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Review Article, Res J Econ Vol: 2 Issue: 1

Upper Semicontinuous Representability of Maximal Elements for Non total Preorders on Compact Spaces

Gianni Bosi1*, Paolo Bevilacqua2 and Magali Zuanon3

1Bruno de Finetti Department of Economics, Business, Mathematics and Statistics (DEAMS), University of Trieste, Italy

2Department of Engineering and Architecture, University of Trieste, Italy

3Department of Economy and management, University of Brescia, Italy

*Corresponding Author : Gianni Bosi
Bruno de Finetti Department of Economics, Business, Mathematics and Statistics (DEAMS), University of Trieste, Italy
Tel: +39-040-558-7115
E-mail:
[email protected]

Received: October 20, 2017 Accepted: January 02, 2018 Published: January 08, 2018

Citation: Bosi G, Bevilacqua P, Zuanon M (2018) Upper Semicontinuous Representability of Maximal Elements for Non total Preorders on Compact Spaces. Res J Econ 2:1.

Abstract

We discuss the possibility of determining all the maximal elements of a preorder on a compact topological space by maximizing all the functions in a suitable family of upper semicontinuous orderpreserving functions.

Keywords: Preorder; Order-preserving function; Weak utility; Maximal element; Upper semicontinuous function

Introduction

White’s theorem [1] is important since, for every maximal element xo relative to a preorder image on set X, it guarantees the existence of an order-preserving function u on the preordered set (X,image ) attaining its maximum at xo, provided that an order-preserving function uˈ on (X, image) exists. So, at least theoretically, every maximal element is obtained by maximizing a real-valued order-preserving function. When this happens, it is clear that every maximal element is potentially optimal in the sense of Podinovski et al. [2] (i.e., for every maximal element there exists a total preorder extending the original preorder with respect to which such maximal element is best preferred).

In this paper, we generalize White’s theorem to the upper semicontinuous case. This means that we present conditions on a preorder image on a topological space (X,τ) under which, for every maximal element xo relative to image , there exists an upper semicontinuous order preserving function u on the preordered topological space (X,τ, image) attaining its maximum at xo, provided that an upper semicontinuous order-preserving function uˈ on (X,τ, image) exists. It should be noted that Bevilacqua et al. [3] already characterized the property according to which every maximal element relative to a preorder on a compact topological space can be obtained by maximizing a transfer weakly upper continuous weak utility for its strict part (see the generalization of Weierstrass Theorem presented by Tian et al. [4]).

It is clear that these results are important due to the well-known fact that every upper semicontinuous (more generally transfer weakly upper continuous) function attains its maximum on a compact topological space, and the nearly obvious consideration that a point xo at which an order-preserving function u for a preorder (or, more generally, a weak utility for its strict part) attains its maximum is also a maximal element for image.

Notation and preliminaries

Let X be a nonempty set (decision space). A binary relation image on X is interpreted as a weak preference relation, and therefore, for any two elements x, y ϵ X, the scripture image has to be thought of as “the alternative y ϵ X is at least as preferable as x ϵ X “. As usual, image denotes the strict part of a binary relation image (i.e., for all x, y ϵ X, ximagey if and only if image and not image. A preorder is a reflexive and transitive binary relation. An anti-symmetric preorder image is referred to as an order. Furthermore, ∼ stands for the indifference relation (i.e., for all x, y ϵ X, x ∼ y if and only if image and image. We have that ∼ is an equivalence relation on X whenever image is a preorder.

For every x ϵ X, we set

l(x)={z ϵ X : zimagex} i(x)={ z ϵ X : ximagez}

Given a preordered set (X,image ), a point xo ϵ X is said to be a maximal element for image if for no z ϵ X it occurs that xoimagez. In the sequel we shall denote by image the set of all the maximal elements of a preordered set (X,image). Please observe that imagecan be empty.

We recall that a function u: (X, image) → (R, ≤) is said to be

i. isotonic or increasing if, for all x, y ϵ X , x image y → u(x) ≤ u(y);

ii. a weak utility for image if, for all x, y ϵ X , x image y → u(x) < u(y);

iii. Strictly isotonic or order-preserving if it is both isotonic and a weak utility for image.

Strictly isotonic functions on (X, image) are also called Richter-Peleg representations of imagein the economic literature (see e.g. Richter et al. [5] and Peleg et al. [6])

A preorder image on a topological space (X, τ) is said to be

i. upper semicontinuous if, for all x, y ϵ X , i(x)={ z ϵ X : x image z} is a closed subset of X for every x ϵ X;

ii. Quasi upper semicontinuous if there exists an upper semicontinuous preorder image on (X , τ) such that image⊂<.

An upper semicontinuous preorder Ward et al. [7] or more generally a quasi-upper semicontinuous preorder Bosi et al. [8, Theorem 3.1] image on a compact topological space (X , τ) admits a maximal element. As usual, for a real-valued function u on X, we denote by arg max u the set of all the points x ϵ X such that u attains its maximum at x (i.e., arg max u={z ϵ X:(u(z)≤u(x))∀zϵX}

If τ is a topology on a set X, and image is a preorder on X, then the triplet (X, τ, image) will be referred to as a topological preordered space. The (natural) (interval) topology on the real line R will be denoted by τnat .

Finally, we recall that a real-valued function u on a topological space (X , τ) is said to be upper semicontinuous if u-1(]-∞, α[)={x ϵ X: u(x) < α} is an open set for all αϵR. A very well know result guarantees that every upper semicontinuous real-valued function u on a compact topological space (X , τ) attains its maximum.

Maximal elements of preorders from maximization of upper semicontinuous functions

The following theorem was proved by White et al. [1]. Given any maximal element xo relative to a preorder image on a set X, it guarantees the existence of some order-preserving function u attaining its maximum at xo, provided that an order-preserving function uˈ: (X, image ) → (R, ≤) exists. Therefore, in order to determine all the maximal elements of a preorder image on a set X, the agent maximizes all the functions u in a family U of bounded order-preserving functions for image . Needless to say, this is a very important opportunity, at least theoretically.

Theorem: (White et al. [1]): Let (X, image ) be a preordered set and assume that there exists an order-preserving function uˈ: (X, image )→(R, ≤). If image is nonempty, then for every xo ϵ image there exists an orderpreserving function u:(X, image )→(R, ≤). Such that arg max u=[x0]={zϵ X:z∼xo}.

We now present a generalization of the above theorem to the “upper semicontinuous case”.

Theorem: Let (X, τ, image ) be a topological preordered space, and assume that image is nonempty. Consider an element xoϵ image. Then the following conditions are equivalent:

i. There exists an upper semicontinuous order-preserving function uˈ:(X, τ, image )→(R, τnat,≤) and [x0]={z ϵ X:z∼x0} is a closed subset of X;

ii. There exists an upper semicontinuous order-preserving function u:(X, τ, image)→(R, τnat, ≤) such that arg max u=[x0]={z ϵ X:z∼x0}.

Proof: Consider a topological preordered space (X, τ, image).

i) ⇒ (ii) Let uˈ be an upper semicontinuous order-preserving function on (X, τ, X). Without loss of generality, we can assume uˈ to be bounded. Consider a point xo ϵ image and define the real-valued function u on X as follows for any choice of a positive real δ:

uˈ(x) if not (x∼xo)

u(x)={

sup uˈ(x)+δ if (x∼xo)

White et al. [1, Theorem 1] proved that the above function u is order-preserving for image as soon as uˈ is order-preserving for image . For the sake of completeness, let us recall here the arguments supporting this consideration. It is clear that uˈ (x) ≤ u (x) for every x ϵ X. In order to show that u is increasing with respect to image , consider any two points x, y ϵ X such that x image y. If y∼xo, then it is clear that u(x) ≤ u (x) from the definition of u, On the other hand, if not(y∼xo), then it must be also not(x imagexo), since xo ∼ x imagey would imply(xo image y ),, and in turn xo∼y due to the fact that xo is a maximal element relative to image . Hence, since neither x ∼ xo nor y ∼ xo, we have that u(x)=uˈ(x)\≤ uˈ(y)= u(y) from the definition of u and the fact that uˈ is increasing with respect to image . In order to show that u is a weak utility for image , consider any two points x, y ϵ X such that x image y. Then we have that not(x∼xo), since xo ∼ x image y implies that xo ∼ y (a contradiction, since xo is assumed to be a maximal element for image ). Therefore, from the definition of u and the fact that uˈ is a weak utility for image, we have that u(x)=uˈ(x)<uˈ(y)≤u(y), which obviously implies that u(x)<u(y). Further, u attains its maximum at xo and actually, since δ is a positive real, it is clear that (i) arg max u=[xo]={z ϵ X:z∼xo} Therefore, it only remains to show that under our assumptions u is upper semicontinuous. It is clear that u is upper semicontinuous at every point x ϵ X such that x ∼ xo. Therefore, consider any point x ϵ X such that not x ∼ xo and α ϵ R such that u(x)< α. We can limit our considerations to the case when α ≤sup uˈ (x) +δ. Since in this case u(x)=uˈ(x), from upper semi continuity of uˈ there exists an open set Uxˈ containing x such that uˈ(z)< α for every zϵUxˈ. Since [xo] is closed, we have that Ux=Uxˈ-[xo] is an open set containing x such that uˈ(z)=u(z)< α for every zϵ Uxˈ. Hence, u is an upper semicontinuous function.

(ii) ⇒ (i) Assume that there exists an upper semicontinuous order-preserving function u:(X, τ, image )→(R, τnat,≤) such that arg max u=[xo]={z ϵ X:z∼xo}. If [xo] is not closed, then there exists an element zϵ X-[xo] such that Uz ∩[xo] ≠Φ for every neighborhood Uz of the element z. But this contradicts the fact that u is upper semicontinuous, since in this case u-1(]-∞,u(xo)[), an open set containing z, should contain an element zˈϵ [xo], for which u(zˈ)=u(xo). This consideration completes the proof.

Remark: It is clear that Theorem 3.2 generalizes White’s theorem, due to the fact that these two results precisely coincide when we consider the discrete topology τ on X.

Since in order to determine a maximal element relative to preorder image on a set X it suffices to maximize a weak utility for the strict part image of image , the following corollary can be considered as useful. Indeed, the reader can easily verify that the implication “(i) ⇒ (ii)” in Theorem 3.2 is still valid if one considers weak utilities for image instead of orderpreserving functions uˈ for X

Corollary: Let (X, τ, image ) be a topological preordered space with τ a compact topology. If there exists an upper semicontinuous weak utility uˈ for image, and [x]={z ϵ X:z∼x} is a closed set for all xϵ image , then for every xo ϵ image there exists an upper semicontinuous weak utility u for image such that arg max [x]={z ϵ X:z∼xo}.

Bosi et al. [8, Theorem 2.11] proved that there exists an upper semicontinuous weak utility for the strict part image of a quasi-upper semicontinuous preorder image on a second countable (i.e., with a countable base) topological space(X, τ). Hence, we get the following corollary.

Corollary: Let (X, τ, image ) be a topological preordered space. Assume that τ is a compact and second countable topology, and that image is quasi upper semicontinuous. If [x]={z ϵ X:z∼x} is a closed set for all xϵ image, then for every xo ϵ image there exists an upper semicontinuous weak utility u for image such that arg max [x]={z ϵ X:z∼xo}.

Conclusion

In this paper, following the spirit of a theorem of White et al. [1], we have presented some results concerning the representation of the set of all maximal elements of a preorder on a compact topological space by means of the maximization of all functions in a suitable family of bounded upper semicontinuous order-preserving functions. The more delicate problem of characterizing the possibility of representing all the maximal elements for a preorder on a compact topological space by means of the maximization of finitely many upper semicontinuous order-preserving functions will be hopefully considered in a future paper.

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