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The Petersburg Paradox at 300

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  • Seidl, Christian

Abstract

In 1713 Nicolas Bernoulli sent to de Montmort several mathematical problems, the fifth of which was at odds with the then prevailing belief that the advantage of games of hazard follows from their expected value. In spite of the infinite expected value of this game, no gambler would venture a major stake in this game. In this year, de Montmort published this problem in his Essay d'analyse sur les jeux de hazard. By dint of this book the problem became known to the mathematics profession and elicited solution proposals by Gabriel Cramer, Daniel Bernoulli (after whom it became known as the Petersburg Paradox), and Georges de Buffon. Karl Menger was the first to discover that bounded utility is a necessary and sufficient condition to warrant a finite expected value of the Petersburg Paradox. It was, in particular, Menger's article which provided an important cue for the development of expected utility by von Neumann and Morgenstern. The present paper gives a concise account of the origin of the Petersburg Paradox and its solution proposals. In its third section, it provides a rigorous analysis of the Petersburg Paradox from the uniform methodological vantage point of d'Alembert's ratio text. Moreover, it is shown that appropriate mappings of the winnings or of the probabilities can solve or regain a Petersburg Paradox, where the use of probabilities seems to have been overlooked by the profession.

Suggested Citation

  • Seidl, Christian, 2012. "The Petersburg Paradox at 300," Economics Working Papers 2012-10, Christian-Albrechts-University of Kiel, Department of Economics.
  • Handle: RePEc:zbw:cauewp:201210
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    References listed on IDEAS

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    1. Brito, D. L., 1975. "Becker's theory of the allocation of time and the St. Petersburg Paradox," Journal of Economic Theory, Elsevier, vol. 10(1), pages 123-126, February.
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    4. Shapley, Lloyd S., 1977. "The St. Petersburg paradox: A con games?," Journal of Economic Theory, Elsevier, vol. 14(2), pages 439-442, April.
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    6. Tversky, Amos & Kahneman, Daniel, 1992. "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 297-323, October.
    7. Kenneth J. Arrow, 1974. "The Use of Unbounded Utility Functions in Expected-Utility Maximization: Response," The Quarterly Journal of Economics, Oxford University Press, vol. 88(1), pages 136-138.
    8. Bentham, Jeremy, 1781. "An Introduction to the Principles of Morals and Legislation," History of Economic Thought Books, McMaster University Archive for the History of Economic Thought, number bentham1781.
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    14. Tibor Neugebauer, 2010. "Moral Impossibility in the Petersburg Paradox : A Literature Survey and Experimental Evidence," LSF Research Working Paper Series 10-14, Luxembourg School of Finance, University of Luxembourg.
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    Cited by:

    1. Daniel Muller & Tshilidzi Marwala, 2019. "Relative Net Utility and the Saint Petersburg Paradox," Papers 1910.09544, arXiv.org, revised May 2020.
    2. Jean Baccelli, 2018. "Risk attitudes in axiomatic decision theory: a conceptual perspective," Theory and Decision, Springer, vol. 84(1), pages 61-82, January.
    3. Christian Gollier & James Hammitt & Nicolas Treich, 2013. "Risk and choice: A research saga," Journal of Risk and Uncertainty, Springer, vol. 47(2), pages 129-145, October.
    4. James C. Cox & Eike B. Kroll & Marcel Lichters & Vjollca Sadiraj & Bodo Vogt, 2019. "The St. Petersburg paradox despite risk-seeking preferences: an experimental study," Business Research, Springer;German Academic Association for Business Research, vol. 12(1), pages 27-44, April.
    5. Ulrich Schmidt & Christian Seidl, 2014. "Reconsidering the common ratio effect: the roles of compound independence, reduction, and coalescing," Theory and Decision, Springer, vol. 77(3), pages 323-339, October.
    6. Bronshtein, E. & Fatkhiev, O., 2018. "A Note on St. Petersburg Paradox," Journal of the New Economic Association, New Economic Association, vol. 38(2), pages 48-53.

    More about this item

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • B16 - Schools of Economic Thought and Methodology - - History of Economic Thought through 1925 - - - Quantitative and Mathematical

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