Von Neumann-Morgenstern
Utilities and Cardinal
Preferences
Chichilnisky, G.
IIASA Collaborative Paper
September 1983
Chichilnisky, G. (1983) Von Neumann-Morgenstern Utilities and Cardinal Preferences. IIASA Collaborative
Paper. IIASA, Laxenburg, Austria, CP-83-038 Copyright © September 1983 by the author(s).
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VON NEUIM-?lORGENSTERN UTILITIES
AND CARDINAL PREFERENCES
Graciela Chichilnisky
September 1 9 8 3
CP-83-38
C o Z Z 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 n o t been
performed s o l e l y a t t h e I n t e r n a t i o n a l I n s t i t u t e f o r
A p p l i e d Systems A n a l y s i s and which h a s r e c e i v e d o n l y
V i e w s o r opinions expressed herein
l i m i t e d review.
do n o t n e c e s s a r i l y r e p r e s e n t t h o s e of t h e I n s t i t u t e ,
i t s N a t i o n a l Member O r g a n i z a t i o n s , o r o t h e r o r g a n i z a t i o n s s u p p o r t i n g t h e work.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
Laxenburg, A u s t r i a
A-2361
FOREWORD
T h i s C o l l a b o r a t i v e Paper i s one of a s e r i e s embodying t h e
outcome of a workshop and c o n f e r e n c e on Economic S t r u c t u r a l
Change: A n a l y t i c a l I s s u e s , h e l d a t IIASA i n J u l y and August
1983.
The c o n f e r e n c e and workshop formed p a r t of t h e cont i n u i n g IIASA program on P a t t e r n s of Economic S t r u c t u r a l Change
and I n d u s t r i a l Adjustment.
S t r u c t u r a l change was i n t e r p r e t e d v e r y b r o a d l y : t h e t o p i c s
c o v e r e d i n c l u d e d t h e n a t u r e and c a u s e s of changes i n d i f f e r e n t
s e c t o r s of t h e world economy, t h e r e l a t i o n s h i p between i n t e r n a t i o n a l m a r k e t s and n a t i o n a l economies, and i s s u e s of o r g a n i z a t i o n and i n c e n t i v e s i n l a r g e economic systems.
There i s a g e n e r a l c o n s e n s u s t h a t i m p o r t a n t economic
s t r u c t u r a l changes a r e o c c u r r i n g i n t h e world economy. There
a r e , however, s e v e r a l a l t e r n a t i v e a p p r o a c h e s t o measuring t h e s e
c h a n g e s , t o modeling t h e p r o c e s s , and t o d e v i s i n g a p p r o p r i a t e
r e s p o n s e s i n t e r m s o f p o l i c y measures and i n s t i t u t i o n a l redesign.
O t h e r i n t e r e s t i n g q u e s t i o n s c o n c e r n t h e r o l e of t h e
i n t e r n a t i o n a l economic system i n t r a n s m i t t i n g such c h a n g e s , and
t h e merits o f a l t e r n a t i v e modes of economic o r g a n i z a t i o n i n
r e s p o n d i n g t o s t r u c t u r a l change. A l l of t h e s e i s s u e s were
a d d r e s s e d by p a r t i c i p a n t s i n t h e workshop and c o n f e r e n c e , and
w i l l be t h e f o c u s of t h e c o n t i n u a t i o n o f t h e r e s e a r c h p r o g r a m ' s
work.
G e o f f r e y Heal
A n a t o l i Smyshlyaev
Ern6 Z a l a i
VON NEUMANN-MORGENSTERN UTILITIES
AND CARDINAL PREFERENCES*
Graciela Chichilnisky**
Abstract
We study the aggregation of preferences when intensities are taken into
account: the aggregation of cardinal preferences and also of von NeumannMorgenstern utilities for cases of choice under uncertainty. We show that
with a finite number of choices, there exist no continuous anonymous aggregation rules that respect unanimity for such preferences or utilities. With
infinitely many (discrete sets of) choices, such rules do exist and they are
constructed here. However, their existence is not robust: each is a limit
of rules that do not respect unanimity. Both results are for economies with
a finite number of individuals.
The results are obtained by studying the global topological structure
of spaces of cardinal preferences and of von Neumann-Morgenstern utilities.
with-a finite number of choices, these spaces are proven-to be noncontractible. With infinitely many choices, on the other hand, they are proven to
be contractible.
*
This research was supported by UNITAR and by NSF Grant No. SES-7914050.
I thank L.J. Billera, G. Bergmann,L. Gevers, 0. Hart, G. Heal, J.F. Mertens,
and a referee for comments and suggestions.
Columbia University, New York, USA.
**
1.
Introduction
The a g g r e g a t i o n o f p r e f e r e n c e s s t u d i e d i n s o c i a l c h o i c e t h e o r y t y p i c a l l y
d e s c r i b e s an i n d i v i d u a l p r e f e r e n c e a s a r a n k i n g among c h o i c e s , i . e . , i n o r d i n a l
terms.
I n most of t h e l i t e r a t u r e f o l l o w i n g Arrow's and B l a c k ' s p i e c e s [l] [3],
i n t e n s i t i e s o f p r e f e r e n c e s a r e not recorded; i n p a r t i c u l a r , i t i s not p o s s i b l e t o
express whether a c h o i c e x i s p r e f e r r e d t o another y more t h a n t o a t h i r d , z .
Since most o f t h e r e s u l t s i n t h e a g g r e g a t i o n o f o r d i n a l p r e f e r e n c e s a r e n e g a t i v e ,
i t seems n a t u r a l t o i n q u i r e whether more p o s i t i v e r e s u l t s can be o b t a i n e d when
t h i s p r o p e r t y is r e l a x e d and i n t e n s i t i e s o f p r e f e r e n c e s a r e recorded.
A s i g n i f i c a n t s t e p i n allowing t h e consideration of preference i n t e n s i t i e s
is i n t r o d u c e d with c a r d i n a l p r e f e r e n c e s .
These p r e f e r e n c e s express p r e c i s e l y
t h e n o t i o n t h a t a c h o i c e x i s p r e f e r r e d t o a n o t h e r y more t h a n t o a t h i r d z .
In t h e c a s e o f choice under u n c e r t a i n t y , t h e s e p r e f e r e n c e s can be shown t o have
t h e same mathematical s t r u c t u r e a s von Neumann-Morgenstern u t i l i t i e s , t h e
numerical r e p r e s e n t a t i o n s o f p r e f e r e n c e s o v e r l o t t e r i e s :
These u t i l i t i e s a r e
denoted NM u t i l i t i e s , and a r e f r e q u e n t l y used i n t h e o p e r a t i o n s r e s e a r c h
l i t e r a t u r e i n t h e f i e l d o f d e c i s i o n theory s i n c e t h e concept was developed by
van kumann and Morgenstern (19); NM u t i l i t i e s a r e a l s o widely used a s a
r e p r e s e n t a t i o n of i n d i v i d u a l behavior i n game t h e o r e t i c models, see Fishburn ( 1 3 ) .
The main d i f f e r e n c e i n t h e d e f i n i t i o n s of o r d i n a l and c a r d i n a l p r e f e r e n c e s
i s t h e i n v a r i a n c e t h e y r e q u i r e from a numerical r e p r e s e n t a t i o n .
Cardinal
p r e f e r e n c e s r e q u i r e much weaker i n v a r i a n c e than o r d i n a l p r e f e r e n c e s :
the
r e p r e s e n t a t i o n of c a r d i n a l p r e f e r e n c e by a numerical f u n c t i o n i s i n v a r i a n t
under (and only under) p o s i t i v e l i n e a r t r a n s f o r m a t i o n s .
For o r d i n a l p r e f e r e n c e s ,
i n s t e a d , t h e r e p r e s e n t a t i o n must be i n v a r i a n t under any p o s i t i v e t r a n s f o r m a t i o n .
1
The weaker t h e i n v a r i a n c e , t h e c l o s e r a r e p r e f e r e n c e s t o numerical
u t i l i t i e s , and numerical u t i l i t i e s have no problem of a g g r e g a t i o n .
Therefore
one may expect t h a t t h e t a s k of a g g r e g a t i n g i s made e a s i e r w i t h c a r d i n a l r a t h e r
than o r d i n a l p r e f e r e n c e s .
However, t h i s i s n o t t h e c a s e .
It was shown i n
C h i c h i l n i s k y [ 4 ] and i n C h i c h i l n i s k y and Heal [ 8 ] t h a t t h e c r u c i a l element i n
our a b i l i t y t o a g g r e g a t e p r e f e r e n c e s i s t h e g l o b a l t o p o l o g i c a l s t r u c t u r e of t h e
space of p r e f e r e n c e s c o n s i d e r e d .
I n o r d e r t o admit a p p r o p r i a t e a g g r e g a t i o n
r u l e s , t h e s e spaces must be c o n t r a c t i b l e , i . e .
topologically t r i v i a l .
However,
t h e t o p o l o g i c a l s t r u c t u r e of spaces of p r e f e r e n c e s may be complex even when
l e s s invariance i s required.
For i n s t a n c e , NM u t i l i t i e s w i t h f i n i t e l o t t e r i e s
a r e shown h e r e t o d e f i n e a n o n - c o n t r a c t i b l e s p a c e , i . e . a space w i t h a nont r i v i a l topological s t r u c t u r e (see Section 2).
By i n v e s t i g a t i n g t h e g l o b a l topology of spaces of c a r d i n a l p r e f e r e n c e s
and of NM u t i l i t i e s , we prove h e r e t h a t w i t h f i n i t e l y many c h o i c e s , t h e r e
e x i s t s no continuous anonymous s o c i a l a g g r e g a t i o n r u l e t h a t r e s p e c t s unanimity.
Aggregation i s i m p o s s i b l e f o r c a r d i n a l p r e f e r e n c e s and f o r NM u t i l i t i e s .
With i n f i n i t e l y many choices we show i n s t e a d t h a t such a g g r e g a t i o n r u l e s
do e x i s t .
-
However, t h e i r e x i s t e n c e i s n o t r o b u s t i n t h e s e n s e t h a t they a r e
t h e l i m i t of r u l e s d e f i n e d on s u b s e t s of f i n i t e l y many c h o i c e s , which do n o t
r e s p e c t unanimity.
The same r e s u l t s apply t o von Neumann-Morgenstern u t i l i t i e s
d e f i n e d over i n f i n i t e l y many l o t t e r i e s . A f i n i t e number of i n d i v i d u a l s i s
considered throughout t h e paper.
The r e s t of t h e paper i s organized a s f o l l o w s :
and d e f i n i t i o n s ;
Section 2 gives notation
S e c t i o n 3 d i s c u s s e s previous l i t e r a t u r e ;
and S e c t i o n 4
gives the r e s u l t s .
2.
Notation and D e f i n i t i o n s
I n t h e c a s e of f i n i t e choices t h e c h o i c e space
p o i n t s i n Euclidean space
x=(xiI, i.1,
..., n,
n a 3 .
X i s a f i n i t e s e t of
2
A preference with i n t e n s i t y o r cardinal preference p i s i d e n t i f i e d
with a positive vector i n R
n
,
pn)
Pi
denotes the u t i l i t y value a t t a c h e d to the choice x
i
.
The t o t a l i n d i f f e r -
ence p r e f e r e n c e i s thus t h e v e c t o r with a l l c o o r d i n a t e s e q u a l .
Our s p a c e of
preferences contains t h i s t o t a l indifference preference a s well.
The f o l l o w i n g s t e p i s t o n o r m a l i z e u t i l i t y v e c t o r s i n o r d e r t o o b t a i n
a unique r e p r e s e n t a t i o n o f e a c h c a r d i n a l p r e f e r e n c e by a v e c t o r i n E u c l i d e a n
This normalization is a standard one; s e e , e . g . ,
space.
K a l a i and S c h m e i d l e r
i t s economic c o n t e n t i s d i s c u s s e d i n t h e f o l l o w i n g s e c t i o n .
[I51 ;
mally, i f p = (pl,..
.,
) i s a u t i l i t y vector i n R
n
,
For-
p i s n o r m a l i z e d by sub-
t r a c t i n g from i t s c o o r d i n a t e s t h e v e c t o r w i t h a l l components i d e n t i c a l t o
t h e minimum u t i l i t y v a l u e
where m = min { p i ) ,
i
and t h e n d i v i d i n g t h e outcome by i t s maximum component M i f H # 0 , i . e . ,
where M = max {pi - m). The t o t a l i n d i f f e r e n c e p r e f e r e n c e i s i d e n t i f i e d
i
t h e r e f o r e w i t h the v e c t o r ( 0 ,
, 0 ) The f a c t t h a t a l l n o r m a l i z e d p r e f e r -
.. .
.
e n c e s have t h e same minimum and maximum u t i l i t y v a l u e s can be c o n s i d e r e d a
3
weak form of i n t e r p e r s o n a l comparison.
I t f o l l o w s t h e r e f o r e t h a t w i t h f i n i t e l y many c h o i c e s t h e s p a c e of c a r d i n a l
preferences i s P = Q
erences,
U
{ o ) , where Q i s t h e s u b s p a c e of non-zero c a r d i n a l p r e f -
2
Q = {~ER~':
pi ~n
i=l
-
1, p j
0 and pk = 1 f o r s o m e k , j ~ ( 1 ,..., n } } ,
r
and {O) d e n o t e s t h e t o t a l i n d i f f e r e n c e p r e f e r e n c e .
I n o r d e r t o d e f i n e con-
t i n u i t y of t h e s o c i a l c h o i c e r u l e , P i s given t h e n a t u r a l topology i t i n h e r n
i t s from R
.
The s p a c e P h a s two connected components, Q and {O)
W e s h a l l now d e f i n e t h e s p a c e of c a r d i n a l p r e f e r e n c e s P
w
.4
f o r t h e c a s e of
i n f i n i t e l y many c h o i c e s .
Assume now t h a t t h e c h o i c e space X i s N ,
p r e f e r e n c e p i s assumed t o be a non-negative
t h e s e t of i n t e g e r s .
A
sequence of numbers, i . e . , a
non-negative r e a l v a l u e d f u n c t i o n on N . S i n c e we a r e concerned w i t h
bounded sequences, w i t h o u t l o s s of g e n e r a l i t y , we may assume t h a t
&P ( n >p(n) <
f o r some f i n i t e measure p o r N given by a d e n s i t y f u n c t i o n p(n)
w
The s p a c e of p r e f e r e n c e s P
.
i s t h e r e f o r e s t r i c t l y contained i n the space
of a l l bounded sequences, and t h i s i s i n t u r n a s u b s e t ( t h e p o s i t i v e cone) of a
weighted P 1 s p a c e . 5
00
Note t h a t we could have embedded P
i s a dense subspace of f
1
t h e s p a c e of
.
However, t h e s p a c e
i=1,2..
w i t h t h e ( f i n i t e ) weight p ( n ) .
T h e r e f o r e , i f one
a l l bounded sequences w i t h t h e sup norm, 1 x
fa
into fa,
Hw
= sup lxnl
.
d e f i n e s an a g g r e g a t i o n map @ f o r P l y one has a u t o m a t i c a l l y d e f i n e d an a g g r e g a t i o n
map f o r f a , given by t h e r e s t r i c t i o n of @ on f w c o n s i d e r e d a s a subspace of 2! 1'
The topology induced by P 1 i s d i f f e r e n t t h a n t h e sup norm on P a ,
b u t s i n c e our
-
aim i s t o prove an e x i s t e n c e theorem, f o r some a d e q u a t e t o p o l o g y , t h i s p r o c e d u r e
seems adequate.
w
I n any c a s e , P
i s a s t r i c t s u b s e t of fa, a s w e l l a s of P I ,
Therefore, neither f w
and i s s i g n i f i c a n t l y s m a l l e r t h a n e i t h e r f a o r f l .
f 1 coincide with p
00
and t h u s t h e c h o i c e of t o p o l o g y i s b e s t made on t h e b a s i s
of m a t h e m a t i c a l adequacy.
spacesf
nor
Theorem 2 shows t h a t f 1 i s a n a d e q u a t e s p a c e ;
o r more g e n e r a l l y f
economic l i t e r a t u r e ;
P
the
( w i t h 15 p $00) have been u s e d p r e v i o u s l y i n t h e
see e.g. Chichilnisky [ 6 ] .
A s i n t h e c a s e of f i n i t e l y many c h o i c e s , we n o r m a l i z e t h e v e c t o r p i n
An e q u i v a l e n c e
o r d e r t o o b t a i n a u n i q u e r e p r e s e n t a t i o n of c a r d i n a l p r e f e r e n c e s .
relation
a
E
fl,
'L
B
E
i s d e f i n e d by p
R+.
1
'L
p2 i f and o n l y i f
=
a + Bp
2
,
A p r e f e r e n c e i s a n e q u i v a l e n c e c l a s s of p o s i t i v e v e c t o r s p
under t h e r e l a t i o n
'L.
A s p a c e which i s i n a one t o o n e c o r r e s p o n d e n c e w i t h t h e
s p a c e of p r e f e r e n c e s i s o b t a i n e d by c o n s i d e r i n g a l l v e c t o r s w i t h c o o r d i n a t e s
s m a l l e r t h a n 1, w i t h a t l e a s t one c o o r d i n a t e z e r o , and w i t h t h e f i r s t nonz e r o c o o r d i n a t e ( i f i t e x i s t s ) e q u a l t o 1.
00
00
s p a c e of c a r d i n a l p r e f e r e n c e s i s P = Q
f i $.
1;
f
W O } , where' Q~ = { f
= 0 and f . + 1 = 1 f o r some j
j
J
and i s a c l o s e d s u b s e t of a Banach s p a c e .
connected components.
T h e r e f o r e w i t h many c h o i c e s t h e
1.
00
P
E
f
+
:
f o r a l l i,
i n h e r i t s t h e t o p o l o g y of f 1
00
As P, P
c o n s i s t s of e x a c t l y two
Assume now t h e r e a r e k a g e n t s , k
2 2.
1
o f c a r d i n a l p r e f e r e n c e s i s a v e c t o r {p ,
of p w i t h i t s e l f k-times
1
is a vector {P ,
.. ., p
repectively.
m
k ~E (P )
k
With f i n i t e c h o i c e s a p r o f i l e
. . . , pk )EPk ,
the cartesian product
With i n f i n i t e c h o i c e s a p r o f i l e
.
A r u l e I$ i s s a i d t o r e s p e c t u n a n i m i t y when @ ( p ,
. . . , p)
= p; i.e.
,
i f a l l v o t e r s have i d e n t i c a l p r e f e r e n c e s o v e r dl c h o i c e s , s o does t h e s o c i a l
preference.
6
A r u l e I$ i s anonymous when t h e outcome i s i n d e p e n d e n t o f t h e o r d e r o f
the voters, i . e . ,
where
i s any p e r m u t a t i o n o f t h e set { l ,
..., k) .
Continuity of a r u l e is d e f i n e d w i t h r e s p e c t t o the u s u a l product topologies
of t h e s p a c e s of p r e f e r e n c e s
nk
k
a s s u b s e t s of R
o r (el)
i n t h e f i n i t e and
i n f i n i t e choice case, respectively
.
We now d i s c u s s c e r t a i n b a s i c t o p o l o g i c a l c o n c e p t s u s e d i n t h e f o l l o w i n g .
A topological space X i s c o n t r a c t i b l e i f t h e r e e x i s t s a continuous
map f : X x [ 0 ,
some x EX.
0
11
+
X such t h a t f ( x , 0) = x Yx i n X , and f ( x , 1) = x
Intuitively,
0'
for
X i s c o n t r a c t i b l e i f i t can be deformed c o n t i n u o u s l y
t h r o u g h i t s e l f , i n t o one o f i t s p o i n t s , xO.
convex s e t a r e c o n t r a c t i b l e .
C l e a r l y e u c l i d e a n s p a c e and any
I n p a r t i c u l a r , the space of continuous r e a l
valued u t i l i t y functions is a c o n t r a c t i b l e space.
Topologically speaking,
t h e s e a r e a l l t r i v i a l s p a c e s , s i n c e they a r e t o p o l o g i c a l l y e q u i v a l e n t t o ( i . e . ,
c o n t i n u o u s l y deformable i n t o ) p o i n t s .
tible.
A hollow s p h e r e i n
R'I
i s not contrac-
As w e s h a l l prove below, n e i t h e r t h e non t r i v a l c o n n e c t e d component
t h e s p a c e of c a r d i n a l p r e f e r e n c e s Q , n o r t h a t of von Neumann-Morgenstern
are contractible.
utilities,
T h i s proves t o be i m p o r t a n t f o r t h e a g g r e g a t i o n r e s u l t s of
t h i s paper.
R e l a t i o n s h i p w i t h P r e v i o u s Work
B e f o r e p r o v i n g t h e r e s u l t s , i t may be u s e f u l t o d i s c u s s t h e r e l a t i o n s h i p o f t h e s p a c e s of p r e f e r e n c e s s t u d i e d h e r e w i t h e a r l i e r c o n c e p t s of c a r d i n a l p r e f e r e n c e s used i n t h e l i t e r a t u r e ,
The s p a c e of
and a l s o e a r l i e r r e s u l t s i n t h i s a r e a .
non z e r o c a r d i n a l p r e f e r e n c e s Q c o r r e s p o n d s t o t h e s p a c e
of c a r d i n a l u t i l i t i e s as s t u d i e d f o r i n s t a n c e by K a l a i and S c h m e i d l e r i1.51:
two v e c t o r s
and p
2
d e f i n e t h e same c a r d i n a l p r e f e r e n c e when t h e r e e x i s t
a p o s i t i v e number f3 and a p o s i t i v e v e c t o r a s u c h t h a t
I t i s e a s y t o check t h a t o u r n o r m a l i z a t i o n o f t h e p r e v i o u s s e c t i o n
i d e n t i f i e s e a c h v e c t o r i n Q w i t h a n e q u i v a l e n c e c l a s s of v e c t o r s under t h e
equivalence r e l a t i o n
2
p l z a + b p ,
f o r all a > 0, b > 0 .
C o n s i d e r now t h e c a s e o f c h o i c e under u n c e r t a i n t y .
I n t h i s case the
s p a c e of von Neumann-Morgenstern u t i l i t i e s , i . e . , n u m e r i c a l r e p r e s e n t a t i o n
of p r e f e r e n c e s o v e r l o t t e r i e s , c o r r e s p o n d s p r e c i s e l y t o o u r f o r m a l d e f i n i t i o n of the spaces of cardinal preferences.
i 1 5 j and [ 1 9 ] .
For f u r t h e r d i s c u s s i o n s e e , e . g . ,
The s p e c i f i c a t i o n of P g i v e n h e r e i s a l s o r e l a t e d t o one of t h e forms
of r e l a x a t i o n of t h e u s u a l o r d i n a l i t y and c o m p a r a b i l i t y a s s u m p t i o n s d i s c u s s e d
i n dtAspremont and Gevers.
non-comparability
T h e i r c o n d i t i o n CN of c a r d i n a l i t y and
requires that i f u
1
and u
a r e two u t i l i t i e s , t h e n t h e y
2
d e f i n e t h e same p r e f e r e n c e whenever
..,R
where j = 1,.
i s t h e index f o r the v o t e r ,
x d e n o t e s a c h o i c e , and
I n our framework CN means t h a t f o r
where { a * ) and { f i e ) a r e p o s i t i v e real numbers.
J
J
1
2
e a c h v o t e r t h e v e c t o r p r e p r e s e n t s t h e same p r e f e r e n c e as a n o t h e r p when
t h e r e e x i s t s a v e c t o r a and a p o s i t i v e number B s u c h t h a t p
1
= a + Bp 2 .
This is
p r e c i s e l y t h e c a r d i n a l i t y c o n d i t i o n d i s c u s s e d above:
Let
and p
2
s a t i s f y p1 = a
+ Op 2 .
Then t h e y y i e l d t h e same e l e m e n t
i n P , s i n c e f o r any p = ( p l , . . . , p n ) , and any j = l , . . . , n
a
+
Max [ a
8p
+
j
-
min ( a
O(pi)
-
i
+ hi)
min ( a + 3pR1
R
C o n v e r s e l y , i f two u t i l i t y v e c t o r s p
1
and p
2
i n R~ y i e l d t h e same e l e m e n t i n
t h e s p a c e of i - p r e f e r e n c e s P ,
then
I
1
Max (pi
i
-
1
min ( p e ) )
R
-
1
2
Max (pi
i
- min
R
2
(pi))
which i m p l i e s t h a t
(min (p:))
-
min ( P
j
2
Max ( p 2 - m i n ( p . ) )
i
J
i
I
Max ( p i
and 6 =
I
m x (pi
i
-
A.
-,
L
Max (pi
i
min ( p i )
J
?
,
-
L
j
(P.))
J
T h e r e f o r e s o c i a l c h o i c e r u l e s t h a t a r e i n v a r i a n t under t h e n o r m a l i z a t i o n o f
P c o r r e s p o n d p r e c i s e l y t o t h o s e s a t i s f y i n g c o n d i t i o n CN.
S e v e r a l a u t h o r s t h a t s t u d i e d t h e problems i n v o l v e d i n a g g r e g a t i n g
c a r d i n a l p r e f e r e n c e s , e . g.
,
Sen [16 ] and K a l a i and Schmeidler [15 1
.
It h a s
been shown [ 1 5 ] t h a t Arrow-like p a r a d o x e s nay e x i s t even w i t h c a r d i n a l p r e f e r e n c e s , p r o v i d e d Arrow-like
c o n d i t i o n s a r e r e q u i r e d of t h e a g g r e g a t i o n pro-
cedure : t h e s e a r e t h e somewhat c o n t r o v e r s i a l independence of i r r e l e v a n t
a l t e r n a t i v e s , P a r e t o and non d i c t a t o r s h i p .
Such c o n d i t i o n s may be t o o s t r o n g .
Also, w h i l e making t h e problem amenable t o a c o m b i n a t o r i a l a n a l y s i s , s u c h c o n d i t i o n s
t e n d t o l e a v e o u t i t s i n t r i n s i c geometry.
Here, i n s t e a d , o t h e r conditions
of t h e a g g r e g a t i o n r u l e a r e s t u d i e d : c o n t i n u i t y , anonymity and r e s p e c t of
unanimity.
These c o n d i t i o n s admit a ready g e o m e t r i c a l i n t e r p r e t a t i o n , and
f u r t h e r m o r e h e l p t o e x h i b i t t h e t o p o l o g i c a l n a t u r e of t h e problem a t hand.
The f o l l o w i n g r e s u l t e s t a b l i s h e s t h e i m p o s s i b i l i t y of a g g r e g a t i o n w i t h
f i n i t e l y many c h o i c e s .
The s p a c e of p r e f e r e n c e s i s t h e r e f o r e P = Q
u
{O},
as d e f i n e d above.
Theorem 1
There e x i s t s no c o n t i n u o u s a g g r e q a t i o n r u l e f o r c a r d i n a l p r e f e r e n c e s
where i n d i v i d u a l and s o c i a l p r e f e r e n c e s may b e i n d i f f e r e n t among a l l c h o i c e s .
Proof:
k
A n a g g r e g a t i o n r u l e f o r c a r d i n a l p r e f e r e n c e s i s a map @ : P
+
P.
Now, as d i s c u s s e d i n s e c t i o n 2 , t h e s p a c e P h a s e x a c t l y two connected comp o n e n t s , Q and {o} ( f i g u r e 1 below i l l u s t r a t e s t h e c a s e of t h r e e c h o i c e s ) .
T h e r e f o r e t h e p r o d u c t s p a c e pk h a s e x a c t l y 2k c o n n e c t e d components.
The map $ i s t h e r e f o r e a c o n t i n u o u s f u n c t i o n from a t o p o l o g i c a l s p a c e
with 2
k components i n t o a n o t h e r w i t h 2 components.
I t f o l l o w s from c o n t i n u -
i t y o f $ t h a t e a c h of t h e c o n n e c t e d components o f pk must be mapped by @ i n t o
is i n d i c a t e d a s f i e union
of the p o i n t ( 0 )with +-he s e t drawn with a heavy l i n e . The non zero
connected component of P I in indicated with t h e heavy l i n e ) is cn a one
t o cne bicontinuous correspondsnce with the boundary aS of t h e simplex
S i n Ft3, and is t h e r e f o r e not c o n t r a c t i b l e in R ~ .
';he space of preferences P w i t h t h r e e choices
one c o n n e c t e d component o f P .
n e c t e d component o f P
k
Consider i n p a r t i c u l a r Q
k
.p
,
which i s t h e con-
c o n s i s t i n g o f a l l non z e r o c a r d i n a l p r e f e r e n c e s .
e i t h e r $(Q ) C Q, o r else $(Q~) C {O}.
o f unanimity p , . .
k
Then
However, by t h e c o n d i t i o n of r e s p e c t
= p for a l l p in
k
Q, i m p l y i n g t h a t $(Q )
$ {0},
i.e.,
$ ( Q ~ )C Q.
T h e r e f o r e , t h e axioms o f c o n t i n u i t y and t h a t of r e s p e c t of unanimity
t a k e n t o g e t h e r r u l e o u t t h e p o s s i b i l i t y t h a t a p r o f i l e w i t h all p r e f e r e n c e s
d i f f e r e n t from t h e z e r o v e c t o r may b e mapped i n t o t h e z e r o outcome v e c t o r .
fore
,
There-
a c o n t i n u o u s r u l e f o r c a r d i n a l p r e f e r e n c e s which r e s p e c t s unanimity
w i l l o n l y a s s i g n t h e t o t a l i n d i f f e r e n c e t o a set of v o t e r s i f a t l e a s t one
o f them i s t o t a l l y i n d i f f e r e n t among a l l c h o i c e s .
$ i n d u c e s t h e r e f o r e a c o n t i n u o u s map
7
7
: Q
k
anonymous and r e s p e c t s unanimity. S i n c e Q and Q
can
now
k
+ Q, which i s a l s o c o n t i n u o u s ,
a r e b o t h connected s p a c e s , w e
use t h e r e s u l t s i n [ 7 ] . These results e s t a b l i s h t h a t t h e e x i s t e n c e o f
such a map $ depends on c e r t a i n t o p o l o g i c a l i n v a r i a n t s o f t h e s p a c e Q.
The n e x t
s t e p of t h e proof c o n s i s t s t h e r e f o r e o f i n v e s t i g a t i n g t h e topology of t h e s p a c e
of non z e r o c a r d i n a l p r e f e r e n c e s , f o r any f i n i t e number of c h o i c e s n
Consider t h e subspace R
j -th
coordinate p
j
= 0.
one c o r r e s p o n d e n c e w i t h T
j'
,
n
c
i=l
j
The set T
i s a n n-2 d i m e n s i o n a l s i m p l e x .
m
P +
n-1
o f Rn+,
.C
J
3.
c o n s i s t i n g of a l l v e c t o r s p w i t h t h e
R ~ - ' d e f i n e d by
j
l o w , t h e set Q
=
Q r/ Rn-I
j
by t h e map m d e f i n e d by
s i n c e p # !0)
2
f o r a l l p i n Q.
j
i s i n a one-to-
S i n c e t h e map m i s
c o n t i n u o u s , and s o i s i t s i n v e r s e , i t f o l l o w s t h a t t h e s p a c e Q i s homeomorphic t o t h e s i m p l e x T
j
.
. ..
fl
Now, f o r any j , L E { l ,
,n} t h e i n t e r s e c t i o n Q . r ) Q = T .
TQ.
J
L
J
n
and Q j n Q Q = T . n T L f o r all j and L , i t f o l l o w s t h a t
Since Q =
Q
J
j =1 j '
()
Qj
j =1
Now,
i s homeomorphic t o
(31 T j .
T h e r e f o r e Q i s homeomorphic t o
j =1
0
T
j
j =l
'
0
T2 i s , i n t u r n , homeomorphic t o t h e boundary of an n-dimensional
s i m p l e x i n R",
i.e .
,
an
n-2
dimensional sphere.
I t f o l l o w s t h a t Q i s homeo-
morphic t o an n-2 d i m e n s i o n a l s p h e r e a n d , t h e r e f o r e , i n p a r t i c u l a r , Q i s s t
contractible.
W e c a n now a p p l y t h e r e s u l t s of [ 7 ] ,
which e s t a b l i s h t h a t f o r any
( p a r a ) f i n i t e CW complex X , t h e c o n t r a c t i b i l i t y o f X i s a n e c e s s a r y c o n d i t i o n
f o r t h e e x i s t e n c e of a c o n t i n u o u s anonymous r u l e I$ : X
> 2.
unanimity, f o r a l l k -
k
+
X , which r e s p e c t s
S i n c e Q i s a f i n i t e CW complex and i s n o t con-
t r a c t i b l e , t h i s completes t h e p r o o f .
S i n c e a s d i s c u s s e d above t h e s p a c e o f NM u t i l i t i e s can be i d e n t i f i e d
w i t h P i f t h e r e a r e f i n i t e l y many c h o i c e s i n e a c h s t a t e , we have t h e r e f o r e o b t a i n e d
from theorem 1:
Corollary 1
The s p a c e o f von-Neumann Morgenstern u t i l i t i e s w i t h f i n i t e l y many l o t t e r i e s
is not contractible.
and
Corollarv 2
With f i n i t e l y many l o t t e r i e s t h e r e e x i s t no c o n t i n u o u s anonymous a g g r e g a t i o n
r u l e f o r von Neumann-Morqenstern u t i l i t i e s which r e s p e c t s u n a n i m i t y .
This
i n c l u d e s c a s e s where i n d i v i d u a l and s o c i a l u t i l i t i e s may be i n d i f f e r e n t
among a l l l o t t e r i e s .
Remark : Even though o u r f r a n e w o r k and c a c d i t i o n s on the a g g r e g a t i o n r u l e
a r e r a t h e r d i f f e r e n t from t h o s e o f K a l a i and S c h m e i d l e r , our i m p o s s i b i l i t y
r e s u l t is consistent with t h e i r s
i n c a s e s of f i n i t e l y many c h o i c e s .
However, t h i s i s n o t t h e c a s e f o r i n f i n i t e l y many c h o i c e s .
Instead,
we obtain i n the next s e c t i o n a p o s s i b i l i t y of aggregation r e s u l t f o r h i s l a t t e r case.
T h i s c o n t r a s t s w i t h t h e r e s u l t o f K a l a i and S c h m e i d l e r
b e c a u s e t h e i r impossi-
b i l i t y r e s u l t i s v a l i d a l s o w i t h i n f i n i t e l y many c h o i c e s .
T h i s i s because
t h e y r e q u i r e t h e axiom o f independence o f i r r e l e v a n t a l t e r n a t i v e s , which e f f e c t i v e l y r e d u c e s t h e problem o f a g g r e g a t i o n w i t h i n f i n i t e l y many c h o i c e s t o one
of a g g r e g a t i o n w i t h f i n i t e l y many c h o i c e s .
The d i f f e r e n c e between t h e two
r e s u l t s a r i s e s from t h e f a c t t h a t d i f f e r e n t sets of axioms are r e q u i r e d :
we
do n o t r e q u i r e independence of i r r e l e v a n t a l t e r n a t i v e s , b u t r e q u i r e c o n t i n u i t y
i n s t e a d , and we do n o t r e q u i r e t h e P a r e t o c o n d i t i o n , b u t r a t h e r a (weaker)
c o n d i t i o n of r e s p e c t o f u n a n i m i t y .
The r e s u l t s g i v e n below a l s o show t h a t t h e
a g g r e g a t i o n w i t h i n f i n i t e l y many c h o i c e s i s n o t r o b u s t a s a l i m i t i n g p r o c e s s
of a g g r e g a t i o n on c e r t a i n f i n i t e s u b s e t s of c h o i c e s .
W e now
t u r n t o t h e c a s e of i n f i n i t e l y many c h o i c e s .
00
cardinal preferences i s therefore P
of i n d i v i d u a l s
,
k
. As
before
,
Our s p a c e of
t h e r e a r e a f i n i t e number
> 2.
Theorem 2
With i n f i n i t e l y many c h o i c e s , t h e r e e x i s t s a c o n t i n u o u s a q q r e q a t i o n
a k
r u l e 4 : (P )
ity
+
pW f o r c a r d i n a l p r e f e r e n c e s r e s p e c t i n g unanimity and anonym-
g i v e n by a c o n t i n u o u s d e f o r m a t i o n o f a Bergsonian r u l e
addition rule
i.e.,
a convex
i
t r a r i l y l a r g e f i n i t e s e t s of c h o i c e s and which do n o t r e s p e c t u n a n i m i t y ;
i n p a r t i c u l a r they a r e not Pareto.
Proof:
A s i n theorem 1, we may c o n s i d e r a c o n t i n u o u s a g g r e g a t i o n r u l e
-
a s s i g n i n g t o each p r o f i l e of k ( n o n - t r i v i a l )
00
of P
, where
preferences i n P
00
, an
element
00
P
00
Q
= Q ~ ~ { o and
) ,
=
+ Pi
EL^:
-< 1 f o r
a l l i , p j = 0 a n d pjtl
= 1 f o r some j } .
00
A s i n t h e f i n i t e d i m e n s i o n a l c a s e one can show t h a t Q i s i n a o n e t t o -
one b i c o n t i n u o u s c o r r e s p o n d e n c e w i t h t h e boundary o f a d i s k i n 2
an i n f i n i t e d i m e n s i o n a l s p h e r e i n k
Now, by c o r o l l a r y 5 . 1 ,
00
Kuiper [ 1 4 ] t h e s p a c e Q
tractible.
are
1'
i . e . , with
1'
p . 109 of Bessaga and P e l c z y n s k i i 2 ] and
i s homeomorphic t o 2
1'
and i n p a r t i c u l a r , i s con-
This i s i n c o n t r a s t t o t h e f i n i t e d i m e n s i o n a l c a s e , where s p h e r e s
not c o n t r a c t i b l e and i n d e e d n s homeomorphic t o e u c l i d e a n s p a c e .
be t h e homeomorphism, H : Q- + Z1.
S i n c e convex a d d i t i o n C i n 2
1
Let H
exists
and i t s a t i s f i e s anonymity, c o n t i n u i t y and r e s p e c t of u n a n i m i t y , t h e compo-
-1
k
s i t i o n map $ = Ho CoH d e f i n e d by
cok
i s a c o n t i n u o u s f u n c t i o n $ : (Q ) + Q~ s a t i s f y i n g anonymity and r e s p e c t of
unanimity.
S i n c e we can r e p e a t t h i s p r o c e d u r e f o r e a c h c o n n e c t e d component
00k
of (P ) , t h i s p r o v e s e x i s t e n c e .
convex a d d i t i o n r u l e C , i . e .
,
C l e a r l y t h e map 4 i s a d e f o r m a t i o n o f t h e
a d e f o r m a t i o n of a B e r g s o n i a n r u l e .
C o n s i d e r now t h e s p a c e of t r u n c a t e d s e q u e n c e s T C Q ,
T
-
{{p)
: ?I0 w i t h pN = O f o r Z.I,M
T h i s s p a c e i s dense i n 2
1
0
).
w i t h a ( f i n i t e ) measure.
Consider the pointwise
convergence t o p o l o g y o f t h e s p a c e F of c o n t i n u o u s f u n c t i o n s F =
f
:
(L1)
k
+
Z1).
8
The sequence of r e s t r i c t i o n maps
f i n i t e dimensional l i n e a r subspaces L
d e f i n e d by t h e r e s t r i c t i o n s of
d
sequence of i n t e g e r s { d l , converges t o
4
to
whose dimensions d e f i n e an unbounded
4.
Note t h a t when r e s t r i c t e d t o any
f i n i t e s u b s p a c e of c h o i c e s ( i . e . , when r e s t r i c t e d t o v e c t o r s o f f i n i t e l e n g t h )
t h e map @d remains anonymous.
I t f o l l o w s by theorem 1 t h a t
unanimity on s u c h s u b s p a c e ; i n p a r t i c u l a r , i t i s n o t P a r e t o .
sequence of maps
converges t o t h e map
4,
4d
cannot r e s p e c t
Since the
t h i s completes t h e p r o o f .
From theorem 2 we o b t a i n immediately t h e a n a l o g u e t o c o r o l l a r i e s 1 and 2
f o r von Neumann-Morgenstern u t i l i t i e s :
Corollarv 3
With i n f i n i t e l y many l o t t e r i e s , t h e s p a c e o f von Neumann-Morgenstern
m
utilities P
is contractible.
and
Corollarv 4
9
a q g r e q a t i o n r u l e f o r von Neumann-Morqenstern u t i l i t v f u n c t i o n s which r e q n e c t s
unanimity.
However, t h i s r u l e i s n o t r o b u s t s i n c e it i s t h e L i m i t of non-Pareto
r u l e s on a r b i t r a r i l y l a r g e sets of l o t t e r i e s .
l ~ h eproblem of a g g r e g a t i o n of o r d i n a l p r e f e r e n c e s i s s i g n i f i c a n t l y
d i f f e r e n t from t h a t of a g g r e g a t i n g u t i l i t y f u n c t i o n s ( e . g . , t h e c l a s s i c a l
B e r g s o n i a n s o c i a l w e l f a r e f u n c t i o n ) because t h e a g g r e g a t i o n of o r d i n a l p r e f e r e n c e s must be i n d e p e n d e n t of t h e c h o i c e o f t h e i r u t i l i t y r e p r e s e n t a t i o n .
For i n s t a n c e , i f u i s a u t i l i t y f u n c t i o n and F i s a s t r i c t l y i n c r e a s i n g numeri c a l . f u n c t i o n , t h e o r d i n a l p r e f e r e n c e a s s o c i a t e d w i t h t h e u t i l i t y u must be
t h e same a s t h a t a s s o c i a t e d w i t h t h e f u n c t i o n F o u. T h e r e f o r e , a r u l e f o r
aggregating u t i l i t i e s w i l l o n l y induce a r u l e f o r aggregating o r d i n a l p r e f e r e n c e s i f i t i s i n v a r i a n t under any such i n c r e a s i n g t r a n s f o r m a t i o n of u t i l i t i e s .
T h i s i s i n d e e d a r a t h e r s t r o n g c o n d i t i o n , and s e v e r a l p o s s i b l e r e l a x a t i o n s
have been s t u d i e d , f o r i n s t a n c e , by Sen [16] [ 1 7 ] , Hammond [ 1 3 ] , K a l a i and
Schmeidler [ 1 5 ] and more r e c e n t l y by dfAspremont and Gevers [ l o ] . Sen [ 1 7 ]
concentrates
on t h e r e l a x a t i o n of t h e a s s u m p t i o n of no i n t e r p e r s o n a l comparisons. dfAspremont and Gevers d i s c u s s and c h a r a c t e r i z e a wide
combination of a s s u m p t i o n s t h a t r e l a x ' b o t h t h e i n t e r p e r s o n a l comparison and
t h e o r d i n a l i t y a s s u m p t i o n s . Our framework h e r e i s most c l o s e l y r e l a t e d t o
axiom (CN) o f dfAspremont and G e v e r s , which assumes t h a t i n d i v i d u a l u t i l i t y
f u n c t i o n s a r e c a r d i n a l and non comparable, and t o t h e c a r d i n a l i t y a s s u m p t i o n s
of Sen [ 1 6 ] and of K a l a i and S c h m e i d l e r .
Respect o f u n a n i m i t y i s s t r i c t l y weaker t h a n t h e P a r e t o c o n d i t i o n .
I t r e q u i r e s t h a t i f a l l i n d i v i d u a l s a g r e e unanimously o v e r a l l c h o i c e s , s o
does t h e a g g r e g a t e . T h i s c o n d i t i o n does n o t imply t h a t i f one c h o i c e x i s
p r e f e r r e d t o a n o t h e r y by a l l i n d i v i d u a l s , t h e n t h e a g g r e g a t e p r e f e r s x t o y .
T h i s n o r m a l i z a t i o n h a s i n p a r t i c u l a r t h e e f f e c t t h a t t h e sum t o t a l
of i n t e n s i t i e s o v e r c h o i c e s i s u n i f o r m l y bounded o v e r a g e n t s ; t h i s was a sugg e s t i o n of L . Gevers.
4 A c o n n e c t e d component o f a t o p o l o g i c a l s p a c e Y i s a maximum connected
subspace of Y . A s p a c e X i s -c o n n e c t e d i f i t c a n n o t be decomposed as a union
X = XI U X 2 , whereX # $, X2 # $, and XI and X2 a r e b o t h open and c l o s e d s e t s .
1
T h i s e x t e n d s t h e n o t i o n t h a t any p o i n t i n X can be j o i n e d t o a n o t h e r i n X by
a path contained i n X.
5(!L1,u) i s t h e Banach s p a c e of i n f i n i t e s e q u e n c e s of r e a l numbers
.
{ ~ n I n = l , 2 , . . such t h a t
co
see [ l l ] .
Note t h a t t h i s c o n d i t i o n i s b i n d i n g o n l y when a l l v o t e r s have i d e n t i c a l p r e f e r e n c e s . I t i s t h e r e f o r e a s t r i c t l y weaker c o n d i t i o n t h a n P a r e t o ,
s i n c e a r u l e $ s a t i s f i e s t h e P a r e t o c o n d i t i o n i f whenever a c h o i c e x E X i s
preferred t o another y E X f o r a l l preferences
. , p k , t h e n $I($,.
,pk)
a l s o p r e f e r s x t o y.
.
..
'
I n a d d i t i o n t o v i o l a t i n g our axioms, r u l e s t h a t a s s i g n z e r o outcomes
t o non z e r o v e c t o r s a r e c l e a r l y u n d e s i r a b l e f o r o t h e r r e a s o n s : It i s e a s y t o
check t h a t i f $ i s a c o n t i n u o u s map from ( R ~ i n) t ~
o Rn t h a t maps a p r o f i l e of
t h r e e v o t e r s w i t h non z e r o v e c t o r s i n t o t h e t r i v i a l ( z e r o ) s o c i a l p r e f e r e n c e ,
i t w i l l n e c e s s a r i l y map t h e e q u i v a l e n t of some Condorcet t r i p l e i n t o t h e
t r i v i a l outcome ( 0 , ...,0 ) . The 'Condorcet t r i p l e ' we a r e r e f e r r i n g t o i s
o b t a i n e d by choosing t h r e e p a i n t s (xyz) i n Rn, so t h a t t h e t h r e e v e c t o r s g i v i n g t h e v o t e r s ' p r e f e r e n c e s r a n k t h e s e c h o i c e s i n t h e o r d e r s (xyz) (zxy) and
Such
a g g r e g a t i o n would g i v e a t r i v i a l ( t o t a l i n d i f f e r e n c e )
(yzx) r e s p e c t i v e l y .
s o l u t i o n t o t h e Condorcet t r i p l e , which i s c l e a r l y n o t an a c c e p t a b l e s o l u t i o n .
-
8 ~ h ep o i n t w i s e convergence topology
rule
see [ I l l .
-
r
on F i s d e f i n e d by t h e c o ~ e r g e n c e
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