What is the St. Petersburg paradox in simple terms?

A coin-toss game has an infinite expected payout on paper, yet people will pay only a few dollars to play. It became the starting point for the idea that people judge outcomes by satisfaction, not dollar amounts.

Did Bernoulli's solution settle the paradox?

No. Menger showed that changing the prize structure revives the paradox even under log utility, and bounded utility, probability weighting and time averaging have all been proposed since.

St. Petersburg Paradox

📈 Economics By CampusDown Wiki Editorial Team Last updated October 27, 2026

The St. Petersburg paradox is a puzzle in probability posed by Nicolaus Bernoulli in 1713 and analyzed by his cousin Daniel Bernoulli in a 1738 paper for the St. Petersburg Academy: a game in which a coin is tossed until it lands heads, with the prize doubling each time, has infinite expected value, yet people will pay only a small amount to play. It shows that maximizing expected value alone cannot explain rational choice.

Contents

1. The game and the paradox
2. Bernoulli's solution and expected utility theory
3. Modern interpretations and applications
4. FAQ
5. References

The game and the paradox

Consider a game that pays $2ⁿ if the first heads appears on the nth toss. Heads on the first toss has probability 1/2 and pays $2, on the second 1/4 and pays $4, and so on, so each term contributes $1 and the sum grows without limit: the expected value is infinite. Yet almost no one would pay a large sum to play. Nicolaus Bernoulli first raised the problem in correspondence with Montmort, and the paradox took its name from the city when Daniel Bernoulli (1738) published a solution in the academy's journal. Jorland (1987) and Martin (2004) traced the more than two centuries of debate that followed.

Bernoulli's solution and expected utility theory

Daniel Bernoulli (1738; English translation 1954) argued that people weigh not amounts of money but their moral expectation, or utility, and proposed logarithmic utility, under which each additional dollar is worth less as wealth grows. The game's expected utility then becomes finite, and a small entry fee is rational. Cramer (1728) had suggested a similar idea in a letter. Menger (1934) pointed out that prizes growing faster revive the paradox even under log utility, so utility must be bounded above. Von Neumann and Morgenstern (1944) axiomatized the underlying intuition as expected utility theory, and Arrow (1974) discussed the need for bounded utility. Samuelson (1977) surveyed the paradox's history and proposed solutions.

Modern interpretations and applications

Practical resolutions point to the finite resources of whoever pays out and to the tendency to ignore very small probabilities. Kahneman and Tversky's (1979) prospect theory offers a framework explaining the paradox through probability weighting and the shape of the value function. Durand (1957) noted that valuing growth stocks has the same structure, since prices depend on huge but distant payoffs. Hayden and Platt (2009) reported that experimental subjects value the game closer to its median than its mean, and Peters (2011) argued that using time-average growth rates instead of ensemble averages resolves the paradox without any utility assumption.

FAQ

What is the St. Petersburg paradox in simple terms?: A coin-toss game has an infinite expected payout on paper, yet people will pay only a few dollars to play. It became the starting point for the idea that people judge outcomes by satisfaction, not dollar amounts.
Did Bernoulli's solution settle the paradox?: No. Menger showed that changing the prize structure revives the paradox even under log utility, and bounded utility, probability weighting and time averaging have all been proposed since.

Sources reviewed: 2026-10-26

References

Bernoulli, D. (1738). Specimen theoriae novae de mensura sortis. Commentarii Academiae Scientiarum Imperialis Petropolitanae, 5, 175–192.
Bernoulli, D. (1954). Exposition of a new theory on the measurement of risk (L. Sommer, Trans.). Econometrica, 22(1), 23–36.
Cramer, G. (1728). Letter to Nicolas Bernoulli. Reprinted in D. Bernoulli (1738).
Menger, K. (1934). Das Unsicherheitsmoment in der Wertlehre. Zeitschrift für Nationalökonomie, 5(4), 459–485.
von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.
Samuelson, P. A. (1977). St. Petersburg paradoxes: Defanged, dissected, and historically described. Journal of Economic Literature, 15(1), 24–55.
Arrow, K. J. (1974). The use of unbounded utility functions in expected-utility maximization: Response. Quarterly Journal of Economics, 88(1), 136–138.
Jorland, G. (1987). The Saint Petersburg paradox 1713–1937. In L. Krüger, L. J. Daston, & M. Heidelberger (Eds.), The Probabilistic Revolution (Vol. 1, pp. 157–190). Cambridge, MA: MIT Press.
Durand, D. (1957). Growth stocks and the Petersburg paradox. Journal of Finance, 12(3), 348–363.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–291.
Hayden, B. Y., & Platt, M. L. (2009). The mean, the median, and the St. Petersburg paradox. Judgment and Decision Making, 4(4), 256–272.
Peters, O. (2011). The time resolution of the St Petersburg paradox. Philosophical Transactions of the Royal Society A, 369(1956), 4913–4931.
Martin, R. (2004). The St. Petersburg paradox. In E. N. Zalta (Ed.), Stanford Encyclopedia of Philosophy. Stanford, CA: Stanford University.

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This article was last updated on October 27, 2026. It is based on widely recognized original works and textbooks; when citing it in a paper, please check the original sources listed in the references.