The St. Petersburg paradox and Bernoulli
Imagine a game that pays 2ⁿ dollars, where n is the number of coin tosses until the first heads. Its expected payoff is infinite, yet almost no one would pay much to play. In 1738 Daniel Bernoulli explained that people judge not by sums of money but by the satisfaction, or utility, money brings, and that each extra dollar adds less utility as wealth grows (he proposed logarithmic utility). The idea anticipated what became known as diminishing marginal utility, and computing expected utility instead of expected value dissolves the paradox.
Axioms and attitudes toward risk
In Theory of Games and Economic Behavior (1944), John von Neumann and Oskar Morgenstern proved that any preferences satisfying a few axioms (completeness, transitivity, continuity and independence) can be represented by expected utility. The result became a foundation of game theory and modern microeconomics, and Leonard Savage (1954) showed that the theory still works when probabilities are subjective rather than objectively given, linking it to Bayesian statistics. The curvature of the utility function captures risk attitudes: concave means risk-averse, convex means risk-seeking. John Pratt (1964) and Kenneth Arrow (1965) devised measures of risk aversion now used throughout insurance, portfolio choice and asset pricing, while Friedman and Savage (1948) tried to explain why people buy both insurance and lottery tickets through the shape of the utility curve.
Paradoxes and alternatives
In 1953 Maurice Allais presented choices in which people systematically violate the independence axiom when comparing sure gains with risky options, and in 1961 Daniel Ellsberg showed that people dislike ambiguity, situations in which probabilities are unknown. Daniel Kahneman and Amos Tversky's prospect theory (1979) proposed that people evaluate outcomes as gains and losses from a reference point, feel losses more keenly and overweight small probabilities, launching behavioral economics. Mark Machina (1987) surveyed generalized expected utility models that relax independence. Even so, expected utility remains the normative benchmark for rational choice and a workhorse model in economics.
FAQ
- How is expected value different from expected utility?
- Expected value weights dollar amounts by their probabilities; expected utility weights the satisfaction those amounts bring. Because utility is not proportional to money, the two can rank options differently.
- Is expected utility theory wrong?
- Plenty of evidence shows it does not perfectly describe actual behavior, but it remains important as a normative account of how rational choices should be made and as a starting point for analysis.
Sources reviewed: 2026-10-26
References
- Bernoulli, D. (1738/1954). Specimen theoriae novae de mensura sortis. Commentarii Academiae Scientiarum Imperialis Petropolitanae, 5, 175–192 (Eng. trans. Exposition of a new theory on the measurement of risk. Econometrica, 22(1), 23–36).
- von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.
- Friedman, M., & Savage, L. J. (1948). The utility analysis of choices involving risk. Journal of Political Economy, 56(4), 279–304.
- Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: critique des postulats et axiomes de l'école américaine. Econometrica, 21(4), 503–546.
- Savage, L. J. (1954). The Foundations of Statistics. New York: Wiley.
- Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics, 75(4), 643–669.
- Pratt, J. W. (1964). Risk aversion in the small and in the large. Econometrica, 32(1/2), 122–136.
- Arrow, K. J. (1965). Aspects of the Theory of Risk-Bearing. Helsinki: Yrjö Jahnssonin Säätiö.
- Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–291.
- Machina, M. J. (1987). Choice under uncertainty: Problems solved and unsolved. Journal of Economic Perspectives, 1(1), 121–154.