From preference to utility: a problem of descriptive set theory.

Abstract

IntroductionSome years ago J. H. Silver proved that a co-analytic equivalence relation on a Polish space has either countably many or continuum many equivalence classes. Later L. Harrington greatly simplified the complicated original proof. The present paper is a sort of footnote to Harrington's lectures on these matters. It will be shown that information developed in his proof settles a problem of (hyper-)theoretical mathematical economics first investigated by Wesley [13] and Mauldin [8]. Namely, it will be shown that any family of closed preference orders that is parametrized in a Borel fashion can be represented by a family of continuous utility functions parametrized in an absolutely measurable fashion. Though the author is greatly indebted to Mauldin's work [8], the treatment of the problem here will be self-contained. Background and motivation for problems of this kind can be found in [6], Section 2.1. Terminology and notation pertaining to descriptive set theory will be as in [9]. DefinitionsThroughout let ψ be a topological space. A preference order on 'ψ is any transitive, connected binary relation <*. Associated are the strict preference and indifference relations given by:Note that Ξ* i s an equivalence relation, and that <* induces a linear order on its equivalence classes, [x]* will denote the equivalence class of x. <* will be

Utility of Sacrifices: Reorientation of the Utility Theory

“…Jeremy Bentham was an English philosopher who introduced the utility concept into social science in 1789 [3]. Utility is the reflection of an individual's value or preference [4][5][6][7], which generates pain and pleasure in an individual from their action [8]. In the early stages, economists used utility theory to explain mainly two concepts -demand behavior and to justify or amend an economic policy [9].…”

Representations of Well-Founded Preference Orders

“…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.…”

“…Case I: a + 1) Suppose that <f yg has been constructed, satisfying(1) and(2), for all ft ^ a. Define the saturated coanalytic subset C of B by C = {(/, x):sup {$ a (uy) + l:y -<,*} ^ a < <£«(/, x) } = { (/, *):*<*(', x) > a and (V j0(y -<, x -> <j> a (t, y) < a}.…”