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

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

Gianni Bosi^1*, 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: gianni.bosi@deams.units.it

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

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

Introduction

White’s theorem [1] is important since, for every maximal element x_o relative to a preorder on set X, it guarantees the existence of an order-preserving function u on the preordered set (X, ) attaining its maximum at x_o, provided that an order-preserving function uË�?�? on (X, ) 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 on a topological space (X,τ) under which, for every maximal element x_o relative to , there exists an upper semicontinuous order preserving function u on the preordered topological space (X,τ, ) attaining its maximum at x_o, provided that an upper semicontinuous order-preserving function uË�?�? on (X,τ, ) 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 x_o 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 .

Notation and preliminaries

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

For every x ϵ X, we set

l(x)={z ϵ X : zx} i(x)={ z ϵ X : xz}

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

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

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

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

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

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

A preorder 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 z} is a closed subset of X for every x ϵ X;

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

An upper semicontinuous preorder Ward et al. [7] or more generally a quasi-upper semicontinuous preorder Bosi et al. [8, Theorem 3.1] 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 is a preorder on X, then the triplet (X, τ, ) will be referred to as a topological preordered space. The (natural) (interval) topology on the real line R will be denoted by τ_nat .

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 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 uË�?�?: (X, ) → (R, ≤) exists. Therefore, in order to determine all the maximal elements of a preorder on a set X, the agent maximizes all the functions u in a family U of bounded order-preserving functions for . Needless to say, this is a very important opportunity, at least theoretically.

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

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

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

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

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

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

uË�?�?(x) if not (x∼xo)

u(x)={

sup uË�?�?(x)+δ if (x∼xo)

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

(ii) ⇒ (i) Assume that there exists an upper semicontinuous order-preserving function u:(X, τ, )→(R, τ_nat,≤) such that arg max u=[x_o]={z ϵ X:z∼x_o}. If [x_o] is not closed, then there exists an element zϵ X-[x_o] such that U_z ∩[x_o] ≠Φ for every neighborhood U_z of the element z. But this contradicts the fact that u is upper semicontinuous, since in this case u^-1(]-∞,u(x_o)[), an open set containing z, should contain an element zË�?�?ϵ [x_o], for which u(zË�?�?)=u(x_o). 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 on a set X it suffices to maximize a weak utility for the strict part of , 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 instead of orderpreserving functions uË�?�? for X

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

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

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

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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