Conditions for the Upper Semicontinuous Representability of Preferences with Nontransitive Indifference ()
Keywords
Interval Order, Upper Semicontinuous Numerical Representation, Semiorder
1. Introduction
An interval order 
on a set X can be thought of as the simplest model of a binary relation on X whose associated preference-indifference relation is not transitive. Indeed, under certain conditions, it can be fully represented by means of a pair 
of real-valued functions on X, in the sense that 
is equivalent to 
for all
. Therefore, interval orders are particularly interesting in economics and social sciences. Whenever the set X is endowed with a topology
, it is interesting to look for representations of an interval order 
on 
that satisfy suitable continuity conditions.
The existence of numerical representations of interval orders was first studied by Fishburn [1] [2] and then by other authors (see, e.g., Bosi et al. [3] and Doignon et al. [4] ).
When the set X is endowed with a topology
, it may be of interest to look for continuous or at least semicontinuous representations of an interval order 
on
. Results in this direction were presented by Bosi et al. [5] [6] , Chateauneuf [7] and, in the particular case of semiorders, by Candeal et al. [8] [9] .
For many purposes, the existence of a representation 
with u and v both upper semicontinuous is satisfactory. In particular, if such a representation exists and the topology 
is compact, then there exist maximal elements for the interval order 
which are obtained by maximizing u or v. Also the existence of undominated maximal elements can be guaranteed by means of an approach of this kind (see, e.g., Alcantud et al. [10] ). This kind of semicontinuous representability of interval orders was first studied by Bridges [11] and then by Bosi and Zuanon [12] [13] .
In this paper, we present different results concerning the representability of an interval order 
on a topological space 
by means of a pair 
of upper semicontinuous real-valued functions.
2. Notations and Preliminaries
An interval order 
on a set X is an irreflexive binary relation on X which in addition satisfies the following condition for all
:
An interval order 
is in particular a partial order (i.e., 
is an irreflexive and transitive binary relation). The preference-indifference relation 
associated to an interval order 
on set X is defined as follows for all
:
It is well know that if 
is an interval order, then 
is total (i.e., for all
, either 
or
). On the other hand, 
is not transitive in general.
Fishburn [2] proved that if 
is an interval order on a set X, then the following two binary relations 
and 
(the traces of the original interval order) are weak orders (i.e., asymmetric and negatively transitive binary relations on X):
The following proposition was proved, for example, by Alcantud et al. ([10] , Lemma 3).
Proposition 2.1. Let 
be an irreflexive binary relation on a set X. Then 
is an interval order if and only if 
is a asymmetric.
An interval order 
on a set X is a weak order if and only if
. The preference-indifference relations 
and 
associated to the binary relations 
and 
are defined as follows for all
:
Therefore, we have that
An interval order 
on a set X is said to be i.o. separable (see Bosi et al. [3] and Doignon et al. [4] ) if there exists a countable subset D of X such that for all 
with 
there exists 
such that
. In this case D is said to be an i.o. order dense subset of
.
From Chateauneuf [7] , an interval order 
on a set X is said to be strongly separable if there exists a countable set 
such that, for every 
with
, there exist 
with
. D is said to be a strongly order dense subset of X. It is clear that strong separability implies i.o. separability. Further, strong separability occurs, for example, whenever an interval order is representable by means of a pair 
of nonnegative positively homogeneous functions on a cone in a topological vector space. This kind of representability, in the more general setting of acyclic binary relations, was studied, for example, by Alcantud et al. [14] and in the case of not necessarily total preorders by Bosi et al. [15] .
If R is a binary relation on a set X, then denote by 
and 
the lower section and respectively the upper section of any element
, i.e., for every
,
A subset A of a related set 
is said to be R-decreasing if 
for every
.
A real-valued function u on X is said to be a weak utility function for a partial order 
on a set X if, for all
,
The following characterization of the existence of an upper semicontinuous weak utility for a partial order on a topological space is well known (see e.g. Alcantud and Rodríguez-Palmero ([16] , Theorem 2)).
Proposition 2.2. Let 
be a partial order on a topological space
. Then the following conditions are equivalent:
1) There exists an upper semicontinuous weak utility function u for
;
2) There exists a countable family 
of open 
-decreasing subsets of X such that if 
then there exists 
such that
,
.
A real-valued function u on X is said to be a utility function for a partial order 
on a set X if, for all
,
If a partial order 
admits a representation by means of a utility function, then 
is a weak order or equivalently the associated preference-indifference relation 
is a total preorder (i.e. 
is total and transitive).
The following proposition is well known and easy to be proved.
Proposition 2.3. Let 
be a weak order on a topological space
. Then the following conditions are equivalent:
1) There exists an upper semicontinuous utility function u for
;
2) There exists a countable family 
of open
-decreasing subsets of X such that if 
then there exists 
such that
,
.
A pair 
of real-valued functions on X represents an interval order 
on X if, for all
,
If 
is a representation of an interval order
, then it is easily seen that u and v are weak utility functions for 
and
, respectively, while it is not in general guaranteed that u and v are utility functions for 
and
, respectively.
We say that a pair 
of real-valued functions on X almost represents an interval order 
on X if, for all
,
An interval order 
on a topological space 
is said to be upper (lower) semicontinuous if
is an open subset of X for every
. If 
is both upper and lower semicontinuous, then it is said to be continuous.
If there exists a representation 
of an interval order 
on a topological space 
and u and v are both upper semicontinuous, then 
is necessarily upper semicontinous, due to the fact that
is open for every
. In this case, also the associated weak order 
is upper semicontinuous, since
is expressed as union of open sets.
On the other hand, the existence of an upper semicontinuous representation does not imply that the weak order 
is upper semicontinuous. The following example, that was already presented in Bosi and Zuanon [13] , illustrates this fact.
Example 2.4. Let X be the set 
endowed with the natural induced topology on the real line and consider the interval order 
on X defined as follows for all
:
If we define 
and 
for every
, then it is clear that 
is an (upper semi) continuous representation of
. We can easily verify that the associated weak order 
is not upper semicontinuous. Indeed, consider for example that 
is not an open set. Notice that 
for all
, 
since 
but for no 
we have that 
because this would imply the existence of 
such that
.
A weak order 
on a topological space 
is said to be weakly upper semicontinuous if for every 
that is not a minimal element there exists a uniquely determined 
-decreasing open subset 
of X such that 
and 
(see Bosi and Zuanon [13] ). This definition was presented by Bosi and Herden [17] in the context of preorders (i.e., reflexive and transitive binary relations). If a weak order 
on a topological space 
admits an upper semicontinuous weak utility u then it is weakly upper semicontinuous (just define, for every
,
). Further, it is clear that an upper semicontinuous weak order is also weakly upper semicontinuous.
If 
is a topological space and S is a dense subset of 
such that
, then we say that a family 
of open subsets of X is a quasi scale in 
if the following conditions hold:
1)
;
2) 
for every 
such that
.
The following proposition is a particular case of Theorem 4.1 in Burgess and Fitzpatrick [18] .
Proposition 2.5. If 
is a quasi scale in a topological space
, then the formula
defines an upper semicontinuous function on 
with values in
.
3. Conditions for the Semicontinous Representability of Interval Orders
In the following theorem we present some conditions that are equivalent to the existence of an upper semicontinuous representation of an interval order on a topological space.
Theorem 3.1. Let 
be an interval order on a topological space
. Then the following conditions are equivalent:
1) There exists a pair 
of upper semicontinuous real-valued functions on 
representing the interval order
;
2) The following conditions are verified:
a) The interval order 
on X is representable by means of a pair of real-valued functions
;
b) 
is upper semicontinuous;
c) There exists an upper semicontinuous weak utility 
for
;
3) The following conditions are verified:
a) The interval order 
on X is i.o.-separable;
b) 
is upper semicontinuous;
c) There exists a countable family 
of open 
-decreasing subsets of X such that if 
then there exists 
such that
,
;
4) There exists a countable family 
of pairs of upper semicontinuous real-valued functions on
almost representing 
such that for every 
with 
there exists 
with 
.
5) There exists a countable family 
of pairs of open subsets of X satisfying the following conditions:
a) 
and 
imply 
for all 
and for all
;
b) 
and 
imply 
for all 
and for all
;
c) for all 
such that 
there exists 
such that
,
;
6) There exist two quasi scales 
and 
in 
such that the family 
satisfies the following conditions:
a) 
and 
imply 
for every 
and
;
b) for every 
such that 
there exist 
such that
, 
,
.
Proof. The equivalence 1) Û 5) was proved in Bosi and Zuanon ([12] , Theorem 3.1), while the equivalences 1) Û 2) and 1) Û 3) were proved in Bosi and Zuanon ([13] , Theorem 3.1).
Let us prove the equivalence of conditions 1) and 4). It is clear that 1) implies 4). In order to show that 4) implies 1), assume that there exists a countable family 
of pairs of upper semicontinuous real-valued functions on 
almost representing 
such that for every 
with 
there exists 
with
. Without loss of generality, assume that 
and 
take values in 
for every index n. Define functions u and v on X as follows:
in order to immediately verify that 
is an upper semicontinuous representation of the interval order 
on the topological space
.
Finally, let us show that also the equivalence of conditions 1) and 6) is valid. In order to show that 1) implies 6), assume without loss of generality that there exists a pair of upper semicontinuous real-valued functions with values in 
representing the interval order 
on the topological space
. Then just define 
, 
, 
for every
, and 
in order to immediately verify that 
and 
are two quasi scales in 
such that the family 
satisfies the above subconditions a) and b) of condition 6).
In order to show that 6) implies 1), assume that there exist two quasi scales 
and 
such that the family 
satisfies the above subconditions a) and b) in condition 6). Then define two functions
as follows:
We claim that 
is a pair of continuous functions on 
with values in 
representing the interval order
.
From the definition of the functions u and v, it is clear that they both take values in
. Let us first show that the pair 
represents the interval order
. First consider any two elements 
such that
. Then, by condition b), there exist 
such that
, 
,
. Hence, we have 
, which obviously implies that
. Conversely, consider any two elements 
such that
, and observe that, for every
, if 
then it must be 
by the above condition a). Hence, it must be 
from the definition of u and v.
Finally, observe that u and v are upper semicontinuous real-valued functions on 
with values in 
as an immediate consequence of Proposition 2.5. This consideration completes the proof. QED It has been noticed that if 
is a representation of an interval order 
on a set X, then not necessarily u is a utility function for the trace
. The following immediate corollary to Theorem 3.1 concerns this particular case.
Corollary 3.2. Let 
be an interval order on a topological space
. Then the following conditions are equivalent:
1) There exists a pair 
of upper semicontinuous real-valued functions on 
representing the interval order 
such that u is a utility function for the associated weak order
;
2) The following conditions are verified:
a) The interval order 
on X is representable by means of a pair of real-valued functions
;
b) 
is upper semicontinuous;
c) 
is upper semicontinuous;
3) The following conditions are verified:
a) The interval order 
on X is i.o.-separable;
b) 
is upper semicontinuous;
c) There exists a countable family 
of open 
-decreasing subsets of X such that if 
then there exists 
such that
,
.
Since Bridges ([11] , Proposition 2.3) proved that an interval order 
on a second countable topological space 
is representable by a pair 
of (nonnegative) real-valued function, we have that the following corollary is an immediate consequence of the previous theorem.
Corollary 3.3. Let 
be an interval order on a second countable topological space
. Then the following conditions are equivalent:
1) There exists a pair 
of upper semicontinuous real-valued functions on 
representing the interval order
;
2) The following conditions are verified:
a) 
is upper semicontinuous;
b) There exists an upper semicontinuous weak utility 
for
.
The following corollary is a consequence of both Corollary 3.2 and Corollary 3.3.
Corollary 3.4. Let 
be an interval order on a second countable topological space
. Then the following conditions are equivalent:
1) There exists a pair 
of upper semicontinuous real-valued functions on 
representing the interval order 
such that u is a utility function for the associated weak order
;
2) The following conditions are verified:
a) 
is upper semicontinuous;
b) 
is upper semicontinuous.
The following corollary is found in Bosi and Zuanon ([13] , Proposition 3.1).
Corollary 3.5. Let 
be a strongly separable interval order on a topological space
. Then the following conditions are equivalent:
1) There exists a pair 
of upper semicontinuous real-valued functions on 
representing the interval order
;
2) The following conditions are verified:
a) 
is upper semicontinuous;
b) 
is weakly upper semicontinuous.
We finish this paper by presenting some applications of the previous results to the semiorder case. We recall that a semiorder 
on an arbitrary nonempty set X is a binary relation on X which is an interval order and in addition verifies the following condition for all
:
If 
is a semiorder, then the binary relation 
is a weak order (see e.g. Fishburn [2] ). The following proposition was proved by Bosi and Isler ([19] , Proposition 3).
Proposition 3.6. Let 
be an interval order on a set X. Then 
is a semiorder if and only if 
is asymmetric.
Clearly, this happens in the particular case when
. More generally, we have that the following proposition holds. The easy proof of it is left to the reader.
Proposition 3.7. Let 
be an interval order on a set X. If there is a real-valued function u on X that is a weak utility for both 
and
, then 
is semiorder.
Since it was already observed that upper semicontinuity of an interval order 
always implies upper semicontinuity of the associated weak order
, we obtain the following corollaries as other immediate consequences of Theorem 3.1.
Corollary 3.8. Let 
be a semiorder on a topological space
. If 
is upper semicontinuous and
, then there exists a pair 
of upper semicontinuous real-valued functions on 
representing 
provided that there exists a pair 
of real-valued functions on X representing
.
Corollary 3.9. Let 
be a semiorder on a second countable topological space
. If 
is upper semicontinuous and
, then there exists a pair 
of upper semicontinuous real-valued functions on 
representing
.