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What is Topological Methods in Cardinal Utility Theory about?

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

Author

Gerard Debreu; Gerard Debreu; Werner Hildenbrand

Language

EN