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Timing contradictions in von Neumann and Morgenstern's axioms and in savage's ‘sure-thing’ proof

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Abstract

At the point of choice, let N be the delay in learning the outcome. Then von Neumann and Morgenstern's postulates contradictorily imply that N = 0 and N > 0. As a consequence, Savage's ‘sure-thing’ proof, which has bestowed on expected utility theory most of its normative appeal, depends on inconsistent assumptions. Further, the validity of Savage's proof cannot be retrieved by minimizing N > 0, by making the delay a mere moment or so. The historical origins of these contradictions are traced to (i) von Neumann and Morgenstern inadvertently limiting their risk model to the certain period, that is the period after gamblers learn the outcome(s), and (ii) Savage's use of the sure-thing principle for analysing “atemporally but also quite formally” compound gambles [Savage, 1954, p. 23].

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References

  • Allais, M., 1952, ‘Fondements d'une Théorie Positive des Choix comportant un Risque’, Econométrie, Colloques Internationaux du Centre National de la Recherche Scientifique, 40, pp. 127–140, 257–332, appendix translated into English: ‘The Foundations of a Positive Theory of Choice Involving Risk and a Criticism of the Postulates and Axioms of the American School’, in M. Allais & O. Hagen (eds), Expected Utility Hypotheses and the Allais Paradox: Contemporary Discussions of Decisions under Uncertainty with Allais' Rejoinder, Reidel, Dordrecht, 1979, 27–145.

  • Allais, M., 1979, ‘The So-Called Allais Paradox and Rational Decision under Uncertainty’, in M. Allais & O. Hagen, (eds), Expected Utility Hypotheses and the Allais Paradox: Contemporary Discussions of Decisions under Uncertainty with Allais' Rejoinder, Reidel, Dordrecht, 437–681.

    Google Scholar 

  • Allais, M., 1984, ‘The Foundations of the Theory of Utility and Risk. Some Central Points risques’ Econométrie, Colloques Internationaux du Centre National de la Recherche in Utility and Risk Theory, Reidel, Dordrecht, 3–131.

    Google Scholar 

  • Arrow, K., 1953, ‘Le rôle des valeurs des boursières pour la répartition la meilleure des and F. Wenstøp (eds.) Foundations of Utility and Risk Theory with Applications, Reidel Scientifique, 40, 41–47, translated into English in The Role of Securities in the Optimal Allocation of Risk Bearing’, Review of Economic Studies, 1963–64, 31(2), 91–96.

  • Arrow, K., 1982, ‘Risk Perception in Psychology and Economics’, Economic Inquiry, 20, 1–9.

    Google Scholar 

  • Arrow, K., 1983, ‘Behavior under Uncertainty and its Implications for Policy’, in B. Stigum and F. Wenstøp (eds.) Foundations of Utility and Risk Theory with Applications, Reidel Dordrecht, 19–32.

    Google Scholar 

  • Baumol, W., 1972, Economic Theory and Operations Analysis, Third edition, Prentice-Hall, Englewood Cliffs, New Jersey.

    Google Scholar 

  • Bell, D., 1981, ‘Components of Risk Aversion’, in J. Brans (ed.), Operational Research '81, North Holland, Amsterdam, 235–242.

    Google Scholar 

  • Bernard, G., 1974, ‘On Utility Functions’, Theory and Decision, 5, 205–242.

    Google Scholar 

  • Bernard, G., 1982, ‘On Utility Functions, Present State’, mimeo, Paris.

  • Blatt, J., 1983, ‘Expected Utility Theory does not apply to all rational men’, in B. Stigum and F. Wenstøp (eds.) Foundations of Utility and Risk Theory with Applications, Reidel, Dordrecht, 107–116.

    Google Scholar 

  • Borch, K., 1968, The Economics of Uncertainty, Princeton University Press, Princeton, New Jersey.

    Google Scholar 

  • Drèze, J. & F. Modigliani, 1972, ‘Consumption Decisions under Uncertainty’, Journal of Economic Theory, 5, 308–335.

    Google Scholar 

  • Ellsberg, D., 1954, ‘Classic and Current Notions of Measurable Utility’, Economic Journal, 64, 528–566.

    Google Scholar 

  • Fishburn, P., 1970, Utility Theory for Decision Making, John Wiley & Sons, Inc., New York.

    Google Scholar 

  • Freeman, D., 1978, ‘Towards an Anthropology both Scientific and Humanistic’, Canberra Anthropology, 1(3), 44–69.

    Google Scholar 

  • Freeman, D., 1979, ‘The Anthropology of Choice’, an ANZAAS presidential address given in Auckland, New Zealand on 24 January 1979, reprinted in Canberra Anthropology, 1981, 4(1), 82–100.

    Google Scholar 

  • Friedman, M. & L. Savage, 1948, ‘The Utility Analysis of Choices Involving Risk’, Journal of Political Economy, 56(4), 279–304.

    Google Scholar 

  • Hagen, O., 1972, ‘A New Axiomatization of Utility Under Risk’, Teorie A Metoda, IV/2, 55–80.

    Google Scholar 

  • Hagen, O., 1979, ‘Introductory Survey’ in M. Allais, & O. Hagen (eds.) Expected Utility Hypotheses and the Allais Paradox: Contemporary Discussions of Decisions under Uncertainty with Allais' Rejoinder, Reidel, Dordrecht, 13–24.

    Google Scholar 

  • Hagen, O., 1983, ‘Paradoxes and their Solutions’, in B. Stigum and F. Wenstøp (eds.) Foundations of Utility and Risk Theory with Applications, Reidel, Dordrecht, 5–17.

    Google Scholar 

  • Herstein, I. & J. Milnor, 1953, ‘An Axiomatic Approach to Measurable Utility’, Econometrica, 21, 291–297.

    Google Scholar 

  • Kendall, M. & W. Buckland, 1982, A Dictionary of Statistical Terms, Fourth edition, Longman Group Ltd., London.

    Google Scholar 

  • Knight, F., 1921, Risk, Uncertainty and Profit, reprinted in 1971 by University of Chicago Press, Chicago.

    Google Scholar 

  • Kreps, D. & E. Porteus, 1979, ‘Temporal von Neumann-Morgenstern and Induced Preferences’, Journal of Economic Theory, 20, 81–109.

    Google Scholar 

  • Libby, R. & P. Fishburn, 1977, ‘Behavioral Models of Risk Taking in Business Decisions: A Survey and Evaluation’, Journal of Accounting Research, 15(2), 272–292.

    Google Scholar 

  • Lopes, L. L., 1984, ‘What Naive Decision Makers can Tell us About Risk’, paper presented to the Second International Conference on Foundations of Utility and Risk Theory, Venice, June.

  • Luce, R. & H. Raiffa, 1958, Games and Decisions, introduction and critical survey, John Wiley & Sons, Inc., New York.

    Google Scholar 

  • MacCrimmon, K. & S. Larsson, 1979, “Utility Theory: Axioms versus ‘Paradoxes’”, in M. Allais, and O. Hagen (eds.), Expected Utility Hypotheses and the Allais Paradox: Contemporary Discussions of Decisions under Uncertainty with Allais' Rejoinder, Reidel, Dordrecht, 333–409.

    Google Scholar 

  • MacCrimmon, K. & D. Wehrung, 1984, ‘Assessing Risk-Taking: A Large-Scale Empirical Study’, paper presented to the Second International Conference on Foundations of Utility and Risk Theory, Venice, June.

  • Machina, M., ‘Temporal Risk and the Nature of Induced Preferences’, Journal of Economic Theory, forthcoming.

  • Markowitz, H., 1970, Portfolio Selection: Efficient Diversification of Investment, first edition second printing, Yale University Press, New Haven.

    Google Scholar 

  • Marschak, J., 1946, ‘Neumanns’ and Morgenstern's New Approach to Static Economics’, Journal of Political Economy, 54(2), 97–115.

    Google Scholar 

  • Marschak, J., 1950, ‘Rational Behavior, Uncertain Prospects, and Measurable Utility’, Econometrica, 18(2), 111–141.

    Google Scholar 

  • Marschak, J. & R. Radner, 1972, Economic Theory of Teams, Yale University Press, New Haven.

    Google Scholar 

  • Morgenstern, O., 1974, ‘Some Reflections on Utility’, posthumously published in M. Allais & O. Hagen (eds.), Expected Utility Hypotheses and the Allais Paradox; Contemporary Discussions of Decisions Under Uncertainty with Allais' Rejoinder, Reidel Dordrecht, 1979, 175–183.

    Google Scholar 

  • Mossin, J., 1969, ‘A Note on Uncertainty and Preferences in a Temporal Context’, American Economic Review, 59(1), 172–174.

    Google Scholar 

  • Peirson, J., 1983, ‘Decision Making Under Risk and Across Time’, Discussion Paper 83.2, Division of Economic Studies, University of Sheffield, Sheffield.

    Google Scholar 

  • Pope, R., 1982, ‘Risk Pooling, Firm Strategy and the Government’, Working Paper 82/ 83–2–5, Department of Finance, University of Texas, Austin.

    Google Scholar 

  • Pope, R., 1983, ‘The Pre-Outcome Period and the Utility of Gambling’, in B. Stigum and F. Wenstøp (eds.), Foundations of Utility and Risk Theory with Applications, Reidel, Dordrecht, 137–177.

    Google Scholar 

  • Pope, R., 1984a, ‘The Utility of Gambling and the Utility of the Outcomes: Inconsistent First Approximations’, in O. Hagen and F. Wenstøp (eds.), Progress in Utility and Risk Theory, Reidel, Dordrecht, 251–273.

    Google Scholar 

  • Pope, R., 1984b, ‘Probabilities, Choice and Quantum Physics’, mimeo, University of New South Wales, Sydney.

    Google Scholar 

  • Ramsey, F., 1950, ‘Truth and Probability’ in R. Braithwaite (ed.), The Foundations of Mathematics and Other Logical Essays, The Humanities Press, New York.

    Google Scholar 

  • Samuelson, P., 1952, ‘Probability, Utility and the Independence Axiom’, Econometrica, 20, 670–678.

    Google Scholar 

  • Savage, L., 1954, The Foundations of Statistics, John Wiley & Sons, Inc., New York.

    Google Scholar 

  • Schoemaker, P., 1982, ‘The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations’, Journal of Economic Literature, 20(2), 529–563.

    Google Scholar 

  • Sen, A.: 1984, “Rationality and Uncertainty”, paper presented to the Second International Conference on Foundations of Utility and Risk Theory, Venice, June.

  • Simon, H., 1979, ‘Rational Decision Making in Business Organizations’, American Economic Review, 69(4), 493–513.

    Google Scholar 

  • Von Neumann, J. & O. Morgenstern, 1944, Theory of Games and Economic Behavior, Princeton University press, Princeton, New Jersey.

    Google Scholar 

  • Von Neumann, J. & O. Morgenstern, 1947, Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey.

    Google Scholar 

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  1. Department of Economics, University of New South Wales, P.O. Box 1, 2033, Kensington, N.S.W., Australia

    Robin Pope

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Pope, R. Timing contradictions in von Neumann and Morgenstern's axioms and in savage's ‘sure-thing’ proof. Theor Decis 18, 229–261 (1985). https://doi.org/10.1007/BF00126593

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