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The mean, the median, and the St. Petersburg paradox

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  • Benjamin Y. Hayden
  • Michael L. Platt
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    Abstract

    The St.~Petersburg Paradox is a famous economic and philosophical puzzle that has generated numerous conflicting explanations. To shed empirical light on this phenomenon, we examined subjects' bids for one St.~Petersburg gamble with a real monetary payment. We found that bids were typically lower than twice the smallest payoff, and thus much lower than is generally supposed. We also examined bids offered for several hypothetical variants of the St.~Petersburg Paradox. We found that bids were weakly affected by truncating the gamble, were strongly affected by repeats of the gamble, and depended linearly on the initial ``seed'' value of the gamble. One explanation, which we call the \textit{median} \textit{heuristic}, strongly predicts these data. Subjects following this strategy evaluate a gamble as if they were taking the median rather than the mean of the payoff distribution. Finally, we argue that the distribution of outcomes embodied in the St.~Petersburg paradox is so divergent from the Gaussian form that the statistical mean is a poor estimator of expected value, so that the expected value of the St.~Petersburg gamble is undefined. These results suggest that this classic paradox has a straightforward explanation rooted in the use of a statistical heuristic.

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    Bibliographic Info

    Article provided by Society for Judgment and Decision Making in its journal Judgment and Decision Making.

    Volume (Year): 4 (2009)
    Issue (Month): 4 (June)
    Pages: 256-272

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    Handle: RePEc:jdm:journl:v:4:y:2009:i:4:p:256-272

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    Keywords: St.~Petersburg paradox; risk; risk aversion; heuristics; median heuristic; expectation heuristic.;

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    1. Vivian, Robert William, 2003. "Solving Daniel Bernoulli's St Petersburg Paradox: The Paradox which is not and never was," MPRA Paper 5233, University Library of Munich, Germany, revised 2003.
    2. Pavlo R. Blavatskyy, 2005. "Back to the St. Petersburg Paradox?," Management Science, INFORMS, vol. 51(4), pages 677-678, April.
    3. Shapley, Lloyd S., 1977. "The St. Petersburg paradox: A con games?," Journal of Economic Theory, Elsevier, vol. 14(2), pages 439-442, April.
    4. Milton Friedman & L. J. Savage, 1948. "The Utility Analysis of Choices Involving Risk," Journal of Political Economy, University of Chicago Press, vol. 56, pages 279.
    5. Kahneman, Daniel & Tversky, Amos, 1979. "Prospect Theory: An Analysis of Decision under Risk," Econometrica, Econometric Society, vol. 47(2), pages 263-91, March.
    6. Schoemaker, Paul J H, 1982. "The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations," Journal of Economic Literature, American Economic Association, vol. 20(2), pages 529-63, June.
    7. Sennetti, John T, 1976. "On Bernoulli, Sharpe, Financial Risk and the St. Petersburg Paradox," Journal of Finance, American Finance Association, vol. 31(3), pages 960-62, June.
    8. Schoemaker, Paul J. H. & Hershey, John C., 1996. "Maximizing Your Chance of Winning: The Long and Short of It Revisited," Organizational Behavior and Human Decision Processes, Elsevier, vol. 65(3), pages 194-200, March.
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