Download Order, topology, and preference

LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
ORDER, TOPOLOGY AND PREFERENCE
Murat
R.
Sertel
565-71
October, 1971
MASSACHUSETTS
"ECHNOLOGY
CAMBRIDGE. MASSAC
(MASS.
If^ST.
DEC 10
TECH^
1S71J
DtWEY LIBRARY
ORDER, TOPOLOGY AND PREFERENCE
Murat
565-71
R.
Sertel
October, 1971
This is part of ongoing research on 'planning and control
for multiple objectives' being conducted within the MIPC
(Management Information for Planning and Control) Group at
The author wishes
the Sloan School of Management, M.I.T.
to acknowledge a helpful discussion with Paul Kleindorfer.
RECEIVED
DEC
8
1971
I
.
TABLE OF CONTENTS
1
Preliminaries
2.
Some Basic Order- Topological Facts
6
^
H
3.
Consequences of Connectivity
4.
Application in the Theory of Preference
1^
^^
4.1
Representation of
4.2
The Structure of Preference Relations and Their
Representations
20
4.3
Social Optimum and Consensus - A Simple Existence Result
33
a
Preference Relation,
633659
ORDER
,
TOPOLOGY AND PREFERENCE
Murat R. Sertel
Massachusetts Institute of Technology
This is a reasonably self-contained paper bringing some standard
order-related and topological notions, facts and methods to bear on a
number of central topics in the theory of preference and, as
a
nat-
ural but henceforth unmentioned correlate, the theory of optimization.
Much of the material brought to bear is well-known to economic theorists and even more so to mathematicians.
Part of it, however, falls
into that growing class of mathematical results motivated by social
analysis, and is,
I
think, new.
spersed throughout Section
[9]
work.
3 as
Such results will be found inter-
extensions of the basics of Nachbin's
Among such results, however, the two theorems under 3.4
and 3.5 are probably the more important to note.
The topic of Section A, the theory of preference, is largely but
not solely the motivation for the study as a whole.
Thus, some of
the facts and notions presented before that section are not used at
all in Section 4.
Sections 1-2 are preparatory.
Section
3
exploits
some consequences of connectivity, especially for normally preordered
spaces, thus extending the early work of Eilenberg [5] and Nachbin
[9]
on order and topology.
Section
4
begins with the subsection 4.1 in which Debreu's
celebrated first representation theorem [1, Theorem 1,
p.
260] is
made obsolete by the more general corollary (4.1.1) to theorems
(3.4) and
(3.5), as a result of which Debreu's assumption of
-4[Thus,
separability for the space of prospects can be dropped.
from the viewpoint of economic theory, 3.4 and 3.5 may be looked
upon as lemmas aimed at 4.1.1, which is then "elevated" to theorem
status.
]
The next subsection, 4.2, uses this and [10] to extend the
foundations of Gorman's [6] insightful characterization of the
structure of preferences.
It
is indicated how this significantly
extends that characterization as a whole by allowing two of Gorman's
postulates (separability and arcwise connectivity) for the space
of prospects to be relaxed to a much weaker postulate (connectivity),
The content of the brief final subsection, 4.3, is described
quite well by its title.
To the reader minimally knowledgeable in topology, this study
In any case, Dugundji's
is mathematically self-contained.
Topology [4] will be our standard reference in this domain.
Dugundj
i
Standing Terminology and Notation
will be denoted by
R,
while
with the usual
a set
X,
(equivalently
a relation on X
of the Cartesian product of
:
The set of real numbers
will denote k-dimensional
E
Euclidean space (k = 0, 1, ...).
R
[N.B.:
uses 'path' for 'arc'.]
Thus,
,
E^
will stand for
will mean a subset
X
Given
the order-) topology.
upon itself.
F
c
X x x
A preference
relation is a complete transitive relation, and a utility
function is simply a real-valued function preserving
ence relation.
a
prefer-
[See further terminology given in Section 1.]
-5A set
X
on which a preference relation is postulated is a
space of prospects
.
The last three underlined terms will
seldom be used from here on.
and
I
(the author)';
'We'
will mean 'you (the reader)
'iff will mean
will mean 'neighborhood'.
'if and only
if;
'nbd'
-6-
Prelimlnarles
Let
:
rcXxX=X^
Tx = {y e X|
be a relation on a set
(x, y)
e
D
^
(x e X).
\
xr = {y e X|
A subset
A
c:
(y, x)
D
ATc
said to be decreasing iff
iff its complement, denoted by
A
A, and it is
CX
A
Clearly,
A.
C
FA
to be increasing iff
is said
X
e
Denote
X.
is
Also, it is
is decreasing.
,
increasing
plain that any intersection and any union of increasing (decreasing)
A
Thus, each set
sets is increasing (decreasing).
determines
X
CI
AA
a unique smallest increasing (decreasing) set, denoted by
which contains
Ac
X^
=
A
{
In fact, if
A.
AA
=
(x, x)
I
x e X}
FA U A,
is
c; X,
^
Z>
(A
6
antisymmetric iff
or decisive ) iff
fi
9.
U Q-i
denotes the converse of
T,
C
A = T U
= x^
Q.
,
9,
it is reflexive and transitive;
iff it is a preorder on
a total order on
X
X
X)
9'^
f)
(where
/^
and we have
[Let
.
^
C
transitive iff
,
,
X^) is the smallest re-
the diagonal of
AA = AT U A
to be reflexive iff
A
T
arising from this notation by setting
flexive relation containing
(AA)
is transitive, then the relation
C
A
where
,
X^
Q.A
.
3
Q
is said
ffiA
for all
and complete (or total
^^
=
{(y, x)
(x,
|
is called a preorder on
it
X
y) e Q]
iff
is called a partial order on
X
and antisymmetric; finally, it is called
iff it is a complete partial order on
X].
The proofs of the following two useful facts are entirely
straightforward and, hence, omitted.
-7-
Proposition
1.1
be a relation on a set
T'. X^
Let
:
A = T
is transitive iff each of
A
i
,
F
\
A
,
T'^
X.
and
Then
V
is
T"-"
transitive.
Proposition
1.2
is
r
irreflexive (i.e.,
r =
ular,
1.2.2
2.
\
(r'^)"^
A =
(r'^)~^,
and
U AU
r\
such that
r'^
T'^
\
a
T"^;
in partic-
.
are pairwise disjoint and
is transitive.
r
if
= 0) and so is
r-^ =
A
T,
X^: X^ = r
X,
Then
A
il
T
and
A
The three relations
exhaust
1.2.3
be a relation on a set
T / X^
Let
is complete and antisymmetric.
r
1.2.1
:
Some Basic Order-Topo logical Facts
:
The facts and notions presented in this section are here
as basic propositions demonstrated or notions used by Nachbin
[
9
,
pp.
A relation
26-27].
said to be semiclosed iff
X c X;
2.1
lation on
2.1.1
:
Let
F
C
and
X^
xF
on a topological space
X
be a topological space and
(x, y)
e
F^
FCJ X^
there exist disjoint nbds
of X and y, respectively, such that either
V
if
a re-
X.
If for every
or
X
are both closed for each
be closed iff it is closed in X^.
it is said to
Proposition
Fx
is decreasing,
then
F
is closed.
U
U
and
us increasing
V
2.1.2
If
is a closed preorder,
r
then for each
there exists an increasing nbd
V of y
2.1.3
If
such that
Cad 2.1.1):
:
Let
(u, v)
e
Then
(ad 2.1.2):
of
(x,
U
U X V Z) U' X V'
e
H V =
Suppose
not meeting
y)
that, indeed,
z
U
T
C
is increasing and
U =
implies that
r
Fu,
i.e.,
(u, v)
F)
e
Given
xT
is closed trivially.
Then, by
x
U'
v'
meets
F
,
v'
so
U
O
V =
e Tu)
z
e Fz)
,
,
and
for if
z
e
V
then transitivity
(since now
v
£
Fz
C FFu
a contradiction.
x £ X,
we show that
is also closed.
So assume
(rx)
If
^
Ix
is closed;
rx = X,
then
and let
y £
2.1.2, there exists an increasing
U of X
x
decreasing, while
But
v
U'
V = V'r,
TU',
V
whereby
T = 0,
choose a nbd
V
y).
with
v e V'
D
U x v
U (so that there exists u e U' with
(ad 2.1.2):
nbd
(x,
y).
be increasing, nor
U
£ T'^,
y)
(x,
so that,
u e vT,
then
and define
is a nbd of
imitation shows that
it
(x,
be as
U, V
is a nbd of
V
,
Thus,
(?.
F,
(so that there exists
of
x
U
(U x v)
e
and let
T*^,
contrary to assumption, neither can
decreasing, since
and a decreasing nbd
U of x
then it is semiclosed.
(x, y)
described in the hypothesis.
Furthermore, if
T
£
V = 0.
is a closed preorder,
r
Proof
O
U
(x, y)
and a decreasing nbd
V of y with
U
D
V = 0.
(rx)
But
Fx
Fx
C
the smallest increasing set containing
is
V = 0.
Fx
and, hence,
U
This shows that
so that
x,
Fx
is
closed, and completes the proof.
From the fact that
closedness of
2.2
Corollary
A
when
Let
:
A = F U A
X
is Hausdorff immediately yields the
X
be
and is reflexive, the
T^
preorder, and if
FCX^.
and
A
is a
A
now becomes a closed preorder on
If
r
is
is closed, then so is
T
transitive, then
A,
whereby
X.
Furthermore, the conjunction of 2.1.1 and 2.1.2 plainly implies
the following
2.3
Corollary
For a preorder
;
F
d X^
on a topological space
X,
being closed is equivalent to the condition that, for each
(x,
y)
r
e
has an increasing nbd disjoint from some decreasing
X
,
nbd of y.
The next proposition relates our earlier observations to
separation properties of
2.4
Proposition
if
Let
:
F
CX^
is semiclosed,
F
X
and
via antisymmetry of
TCI X^.
be a partial order on a top-
X
is
(i.e., a Hausdorff
T
2
space)
if
Proof
Suppose
(i)
:
x ^ Ty
F
or
is closed.
x, y c X
(ii)
are distinct, so that either
y ^ Tx.
Assume that
F
is semiclosed,
-10-
Suppose that
X
holds.
(i)
belongs and
Then
belongs and
then
not, and
(yF
does not.
This shows that
is closed,
X
so that
then, whether
and
X
T,
(ii)
holds,
while
x does
x belongs while
(ii) holds,
or
(i)
have disjoint nbds, i.e.,
y
of
y
proving the first half
Tj,
is
{y}
To prove the second half, we note that, if
of the proposition.
r
y belongs
which
is an open set to
)
y
is an open set to which
)
Also similarly, if
does not.
x
(xT
is an open set to which
(Fx)
Ty = Fy
by the reflexivity
y,
does not; similarly, we see that
y
is an open set to which
(ry)
X
2.1.2 applies,
T2.
is
This
completes the proof.
Given a relation
said to be
X e yr.
T
T-convex iff
Tx H yf
a subset
X,
CA
whenever
be a topological space.
X
Let
on a set
Then
A
C
x,
X
will be
y e A
and
X (or its
topology) will be said to be locally F-convex (l.T-c.) iff, for each
X
e X,
the set of
nbd system of
F-convex nbds of
x
forms a base for the
X (or its topology) will be said to be weakly
x.
F-convex (w.F-c.) iff the set consisting of the open
subsets of
X
equipped with
forms a base for the topology of
F
C
X^
,
X.
F-convex
Finally,
X (or its topology) will be said to be
F-convex (F-c.) iff the set consisting of the open increasing
and the open decreasing subsets of
X
forms a subbase for the
topology of
mention of
2.5
in the above may be omitted.)
T
Proposition
Proof
T-c. => w.T-c. => l.I'-c.
:
F-c. => w.F-c.):
(ad
:
(When there can be no consequent confusion,
X.
The intersection of any number of
Thus,
increasing (decreasing) sets is increasing (decreasing).
the intersection of a finite number of increasing (decreasing)
open sets is increasing (decreasing) and open.
To finish the
proof, it suffices to note that the intersection of an open
increasing with an open decreasing set is open and convex.
(ad w.r-c. =>
3.
l.r-c):
Obvious
Consequences of Connectivity
Connectivity plays a great role in the pioneering work of
Eilenberg [5]
on ordered topological spaces, and we open this
section with, essentially, a rewording of one of his early results,
including proof for the sake of completeness.
3.1
Proposition
;
X
Let
(see also 1.1), hence
Proof
Then
:
Suppose
F'^y
CX
z
e
\{z},
F
C
Then
F
be a connected space and
semiclosed complete antisjonmetric relation.
F
Fy
X^
a
is transitive
is a total order.
and
y e Fx
holds for some
while 1.2.2 implies that
which is the union of two disjoint
sets
Fz
X\{z}
and
zF,
x, y,
= Fz
z
£
(j
zF
each of
X.
which, by semiclosedness of
so is
hence, either
r y;
is open.
T
or
F y CI Fz
by reflexivity (from completeness) of
by assumption, we must have
X
e
yF = F'^'y,
z
e
Fx,
that
3.2
yC
F
C
yd
and since
,
y e zF
x e zF, i.e.,
Hence,
F y.
Then 1.1 implies
is seen to be transitive.
F
y e F y
As
zF.
But, also by assumption,
zF.
F~-'y
F
is connected,
X
is transitive, hence a total order, as to be shown.
Proposition
Let
:
F
dX
be a preorder on a connected space
X.
is closed iff it is semiclosed.
F
Proof:
and 1.2.1 has
whereby
F
Then
F
As
We state "only if" merely for completeness, as it
is already given by 2.1.3.
To see "if", assume
semiclosed, and suppose (x, y)
F
Combining 2.1.2 with the fact that
Fx
F =
Define
Z =
(Fx)
imply
X = Z
since
Fx
and
demonstration.
By definition of
open.
f]
= Fx
U
(yF)
yF,
Z
.
a
x
while
yF
is
we see that FxHyF
y,
= 0.
cannot be empty, since this would
contradiction of
X
being connected,
are closed by assumption and disjoint by
yF
Let
Z,
z
c
Z,
and define
x e U
and
y
e
U =
V,
and
(zF)
while
U
and
V =
Thus, 2.1.1 applies, so that
This completes the proof.
(Fz)
are
V
As the complement of a decreasing (increasing) set,
increasing (vis decreasing).
is closed.
.
is reflexive and thus
A
is the smallest increasing set containing
the smallest decreasing set containing
e F
is
U
F
.
3.3
Corollary
Let
:
be connected and
X
Then
antisymmetric relation.
Furthermore, each point of
is Hausdorff.
infimum or the supremum of
X
a tree.
is
cutpoint iff
F CZ X
[
N.B
Given
.
X\{x}
connected,
X
said to be separated by (a cutpoint)
compact connected
semiclosed complete
X\{x}.
X
cutpoint, the
is a
is
compact, then
is called a
x c X
Points
is not connected.
distinct components of
X
X
and, if
X,
a
is a closed total order and
F
are
y, z e X
iff they belong to
x
A tree is a continuum (i.e., a
space) whose each two distinct points are
1^
separated by some point.]
Proof
:
That
is a closed total order follows directly from the
F
conjunction of 3.1 and 3.2.
Using 1.2.2,
X\{x}
Then 2. A implies that
U
= r*^x
for each
xF'^
is Hausdorff.
X
where this is
x e X,
a decomposition into two disjoint sets which are open by semi-
closedness of
If
F X = 0,
F.
then
by antisymmetry of
Thus,
is lost by
x
assuming
and
X
If
X.
x
then
x
is seen, similarly,
is compact, then it is a con-
no generality
X,
and semiclosedness of
X
e
.
from which it is straightforward to
x e F y,
z
^ xF
F x ^
is an infimum and is unique
are distinct points of
y
show that connectedness of
the existence of a point
,
= 0,
xF
If
F.
to be the supremum of
tinuum, and if
is a cutpoint if
x
X = Fx, i.e.
X
with
x
lying in
F
F
z
implies
and
y
in
zF
showing that
,
is a tree and completing the proof.
X
Actually, the very first consequence in the last corollary
implies that if
space
and
is disconnected
/T
(since
is now closed
T
is the union, by 1.2.2, of the two disjoint sets
/T
and
is a semiclosed total order on a connected
T
then
X,
)~
(r
each of which is open by closedness of
T
This
F).
is a rewording of a "half" of the first of three main theorems
of Eilenberg
Theorem
[5,
states, conversely, that
space
X
only if
I,
p.
40], the other "half" of which
is disconnected for a connected
A
can be endowed a semiclosed total order.
X
For what follows, we will need Nachbin's
28]
p.
[9,
generalization of the familiar notion of a normal space, namely,
that of a normally preordered space.
FC
equipped with a preorder
(by
F)
that
P
iff, for every two disjoint closed sets
is decreasing and
disjoint open nbds
creasing and
Let
F
A topological space
Uj
and
U
increasing,
P
U^
X
is said to be normally preordered
X^
P
,
P^C
and
P
respectively, with
,
have
de-
U
We need some further terminology.
increasing.
be a preorder on a topological space
If
X.
C
Y
write
Y << Z
V,
Y < Z
I(Y)
(respectively,
to mean that
to mean that
Y
D(Y)
and
respectively, such that
Z
U
D(Y)).
I(Z) = 0;
If
Y,
Z
C
then
X,
the smallest increasing (decreasing) closed set containing
be denoted by
such
X
P
Y
will
we will
X,
we will write
have disjoint open nbds
is decreasing and
V
U
and
increasing.
We will use the following simple characterization theorem [9,
p.
29]:
-15-
A topological space
preordered iff
3.4
Theorem
X
equipped with a preorder is normally
Z
whenever
a semiclosed relation
by
Y < Z
F
C
X^
Then
.
is normally preordered
X
r.
Proof
Let
:
A, B
for otherwise
reflexivlty of
y t D(A)
trivially.
B
F),
U 1(B).
y
taining
yF.
y
is
I({y}) = Fy.
there is a point
y e X
F
n
Fy =
is
As
= yF
being semiclosed,
F
preordered by
F,
F.
by
such that
U
implies that
1(B)
A
<
{y}
<
Thus,
B.
is an open decreasing nbd of
Hence,
A
B, and
«
B,
and
decreasingness of D(A)
n 1(B), i.e., that
and open increasing nbd of
transitivity of
Cl(B),
B
D({y}) = yF
is semiclosed,
1(B), y i D(A)
F y
are
preorder, the smallest increasing
a
By transitivity of
(?
and
B,
^^
(i)
1(B)
and the smallest decreasing set con-
Fy
is
A ^
and
D(A)
d D(A)
and increasingness of
D(A)
Assume
< B.
Since
(A
As
set containing,
A
and suppose
CZX
«
A
closed and disjoint nonempty sets
yF
ZCX).
(Y,
be a connected space completely preordered by
X
Let
:
«
Y
F y
f)
Y^
showing that
X
and
A
=0
F
by
is normally
F.
The following theorem of Nachbin
[9
,
Theorem
2,
p.
36]
re-
duces to Urysohn's famous extension theorem for continuous real-
valued functions on a normal space when it is noted that a
normal space is simply a space
discrete partial order
ACX^.
X
normally preordered by the
be a space normally preordered by
X
Let
Theorem:
be a closed subset such that fp: P
P C: X
->
real-valued continuous function preserving
y e Tx => fp(y) > fp(x)].
A(A) = {x e P|
Then
X
[i.e.,
T
denote
,
B(A) = {x e P|
and
A}.
fp(x) >
can be extended to some bounded (real-valued) continuous
f
function
A
For each
fp(x) < A}
e E^
and let
T,
bounded
is a
E^
preserving
X->- E^
f:
V
whenever
A(A) < B(A')
iff
< A'.
Our immediate motivation for recording this theorem is its use
in proving the following
3.5
Theorem
r
e
Let
;
be a space normally preordered by a relation
X
such that
X^
= T
S
a bounded continuous real-valued function
the preorder
Proof
point
Arbitrarily choose
the constant
and
=
f
r-preserving.
Defining
P =
Define
p £ X.
closed.
Then
tt.
Ep.
As
£ E^
tt
->-
preserving
E^
is semiclosed,
E
and define
<
TT
case of
(i)
case of
(ii),
or
A
or
(ii)
A(A) = 0,
as
A, A'
e
such that
E
A
<
A'
as in Nachbin's theorem above, the
B(A)
(iii),
^ E
P
fp:
is
P
is trivially bounded, continuous
fp
Now choose
and
A(A)
(i)
X
and choose an arbitrary
X ^
theorem requires only for us to show that
either
f:
T.
To avoid triviality, assume
:
Then there exists
is semiclosed.
I"''-
A'
<
tt
B(A') = 0,
so that
or
Now
A(A) < B(A').
(iii) A <
so that
tt
<
A
'
.
I(B(A')) = 0.
D(A(A)) = 0.
In
In
Thus, in all
D I(B(A'))
D(A(A))
cases
= 0,
showing that
A(A) < B(A').
We
conclude that there exists a bounded, continuous, real-valued,
r-preserving
Corollary
3.6
f:
X ^
f(P) =
with, in fact,
E"'
(See 4.1.1 below)
:
Applications in the Theory of Preference
4.
tt.
:
This section will illustrate how the methods so far presented
may fruitfully be applied in social analysis.
The chosen specific
area of application is the theory of preference, otherwise known
to economists as "utility theory".
4
.
1
Representation of a Preference Relation
:
A celebrated result in this theory is Debreu's
Theorem
Theorem
I,
:
p.
[1
,
162] following first "representation"
X
Let
be a separable connected space completely pre-
ordered by a semiclosed relation
continuous real-valued function
fact, representing
X ^ E^
f:
i.e., obeying
V,
Then there exists a
F (Z X^.
preserving
iff
y e Fx
f(y)
T
(in
^
f(x)).
The first contribution of our results in the previous sections
to our present area of application consists in subsuming Debreu's
just-stated theorem as a direct corollary of the more general
4.1.1
Corollary
;
Let
by a relation
X
be a connected space completely preordered
F CZ X^
.
Then there exists a bounded continuous
real-valued representation
f
:
X
-^
E^
of
F
iff
F
is
semiclosed.
Proof
By 3.4,
:
is normally preordered by
X
semiclosed, then so is
F~\ whereby
is
F,
If
F.
F
f:
X
->
E
in fact, a representation of
,
is
3.5 directly yields a
F-preserving function
bounded, continuous,
by completeness of
F
fl
(which,
F)
The converse is obvious.
as sought.
In comparing 4.1.1
with Debreu's indicated theorem, it will be found
that the hypothesis of 4.1.1 is weaker, missing the separability
of
X,
bounded
while its consequence appears stronger, guaranteeing a
f
Of these differences, it must
of the desired sort.
be remarked, the latter should not be considered important
or,
for that matter, real, as we can always use the bounded function
instead of
f
f/f+1
if
f
>.
f/f-1
if
f
<
whenever
f
happens not to be bounded, and g ob-
viously has all properties desired of
separability assumption for
X
The absence of the
f.
in 4.1.1, however, must be viewed
as a strict improvement with some important "practical" con-
sequences - from a technical viewpoint - for the theoretician
concerned with matters of preference.
ological fact that a product space
topology of
{X
I
a
e
a})
[It is a standard top-
(with the product
A
is coniff each X
X =
IIX
is connected
nected but that the following restrict the product invariance of
separability and 2° countability
:
(1)
X
is separable iff each
yt
is separable and all but at most 2
X
single points, where
("aleph naught");
numbers
is 2°
the cardinality of the set of natural
is
if^
8
[
]
#
is 2°
X
(2)
countable and all but
(See Marczewski
X 's consist of
of the
°
countable iff each
of the
o
X 's
a
concerning (1).)
X
are indiscrete.
Thus, for instance, a
topological vector space obtained as the product of more than
2
2°
°
copies of the real line is neither separable, nor, indeed,
countable, while every convex subset is (in fact arcwise-)
connected, so that, in this case Debreu's second representation
theorem
1
[
,
Theorem 2,
p.
163] is just as inapplicable as his
first, while 4.1.1 can be used.
and Milnor's
7
[
]
Furthermore, interpreting Herstein
mixture set in the natural sense of convex
in this vector space, notably
set here, there are connected sets
(from the viewpoint of generalized Kuhn-Tucker theory) the star-
shaped sets, which will not be mixture sets, so that an instance
is
found where the representation theory of
cidentally, deals with the case where
[
7
]
- which,
is a total order
T
in-
will
not apply while 4.1.1 will.]
Applications of 4.1.1 extend also into the next subsection.
Another simple fact to be used there but properly belonging under
the present heading is the following
4.1.2
Lemma
:
Let
F
connected space
C
X^
be a complete transitive relation on a
X,
and let
real-valued function.
F) iff
r
f
is
is semiclosed and
f
:
(a)
X
->
E
be a
continuous
F-preserving
(representation nf
f(X) is connected.
-20-
Proof
r
(and stated merely for completeness) that
is obvious
It
:
is semiclosed and
is connected if
f(X)
W
f~^(W)CI X
is open if
CI E
W={weE^|w>w}
we take
of every subbasic open set
is semiclosed and
T
continuous.
is
For this it suffices to show that
We prove only the converse.
the inverse image
f
f (X)
connected, and
for some arbitrary
w
e\
e
remarking that the argument will be entirely similar for
W'
= {w c E^
(ii)
w
I
<
w
w
>
w
Now consider
f~^(W) = X r
serving.
w
for all
and if (ii), then
open.
If
}.
implies that either
f(X)
,
f~
^
t
w
f(X).
e
= 0;
(W)
w
w
(i)
f(X),
e
by the fact that
Therefore, if
T
then connectivity of
fix),
<
w
for all
If (i),
w
T
f(X)
e
=
f
~
•"
f(x
(W)
then
is
Then
).
is complete and
is semiclosed,
or
f-'(W) = X;
then
in either case
and let
w
f
f~^(W)
F-preis
open, and this completes the proof.
4.2
The Structure of Preference Relations and Their Representations
:
Otherwise stated, the topic of the present subsection is
that of "aggregation" and, in particular, the "separability" -
additively or in general - of utility functions.
The immediate
motivation is to extend the complete characterization by Gorman
[6]
of the "separability" and, in general,
the structure of
utility functions.
This characterization was given by Gorman under the assumptions, among others, that the space of prospects was
ically) separable and arcwise connected.
(topolog-
Arcwise connectivity
-21-
was effectively shown [10] to be relaxable to connectivity for
Gorman's results, being so for his underlying Lemma
p.
387].
It
1
[6,
appears that the only reason for postulating
separability as an assumption was to ensure the existence of
continuous utility functions, invoking Debreu's first rep-
resentation theorem stated in 4.1.
As 4.1.1 now outrules any
need for this theorem and for the space of prospects to be
separable, one quickly intuits that Gorman's separability
assumption may also be eliminated.
Lemma
1
Here we show that his
can be extended so as to apply whether or not the
space of prospects is separable (or arcwise connected), so
long as it is connected, and encourage the reader to check that
this actually yields a corresponding extension of the whole of
Gorman's results in [6], so that his assumptions of separability
and arcwise connectivity can, in fact, be diminished to con-
nectivity throughout.
I
consider this as opportune a moment as any to indulge
in the premature expression of a thought, as fuzzy as it is in
my mind, that what we. Including Professor Gorman, are looking
at is a topic of interest in its own right as having to do
with the "structure", in general, of relations and of maps preserving them, deserving at least a glance by specialists in
functional equations and semigroups - if, indeed, they had not
-22And, continuing to bleed out
already seen through the matter.
this thought. Professor Gorman's analysis of the structure of
(see A. 2.1)
a|
is probably a key to
more doors than meet the
eye in this dimly illuminated hallway.
Throughout,
We now turn to more conrrete matters.
X =
X^
n
„ill be the product of a family
{X^
|
a
c
of
A}
A
spaces, and
c
B
will be a relation on
X^
onto a factor
jection of
X2
for each
C
B
onto
A,
= H X
X
and
Given any
X.
Projection
= X c-
X
X
will be denoted by
tt
,
and pro-
X^
will be denoted by
tt^.
Finally,
we define two relation-valued maps
and
y
by
X
on
Yg
C
we will denote
A,
X
of
F
y(x^) =
r
n [X„
X
and
{x®}]2
D
Yb(x^) = ^B (Y(x^)),
lower case Latin denoting, as from here on, a generic element
of the respective capital
and
4.2.1
denoting
x„
D
Definition
that
B
:
is
D
by
D
e
X
x
,
a sector of
is semidivisible by
A
We say that
and we write
A,
e
X
,
etc.],
a|b,
B
or
iff
y
(in which case its constant value on
clearly coincides with
TT^(r),
which latter we denote
rS
We say that
r„).
D
divides
of
x
etc.
tt_,(x),
is a constant function
X^
[e.g., x e X,
A,
A,
or that
and we write
we define
A
B
divisible by
is
is
A||b,
a
B,
that
B
complemented sector or factor
iff
a|b
and
aIb'^.
Finally,
a|
= {b|
a|b},
a||
= {b|
a|b}.
The following are clear.
4.2.2
Proposition
4.2.3
Proposition
A
a||c;a|.
:
0,
:
For any
c
B,
CA
C
x = (x
and any
,
x
=
)
(x^, x^) z X,
Y(x
,
Y(x
)
B
,
C A)
4.2.4
D =
^Q
Proposition
i.e.,
B
:
fZ k\
partial order
(a|,C)
a|
_
CZ
,
y(x
B
)
n y(x
B, ^
)
,
n Y(x
C2
).
denoting
,
D
and
B
f].
(x) = x^
Y(x^)3Y(x^),
a|
is closed under arbitrary intersection,
=>
Q
C
B e a|.
[Thus,
of containment is a complete lower semilattice
Inf(Al) =
with
(a|,
together with the
a|
H
a|
=
D
)
Sup(Al) =
and
intersection is a commutative band (band
Proof:
tt
E = ^l^ B.
together with the binary operation
potents)
and
),
C.
,
Y(x°)
where
C^
for any subset
In fact,
(x e X
Z>y(x
)
U
B
Q
:
•
a|
-*
with identity element
A
Merely observe that the containment
Yn
^^ ^ constant
of
semigroup of idemand zero
Y (x
)
^
of the last proposition becomes an equality wlienover
In that case
Thus,
A.
A|
function, as
y
0.]
I.Y(x')
BcL'
B
(.
a|
is so for
]
-2Aeach
thus,
B e S;
D e a|
[The parenthetical note is
.
now obvious.
S
Thus, if we know that
sectors of
is a set of
then
A,
we know that the intersection of the -members of any subset
is also a sector of
C CLB
4.2.5
Proposition
B
if
and
are factors of
C
as well as
B*^
sectors of
A.
,
we have the obvious
a||
is closed under complementation,
a||
:
For
A.
c'^
,
B
.0
A,
and
C
then
B*^ fl
c'^
B
=
\C
(B
U
and
C)'^
We now focus our attention on the case where
order.
T
so that,
C
\
B,
are
is a pre-
In this connection, the following two propositions
collect some elementary facts, the proofs of which are straightforward.
4.2.6
Proposition
:
If
a relation
T
C X^
on a product
X =
11
X^
A
satisfies any one of the properties transitivity /reflexivity/
symmetry /antis3mimetry/completeness
T^ = TT^(r)
projection
4.2.7
Proposition
:
Let
F
c
(B
C X^
A)
,
then so does each
.
be a relation on a product space
Then T is transitive/
X = n X
such that F = T^ x T^.
C
a
B
A
and
reflexive/symmetric/antisymmetric, respectively, if T
,
Thus, in particular, if
so is each projection
T
T
is a
(BCA).
(complete) prcordcr, then
The relation between
T
-25and its projections
is stronger for sectors
T
may see that the structure of
In fact, one
of
B
and of
a|
bear a systematic kinship with the "structure" of
T
,
A.
a||
by
the latter of which we refer, broadly, to the set of relation-
ships obtaining between
T
and certain of its projections
Both the "structure" of
T
and that of the collections
and
are related to the "structure" of functions
a||
preserving
T,
u:
F^.
a|
X ^ E
more particularly to the forms in which such
functions can be written in terms of certain functions
u„: X„
Information gained about either of these "structures"
R.
-»-
For this reason we now set
seems to help illuminate the others.
V- preserving
up some apparatus to deal with the "structure" of
functions
CX^,
r
u: X
and let
clear that
X„
D
{x
by
be such a function for a preorder
u
Let
.
We use
x e X.
u^: X^ ^ R
define
X
E^
->-
D
reference point " to
Ug(Xg) = u(Xg, x^)
In the case where
however, varying
"
as a
preserves the preorder
u„
}.
x
B £ aI
T
(B
fl
(X
Some clues involving the form of
families of sectors of
:
Let
product
X
T
= n
A
tation of
F,
CX
X
«
It
is
on
{x })
,
does not alter (the projection onto
x
this case, and only in this case, we abbreviate
Lemma
x
D
.
and only in this case,
of) this preorder considered as a relation on
4.2.8
C A)
,
x
A
x
let
and let
u
X
.
u
X
Thus,
to
in
u
.
and pairwise disjoint
are furnished by the following
be a complete preorder on a nonempty
u: X
{B
|
^ R
n £ N}
be a real-valued represenbe a partition of
A \
C
CCA.
for some
4.2.8.1
Then the following are equivalent,
There exists a family
{v
X
:
^
R|
x
11
of real-valued
N}
£
n
n
functions and a function
f
:
X
v (X
in each
v
(n e N)
such that
,
R
->
)
N
increasing
n
can be expressed as
u
u(x) = f(x^, (v (X^ )}
).
n
4.2.8.2
For each
Proof
n e N,
(ad 4.2.8.1 => 4.2.8.2):
:
attention to an arbitrary
erence point")
x =
(x^
x
x n)
,
,
i.e.,
F y,
e
y =
in
(y„
1
As
increasing in
is
f
v
v
n
V (y^
x = (x^
Suppose
).
n
x n)
,
y =
,
(y^
x n) £ X,
,
n
we then have
,
("ref-
and an arbitrary
Suppose that
X e X.
for some
u(x) > u(y),
Assume 4.2.8.1, and fix
n e N
(x
x]
,
x
B^
n)
)
->
e
X.
Clearly, the proof rests on being able to show that
u(x)
from
Now
u(y).
>_
(x
{v
,
since the
V (y^ ),
B^
B
n
But
(x
)}
's
)
can differ from
are pairwise disjoint.
y
> V (y^ )
V (x„ ) —
T\
n
a
ij
n
n
is increasing in
v
has already been established.
As
'
f
i.e., that
x
e
this implies that
,
Thus,
Fy.
u(x)
is identically
Yt,
F
a|b
.
As
n £
N
,
n
n
whereby
u(y),
>^
was arbitrary, we have shown
4.2.8.1 => 4.2.8.2.
(ad 4.2.8.2 => 4.2.8.1):
v(x*^)
= {u
(x
and define
)}
'
n
n
Assume 4.2.8.2, denote
'
J.,
w: X
->
X
x
v(X^)
by
-27-
for each
w~-'(a)*)
(x„
{u
03'
u
=
(x,
D,,
y'
,
that of
"^n
Where
}_,,,„^),
rifN(y)
w
For given such an
^(o)').
A
may select
co
e
oj'
w'
there
,
coincides with
X
with that of
N(y),
c N)
(A
)
"A
'A
and whose projection into
w'
u^ (X^
e
p,
whose projection into
w
for each
,
{P
is constant on
exists an
(x„^,
and show that, given an arbitrary
„/ x)
)}
=
w
of the form
co*
coincides
u„ (X^ ),
^A ^A
furthermore, we
;
so that its projection into
X
is a
y
point
such that
x„
a
is some
u(w~-'(a) ))
which
u^ (x„
D
D
)
(constant)
= p
u°
By hypothesis,
.
y
and, by the way in
e R,
was selected, there is a point
(jo'
u(x') = u°
such that
.
x'
e
takes the (constant) value u°
x
on each point
w"
(w'
a|b
But, by the fact that
e
u
,
w
(oj')-
By application of the principle of transfinite induction,
for each
w
e
w(X),
u
is constant on
w
^
(oj)
,
and this
completes the proof.
To economize on proofs which are either obvious or both
straightforward and tedious, some further facts are given in
the form of an
4.2.9
Exercise
;
in 4.2.8 can be expressed in the form
u
If
indicated in 4.2.8.1, then
4.2.9.1
for each
represents
ri
e
N, v
T
is
increasing in
u
,
so that
;
n
4.2.9.2
f
is "strictly increasing" in
p =
^P^L^j^'
where
then
it
r(x)
=
(x
v(x
,
All we need to show is that the diagram
)).
^
w(X)
u(X)
commutes for some function
show that
of each
defines
on
f:
(J
as the desired function.
set equal to, say,
u
since sending
w(X),
e
f
w
is constant on the inverse image
u
co
and for that it suffices to
f,
w~''((jj)
w
i.e., for arbitrary
in general,
N
[If
then
were a finite
would be clear as a con-
w(X)
for each
a|b
sequence of having
c
u(w~Va)))
l->
then the constancy of
M = {O, ..., m},
for each
(to)
To show this
n e N.]
we use transfinite
N,
induction.
N,
and denote
N(ri).
For each
Thus, consider some well-ordering of
y e N,
define
U
=
B
^
and
B
N(y)
and the functions
w (x) = (X
C^ =
C^\B^.
\
^'
and
w
= A
C
^
w^(x) = (x^*
where
by
n c N
initial segments of elements
{ug^
,
,
w
u
(X
B
.
^
on
X
by
(-B^)^neN(y)^
),
{u
(xg
Now we suppose that
^"'^
)}^,N(y)^'
u
is constant on
-29p
V (X
e
n
n
P^ > P^
B^
c
for each
)
^n^B
f(x^, p')
(a) holds with,
then
4.2.9.3
u
^°^ ^^^^
^
for each
f(X|.,
p)
in the sense that
n e N,
x^
,
furthermore,
> f(x^,
then
n e N,
X
e
and
p,
>
p)
if
>
if the hypothesis of
(b)
for some
p'
for each
p')
f (x^,
(a)
A e N,
x^ ^ \'>
can be expressed as
u(x) = g(x^,
{u^ (X
n
)}
)
n
for some g;
So far in this subsection,
the discussion invoked no
The next lemma, also proved in [10],
topology.
is concerned
with the continuity of the "macroscope" functions
(and g)
f
For the purposes of that lemma and
as in 4.2.8.1 (4.2.9.3).
some later developments, it is useful to agree on some notation.
M
Accordingly, from now on
of the first m +
1
M^
M \{i}.
will denote
sets indexed by
Given a family
we will denote
M,
4.2.10
Lemma
Let
:
X =
11
.
i
for each
e
M°} Cm|.
i e
M
,
{X
|
.
i
M}
E
.
i
.
e
.
,
m}
M,
of
for
X"^ = 11. X.
M^ J
M ^
for generic elements.
11
X.,
be a connected space completely pre-
X
M ^
ordered by a semiclosed relation
{{i}|
X =
eX,xeX
eX.,x
x.
{O, 1,
non-negative integers, and, for each
.
products, and
will denote the set
Let
let
u:
X
11
v.: X.
F CZ
X x X,
such that
->
E^
preserve
T,
-*
E^
preserve
^
T.
1
and,
,
denoting
by
v(x) = {v.(.x.)}.
and let
.
be a function for which
f
the diagram
> v(X)
u(X)
u(X)
If
commutes.
is connected, then
and
u
f
are
continuous.
Proof
Assume
;
is continuous it suffices to show that
f
is open for each subbasic open
W
C e\
W={weE-'|w>w}
w
e E^
f~^(W)
or
for some
w
I
< w*}.
If
implies that either
(ii)
w*
u~i(w) = X,
<
u(x)
w*
i u,
(i)
w*
x
for all
(w)
and show that
case of
f"Vw)
(ii),
u-^(W) =
is again open.
w* = u(x
).
then connectivity of
>
for all
u(x)
If
e X.
(i)
x e X
holds, then
so that commutativity of the diagram yields
As
u
u'-'CW) = x*r^,
(or by continuity of
f"^(W)
whereby
f~^(W) = v(u~^(W)) = v(X),
have
f
We take
is open, remarking that the argument is similar for
W^ = {w e E^
u(X)
is continuous by
u
Using the apparatus of the proof of 4.1.2, to show
4.1.2.
that
Then
connected.
u(X)
=
vCu'^W))
So, assume
preserves
T
is open.
= f~^(W),
w
e
and
Y
u(X)
In
so that
and, say,
is complete, we
which is open by semiclosedness of
T
u) and is connected by connectivity of
-31Hence, for each
X.
P. =
IT
i
ex.
(u~^(W))
is open trivially.
V.
1
preserves
^
M, the projection
e
On the other hand, for
1
^
11
1
v(u"^(W)) =
Thus,
)
as
,
1
IT
v
'
(P
.
is
)
M
f~^ = v
open, so that the commutation
f~
M
e
i
v (P
v. (P.) = {w c v.(X.)| w >
we have
T.,
which is open.
v.(x*)},
Now
is open and connected.
open, from which we conclude
(W)
o
yields
u~^
to be continuous.
f
This completes the proof.
4.2.11
Corollary
:
^
u: X
Let
E''
be a continuous representation
of a complete semiclosed relation
product space
=
X
II
X..
,2.11.2
for each
i
£
tinuous
v^ (i
e
{v.: X.
f:X xn
M°),
Then the following are equivalent.
{i} e m|;
M°,
for some family
-^
E^|
v.(X.)->E'^
Furthermore, in this case,
(see 4.2.9.2) in
i £
M°,
representation of
Proof
:
i
and some con-
M°}
e
increasing in each
u can be expressed as
u(x) = f(x^, v^(x^),
for each
on a connected
X x X
^
M
4.2.11.1
F c:
p =
f
...,
v^(xj).
is "strictly increasing"
^P^\^°'
and, for each
where
i £
p^
M
,
£
v_j^
"^i^^i^
is a
F..
The stated equivalence directly follows from the
conjunction of 4.2.8 and 4.2.10.
The rest directly follows
from 4.2.9.1-2.
This last corollary extends the fundamental Lemma
1
on which
-32-
Gorman erects his formidably complete characterization of "The
Structure of Utility Functions" 16] in the case where
arcwise connected and separable.
p.
22].]
extension here consists of relaxing the assumptions on
It
is
also extends the result
[It
which Debreu set out to prove in [2, lines 5-18,
connectivity alone.
X
X
Our
to
was shown earlier [10] that arcwise
connectivity could be relaxed to connectivity and
promised
there that the separability assumption, too, could be deleted,
as its role consisted of allowing application of Debreu' s first
representation theorem, stated above as a corollary to 4.1.1.
Given 4.1.1, we now have no need for the separability assumption
on X,
and the earlier promise is met.
The main question, however, is whether the assumptions of
arcwise connectivity and separability, made throughout by
to connectivity alone.
Gorman [6] can be relaxed throughout
[6]
Upon studying the mentioned paper,
find it safe to state that
my conjecture is Yes.
I
If the proofs of Gorman had to be mod-
ified extensively to support this conjecture, then here would
be a good place to do that.
to be the case, so that the
Fortunately, this does not appear
(well-advised) reader who also
reads Gorman's paper will find,
I
think, that Gorman's
characterization applies so long as
X
is
if Gorman's Lemma 1 is replaced by 4.2.11,
modification
I
connected.
[In fact,
the only necessary
can find in his proofs is in that of his basic
theorem (Theorem 1), where 'arc connected' in the first line
of 2.18 (p.
372) should be replaced by
'connected', in which
-33-
case the theorem, and hence the paper as a whole, will be seen
Thus, the best service to be offered here to
to be preserved.]
the reader seems to be to recommend Gorman's mentioned paper.
Hence, we let this last sentence be a pointer toward the cited
study by Gorman which has been the source of motivation for this
subsection and the validity of which has now been extended.
Social Optimum and Consensus - A Simple Existence Result
4. 3
:
The result (4.3.2) we aim to demonstrate here is, indeed,
We are given a family
very simple.
transitive relations
subset
C
B
A,
F,
C
each on the same set
,
a'
B
Fr
to
.
and
F
to
F
of
X^| a e A}
F.
For each
X.
now denotes the intersection
F
and we simplify
to
T
{F
F
=
Max X
(Sup X),
Max X
=
F
g
a
,
With respect
the set of maximal points (supremal points) in
denoted by
fl
B
X
is
where the standard definitions
{x £ X|
(x,
y)
£ F
=> (y, x) c F},
Sup X = {x e X| X X {x} C; F}
apply.
4.3.1
Obviously,
Proposition
:
X3
Max
Sup X.
The economist will recognize that
Pareto-optimal points.
Sup X,
Max X
is
the set of
on the other hand, is a rather
more interesting set, consisting of all those points which are
"superior" to every point in
of each
F
(a e A)
.
X
Of course.
unanimously from the viewpoint
Sup X
may very well be
-34erapty,
Max X
even if
is not.
In 4.3.2 we show conditions
under which both are non-empty.
Toward that, we agree to say that a relation
a topological space
X
is closed for each
in the Small (CS)
CS
x
semiclosed
is upper
T x
By the condition of Consensus
X.
e
on
T
iff
we mean
,
For each pair (B, Y) of nonempty finite
;
BCA
subsets
n
r
yeY
^
and
YdX,
y y 0.
Finally, we are able to state and prove the intended
4.3.2
Proposition
r
CX^
:
is upper semiclosed
As each
Sup X =
Or
T
a
is transitive, so is
Clearly,
X.
while the condition
(a e A)
Sup X ^ 0.
CS is satisfied, then
Proof:
(transitive relation)
is compact and each
X
If
F.
Hence,
CS implies the finite intersection
XXA°'
property that
f]T
F
r
X
^ ¥
a
is closed for each
with compactness of
X
for each finite
(x, a)
e
X
implies that
x
a,
f~]T
F
CX
x A.
then
x ¥ 0,
If
CS combined
as to be
XXA
shown.
We may refer to
in the Large (CL).
Sup X ¥
In that case, 4.3.2 reads to yield a suf-
ficient condition, namely that
upper semiclosed
as the condition of Consensus
(a e A),
X
is compact and each
under which
CS => CL,
T
i.e..
35-
CS <=> CL,
as
CL => CS
often compact while the
is always true.
T
semiclosed complete preorders
transitive relations.
to 4.3.2.
I
In practice,
X
is
are accustomed to be assumed
's
,
hence certainly upper semiclosed
Perhaps this yields a practical relevancy
do not know whether it is worthwhile obtaining
corresponding results by considering the case with a measure
space of agents, following Debreu [3] and others.
-36-
REFERENCES
[1]
Debreu, G. "Representation of a Preference Ordering by a
Numerical Function," Ch. XI in R.M. Thrall, C.H. Coombs
and R.L. Davis (eds.)> Decision Processes (New York, 1954),
159-166.
[2]
Debreu, G. "Topological Methods in Cardinal Utility Theory"
in K.J. Arrow, S.
Karlin and P. Suppes (eds.). Mathematical
Methods in the Social Sciences
1959 (Stanford University
,
Press, 1960).
[3]
Debreu, G. "Preference Functions on Measure Spaces of
Economic Agents", Econometrica 35 (1967), 111-122.
[4]
Dugundji, J. Topology
[5]
Eilenberg,
[6]
Gorman, W.M. "The Structure of Utility Functions", Review
S.
of Mathematics
,
63
of Economic Studies
[7]
,
(Allyn and Bacon, Boston, 1966).
"Ordered Topological Spaces", American Journal
(January, 1941), 39-45.
,
35
(1968), 367-390.
Herstein, I.N. and J. Milnor
,
"An Axiomatic Approach to
Measurable Utility", Econometrica , 27 (1953), 291-297.
[8]
Marczewski,
E.
"Separabilite et Multiplication Cartesienne
des Espaces Topologiques", Fund
[9]
Math
.
34
(1947), 127-143.
Nachbin, L. Topology and Order , Van Nostrand Mathematical
Studies #4,
[10]
.
(Princeton, New Jersey), 1965.
Sertel, M.R. "A Four-Flagged Lemma", Alfred P. Sloan School
of Management Working Paper 564-71
(October, 1971).
Date Due
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