Topological methods in cardinal utility theory

Abstract

In this paper we shall study the concept of cardinal utility in three different situations (stochastic objects of choice, stochastic act of choice; independent factors of the action set) by means of the same mathematical result that gives a topological characterization of three families of parallel straight lines in a plane. This result, proved first by G. Thomsen [24] under differentiability assumptions, and later by W. Blaschke [2] in its present general form (see also W. Blaschke and G. Bol [3]), can be briefly described as follows. Consider the topological image G of a two-dimensional convex set and three families of curves in that set such that (a) exactly one curve of each family goes through a point of G, and (b) two curves of different families have at most one common point. Is there a topological transformation carrying these three families of curves into three families of parallel straight lines? If the answer is affirmative, the hexagonal configuration of Figure l(a) is observed. Let P be an arbitrary point of G, draw through it a curve of each family, and take an arbitrary point A on one of these curves; by drawing through A the curves of the other two families, we may obtain B and B' and from them C and C'.

Keywords

Affiliated Institutions

• University of California, Berkeley US

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