Revista Española de Economía, Vol. 15, nº 1, 1998
15-21
A Note on Numerical Representations for
Weak-Continuous Acyclic Preferences
*
Begoña Subiza
Josep E. Peris
Universitat d'Alacant. Departament de Fonaments de l'Anàlisi Econòmica.
Recibido:
diciembre de 1996
Aceptado:
mayo de 1997
Abstract
In this note we obtain a continuous utility function for acyclic preferences by using a
weaker than usual continuity condition. Then we extend some results on representability of
continuous quasiorders and acyclic preferences.
Key words:
utility functions, weak-continuity, acyclicity.
Resumen.
Una representación numérica para preferencias acíclicas débilmente continuas
En este trabajo se extienden algunos resultados sobre representaciones numéricas de órde-
nes parciales y relaciones acíclicas, utilizando una condición de continuidad más débil que
la usual.
Palabras clave:
funciones de utilidad, continuidad débil, aciclicidad.
1. Introduction
It is well known that classical utility functions (
u
:
X
→ℜ
such that
x P y
⇔
u
(
x
) >
u
(
y
)) only exist for preorders (complete, reflexive and transitive binary relations);
however, several authors consider that transitivity (especially transitivity of the
indifference relation) is very unrealistic [see, for instance, the seminal paper of
Luce (1956)]. More general binary relations as quasiorders or acyclic relations,
can only be represented in a weaker way, being the most usual a real-valued func-
tion,
u
:
X
→ℜ
such that
x P y
⇒
u
(
x
) >
u
(
y
)
(*) We thank Professor E. Induráin-Eraso for helpful comments in an earlier version. Financial su-
pport from the Spanish DGICYT, under project PB92-0342, and from the Instituto Valenciano de
Investigaciones Económicas (IVIE) is also acknowledged.
16
Vol. 15, nº 1, 1998 Begoña Subiza; Josep E. Peris
this function is also called a utility function [see, for instance, Peleg (1970) for
quasiorders, Subiza (1994) and Peris and Subiza (1995) for acyclic preference
relations]. Although this general type of utility functions does not give complete
information about the binary relation, they are useful because they exist for a
wide class of binary relations and the maxima of such functions are their maximal
elements.
It should be noticed that we are considering numerical representations based
on a real-valued function. There are other kind of numerical representations such
as set-valued functions, hemisymmetric functions, two real-valued functions,
which exist for more general relations than preorders [see Subiza (1993), for a
survey on numerical representations of preferences].
In this paper we obtain a continuous utility function by measuring the lower
contour sets of the relation. The concept of measure has already been used in the
literature to obtain utility representations (Neuefeind (1972), Chichilnisky (1980),
for continuous separable preorders; Candeal-Haro and Induráin-Eraso (1993,
1994) for continuous and separable quasiorders). By using this idea we weaken
the transitivity and continuity conditions, for the case of weak-continuous acyclic
binary relations. The result obtained generalizes Theorem 3 in Candeal and
Induráin (1993) and moreover it provides numerical representations for acyclic
preferences which are not spacious and then the result in Peris and Subiza (1995)
cannot be applied.
2. Preliminaries
Throughout the paper let
X
be a topological space,
P
an asymmetric [
x P y
implies
not (
y P x
)] binary relation defined on
X
,
R
the weak relation associated with
P
[
x R y
if and only if (not
y P x
)] and
I
the indifference relation [
x I y
if and only if
x R y
and
y R x
]. Note that, from the way in which it has been defined, the relation
R
is a complete [
∀
x
,
y
,
x R y
or
y R x
] and reflexive [
∀
x
,
x R x
] binary relation.
The following type of binary relations will be used: we will say that P is
acyclic
if for any
x
1
,
x
2
, …,
x
n
,
x
1
Px
2
,
x
2
Px
3
…,
x
n
–1
Px
n
implies
x
1
Px
n
,
a
quasiorder
if it is transitive [
x P y
and
y P z
implies
x P z
]
We will denote by
L
P
(
x
) and
U
P
(
x
) the lower and upper contour sets of
x
with
respect to the relation
P
:
L
P
(
x
) = {
z
∈
X
|
x P z
}
U
P
(
x
) = {
y
∈
X
|
y P z
}
The relation
P
is said to be
continuous
if
L
P
(
x
) and
U
P
(
x
) are open sets for all
x
∈
X
;
P
is said to be
separable
if there is a countable subset D = {
d
i
,
i
∈
N} of
X
such that whenever
x P y
then there is some
d
i
∈
D
such that x P
d
i
,
d
i
P y
.
We will denote by
PP the transitive closure of P , defined by
x PP y ⇒ ∃ z 1 , z 2 , …, z n ∈ X | x = z 1 Pz 2 P … Pz n = y
When P is an acyclic relation, PP is a quasiorder.
A Note on Numerical Representations for Weak-Continous Acyclic… Vol. 15, nº 1, 1998 17
When the set of alternatives is a topological space, the continuity of the
function which represents the binary relation is a very desirable property. Some
authors define utility functions from this point of view, as numerical representa-
tions satisfying continuity [see, for instance, Peleg (1970)]. We also consider this
point of view and we define a utility function representing the binary relation P as
a continuous real-valued function u : X → R such that x P y implies u ( x ) > u ( y ).
In order to obtain the existence of a utility function a usual requirement is the
continuity of this relation. We will introduce a weaker property with the same
purpose.
Definition 1
A binary relation P defined on a topological space X is said to be weak-continu-
ous if whenever x P z P y there are z 1 , z 2 ∈ X and U ∈ E ( x ), V ∈ E ( y ) such that
∀ a ∈ U a P z 1 P y
∀ b ∈ V x P z 2 P b
where E ( x ) stands for the family of open neighborhoods of x .
It is clear that continuity implies weak continuity. The converse, even by add-
ing separability, is not true. The next example shows this fact, and the following
Lemma establishes the relationship between both notions of continuity.
Example 1
Let X = { x ∈ R 2
+ | Σ x i > 10 } and let P be the binary relation defined by
x, y ∈ X x P y ⇔∑ x
i
> ∑ y i and || x – y ||
≥
1
where || a || denotes the euclidean norm of a vector. This relation is acyclic, separa-
ble ( D = X ∩ ( Q × Q )) and weak-continuous. Nevertheless it is not continuous.
Lemma 1
Let X be a topological space and let P be a separable weak-continuous binary
relation defined on X . Then PP is continuous.
Proof:
Let y ∈ L PP ( x ). Thus there exist z 1 ,…, z n– 1 , z n such that
x = z 1 P … Pz n– 1 P z n = y
Note that, by separability, we can obtain such a chain with a length of at least 3
and then applying weak-continuity to the sequence
z n– 2 Pz n– 1 Pz n = y
there is some z ∈ X and some U ∈ E ( y ) such that ∀ b ∈ U
z n– 2 P z P b
so x PP b , which implies U ⊆ L PP ( x ), and the lower contour set of PP is open.
With an analogous argument, U PP ( x ) is open and PP is continuous. ■
18 Vol. 15, nº 1, 1998 Begoña Subiza; Josep E. Peris
From the above Lemma, as every quasiorder coincides with its transitive clo-
sure, it is deduced immediately that for a separable quasiorder continuity and
weak-continuity are equivalent conditions.
3. The Result
In order to get a utility function we assume that there is a finite measure µ defined
on a topological space such that any open set U is a measurable set and µ ( U ) > 0,
for any non-empty open set. Formally, given a binary relation P , we define the
real-valued function α : X → R in the following way:
for any x ∈ X , α ( x ) = µ [int( L P ( x ))]
where int ( A ) stands for the interior of this set. Since open sets are measurable, this
function is well-defined.
To obtain the continuity of this function we need to introduce an additional
property about the behavior of the lower contour sets. Similar conditions are
introduced in the literature in order to avoid the fact that the indifferent classes
have positive measures (Neuefeind (1972) directly requests this condition,
µ [{ a | a I x }] = 0 , ∀ x ). The condition we are going to use is taken from Candeal-
Haro and Induráin-Eraso (1993).
Definition 2
Let X be a topological space with a finite measure µ , and P a binary relation
defined on X . µ is said to satisfy the regularity condition if for every net { x j } j ∈ J ,
, the net of real numbers
α j = µ [int ( L P ( x )) ∆ int( L P ( x j ))]
converges to 0, where ∆ denotes the symmetric difference of two sets,
A ∆ B = ( A ∪ B ) – ( A ∩ B ).
Candeal-Haro and Induráin-Eraso (1993) use the regularity condition in order
to obtain utility functions for continuous quasiorders; Bosi and Isler (1995) have
used this property to prove the existence of a pair of continuous real-valued func-
tions representing a strongly separable interval order.
The following result proves the continuity of the function α ( x ) when P is a
quasiorder satisfying the regularity condition.
Lemma 2
Let P be a quasiorder defined on a topological space X with a finite measure µ sat-
isfying the regularity condition. Then the real function α : X → R defined as
α ( x ) = µ [int( L P ( x ))]
is continuous.
x j
jJ ∈
lim x =
A Note on Numerical Representations for Weak-Continous Acyclic… Vol. 15, nº 1, 1998 19
Proof:
For any convergent net { x j } j ∈ J , , then by using measure properties,
| α ( x ) – α ( x j ) | = | µ [int( L P ( x ))] – µ [int( L P ( x j ))] | =
= | µ [int( L P ( x )) – int( L P ( x j ))] – µ [int( L P ( x j )) – int( L P ( x ))] | ≤
≤ µ [int( L P ( x )) – int( L P ( x j ))] + µ [int( L P ( x j )) – int( L P ( x ))] =
= µ [ ( int( L P ( x )) – int( L P ( x j )) ) ∪ ( int( L P ( x j )) – int( L P ( x )) ) ] =
= µ [int( L P ( x )) ∆ int( L P ( x j ))] = α j
and the regularity condition implies that . Therefore α ( x ) is con-
tinuous. ■
If we consider, in the previous Lemma, that the quasiorder P is continuous,
then the function α ( x ) coincides with the measure of the lower contour set, since
this set is open. The main result is now obtained as a consequence of Lemma 1
and 2.
Theorem 1
Let P be an acyclic, weak continuous and separable binary relation defined on a
topological space with a finite measure µ satisfying the regularity condition with
respect to the transitive closure of P . Then the function
u ( x ) = µ [ L PP ( x )]
is a utility function representing P .
Proof:
By using Lemma 1, relation PP is a continuous quasiorder; the separability of P
implies that PP is also separable.
As PP is continuous Lemma 2 provides the continuity of u ( x ). Let x , y ∈ X
such that x PP y . The continuity and separability of PP imply that there is an open
set U such that
U ⊆ L PP ( x ) – L PP ( y )
and then, by using measure properties
u ( y ) < u ( y ) + µ [ U ] = µ [ L PP ( y ) ∪ U ] ≤ µ [ L PP ( x )] = u ( x ).
Then u ( x ) is a utility for the relation PP . It is immediate that u ( x ) is also a utility
function for P . ■
Theorem 3 in Candeal-Haro and Induráin-Eraso (1993) provides a utility
function for continuous quasiorders when the regularity condition is satisfied;
therefore our result is a generalization of that theorem. This fact can be shown
with binary relation in Example 1 which satisfies the conditions in our Theorem 1
x j
jJ ∈
lim x =
α x j
()
jJ ∈
lim α x () =
20 Vol. 15, nº 1, 1998 Begoña Subiza; Josep E. Peris
and it is not a quasiorder. Easy computations show that the utility function of the
relation in Example 1 is
On the other hand, Peris and Subiza (1995) prove the existence of a utility
function for preferences which are separable, continuous and spacious (for every
x , y ∈ X such that x P y , Cl ( L P ( y ) ⊂ L P ( x )); as Example 1 shows, in this context,
the weak-continuity condition is a weaker property. Moreover, Candeal-Haro and
Induráin-Eraso (1993) provide an example of an acyclic binary relation which
satisfies the hypotheses in Theorem 1 but it is not spacious. So Theorem 1 covers
situations where the results in Candeal-Haro and Induráin-Eraso (1993) or in
Peris and Subiza (1995) cannot be applied.
4. Final Comments
It would be interesting to obtain a «regularity» condition independent of the
measure used in the construction of the utility function. In order to do so, the fol-
lowing condition about convergence of sets could be used:
If A 1 , A 2 , … are a sequence of subsets we define the following set operations:
If lim sup A n = lim inf A n = A , A is said to be the limit of the sequence and we
write A = limit A n .
Note that this definition of set convergence does not coincide, in general, with
the usual notion of set convergence in the Hausdorff topology. By using this set
convergence, we will say that a binary relation defined in a metric space X has
convergent lower sets if for any convergent sequence { x n }, lim x n = x , then
limit [ L P ( x n )] = L P ( x ).
It is now obvious, from the properties of the measure (see, for instance Ash
(1972)), that if a binary relation P has convergent lower sets then for any finite
measure µ defined in the space of alternatives X the function
α ( x ) = µ [int( L P ( x ))]
is continuous.
Requiring this property is, in general, a much stronger assumption than the
regularity condition. Although the result we obtain is presented using the regular-
ity condition, it can obviously be rewritten in terms of the convergence of lower
sets.
ux y , () x
i
∑
()
2
100 –
2
---------------------------- - =
lim sup A n A k
kn =
∞
∪
n 1 =
∞
∩
=
lim inf A n A k
kn =
∞
∩
n 1 =
∞
∪
=
A Note on Numerical Representations for Weak-Continous Acyclic… Vol. 15, nº 1, 1998 21
References
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Mathematical Social Sciences , 28: 67-69, 1994).
C HICHILNISKY , G. (1980). «Continuous Representations of Preferences». Review of Eco-
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L UCE , R.D. (1956). «Semiorder and a Theory of Utility Discriminations». Econometrica ,
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— (1994). «Numerical Representation for Acyclic Preferences». Journal of Mathematical
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