© 1998

Axiomatic Utility Theory under Risk

Non-Archimedean Representations and Application to Insurance Economics

  • Overview on the development of axiomatic utility theory under risk since von Neumann and Morgenstein.

  • New approaches in this field, with important consequences on insurance economics


Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 461)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Ulrich Schmidt
    Pages 1-67
  3. Ulrich Schmidt
    Pages 69-121
  4. Ulrich Schmidt
    Pages 123-161
  5. Back Matter
    Pages 163-201

About this book


The first attempts to develop a utility theory for choice situations under risk were undertaken by Cramer (1728) and Bernoulli (1738). Considering the famous St. Petersburg Paradox! - a lottery with an infinite expected monetary value -Bernoulli (1738, p. 209) observed that most people would not spend a significant amount of money to engage in that gamble. To account for this observation, Bernoulli (1738, pp. 199-201) proposed that the expected monetary value has to be replaced by the expected utility ("moral expectation") as the relevant criterion for decision making under risk. However, Bernoulli's 2 argument and particularly his choice of a logarithmic utility function seem to be rather arbitrary since they are based entirely on intuitively 3 appealing examples. Not until two centuries later, did von Neumann and Morgenstern (1947) prove that if the preferences of the decision maker satisfy cer­ tain assumptions they can be represented by the expected value of a real-valued utility function defined on the set of consequences. Despite the identical mathematical form of expected utility, the theory of von Neumann and Morgenstern and Bernoulli's approach have, however, IFor comprehensive discussions of this paradox cf. Menger (1934), Samuelson (1960), (1977), Shapley (1977a), Aumann (1977), Jorland (1987), and Zabell (1987). 2Cramer (1728, p. 212), on the other hand, proposed that the utility of an amount of money is given by the square root of this amount.


Entscheidungstheorie Insurance Economics Nutzentheorie Risikoaversion Risk Aversion Versicherungsökonomik decision theory economics utility theory

Authors and affiliations

  1. 1.University of KielInstitut für Finanzwissenschaft und SozialpolitikKielGermany

Bibliographic information

  • Book Title Axiomatic Utility Theory under Risk
  • Book Subtitle Non-Archimedean Representations and Application to Insurance Economics
  • Authors Ulrich Schmidt
  • Series Title Lecture Notes in Economics and Mathematical Systems
  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 1998
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Softcover ISBN 978-3-540-64319-7
  • eBook ISBN 978-3-642-58877-8
  • Series ISSN 0075-8442
  • Edition Number 1
  • Number of Pages XV, 228
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Economic Theory/Quantitative Economics/Mathematical Methods
    Operations Research/Decision Theory
  • Buy this book on publisher's site
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