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

## Author

Listed:
• Benjamin Y. Hayden
• Michael L. Platt

## 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.

## Suggested Citation

• Benjamin Y. Hayden & Michael L. Platt, 2009. "The mean, the median, and the St. Petersburg paradox," Judgment and Decision Making, Society for Judgment and Decision Making, vol. 4(4), pages 256-272, June.
• Handle: RePEc:jdm:journl:v:4:y:2009:i:4:p:256-272
as

File URL: http://journal.sjdm.org/9226/jdm9226.pdf

File URL: http://journal.sjdm.org/9226/jdm9226.html

## References listed on IDEAS

as
1. Shapley, Lloyd S., 1977. "The St. Petersburg paradox: A con games?," Journal of Economic Theory, Elsevier, vol. 14(2), pages 439-442, April.
2. Kahneman, Daniel & Tversky, Amos, 1979. "Prospect Theory: An Analysis of Decision under Risk," Econometrica, Econometric Society, vol. 47(2), pages 263-291, March.
3. Sennetti, John T, 1976. "On Bernoulli, Sharpe, Financial Risk and the St. Petersburg Paradox," Journal of Finance, American Finance Association, vol. 31(3), pages 960-962, June.
4. 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.
5. Milton Friedman & L. J. Savage, 1948. "The Utility Analysis of Choices Involving Risk," Journal of Political Economy, University of Chicago Press, vol. 56, pages 279-279.
6. Pavlo R. Blavatskyy, 2005. "Back to the St. Petersburg Paradox?," Management Science, INFORMS, vol. 51(4), pages 677-678, April.
7. 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 529-563, 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.
Full references (including those not matched with items on IDEAS)

## Citations

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Cited by:

1. Smith, Robert Elliott, 2016. "Idealizations of Uncertainty, and Lessons from Artificial Intelligence," Economics - The Open-Access, Open-Assessment E-Journal, Kiel Institute for the World Economy (IfW), vol. 10, pages 1-40.
2. Da Silva, Sergio & Matsushita, Raul, 2016. "The St. Petersburg paradox: An experimental solution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 445(C), pages 66-74.
3. Smith, Robert Elliott, 2015. "Idealizations of uncertainty, and lessons from artificial intelligence," Economics Discussion Papers 2015-50, Kiel Institute for the World Economy (IfW).

### Keywords

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

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