is successful in t h e a g g r e g a t e , and t h e r e exist examples where
t h e conclusions oscillate with t h e level: t h e new d r u g i s successful a t e a c h of t h e
facilities, it i s unsuccessful in e a c h community, and i t i s successful in t h e a g g r e gate, etc.
The a p p r o p r i a t e r a t i o of s u c c e s s t o f a i l u r e in e a c h of t h e s e examples can b e
made t o e x c e e d a n y predetermined constant. This means t h a t t h e r e e x i s t examples
of d a t a w h e r e at e a c h facility t h e probability of regaining h e a l t h by u s e of t h e
d r u g i s at l e a s t twice t h a t obtained by t h e s t a n d a r d t r e a t m e n t ; in e a c h community,
t h e s t a n d a r d t r e a t m e n t i s at l e a s t t h r e e times b e t t e r t h a n t h e d r u g ; a n d a t t h e t o t a l
a g g r e g a t e d level, t h e d r u g is at least f o u r times b e t t e r t h a n t h e s t a n d a r d t r e a t ment!
1.2 Voting and Ranking Methods
The a g g r e g a t i o n of p r e f e r e n c e i s a c e n t r a l i s s u e in t h e social s c i e n c e s . A simple system i s voting. H e r e , s e v e r a l p a r a d o x e s o c c u r when t h e v o t e r s r a n k a s e t of
t h r e e a l t e r n a t i v e s la , b ,c 1 by using t h e stanC t r d plurality voting system. Suppose
t h e r e are nine v o t e r s where f o u r of them giv t h e ranking
c
> a > b , three
give
t h e ranking b
c
>b >a
> a > c , and two give t h e
ranking a
> b > c.
The g r o u p ' s r a n k i n g i s
with a tally of 4 : 3 : 2. This ranking is inconsistent with t h e f a c t t h a t a
majority of t h e v o t e r s (five of them) p r e f e r t h e bottom r a n k e d a l t e r n a t i v e a t o t h e
top ranked alternatives c .
I t might b e s u s p e c t e d t h a t a n election ranking of N a l t e r n a t i v e s must have
some relationship t o how t h e group r a n k s at l e a s t one of t h e p a i r s of a l t e r n a t i v e s .
This need not b e t h e case. F o r e a c h of t h e N(N-1)/2
p a i r s of a l t e r n a t i v e s desig-
n a t e , in a n a r b i t r a r y fashion, one of t h e a l t e r n a t i v e s . W e show t h a t t h e r e e x i s t
examples of v o t e r s ' rankings of a l t e r n a t i v e s s o t h a t (1) t h e plurality election
result is al
> a 2 > ... > a ~ and
, (2) f o r e a c h p a i r of t h e a l t e r n a t i v e s , a majority of
t h e s a m e v o t e r s p r e f e r t h e designated a l t e r n a t i v e .
2. THE GENERAL RESULT
The simple geometric p r o p e r t y w h e r e open s e t s are mapped t o o p e n sets i s t h e
unifying explanation f o r all t h p a r a d o x e s d e s c r i b e d in t h i s p a p e r . The following
s t a n d a r d statement (see Warner, 1970) suffices f o r what follows.
Proposition 1
Let F b e a smooth mapping from a n m-dimensional manifold M t o a n n dimensional manifold N where m
> n.
Let c b e a n i n t e r i o r point of N. Assume t h a t
p in F - l ( c ) i s a n i n t e r i o r point of M . If t h e Jacobian of F at p h a s maximal r a n k ,
t h e n t h e r e i s a n o p e n neighborhood of p t h a t i s mapped onto a n o p e n neighborhood
of c .
The proof of t h e following theorem i l l u s t r a t e s why t h e a b o v e p r o p e r t y i s t h e
s o u r c e of t h e paradoxes.
Theorem 1
Consider t h e example in Section 1.1where a d r u g i s compared with a s t a n d a r d
treatment.
C
Let
A
be
+ C',C,C1,CU,CU',C'U,
the
and C'U'.
variable
representing
the
seven
sets
F o r e a c h c h o i c e of A , designate which t e r m
from t h e p a i r (P(H:TA),P(H:TtA)) i s t o h a v e t h e l a r g e r value. Choose a c o n s t a n t
clA g r e a t e r t h a n unity and e x p r e s s e a c h p a i r as a r a t i o t h a t i s bounded below by
clL. T h e r e e x i s t finite examples of d a t a t h a t simultaneously satisfy all t h e s e v e n
specified inequalities.
To p r o v e t h e theorem, i t suffices t o show t h a t , f o r any c h o i c e of signs f o r t h e
seven quantities P(H:TA) - P ( H : T r A ) ,t h e r e e x i s t sample points t h a t r e a l i z e them.
View t h e s e quantities as defining t h e seven components of a mapping F into R7. The
choice of t h e signs identifies a n o r t h a n t B of R7.
The origin 0 of R 7 i s a boundary point f o r e a c h of t h e o r t h a n t s . This "comparison point" i s used in t h e following way. F i r s t , a point p in F - ~ ( o )i s found s o
t h a t (i) i t i s a n i n t e r i o r point of t h e domain, a n d (ii) t h e Jacobian of F at p h a s
maximal r a n k . According t o Proposition 1,F maps a n open neighborhood of p o n t o
an open neighborhood of 0 . This o p e n image set meets e a c h of t h e o r t h a n t s ; in
p a r t i c u l a r i t meets o r t h a n t B. T h e r e f o r e , t h e r e are sample points t h a t satisfy all
seven conditions simultaneously. The technical p a r t of t h e proof i s t o define t h e
domain s o t h a t F c a n b e r e p r e s e n t e d by a smooth mapping.
Proof
T h e r e are eight sets determined by t h e intersections of t h e sets T , C , U a n d
t h e i r compliments. They are:
S 1 = TCU S 2 = TCU' 3.3 = TC'U S 4 = TC'U'
S 5 = T'CU S 6 = T'CU' S7 = T'C'U S 8 = T'C'U'
T r e a t e a c h of t h e s e sets as a disjoint s p a c e . Let Xj designate t h e c h a r a c t e r i s t i c
function
of
H
in
S .
5 = X Z j + X 2 j - 1 ,j = 1,...,4, and
v a r i a b l e 5 , which i s t h e c h a r a c t e r i s t i c
Define
-
Zj = Y21 + Y 2 j - l , j = 1 3 . The random
function of H o v e r S z j + S 2 j - 1 , r e p r e s e n t s t h e r e s u l t s at t h e community level,
while Zj
= Y Z j + Y Z j r e p r e s e n t s t h e final a g g r e g a t e d r e s u l t s .
If z j denotes t h e value of P(Xj
I,j = 1,..., 8 . Let
= I), t h e n
z j i s in t h e unit i n t e r v a l
9 designate P ( S j ) in t h e s p a c e usk.The 9 v a r i a b l e s d e s c r i b e
a simplex in R 8 which i s denoted by S i ( 8 ) and defined by
These 1 6 v a r i a b l e s are in t h e 15-dimensional s p a c e M = l8X S i ( 8 ) .
By use of t h e s t a n d a r d relationship, f o r any set E ,
it follr u s t h a t t h e probabilities yl = P ( 5 = 1 ) a n d z j = P ( Z j = 1 ) a r e t h e r a t i o n a l
functions
and
C o m p a r i s o n m a p : Let F:M
+
R~ b e
where e j i s t h e unit v e c t o r in R7 with unity in i t s j t h component. The components
of F r e p r e s e n t P(H:TA )
- P ( H :TA ') as A
r a n g e s t h r o u g h i t s seven values.
O p e n m a p p i n g : Clearly, F i s a smooth mapping. That t h e Jacobian of F h a s maxi-
mal r a n k at some preimage of 0 i s a d i r e c t computation. Indeed, t h i s r a n k condition holds e v e r y w h e r e e x c e p t on a c e r t a i n lower-dimensional s u b s e t of M . These
points of lower r a n k are where e i t h e r t h e y values o r t h e z values are uniquely
determined b e c a u s e t h e corresponding p a i r s of x o r y are equal.
The signs chosen f o r t h e seven quantities determine a n o r t h a n t of R7,denoted
by B. By construction, all t h e sample points with t h i s behavior are in U = F-'(B).
By t h e continuity of F,U i s a n o p e n s e t ; w e must show t h a t i t i s nonempty. The Jacobian of F h a s maximal r a n k at some i n t e r i o r point of M in F-'(o), s o F maps a n open
s e t from M o n t o a n open set of 0. This open set meets B . Consequently F-'(B) i s
nonempty .
?\Text w e show t h a t U contains points t h a t c a n b e identified with a finite d a t a
s e t . Any r a t i o n a l point will suffice. A multiple of t h e common denominator of d is
t h e t o t a l number of subjects. The same multiple of t h e numerator of d c o r r e s p o n d s
t o t h e cardinality of S f , and i t s e r v e s as a multiple of t h e denominator of z j . Bec a u s e t h e r a t i o n a l points are dense, t h e r e i s a n infinite set of r a t i o n a l points in
F - ~ ( B ) . E a c h point c a n b e identified with a n infinite number of d i f f e r e n t finite
d a t a sets.
The s e t F-'(o): It remains t h a t t h e inequalities can b e bounded below by t h e designated constants. Once t h e values of
type z j
> dAXj +4
4
a r e specified, t h e inequalities are of t h e
with a similar relationship y and z . Let q = ( q l ,. . . , q , ) b e a point
in B. The set F-'@ ) i s given by equations of t h e form z j
- z j +4 = lqjl with
similar
equations f o r y and z . These equations define lower-dimensional hyperplanes in
t h e domain, s o t h e r e are points in U satisfying inequalities of t h e form
The assertion follows if t h e r e i s a point in U such t h a t t h e right-hand sides of
t h e s e inequalities are bounded below by d A , with similar statements f o r y and z .
These inequalities are satisfied if t h e values of t h e denominators on t h e right-hand
sides c a n be chosen t o be a r b i t r a r i l y small. This involves a d i r e c t computation
t h a t i s easily done because F-'(o) contains t h e intersection of 0
X
Si ( 8 ) and t h e
boundary of U. The points are chosen a r b i t r a r i l y close t o this set.
Comments:
( 1 ) The basic idea of t h i s proof extends t o all t h e paradoxes discussed h e r e .
Individual comparisons are one dimensional. When s e v e r a l comparisons
are made, they m u s t b e viewed as defining a comparison mapping F with a
higher-dimensional r a n g e space. A higher-dimensional s p a c e admits symmetries and cycles, s o it should be expected t h a t t h e s e cycles are manifested as paradoxes by t h e comparisons. To prove t h a t all t h e symmetries are admitted, locate a "comparison point" on t h e boundary of
e a c h of t h e comparison regions. Next, show t h a t t h e image of F includes
an open set about t h e comparison point. The intersection of t h i s open set
with e a c h comparison region is a nonempty open set. Because F i s continuous, t h i s means t h a t t h e r e i s a nonempty set of points in t h e domain
with t h e desired p r o p e r t i e s . To complete t h e proof, impose conditions s o
t h a t in each of t h e s e sets in t h e domain t h e r e exist points t h a t are identified with sample points from t h e model. This simple idea i s t h e essence of
o u r explanation f o r all t h e paradoxes in t h i s p a p e r .
(2) The number of possible, paradoxial relationships i s determined by t h e dimension of t h e domain f o r a comparison mapping.
If t h i s dimension
exceeds t h a t of t h e range, then t h e comparison mapping i s not "complete"; t h e r e exist additional relationships t h a t may define more complex
paradoxes.
A s a c o r o l l a r y , t h e a b o v e illustration and extension of
Simpson's p a r a d o x i s not t h e "best possible" r e s u l t . The domain of F i s
15 dimensional while t h e r a n g e i s only seven dimensional; e i g h t more com-
p a r i s o n s using t h e s e v a r i a b l e s c a n b e added. (They may involve d i f f e r e n t
levels of aggregation, waiting times, e t c . )
(3) O t h e r conclusions a r e derived from t h e p r o p e r t i e s of
(i)
F-l.
I t i s n a t u r a l t o determine t h e limits of a paradox. (This i s illustrated in Theorem 1 with t h e a s s e r t i o n t h a t dA i s not bounded above.)
Often, as f o r t h i s model, t h e s e limits are determined b y t h e p r o p e r t i e s of t h e points n e a r t h e intersection of F-'(o)
a n d t h e boundary
of t h e domain.
(ii) In o r d e r f o r a p a r a d o x (described by B) t o o c c u r , we may need a
c e r t a i n number of d a t a points. The minimal size i s given by t h e smal-
lest
"lowest
common denominator"of
t h e admissible points in
F - ~ ( B ) = U.
(iii) The probability t h a t a p a r a d o x (described by B) o c c u r s is given by
t h e measure of a probability distribution o v e r t h e o p e n set F-~(B).
(4) O t h e r conclusions are d e r i v e d from t h e s t r u c t u r e of t h e image of F. F o r
instance, t h e image contains a n open set a b o u t t h e origin, so it meets any
sector defined by a specified r a t i o of t h e outcomes; e.g.,
The a b o v e shows t h a t t h e r e are sample points t h a t satisfy t h e s e conditions.
(5) To avoid t h e a b o v e behavior, t h e Jacobian of F cannot b e of maximal
r a n k . This singularity c o n s t r a i n t becomes a necessary condition t o avoid
a p a r a d o x . Often, as f o r t h e a b o v e model, t h e s e lower-dimensional singul a r i t y conditions c o r r e s p o n d t o familiar c o n s t r a i n t s such as t h e "independence of random variables".
This a p p r o a c h c a n b e used as long as t h e components of a comparison mapping
are smooth functions. These components could b e functional combinations of p r o babilities, e x p e c t e d values, t h e v a r i o u s moments, waiting times, loss functions, decision r u l e s , c o r r e l a t i o n indices, s c a t t e r i n g indices, covariance, etc. If t h e open
mapping condition holds a t a comparison point, t h e n all possible comparisons c re
realized. In t h i s way i t is e a s y t o show t h a t t h e r e e x i s t examples illustrating, ' o r
instance, t h a t t h e expected value may satisfy E (X)
> E (Y),
yet E (J' (X))
< E (J' (Y))
f o r some monotonically increasing function 'j , and t h a t c e r t a i n decision r u l e s may
b e inconsistent with o t h e r measures. (Indeed, a d i s c r e t e version of this can b e
used t o explain t h e Arrow social choice paradox.) Theorem 2 i s t h e formal statement t h a t c o v e r s all t h e s e situations.
Before stating Theorem 2, w e formally define t h e s t r u c t u r a l relationship
between a comparison point and a comparison region.
Definition
Let a topological s p a c e N b e partitioned. A comparison point is a boundary
point f o r each partition set. For a given comparison point p , a comparison reg i o n is a partition set such t h a t t h e closure of its i n t e r i o r contains p .
Definition
Let F : M -,N b e a comparison mapping f o r a given model. A point in M is a n
admissible point if i t can b e identified with a sample f o r t h e model.
Theorem 2
Let F:M -,N b e a smooth comparison mapping where t h e dimension of M i s
bounded below by t h e dimension of N. Assume t h a t t h e admissible points form a
dense set in M . Let c b e a comparison point in N. If p in F-'(c) i s a n i n t e r i o r
point of M such t h a t t h e Jacobian of F at p h a s maximal r a n k , then t h e behavior
characterized by any comparison region of N is admitted.
Example
Consider t h e following dice game. Each of t h e players rolls his own weighted
die. (Each die i s marked in t h e standard fashion.) On each roll, t h e winner i s t h e
player t h a t rolled t h e l a r g e r f a c e value. For each choice of k = 1,...,4, t h e losing
player pays t h e winning player t h e difference between t h e f a c e values raised t o
t h e k t h power. For each of t h e f o u r choices of k , a r b i t r a r i l y s e l e c t a die t o have
t h e l a r g e r expected payoff, and then a r b i t r a r i l y select a die t o have t h e higher
probability of winning a roll. It i s a d i r e c t consequence of Theorem 2 t h a t t h e dice
can b e weighted in such a fashion t h a t all five selected conditions are satisfied
simultaneously. This illustrates t h e possible incompatibili .y among reward functions and t h e distributions.
A s a s p e c i a l case (K1 = I ) , t h e following demonstrates t h a t t h e more p r o b a b l e
of two e v e n t s may h a v e t h e longer waiting time.
Corollary 2.1
Let e a c h of t h e two u r n s U1 and U 2 contain r e d and black balls. Each u r n i s
randomly sampled without replacement. F o r a positive i n t e g e r k , l e t w j (k ) b e t h e
probability t h a t i t t a k e s at l e a s t k t r i e s b e f o r e a r e d ball is s e l e c t e d from u r n U j ,
j
= 1 , 2 . Let k l
# k 2 . F o r e a c h of t h e two p a i r s ( w l ( k S ) , w 2 ( k S ) ) ,=
s 1 3 , choose
t h e value t h a t is t o b e t h e l a r g e r . T h e r e e x i s t examples of d a t a s o t h a t both conditions are satisfjed simultaneously.
Outline of the proof
W e proof t h e s p e c i a l case where K1 = 1 and K2 = 2 . The g e n e r a l case follows
in much t h e same manner.
The domain of t h e comparison mapping F i s I x I x R + where R + i s t h e halfline of positive numbers. A point in t h e domain i s denoted by ( z ,y , z ) . Let
b e a mapping into R'.
t h e mapping
(W
A t t h e r a t i o n a l points in t h e domain, F c a n b e identified with
-
i ( l ) w ' ( 1 ) , w1 ( 2 ) - w ' ( 2 ) ) . This identification follows by assuming
t h a t t h e r e are zzn r e d a n d zn ( 1
- z) black
balls in U 1 and y n r e d and n (1--y )
black balls in U z , and by choosing a n a p p r o p r i a t e value f o r t h e p a r a m e t e r n .
The comparison point i s 0 = (0,O). Any domain point p of t h e form ( z , z , l )i s
in F - l ( 0 ) . The g r a d i e n t of t h e f i r s t component of F i s ( 1 ,
- 1,O); t h e g r a d i e n t of
t h e second component evaluated at p i s
[l + (l-Z=)zn, zn -1
where z
+ ( 1-22
)m;,
z n -1
(2 -1)z
( z n -1)'
I
= 1 . Clearly, t h e s e two v e c t o r s are linearly independent. This completes
t h e proof.
The conclusion of t h i s c o r o l l a r y holds even if o n e of t h e s e p a i r s is r e p l a c e d
with t h e p a i r of e x p e c t e d waiting times. However, t h e conclusion does n o t hold if
t h e sampling i s with replacement, o r if t h c number of balls i s t h e same f o r e a c h
urn. F o r e a c h of t h e s e models, t h e z terr.1 does n o t a p p e a r in t h e definition of F.
A s a r e s u l t , t h e t h i r d component of t h e I -adient is z e r o , a n d t h e second is t h e
negative of t h e f i r s t . Therefore, t h e Jacobian of t h e comparison mapping i s singul a r . This illustrates comment (5).
A more interesting paradox i s obtained by combining t h e model in Theorem 1
with t h e one given above. Here, s e v e r a l p a i r s of u r n s with r e d and black balls are
used. A s t h e contents of t h e u r n s a r e combined in a specified way, t h e urn with t h e
higher probability of selecting a r e d ball may change with t h e level of aggregation.
Furthermore, t h e waiting time t o select a r e d ball may vary. However, as demons t r a t e d above, t o obtain t h e s e examples, often w e need t h e e x t r a d e g r e e s of freedom o f f e r e d by varying t h e number of balls p e r urn. Also, t h e number of independent comparisons is bounded by t h e dmension of t h e domain.
S e v e r a l interesting paradoxes from population dynamics involve only a small
number of comparisons, so i t i s t o use Theorem 2 t o explain and extend them (see,
f o r example, t h e p a p e r by Vaupel and Yashin, 1985). However, often t h e s e examples, as given by Vaupel and Yashin, a r e based on continuous random variables. To
use Theorem 2, t h e continuous variables are approximated by d i s c r e t e valued random variables.
Alternatively, Proposition 1 can b e extended, in t h e obvious
fashion, t o permit M t o b e a function space. In this way, t h e examples of Vaupal
and Yashin c a n b e t r e a t e d directly.
Another s o u r c e of paradoxes subsumed by Theorem 2 i s Blyth's p a p e r (1972
b). One of his paradoxes with random variables X and Y h a s P(X
unity as desired, even though P ( X
< a ) <P(Y < a )
> Y)
as close t o
f o r all choices of a: This, of
course, i s a n example of t h e boundary behavior of t h e comparison mapping. Both
Blyth (1972 b) and Vaupel and Yashin (1985) d e s c r i b e t h e paradoxes in terms of examples. The above treatment explains and unites them, i t shows t h a t they can b e
extended in s e v e r a l ways, and i t proves t h a t t h e paradoxes are "robust" in t h a t
they are satisfied by open sets of examples.
Theorem 1 and its generalization to a set of N c h a r a c t e r i s t i c functions are
corollaries of Theorem 2. The only surprising f e a t u r e of t h e generalization i s t h a t
t h e dimension of t h e domain f o r a comparison mapping c a n b e v e r y large. To s e e
this, let t h e f i r s t N-1 c h a r a c t e r i s t i c functions define
zN-l sets.
The last random
variable i s t r e a t e d as a c h a r a c t e r i s t i c function on each set. Thus, t h e domain of a
comparison mapping has t h e dimension
comment (2) this means t h a t up t o
zN-l + (zN-l - 1 ) = zN - 1.
zN - 1 functional
According t o
relationships can b e defined
from t h e s e random vari3bles with possible concomitant unexpected behavior.
W e conclude t h i s section with a p a r t i a l c o n v e r s e f o r Theorem 2. I t asserts
t h a t if a c e r t a i n set of examples c a n b e found, t h e n examples of all t y p e s exist.
Such a r e s u l t i s of value b e c a u s e when t h e dimension of t h e r a n g e s p a c e i s sufficiently l a r g e i t may b e difficult t o verify t h e r a n k condition. However, t h e specified set of examples might b e identified by a computer s e a r c h . F o r simplicity, w e
r e s t r i c t a t t e n t i o n t o l i n e a r comparison maps.
Corollary 2.2
Suppose t h a t F i s a l i n e a r comparison mapping from a l i n e a r s p a c e t o a r a n g e
s p a c e R k . Assume t h a t t h e Zk o r t h a n t s of R k are comparison r e g i o n s . If t h e r e exi s t 2(k
+ 1examples,
e a c h in a d i f f e r e n t comparison region, t h e n F h a s maximal
r a n k and a l l possible comparisons are admitted.
Proof
The image of a l i n e a r s p a c e u n d e r a l i n e a r mapping i s a l i n e a r s p a c e . If t h i s
image s p a c e h a s dimension k , t h e n t h e conclusion follows. By assumption, k image
points c a n b e found t h a t do n o t lie in t h e same (k -1)-dimensional
subspace. This
completes t h e proof.
Extensions are obvious. F o r instance, t h e proof r e q u i r e s only k examples
t h a t are not in t h e same (k -1)-dimensional plane. F o r o t h e r c h o i c e s of comparison
regions, t h e emphasis i s placed on t h e geometry defined by t h e image points with
r e s p e c t t o t h e p r o p e r t i e s of t h e image set.
3 RANKING PARADOXES
A r i c h e r assortment of p a r a d o x i c a l behavior emerges from multivalued r a n dom variables. (This i s b e c a u s e t h e dimension of t h e domain f o r a comparison mapping i n c r e a s e s with t h e number of values admitted by a random variable.) W e illus-
trate t h i s with s e v e r a l new r e s u l t s a b o u t ranking and voting p r o c e d u r e s .
Our main r e s u l t s c o n c e r n weighted o r positional voting. This i s defined in t h e
following way.
To r a n k t h e N a l t e r n a t i v e s , a l , ...,aN, choose N s c a l a r weights
(zlil, ...,wN) where w j 2 wk if and only if j
<k
and where w
> w~
2
0. Each v o t e r
lists his ranking of t h e A! a l t e r n a t i v e s on a ballot. To tally a ballot, wj points are
assigned t o t h e j t h r a n k e d a l t e r n a t i v e s , j = 1,...,N.
In t h e obvious way, t h e
grq.rp's ranking of t h e a l t e r n a t i v e s i s determined by t h e sum of t h e assigned
w e ghts.
The weights define a voting v e c t o r WN = (wl, ...,wN) in RN. For plurality voting, t h e voting v e c t o r is (1,0, ...,0). Another well-known voting system, called t h e
Borda count, is defined by t h e voting vector BN = (N,N
- 1,...,1).
(If a voting vec-
t o r is a linear combination of BN and EN = (1,...,I ) , then we call t h e system a Borda
system. This i s because t h e election r e s u l t f o r any Borda system always a g r e e s
with t h e r e s u l t when BN i s used t o tally t h e ballots (see S a a r i (1982.))
This tallying p r o c e s s can b e identified with t h e expected value of multivalued
random variables. For N alternatives, t h e r e are N! different ways t o r a n k t h e N
alternatives. Since t h e sum assigned t o any alternative is a l i n e a r relationship,
t h e group's ranking i s not a l t e r e d should each sum be divided by t h e t o t a l number
of voters. This means t h a t t h e number of v o t e r s i s replaced with t h e fraction of
t h e v o t e r s with each ranking. In t h i s way t h e domain f o r this problem can b e identified with (the rational points in) t h e simplex Si(N!). The simplex i s in t h e positive o r t h a n t of a n N!-dimensional space. If Aj i s t h e random variable assigned t o
alternative a j , then P(A,
= w k ) denotes t h e fraction of t h e v o t e r s who r a n k t h e
j t h alternative in t h e K~~ place. Let A b e t h e v e c t o r valued random variable
(A1, ...,AN). A point in Si(N!) can b e viewed as being a probability distribution, and
s o t h e tally of t h e ballots c a n b e identified with t h e expected value E(A).
The following definitions are used in what follows.
Definition
A voters' profile i s a listing of each v o t e r ' s ranking of t h e N alternatives.
Definition
The
WN
voting
+ (wN,...,w
vector
WN
defines
a
reverse
neutral
system
if
= CEN = (C ,...,C) f o r some s c a l a r C.
A Borda system is always r e v e r s e neutral. An easy algebraic argument demon-
s t r a t e s t h a t t h e s p a c e of r e v e r s e n e u t r a l systems i s a hyperplane of R N with dimension 1 + [N/2] where
[I
denotes t h e "greatest integer function". A basis f o r
this hyperplane can b e computed directly. For N
r e v e r s e neutral.
For N
= 3 only t h e Borda systems are
= 4, a basis f o r t h e hyperplane i s given by E 4 , and
(2,1,1,0). F o r N = 5, a basis is E5,B5 and (2,1,1,1,0); etc.
Although weighted voting systems a r e a n important class of voting methods,
t h e i n t e r p r e t a t i o n of election results i s problematic. This i s illustrated by t h e following theorem which includes Example 2 a s a special case.
Theorem 3
Let N 2 3. Let ALk b e t h e subset [ a L...,
, ak 1. Let Wk b e t h e voting v e c t o r used
t o r a n k ALk. Assume t h a t
N(N
- )/2
wk
i s not r e v e r s e neutral, k = 3,...,N. For each of
p a i r s of alternatives, a r b i t r a r i l y designate one of t h e alternatives. A r -
bitrarily choose a ranking RKk f o r t h e set ALk,k
= 3, ...,N. There exist profiles of
v o t e r s such t h a t (i) t h e election r e s u l t f o r ALk is RKk ,k = 3, ...,N, and (ii) f o r each
p a i r of alternatives, a majority of t h e s e same v o t e r s p r e f e r t h e designated alternative.
The proof of this theorem and a n extension a r e given in Section 4.
A simple consequence of this theorem i s t h a t , if a Borda system is not used,
then t h e r e are profiles of v o t e r s where most of t h e v o t e r s p r e f e r a l t o a 2 , most
p r e f e r a 2 t o a s , most p r e f e r a l t o a 3 , y e t t h e election r e s u l t i s a 3 > a 2 > a l . The
implied ranking obtained by majority vote o v e r t h e p a i r s of alternatives i s t h e rev e r s a l of t h e election result! (In t h e example in Section 1.2, such a profile i s given
= 5 t h e r e i s a profile
of v o t e r s such t h a t majority votes determine t h e rankings a, > a, +l f o r j = 1 ,...,4,
f o r plurality voting.) A more striking example i s t h a t f o r N
a 5 > a l [ t h e s e five alternatives from a cycle1 a 4 > a j f o r j = 1,2,al > a 3 , and
aj
al
> a 5 f o r j = 2,3, and t h e plurality election r e s u l t s of A L j , j = 3,4,5, a r e
> a 3 > a 2 , a 2> a 3 > a l > a 4 , and a 3 > a l > a 5 > a 2 > a 4 respectively. Other
examples are limited only by t h e imagination of t h e designer.
By use of different techniques, Fishburn (1981) proved Theorem 3 f o r t h e special case N = 3. (More accurately, Fishburn gave a proof only f o r t h e f i r s t example above. However, i t i s possible t h a t his approach extends t o include o u r gen-
eral statement f o r N = 3.) F o r N
> 3,
t h e f i r s t conclusion without t h e second i s a
special c a s e of a r e s u l t given by S a a r i (1984).
The second p a r t of t h e theorem i s of independent interest. Essentially, i t as-
serts t h a t if t h e p a i r s of alternatives are ranked by majority voting, then any type
of cycle, subcycle, etc., c a n occur. To highlight this r e s u l t , we restate it.
Definition
Let N
=> 3.
F o r 1S k
<j
S
N, l e t Rkj b e t h e set lak
> a j , a j > ak j.
Let t h e
s p a c e of b i n a r y rankings BR b e t h e c a r t e s i a n p r o d u c t of t h e N(N -1)/2 sets R k j .
An element of BR i s a sequence t h a t imposes a n o r d e r i n g f o r e a c h of t h e N(N -1)/2
p a i r s of a l t e r n a t i v e s . These binary rankings need not b e t r a n s i t i v e , n o r need t h e y
satisfy a n y o t h e r consistency requirement.
Corollary 3.1
Let q b e a n element BR. T h e r e e x i s t examples of v o t e r s ' p r o f i l e s such t h a t ,
f o r e a c h p a i r of a l t e r n a t i v e s , a majority of t h e same v o t e r s h a v e t h e ranking
specified by q .
The remainder of t h i s section i s devoted to e x t r a c t i n g some of t h e consequences of
Theorems 3 and Corollary 3.1. We start by obtaining new r e s u l t s a b o u t t h o s e
schemes t h a t depend on majority v o t e s o v e r p a i r s of a l t e r n a t i v e s ( s e e t h e exposit o r y a r t i c l e by Nierni a n d R i k e r (1976).) F o r example, a n a l t e r n a t i v e is called a
Condorcet w i n n e r i t i s r e c e i v e s a majority v o t e when compared a g a i n s t e a c h of
t h e o t h e r a l t e r n a t i v e s . A Condorcet winner d o e s n o t always e x i s t (e.g., t h e a b o v e ,
second example), so o t h e r schemes h a v e been proposed to determine t h e winning
a l t e r n a t i v e . The following definition a p p e a r s to include a l l methods based on t h e
ordinal rankings.
Definition
A b i n a r y r a n k i n g method i s a nonconstant mapping from a s u b s e t of BR into
lal,...,aNj. That is, based on t h e ordinal rankings of p a i r s of a l t e r n a t i v e s , o n e of
t h e N a l t e r n a t i v e s is selected.
Examples
(1) A Condorcet winner i s a binary ranking method. The s u b s e t is t h e set of
a l l elements of BR where some o n e a l t e r n a t i v e i s p r e f e r r e d to all o t h e r
alternatives.
(2) An obvious extension of t h e Condorcet winner is t o select t h e a l t e r n a t i v e
t h a t wins t h e l a r g e s t number of pairwise comparisons. This extensio-.. admits a l a r g e r s u b s e t of elements from BR.
(3) Suppose N
- 1a l t e r n a t j v e s
are proposed t o r e p l a c e t h e s t a t u s quo, a l .
The s e l e c t e d a l t e r n a t i v e i s a l if and only if a l i s a Condorcet winner. If
not, t h e n from t h e set of t h o s e a l t e r n a t i v e s t h a t b e a t a l , s e l e c t t h e one
t h a t wins t h e most pairwise comparisons.
(4) A commonly used binary ranking method i s a n agenda.
Definition
Let N
> 3.
An agenda i s a n o r d e r e d listing of t h e N a l t e r n a t i v e s . The f i r s t two
listed a l t e r n a t i v e s are voted upon. The a l t e r n a t i v e receiving t h e majority v o t e i s
t h e n compared with t h e t h i r d listed a l t e r n a t i v e . This i t e r a t i v e , pairwise comparison p r o c e d u r e i s continued t o t h e end of t h e listing. The remaining a l t e r n a t i v e i s
t h e selected alternative.
The following statement extends s e v e r a l r e s u l t s from "agenda manipulation"
( s e e , for example, McKelvey (1976) and P l o t t and Levine (1978)). I t implies t h a t
t h e r i g h t t o set a n agenda f o r a meeting i s a potential s o u r c e of power ( t h e f i r s t
conclusion) t h a t may lead t o a n undesired outcome ( t h e second conclusion).
Corollary 3.2
Let N
> 3.
T h e r e e x i s t v e c t o r s ' p r o f i l e s and N agendas such t h a t , when t h e
same v o t e r s u s e t h e j t h agenda, t h e outcome i s a,. j = 1.....N. F o r N
> 3,
there
e x i s t v o t e r s ' profiles and N agenda so t h a t t h e above holds even though all the
voters p r e f e r a g lto a 4 ,a 4 t t o ] a s ,..., land] a
~ to -a ~
~ .
An i n t e r e s t i n g f e a t u r e of t h i s c o r o l l a r y i s t h a t f o r a fixed p r o f i l e of v o t e r s ,
t h e winning a l t e r n a t i v e v a r i e s o v e r a l l possible outcomes as t h e "seeding", o r t h e
choice of t h e agenda, changes. The proof depends on t h e fact t h a t majority, p a i r wise voting c a n define a n y d e s i r e d c y c l e and subcycle. Thus, t h i s conclusion extends t o t h e o t h e r binary ranking methods t h a t depend on t h e initial seeding. This
includes tournaments, w h e t h e r single, double, o r k -fold elimination, c e r t a i n
h i e r a r c h i c a l methods, etc. In p a r t i c u l a r , because a change in t h e seeding c h a n g e s
t h e definition of a binary ranking method, Corollary 3.2 i s a special case of t h e
following.
Corollary 3.3
Let Pl and P2 b e different mappings from BR to lal,...,aN1. There exist profiles of v o t e r s s o t h a t t h e outcome of t h e two binary voting methods differ.
The next r e s u l t compares t h e outcome of a binary ranking method with t h e
election r e s u l t s of a weighted voting system. A s special cases, i t shows t h a t a Cond o r c e t winner, or t h e r e s u l t of a n agenda, need not a g r e e with a n election ranking.
Corollary 3.4
Let N
> 3.
Let t h e s e t Alk b e ranked with t h e voting v e c t o r Wk ,K
= 3,...,N.
Let a binary ranking method b e given. Assume t h a t Wk,k = 3,...,N, is not r e v e r s e
neutral. Let Rkk , k
= 3,...,N, by any ranking of Alk, and l e t a j b e a n a r b i t r a r y ele-
ment in t h e image of t h e binary ranking method. There exist voters' profiles such
t h a t (i) t h e election r e s u l t of Alk,K = 3,...,N, and (ii) t h e binary ranking method
selects aj.
A s a consequence of this theorem, t h e r e is a profile of v o t e r s s o t h a t t h e i r plurality election ranking of Sk i s a l
> a 2 > ... > ak if
k is even and t h e r e v e r s e of t h i s if
k is odd, and t h e Condorcet winner is a l .
This chaotic s t a t e of a f f a i r s cannot b e eliminated if t h e selection method is
defined to combine, in some way, t h e election r e s u l t s o v e r all of subsets
Alk,K
= 2,...,N. For instance, in a run-off election, t h e lower ranked alternatives
are dropped, and t h e remaining set is r e r a n k e d in a s e p a r a t e election. The following definition extends this notion.
Definition
A d y n a m i c a l selection p r o c e s s consists of (i) a set of voting v e c t o r s
1WN,...,W3,(1,O)1,
(ii) r u l e s t h a t eliminate a specified, positive number of alterna-
tives from a s e t of k alternatives, k = 2,...,N, and (iii) a selection function. The
p r o c e d u r e is defined in t h e following way. The set AIN i s ranked by using WN.
Then, based on t h e elimination r u l e f o r t h e N alternatives, N-s
alternatives are
eliminated. The remaining set of s alternatives is ranked by using W,.
Iteratively,
this procedure i s continued. Based on t h e s e election rankings, t h e nonconstant
selection p r o c e d u r e selects one alternz tive.
Examples
(1) The s t a n d a r d "run-off" election i s a dynamical p r o c e d u r e . At e a c h s t e p ,
t h e bottom r a n k e d a l t e r n a t i v e i s eliminated. The s e l e c t e d a l t e r n a t i v e i s
t h e one remaining at t h e end of t h e p r o c e s s .
(2) This run-off p r o c e d u r e c a n b e generalized in t h e following way. Choose a
positjve i n t e g e r k < N. The elimination p r o c e d u r e i s t h e same as in (I),
but t h e selection r u l e s e l e c t s t h e t o p r a n k e d a l t e r n a t i v e from t h e election ranking of k alternatives. If k
k
= 2, t h i s i s t h e a b o v e p r o c e d u r e . If
= N , t h i s i s a s t a n d a r d election p r o c e d u r e .
(3) The run-off p r o c e d u r e c a n eliminate more t h a n one a l t e r n a t i v e at e a c h
s t a g e . F o r instance, a f t e r t h e N a l t e r n a t i v e s are r a n k e d , all b u t t h e t o p
two a l t e r n a t i v e s may b e dropped.
(4) Let a l r e p r e s e n t t h e s t a t u s qua, and let aj,
j = 2,...,N r e p r e s e n t t h e contending a l t e r n a t i v e s . Use WN t o r a n k t h e N a l t e r n a t i v e s . If a l i s t h e t o p
r a n k e d a l t e r n a t i v e , i t i s d e c l a r e d t h e winner. If i t i s not, t h e n eliminate
a l a n d r a n k t h e remaining a l t e r n a t i v e s with WN
The t o p r a n k e d a l t e r -
native from t h i s election i s d e c l a r e d t h e winner.
(5) The elimination r u l e may depend o n t h e a l t e r n a t i v e s . F o r instance, t h e
p r o c e s s d e s c r i b e d in (4)c a n b e modified t o eliminate not only a l b u t also
a l l a l t e r n a t i v e s r a n k e d below a l in t h e f i r s t election.
As a s p e c i a l c a s e , t h e following asserts t h a t t h e winner of a run-off election need
not b e a Condorcet winner.
Corollary 9.5
Let N
> 3.
Assume t h a t a binary ranking method and a dynamical selection
p r o c e s s are given. Suppose t h a t , f o r e a c h K r 3, t h e weight v e c t o r Wk i s not rev e r s e n e u t r a l . A r b i t r a r i l y s e l e c t aj from t h e r a n g e of t h e b i n a r y ranking method
and ak from t h e r a n g e of t h e dynamical process. 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 t h e b i n a r y ranking outcome i s aj while t h e dynamical method outcome i s at.
Recently t h e r e h a s b e e n i n t e r e s t in election p r o c e d u r e s where a v o t e r c a n
choose a voting v e c k r t o tally h i s ballot.
Definition
A m u l t i p l e voting s y s t e m used t o r a n k N alternatives i s determined by a set
MN of voting v e c t o r s where at least two of t h e s e v e c t o r s and EN are linearly independent. Each v e c t o r r a n k s t h e N alternatives on his ballot, and then h e selects
a v e c t o r from MN t o tally his ballot.
Examples
(1) Bullet
M
voting:
The
= (2.0,.0 (1..0
set
defining
( 1 1 0. 0
.
of
voting
vectors
is
This procedure w a s used during
t h e 1970s f o r some legislative offices in Illinois.
(2) Cardinal voting: The set MN contains all voting v e c t o r s where t h e components sum t o unity.
Occasionally, cardinal voting is used t o define
rankings f o r methods from decision analysis.
(3) Approval
voting:
The
defining
I (l,O,...,O),(l,l,O,...,O),..., (1,1,...,1,0) j.
set
of
N-1
vectors
is
For t h i s method, which w a s intro-
duced by R. Weber, among o t h e r s , and ithas been analyzed by Brams and
Fishburn (1982), a v o t e r indicates e i t h e r approval o r disapproval of
each alternative.
Often t h e r e s u l t s of multiple voting systems are compared with t h e Condorcet
winner. The following shows t h a t t h e r e s u l t s can b e incompatible.
Corollary 3.6
Let N r 3. Let a binary ranking method b e given. Let Mk define a multiple
voting system f o r dlk,k
= 3, ...,N. Assume t h a t , f o r each k ,Mk contains at l e a s t one
v e c t o r t h a t i s not r e v e r s e neutral. Let Rkk b e a ranking f o r dlk,and l e t aj b e a n
alternative in t h e r a n g e of t h e binary ranking method. There e x i s t voters' profiles s o t h a t (i) t h e multiple election r e s u l t f o r A l k i s Rkk,k = 3,...,N, and (ii) t h e
binary ranking outcome is aj.
A consequence of this r e s u l t is t h a t , f o r any choice of s , t h e r e e x i s t examples
where t h e alternative ranked in s t h place in an approval voting election i s t h e
Condorcet winner, and t h e r e e x i s t examples where t h e results based upon approval
v3ting are a4 > a 3 > a 2 > a l f o r t h e set of four alternatives, a l > a 2 > a f o r t h e
Q.
~ b s e of
t t h r e e alternatives, and a 2 i s a Condorcet winner.
I t follows from this a p p r o a c h t h a t t h e p r i n c i p a l cause of the social choice
paradoxes i s the difference between the d i m e n s i o n s of the d o m a i n a n d the
range of a comparison mapping (see Section 2 , comment ( 2 ) ) . To model a weighted
election f o r N alternatives, t h e domain S i ( N ! ) h a s dimension N!-1. The image i s
E ( A ) . Because t h e domain i s S i ( N ! ) , this image i s in t h e simplex Si ( N ) in R ~ (The
.
sum of t h e components of
wN define this
sume t h a t t h i s sum i s unity.)
simplex. Without loss of generality, as-
Thus, t h e r a n g e s p a c e h a s dimension N-1.
This
difference of N{(N-I)!-11 i s z e r o if and only if t h e r e are only two alteratives.
Therefore, if N
2
3, o t h e r relationships with resulting paradoxes can b e added.
This is illustrated by Theorem 3.
This dimensional argument also proves t h a t , f o r N
2
4, Theorem 3 i s not t h e
"best possible" r e s u l t . To see this, w e need t o d e s c r i b e t h e comparison mapping L
f o r Theorem 3. The f i r s t N components of L a r e given by E ( A ) , t h e next N-1 by
t h e expected value of t h e weighted voting method defined by W N - ~ e, t c . The last
N ( N - I ) / 2 components are given by t h e expressions P ( A k ) - P ( A , ) , k
<j .
Thus,
the range space f o r L is
Si ( N )
where
N
J
is
the
+ ( N -1)+...+3 +
X
interval
S i ( N -1) x.. .
[-ll]
I(N)(N - 1 ) / 2 1 = N~
X
S(3) x J
This
~-I)' (
range
-N - 1.
space
~
has
the
dimension
For t h e model described in t h e
theorem, t h e difference between t h e dimensions of t h e domain and t h e r a n g e is
N!
- N~ + N.
This value i s positive if and only if N
2
4.
Corollary 3.7
Let N
N!
2
- N~ + N
4. In addition t o t h e subsets of alternatives described in Theorem 3,
additional relationships involving t h e rankings of t h e N alternatives
can b e defined in such a way t h a t , f o r certain profiles of v o t e r s , t h e r e s u l t s are
independent of t h e rankings obtained in Theorem 3.
A simple dimensional argument shows t h a t even if Theorem 3 can b e extended from
nested sets of t h r e e o r more alternatives t o all possible subsets of alternatives,
f o r N t 4 additional relationships can still b e found.
To complete o u r description of L , notice t h a t t h e comparison value on each
simplex in t h e r a n g e i s t h e point of complete indifference N - ~ E ~
For
. each of t h e
intervals J , t h e comparison value i s 0. Thus, t h e comparison point i s
Therefore Theorem 3 is a n example of Theorem 2 where M and N are manifolds and
where t h e comparison point is not t h e origin of a Euclidean space.
The comparison mapping L i s linear. Therefore L -'(0)
space with dimension N!
of Si(N!).
- N~ + N.
If N
> 3 this s p a c e must
must b e a linear subi n t e r s e c t t h e boundary
A boundary point of Si(N!) corresponds to a profile of v o t e r s where
none of t h e v o t e r s r a n k t h e alternatives in c e r t a i n ways. This extreme boundary
behavior describes t h e limits of t h e voting paradox. A special case is described in
Corollary 3.2.
Corollary 3.8
Let N
2
4. The r e s u l t s in Theorem 3 can b e obtained with profiles of v o t e r s
where no v o t e r h a s c e r t a i n rankings of t h e alternatives.
W e have not t r i e d to find a general characterization of t h e boundary behavior.
The proof of Theorem 3 involves showing t h a t t h e linear comparison map has
maximal rank. The r a n k condition does not hold if Wk,k = 3,...,N are Borda vect o r s . I t t u r n s out t h a t L h a s corank (with r e s p e c t t o t h e r a n g e space) of at least
IN(N
- 1 ) / 2 ] - 1. This means t h a t although a Borda election ranking admits incon-
sistencies with r e s p e c t t o a given binary ranking method, not all possible inconsistencies are admitted. In p a r t i c u l a r , Theorem 3 does n o t hold if e v e n one of t h e
Wk is a Borda v e c t o r . A d i r e c t verification of this f o r N = 3 i s given in Section 4.
With only slight modifications, t h e s e r e s u l t s can b e used to d e s c r i b e c e r t a i n
ranking p r o c e d u r e s coming from probability and statistics. For instance, suppose
N forms are making t h e same product, and they are t o b e ranked based on t h e quality of t h e i r products. In Theorem 3, identify t h e "ithalternative" with t h e " i t h
firm" t h e " t th v o t e r " with t h e "jthv e c t o r sample" of t h e product taken from each
of t h e N firms, and t h e "jthv o t e r ' s p r e f e r e n c e ranking" with t h e linear "quality
ranking" of t h e products in t h e jth sample. The relationship ak
> a j means t h a t ,
based upon t h e samples, form k ' s product a p p e a r s t o b e s u p e r i o r to firm j ' s . It
follows from Corollary 3.1 t h a t any possible choice of binary rankings is realized
by an open set of d a t a points. Binary sampling a p p r o a c h e s need not lead t o a
linear ordering of t h e "quality of t h e firms". Indeed, in this way, t h e well-known
Steinhaus-Trybula paradox (Steinhaus and 'Trybula, 1959), where t h e final ranking
of t h r e e forms is a l
> a 2 , a 2 > a 3,
but a 3 > a l , becomes a special case of Corol-
l a r y 3.1.
I t follows from Theorem 3 that, even if t h e firms are ranked by use of weighted ranking methods, t h e r e s u l t s could b e difficult t o i n t e r p r e t . For instance, t h e
weight v e c t o r s W, = (LO, ...,0 ) correspond t o t h e natural ranking method based on
P ( q = maxIXf:j
E Ak
j). I t follows from t h e above t h a t , should some one firm b e
deleted, t h e revised ranking could drastically change. Other measures experience
similar problems. A similar effect o c c u r s f o r t h e scoring of athletic events where
a voter's ranking corresponds t o how t h e various teams are placed in a p a r t i c u l a r
event, etc.
A s a final amusing example, note t h a t a connoisseur is often described as a
person whose taste p r e f e r e n c e s a r e based upon s e v e r a l a t t r i b u t e s (e.g., t h e color,
t h e t a s t e , and t h e bouquet of a wine), and whose rankings are based on a n aggregation of them. If so, w e should not expect his binary comparisons t o define a transitive ordering. This is, of course, an N alternative version of t h e famous folklore
"pie" example (I p r e f e r "apple" t o "cherry", but if "blueberry" is available, then
my choice i s "cherry").
A s in Section 2, t h e r e e x i s t open sets in t h e domain which exhibit e a c h of t h e
above behaviors. Consequently, t h e s e examples cannot b e dismissed as being isolated; t h e behavior is robust. A s t h e number of agents increases ( t h e denominat o r s of t h e rational points become l a r g e r ) , so d o t h e number of t h e possible examples, which leads us t o t h e following corollary.
Corollary 3.9
Consider a system of weighted voting methods as described in Theorem 3. Let
Q denote a n outcome o v e r t h e various sets as described in Theorem 3. Let n (Q,m)
b e t h e probability t h a t t h e election r e s u l t f o r a group of m v o t e r s i s Q. Assume
t h a t t h e profiles of v o t e r s are uniformly distributed. Then, as m -, w , n ( Q , m ) app r o a c h e s t h e r a t i o of t h e area of L -'(Q) to t h e area of t h e simplex Si (N!).
For elementary number t h e o r e t i c reasons, t h e sequence [n(Q ,m )
monotone. The limit i s positive if L -'(Q)
may not b e
contains a n open set; t h i s i s t r u e whenev-
e r Q does not admit ties. For o t h e r distributions, t h e r a t i o is determined in a simil a r fashion, but with a different measure.
4. PROOFS
The proof of Theorem 2 i s obvious. To prove Theorem 3, w e f i r s t prove Coroll a r y 3.1.
Proof of Corollary 3.1
List t h e p a i r s of alternatives in t h e following o r d e r : t h e f i r s t p a i r is ( a l , a z ) ,
t h e second set of two p a i r s i s given by ( a j , a t ) . j
is given by (a, , a k +1), j
= 1.2, and t h e
j t h set of k p a i r s
= 1,...,k ; k = 3,...,N -1. A ranking of t h e N alternatives de-
fines a n tN(N -I)/ 2 ]-dimensional v e c t o r in t h e following way. The j
th
component
is determined by t h e ranking of t h e j t h p a i r of alternatives. This component i s 1 if
t h e f i r s t listed alternative i s p r e f e r r e d t o t h e second; otherwise, i t is -1. F o r example, t h e v e c t o r associated with t h e p r e f e r e n c e ranking a l
> a z > ... > a~ h a s
t h e value 1in all t h e components.
Because t h e N alternatives can b e ranked in N! different ways, t h e comparison map is a linear mapping from Si (N!) t o
J ~ ( ~ -where
~ ) J
' ~
is
t h e interyal
[-1.11
and t h e comparison point i s 0 . (this map defines a convex combination of t h e
above N! vectors.) We must show t h a t t h e r e i s a point p in t h e i n t e r i o r of Si (N!)
such t h a t (i) p i s in t h e preimage of 0 and (ii) t h e Jacobian of t h e comparison map
at p h a s full rank. Let p = (~!)-'(l,
...,1).
Because p i s t h e profile where t h e r e
are equal numbers of v o t e r s f o r e a c h of t h e N! possible ways t o r a n k t h e alternatives, p i s mapped t o 0.
The comparison mapping i s linear, s o
jt
h a s a matrix representation. The ma-
t r i x i s t h e Jacobian, and i t consists of t h e N! column v e c t o r s defined above. I t
remains t o show t h a t t h i s set of N! v e c t o r s includes N(N-1)/2
linearly indepen-
dent vectors.
Consider t h e v e c t o r s Vj ,j
f i r s t IN(N -1)/Z J
- ( j-1)
= 1,...,N(N -1)/2, where
5 has t h e value 1f o r t h e
component and -1 f o r t h e remaining components. This
set of v e c t o r s i s linearly independent. This is because they form a s q u a r e a r r a y
where t h e e n t r i e s on and above t h e diagonal from t h e lower left-hand c o r n e r t o t h e
u p p e r tight-hand c o r n e r are all equal t o 1. A l l t h e o t h e r entries are -1.
There a r e
zN v e c t o r s with
e n t r i e s of e i t h e r 1o r -1. Most of them are not re-
lated t o t h e described ranking method. So, t o complete t h e proof, i t remains t o
5 is associated with one of t h e N! rankings of t h e alternatives. The
choice of t h e components and t h e v e c t o r s 5 makes thi; fairly simple. The v e c t o r
show t h a t e a c h
Vl corresponds t o t h e ranking a
> a 2 > ... > a,,,.
Vectr
.- V2 has
-1 only in t h e last
component; this corresponds t o a transposition of aN and a N - , . These two alternatives are adjacent in t h e f i r s t ranking, s o t h e ranking f o r V2 c a n b e obtained from
t h e ranking f o r Vl by transposing these alternatives. This defines t h e ranking
a,
> a 2 > ... > U N - Z > a~ > a ~ - ~ .
Indeed, t h e only difference between Vj and
is in one component. This
component r e f l e c t s a change in t h e ranking of precisely one p a i r of alternatives.
By construction, t h e s e two alternatives are adjacent in t h e ranking Rkj t h a t is associated with
?.
Therefore, t h e ranking f o r V,
is obtained by transposing these
two adjacent alternatives in R k j . This completes t h e proof.
This proof is based on t h e f a c t t h a t the -1's in t h e square a r r a y correspond t o
the N-1
adjacent transpositions required t o move aN from last place in
> a 2 > ... > a~ t o first. This defines N -1 rankings where
aN > a l > a 2 > ... > a ~ - Next,
~ . move aN-l from what is now last
al
t h e last one is
place t o second,
etc.
Proof of Theorem 3
Let t h e weight v e c t o r s Wk,k = 3 ,...,N , b e as specified in t h e statement of t h e
theorem. With each ranking of t h e N alternatives, w e associate a v e c t o r with
[ N ( N -1) / 2 J
+ 3 + ... + N components.
The f i r s t N ( N -1) / 2 components a r e defined
as above. The next t h r e e are given by t h e a p p r o p r i a t e permutation of Wa t o
correspond t o t h e specified ranking. For instance, t h e ranking a 2 > a g > al i s
identified with t h e vector ( w 3 , w l l w 2 ) .In general, t h e s e t of k components is t h e
a p p r o p r i a t e permutation of Wk t o r e f l e c t t h e ranking of t h e k alternatives,
k = 3 ,..., N .
P (N-1)'2
X
The comparison mapping L which is a mapping from Si ( N ! ) t o
Si ( 3 ) X...
X
Si ( N ) is described in Section 3 . The point p described
above is mapped t o t h e comparison point ( 0 , O,..., 0 ;( 1/ 3)Ea ,...,( 1/ N)EN).
Because L is linear, i t s matrix representation defines t h e Jacobian. This mat r i x h a s N ! column' v e c t o r s with IN (N - I ) / 2 j
+ 3 + ... + N
= N' - 3 components.
(The dimension of t h e r a n g e space is smaller; i t h a s dimension N'-N
- 1.
The
difference results from t h e constraints defining t h e N -2 simplices Si ( k ) in the ima g e space.) W e must show t h a t t h e r e are N'
In t h e proof of the corollary,
in t h e f i r s t N ( N - 1 ) / 2
:i set
- 3 linearly independent vectors.
of N ( N -1) / 2 v e c t o r s t h a t are independent
compo~:~ntsw e r e found.
To obtain t h e remaining
! N ( N + 1 ) /2 1-3 independent vecto! s, t a k e t h e v e c t o r associated with each of t h e N !
rankings and add i t t o t h e v e c t o r associated with t h e r e v e r s a l of this ranking.
Each of t h e f i r s t N(N -I)/ 2 components of t h e v e c t o r associated with t h e r e v e r s e d
ranking will differ in sign from t h e original vector. Therefore, t h e sum v e c t o r s
will have z e r o s in each of t h e s e f i r s t N(N-1)/2
components. Consequently, t h e s e
new v e c t o r s are orthogonal t o t h e range s p a c e used in t h e proof of Corollary 3.1.
All w e need t o do is t o show t h a t t h e s e new v e c t o r s contain a set of 1N(N
+ I)/ 21-3
independent vectors.
For a r e v e r s e neutral v e c t o r , t h e s e new v e c t o r s are all multiplies of
all o t h e r c a s e s , t h e j t h component h a s t h e value w j
+ W ~ - j + l For
.
EN.
In
instance, f o r
t h e voting v e c t o r (4, 3, O), t h e new v e c t o r corresponding t o t h e rankings
a
b
>b >c
>c >a
and c
>b >a
i s (4, 6, 4). The v e c t o r corresponding t o a
>c >b
an
i s (4, 4, 6). In general, t h e s e new v e c t o r s would correspond to a voting
vector e x c e p t t h a t they do not satisfy t h e monotonicity condition. However, t h e
r e s u l t s given by S a a r i (1984) hold even f o r v e c t o r s t h a t do not satisfy t h e s e monotonicity p r o p e r t i e s . Therefore, t h e above reduces t o a special case of t h e one
given by S a a r i (1984). This completes t h e proof.
Proof that a Borda weight vector does not work for N = 3
Assume t h a t t h e alternatives are a , b , and c . assume t h a t t h e Borda weight
vector i s B3 = (3,2,1). The comparison mapping i s linear and its image includes t h e
comparison point (0,0,0;6,6,6). (B3 i s not normalized, and s o t h e sum of t h e components of S
i (3) i s 6.)
The comparison regions in S
i (3) are identified with t h e linear rankings of t h e
t h r e e alternatives. To obtain them, note t h a t if t h e a x e s of R~ are labeled in t h e
> y corresponds to a > b , y > z
to c > a , etc. In this way, t h e simplex
usual z,
y , z notation, then t h e region z
corresponds t o b
> c, z >z
corresponds
S
i (3) i s divided into six open sets which are defined by t h e intersection of t h e simplex with t h e t h r e e hyperplanes z = y ,y
= z , and
z
= z (see S a a r i (1978, 1982)).
Suppose t h a t t h e different behaviors described in t h e theorem hold for B3.
This means t h a t t h e image of t h e comparison mapping meets each of t h e six regions
of Si (3) as well as all of t h e open regions in J ~ In
. all, i t would meet 48 open regions. If this happens, then, by t h e linearity of t h e mapping and a comparison of
t h e dimensions of t h e domain and range, i t follows t h a t t h e mapping i s onto a neighborhood of t h e c ~ m p a r i s o npoint. This f o r c e s t h e matrix t o b e of r a n k five. To
show t h a t this
id
not so, w e list all six of t h e v e c t o r s and then e x t r a c t a four-
dimensional basis.
The v e c t o r s are as follows:
These six v e c t o r s admit a basis consisting of t h e f i r s t t h r e e v e c t o r s and t h e v e c t o r
(0,0,0; 1,1,1). Thus, t h e system has corank 2. Since t h i s last v e c t o r is orthogonal
t o t h e image space, t h e system has corank 2 with r e s p e c t t o t h e image space. This,
and t h e linearity of t h e mapping, means t h a t t h e comparison mapping h a s a nonzero
intersection with 1 2 of t h e 48 admissible comparison regions. If t h e mapping were
always consistent, then t h e mapping would meet only 3! = 6 regions. Thus, t h e mapping still admits s e v e r a l "inconsistent" conclusions. (In a p a p e r being p r e p a r e d ,
we c h a r a c t e r i z e t h e election rankings admitted by a Bol;da count.)
Extension of Theorem 3
The last p a r t of t h e proof of Theorem 3 is based on t h e work of S a a r i (1984)
which admits a wider variety of results. For example, f o r k alternatives, suppose
t h e r e a r e k -1 weight v e c t o r s W k ,which form, with Ek,a linearly independent set.
Arbitrarily choose k -1 rankings of t h e k alternatives. The theorem asserts t h a t
t h e r e exist voters' profiles s o t h a t when t h e s a m e voters rank t h e s e t of k alterna-
= 3,...,N with t h e i t h voting vector, then t h e outcome is t h e i t h specified
ranking of t h e alternatives. This is t r u e f o r all choices of i = 1,...,k -1 and k.
tives, k
A similar extension holds f o r Theorem 3. For each k = 3,...,N, choose
k
- 1- k / 2
voting vectors with t h e following property: (i) t h e voting v e c t o r is
not r e v e r s e neutral, and (ii) t h e s e t of k -k / 2 vectors, defined by Ek and t h e vect o r s formed by adding each of t h e k
- 1 - k / 2 voting
vectors t o i t s r e v e r s a l , is a
linearly independent s e t . For each voting vector, a r b i t r a r i l y choose a ranking of
t h e k alternatives. For each p a i r of alternatives, designate one of them. There
exist voters' profiles s o t h a t when t h e i t h voting vector is used t o r a n k t h e k alternatives t h e outcome is t h e assigned ranking i
= l , ...,k - l[/21, k = 3,...,N. For
each p a i r , a majority of t h e same v o t e r s p r e f e r t h e designated alternative. The
proof of this statement is a straightforward modification of t h e proof of Theorem
3.
Proof of Corollary 3.2
From Corollary 3.1, i t follows t h a t t h e r e is a n open set of v o t e r s ' profiles
where t h e outcome in pairwise elections i s t h e cycle
Consider t h e r e v e r s e d c y c l e a l
< aN < a ~ < ...
- <~a 2 < a l .
The following defines
a n agenda where a j will b e t h e winner. Let t h e agenda b e t h e N t e r m s in t h e rev e r s e d cycle t h a t start with t h e a l t e r n a t i v e immediately following a j a n d e n d s with
a j . F o r example, a 3 wins with t h e agenda [ a 2 , a l , a N , a-l
, ,..., a 4 , a 3 ] .
The second p a r t of t h i s c o r o l l a r y i s a consequence of t h e boundary p r o p e r t i e s
of L -l(0). The second p a r t of t h i s c o r o l l a r y i s a consequence of t h e boundary
p r o p e r t i e s of L-l(0). The profile, w h e r e a n equal number of v o t e r s h a v e e a c h of
the
three
rankings
a 3 > a q > ... > aN > a l
> a2 has
al
> a 2 > ... > a
~ , >aa 3~> ... > a ~ > a l l
and
t h e d e s i r e d p r o p e r t i e s . Note t h a t in e a c h p a i r -
wise comparison t h e winning a l t e r n a t i v e r e c e i v e s e i t h e r
2
of
t h e vote, o r all of it!
3
This i s t r u e f o r whichever agenda i s used and whichever a l t e r n a t i v e wins.
The dice example
This i s a s t r a i g h t f o r w a r d computation. However, t h e domain point used in t h e
image of t h e comparison point should c o r r e s p o n d t o two identical weighted, b u t not
f a i r , dice. The probability t h a t a p a r t i c u l a r f a c e will s u r f a c e is l e f t t o t h e end of
t h e computation. In o t h e r words, t h e r e are some complications in t h e computation
with two f a i r dice.
ACKNOWLEDGEMENT
This r e s e a r c h w a s s u p p o r t e d by NSF Grants IST-8111122 and IST-8415348.
I
a m pleased t o acknowledge conversations with W.W. Funkenbusch, Tom Miles, and
Hans Weinberger on t h e s e and r e l a t e d topics. Some of t h i s work w a s done while I
w a s visiting t h e Institute f o r Mathematics and i t s Applications (IMA) at t h e University of Minnesota. I would like t o thank my hosts, Hans Weinberger, George Sell,
and Leo Hurwicz f o r t h e i r kind hospitality during my stay. These r e s u l t s were
presented at a n IMA c o n f e r e n c e on mathematical political s c i e n c e in July 1984.
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