Expected utility theory
The classical model: choose the option with the highest average utility.
What it means
Expected utility theory is the normative model holding that a rational agent facing risky options should choose the one that maximizes the probability-weighted sum of the utilities of its possible outcomes, with utility defined over final states of wealth. Its mechanism rests on a small set of axioms — completeness, transitivity, continuity, and especially independence — from which von Neumann and Morgenstern proved that consistent preferences can be represented by such a utility function. The theory is elegant, tractable, and prescriptively compelling, and curvature in the utility function neatly captures risk aversion. Its weakness is descriptive: real choices systematically violate its axioms, as the Allais and Ellsberg paradoxes show, and people evaluate changes from a reference point rather than final wealth, weight probabilities nonlinearly, and treat gains and losses asymmetrically. This persistent gap between the theory's predictions and observed behavior is exactly what prospect theory was constructed to describe, so expected utility remains the benchmark of rationality even as behavioral models do the explanatory work.
The axioms, and why independence carries the weight
Von Neumann and Morgenstern's four axioms do more than tidy a preference ordering; they force a specific numerical structure. Completeness and transitivity make preferences a consistent ranking, continuity rules out lexicographic quirks, and independence, the demanding one, says that mixing two prospects with a common third outcome, at common odds, cannot reverse which you prefer. Grant all four and preferences must be representable by a utility that is unique up to a positive affine transformation: a genuinely cardinal quantity, not merely an ordering. Defenders justify the axioms pragmatically, arguing that anyone who violates transitivity can be turned into a 'money pump,' cycled through trades until broke. That normative pull is exactly why the theory survives its descriptive defeats rather than being discarded.
From objective to subjective odds
Von Neumann and Morgenstern assumed the probabilities were handed to you, as a fair coin hands you its odds. Leonard Savage's 1954 extension dropped that crutch, deriving both a utility and the agent's own subjective probabilities from choices alone, governed by a 'sure-thing principle': if an option is better whether or not some event occurs, you should prefer it regardless. This subjective expected utility became the workhorse of economics and statistics. It is also where Ellsberg aimed. His urns show people paying to avoid bets whose odds are merely unknown rather than unfavorable, a preference for known risk over ambiguity that no single subjective probability can rationalize, and that the sure-thing principle explicitly forbids.
The small-stakes problem
Because utility here is defined over total wealth, a small bet registers as a tiny wiggle on the whole wealth curve, and Matthew Rabin turned this into a formal embarrassment. His calibration theorem shows that if a person would reject a modest, actuarially favorable 50-50 gamble, say to lose $10 or gain $11, at every level of wealth, then to stay internally consistent they must also reject a 50-50 gamble to lose $100 against a gain of any size at all, even millions. No plausible curvature of a wealth-based utility function can produce the ordinary caution people show over small stakes without implying insane timidity over large ones. Reference-point models such as prospect theory sidestep this trap by construction, though they were not built for that purpose.
Where it still earns its keep
None of this has retired the model, because prescription and description are different jobs. Where the goal is to decide well rather than to predict a shopper, expected utility remains the tool of choice: medical decision analysis weighs treatments by expected quality-adjusted survival, insurers and portfolio theorists price risk through a concave utility that yields a risk premium, and cost-benefit analysts fold uncertainty into a single comparable number. Analysts reach for it knowing full well that people violate it, precisely because it states what a coherent choice would be. The honest division of labor, then, is expected utility as the yardstick of rationality and prospect theory as the map of behavior: one benchmark, one description.
Examples
By the theory, you should value a 50% chance of $100 at exactly the utility of $50 for sure — but most people don't.
The theory says insure the house and skip the extended warranty on the phone, since only one loss dents your final wealth — yet the shop sells phone cover all day long.
People who take a certain million over a gamble at a bigger prize will then pick the riskier of two long shots, a pair of choices no single utility function permits.
A hospital ranks two treatments by multiplying each outcome's probability by its quality-of-life value and summing; expected utility, not the best-case cure, decides which protocol becomes the standard of care.
A fund prices a portfolio by the expected utility of its returns, with a concave utility function penalizing volatility, so that a steadier 6 percent can outrank a jumpy 8 percent.
First described in Bernoulli (1738); von Neumann & Morgenstern (1944).
Key references
- Rabin, M. (2000). Risk aversion and expected-utility theory: A calibration theorem. Econometrica, 68(5), 1281-1292. doi.org/10.1111/1468-0262.00158
- Starmer, C. (2000). Developments in non-expected utility theory: The hunt for a descriptive theory of choice under risk. Journal of Economic Literature, 38(2), 332-382. doi.org/10.1257/jel.38.2.332
- Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263-291. doi.org/10.2307/1914185
- Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics, 75(4), 643-669. doi.org/10.2307/1884324
- Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'ecole americaine. Econometrica, 21(4), 503-546. doi.org/10.2307/1907921
- von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton University Press. press.princeton.edu/books/paperback/9780691130613/theory-of-games-and-economic-behavior