In this note, we prove that for every non-separable metric space there is a continuous preference ordering which is non respresentable by an utility function

Arias de Reyna, Juan

Estévez Toranzo, Margarita

Hervés Beloso, Carlos

English

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On non representable preferences

Universidad Carlos III de Madrid e-Archivo

Working paper 93-17 Departamento de Economia Economy Series 04 Universidad Carlos III de Madrid September 1993 Calle Madrid, 126 28903 Getafe (Madrid) Fax (34 1) 624-9875 ON NON REPRESENTABLE PREFERENCES Juan Arias de Reyna, Margarita Estevez Toranzo and Carlos Herves Beloso· Abstract ---, _ In this note, we prove that for every non-separable metric space there is a continuous preference ordering which is non representable by an utility function. Key words Preference Ordening; utility Function; Non Separable Metric Space. ·Arias, Departamento de Analisis Matematico, Universidad de Sevillaj Estevez, Departamento de Matematicas, Universidad de Vigoj Herves, Departamento de Economia, Universidad Carlos III de Madrid. The work of M. Estevez Toranzo and C. Herves Beloso is partially supported by Research Grant PS91-0042 from the Direccion General de la Investigaci6n Cientifica y Tecnica, Spanish Ministry of Education. 1 1. Introduction This work is concerned with the numerical representation of all continuous pref-erence orderings on a topological space. As it is well known, if X is a cOllnected and separable topological space, then continuous preference orderings on X al-ways have utility representations (see Eile-nberg (1941) and Debreu (1954». The.. assumption of connectedness is not necessary in the setting of metric spaces: if X is perfectly separable, every continuous preference ordering is representable by an utility function (Debreu (1954»). However, we show here that separability is also a necessary condition for the representability of all continuous preference orderings on a metric space. That is, if X is a llon separable metric space, there exists a continuous preference ordering which does not admit an utility representation. This is relevant since consumption sets in infinite dimensional commodity spaces are not separable, in general. 2.  Definitions A preference ordering on the set X is, to be precise, a binary relation on X, say ~, which is reflexive, transitive and complete. An utility representation for the preference ordering ~ on X is a function u : X ~ R such that ;r ~ y if and only if u(x) :5 u(y). Let X be a topological space. We say that X is separable if it contains a countable subset whose closure is X. We say that X is perfectly separable (or that X satisfies the second countability axiom) if there is a countable class of open subsets such that everjr open subset in X is the union of sets of that class. Every perfectly separable topological space is separable. Every separable metric space is perfectly separable. A topologlcal space X is connected if there is no partition of X into two disjoint, non-empty closed sets. We say that X is path connected if for all x,y in X there is a continuous function f : [0,.1] ~ X with f(O) = x and f(l) = y. Note that every path connected space is connected and every convex set in a linear topological space is path connected. A preference ordering ~ on a topological space X is continuous if the sets {x E X ; x ~ x'}, {J~ EX; x' ~ x} are closed for all x' E X. A subset B C X bounds ~ if for every x E X there are a, b in B with a ::; x ~ b. A preference ordering ~ is countably bounded if there exists a countable set B C X that bounds::;. Any preference ordering which has an utility representation is countably bounded. 2 3. The existence theorem THEOREM: Let X be a non separable metric space. Then there is a continuous preference ordering on X which cannot be represented by an utility function. To prove the theorem we shall make use of an auxiliar space L called the long line (see Monteiro (1987), example 5, p. 151). Let 0 1 be the least non-countable ordinal. We denote by 0 the set of all ordinals et such that et < 0 1 , That is to say that 0 is the set of all countable ordinals. Note that n is a well ordered set, non-counta.ble and such that for all 0 E 0, {,8 E 0 j ,8 :5 et} is countable. Between each et E 0 and its follower et + 1 put one copy of the real interval (0,1). The space L that we get, ordered in the obviuous way, is called the long line. We consider on L the order topology. The details on the topological space L can be seen in Steen and Seebach (1970, pp.71,72). LEMMA: For each a E L, a :F 0 the order interval [0, a] = {x E L j 0:5 x :5 a} is a compact set homeomorphic to the real interval [0,1]. Proof. It is clear that it suffices to prove the result when a = 0 E O. As {I' EO; I' :5 et} is a well ordered countable set, there is an order preserving 1 : {O, 1. ... ,o} -+ [0,1] such that f(O) = 0 and I( et) = 1. We define j : [0,0] -+ [0,1] by j(b) = I(b) if bE 0 and j(b) = 1(1') + t(f(1J + 1) - 1({3)) if b =,8 + t, ,8 EO, t E (0,1). It is clear that j is an isomorphism of the order structures.  PROOF OF THE THEOREM: Let X be a non separable metric space. Non separable metric spaces are ca.racterized by the following property: There are c > 0 and an uncountable set D C X such that for all x, yE D, x :F y implies d(x, y) ~ 3e. (1 ) Otherwise, for each € = ~, n EN, there exists a co~ntable set D n verifying (1) such that X = UaED B(a, i), where B(a,~) = {x E X,d(x,a) < '!;J. Then the n set D = UD n will be countable and dense. As D is uncountable, for each 0 E 0 we can choose an x 0 E D in such a way that 0 '# {3 implies xo':F Xf3. By the lemma, for each et E 0 there exist 'Po : [0, e] -+ L, which is an isomorphism between the order structures of [0. .::] C R and [CPo(O), l'o(c)] = [0,0] C L 3 Let U : X -+ L be defillf'd by o if x ~ UoEo B(xo,e)U l' -( ) - { c.po(e: - d(x o, x)) if x E B(xo,e) It is clear that U is continuous in B( x0' e) because c.po and d are continuous. If x E X is such tha.t d(;ro,.T) = e, we have U(x) = 0, and c.po(O) = 0, then U is continuous in x. As the intersection of two different balls is empty and U is constant in the exterior of UoEo B(x o , e), we have that U is continuous in X. For x,y in X, we define x :$ y if and only if U(x) $ U(y). It is clear that :$ is a continuous preference ordering on X, but has no utility representa.tion because is not countably bounded. To see it, note that given a countable set B C X there exists OB E n such that sUPbEB U(b) < ° and then there is not a countable set B C X that bounds :$. 4. Final remark \Ve remark that separability is not a necessary condition for the representability of all preference orderings on a general topological space X. Monteiro (1987) proves that a continuous preference ordering on a path connected topological space X is representable if and only if it is numerably bounded. A continuous preference ordering on a compact topological space has one best and one worst point. Then any continuous preference ordering on a compact or O'-compact (an union of a countable family of compact sets) path connected topological space is representable by utility functions. Note that any compa.ct or O'-compact metric space is separable but compact topological spaces in general need not to be separable. -------------------- ' 4 References [1] Debreu, G; (1954). Representation of a Preference Ordering by a Numerical Function, in Decision Procesess, R.M. Thrall, C.H. Coombs and R.L. Davis eds., Wiley N.Y. pp 159-165 and also in Mathematical Economics: Twenty papers of Gerard Debreu. Cambridge University Press. Cambrid~e 105-110. [2] Eilemberg,S; (1941). Ordered Topological Spaces. American Journal of Mathematics, 63, 39-45. [3] Monteiro, P.K. (1987). Some results on the existence of utility functions on path connected spaces. Journal of Mathematical Economics, 147-156. [4) Steen, L.A. and J.A. Seebach Jr. (1979). Counterexamples in topology. Springer Verloe:. N.Y. 

Ordered Topological Spaces.

Representation of a Preference Ordering by a Numerical Function,

Some results on the existence of utility functions on path connected spaces.

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On non representable preferences

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