Non-Archimedean Utility Theory

My interest in non-Archimedean utility theory and the problems related to it was aroused by discussions which I have had with Professors Werner Leinfellner and Günter Menges. On the occasion of the Second Inter­ national Game Theory Workshop, Berkeley, 1970, which was sponsored by the National Science Foundation, I had the opportunity to report about a result on non-standard utilities. Work on this subject continued when I was a research assistant of Professor Günter Menges at the Uni­ versity of Heidelberg. The present mono graph is essentially a translation of my habilitation thesis which was accepted on February 15, 1973 by the Faculty of Economics and Social Sciences at the Universtity of Heidelberg. On translating my thesis I took up some suggestions made by ProfessorWerner Böge from the Faculty of Mathematics at the Uni­ versity of Heidelberg. Through lack of time many of his ideas have not been taken into consideration but I hope to do so in a future paper. The first chapter should be considered as a short introduction to pref­ erence orderings and to the notion of a utility theory proposed by Dana Scott and Patrick Suppes. In the second chapter I discuss in some detail various problems of ordinal utility theory. Except when introducing non-standard models of the reals no use is made of concepts of model theory. This is done in deference to those readers who do not wish to be troubled by formal languages and model theory.

I. Preference Orderings and Utility Theory.- 1. Relational Systems.- 2. Preference Relations.- 3. Some Remarks on Utility Theory.- 4. Linear Inequalities.- II. Ordinal Utility.- 1. Some Classical Representation Theorems.- 2. Lexicographic Utility.- 3. Utility Theories with Respect to n?-Sets.- 4. Ultraproducts and Ultrapowers.- 5. Approximating an r*-Valued Utility Function by a Real Valued Function.- 6. Non-Standard Utility Functions Always Exist.- 7. Utility Functions for Partial Orderings.- III. On Numerical Relational Systems.- 1. First-Order Languages.- 2. Some Preliminary Considerations.- 3. Universal and Homogeneous Relational Systems.- 4. Saturated Relational Systems.- IV. Utility Theories for More Structured Empirical Data.- 1. Some Remarks.- 2. The Empirical Status of Axioms.- 3. Utility Theories which are Axiomatizable in an Ordinary First Order Language.- 4. Extensive Utility.- 5. Conjoint Measurement of Utilities.- 6. On Certain Mean Systems.- V. On Utility Spaces, The Theory of Games and the Realization of Comparative Probability Relations.- 1. A Generalization of the Von Neumann/Morgenstern Utility Theory.- 2. Non-Standard Utilities in Game Theory.- 3. Some Aspects of the Realization of Comparative Probability Relations.- Appendix I. Ordinal and Cardinal Numbers.- Appendix II. Some Basic Facts about Filters and Ultrafilters.- Index of Names.- Index of Subjects.