Representations of Well-Founded Preference Orders

Abstract

A preference order, or linear preorder, on a set X is a binary relation which is transitive, reflexive and total. This preorder partitions the set X into equivalence classes of the form . The natural relation induced by on the set of equivalence classes is a linear order. A well-founded preference order, or prewellordering, will similarly induce a well-ordering. A representation or Paretian utility function of a preference order is an order-preserving map f 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]. 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.

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.…”