Measurable Representations of Preference Orders

Abstract

Abstract. A continuous preference order on a topological space Y is a binary relation < which is reflexive, transitive and complete and such that for each x, {y: x < y) and {y: y < x) are closed. Let Tand Xbe complete separable metric spaces. For each t in T, let B, be a nonempty subset of X, let <t be a continuous preference order on B, and suppose E = {(l, x, y): x **,>>} is a Borel set. Let B = ((t, x):x e B,}. Theorem 1. There is an ¡$(T) <S> ^(Xymeasurable map g from B into R so that for each t, g(t, ) is a continuous map of B, into R and g(t, x) =ï g<t, y) if and only if x <, y. (Here o>(T) forms the C-sets of Selivanovskii and <S>( X) is a Borel field on X.) Theorem 2. If for each t, B, is a a-compact subset of Y, then the map g of the preceding theorem may be chosen to be Borel measurable.The following improvement of a theorem of Wesley is proved using classical methods.Theorem 3. Let g be the map constructed in Theorem 1. If' ¡lis a probability measure defined on the Borel subsets of T, then there is a Borel set N such that fi(N) = 0 and such that the restriction of g to B D ((T -N) X X) is Borel measurable.

The Threshold Decision Making Effectuated by the Enumerating Preference Function

“…The existence of utility representations with di®erent properties for preference orders was treated in a number of papers. [2][3][4][5][6] In the simplest case when X is nite one may explicitly set 'ðxÞ ¼ jfy 2 X : ðx; yÞ 2 Pgj, where jAj denotes the number of elements in the set A & X.…”

Representations of Well-Founded Preference Orders

“…A representation or Paretian utility function of a preference order is an order-preserving map / from X into the R of real numbers (provided with the standard ordering). Mathematicians and economists have studied the problem of obtaining continuous or measurable representations of suitably defined preference orders [4,7]. Parametrized versions of this problem have also been studied [1,7,8].…”

“…Mathematicians and economists have studied the problem of obtaining continuous or measurable representations of suitably defined preference orders [4,7]. Parametrized versions of this problem have also been studied [1,7,8]. Given a continuum of preference orders which vary in some reasonable sense with a parameter t, one would like to obtain a continuum of representations which similarly vary with t.…”

“…We will say that E is section-wise closed if, for each t, E t is closed with respect to B t X B t \ in this case, each preference order < t will possess a continuous representation by a result of Debreu [4]. The second author showed in [7] that if E is section-wise closed, then there is an S(T X X)measurable map/of B into R such that; for each t,f(t, •) is a continuous representation of <. (Here S(T X X) form the C-sets of Selivanovskii or the husin heirarchy [6, p.…”

“…468].) Under the further assumption that each B t is a-compact, it was also shown in [7] that the map/may be taken to be Borel measurable.…”