What is the Allais paradox in simple terms?

People avoid risk when a large prize is certain but chase the bigger prize when both options are uncertain, a combination that breaks the consistency conditions of expected utility theory.

Does the paradox show people are irrational?

Interpretations differ. Those who treat the independence axiom as normative see an error, while Allais and others argued that a preference for certainty can itself be rational.

Allais Paradox

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

The Allais paradox is a choice problem devised by the French economist Maurice Allais in 1953 in which many people make inconsistent choices across two pairs of lotteries, violating the independence axiom of von Neumann–Morgenstern expected utility theory.

Contents

1. Structure of the paradox
2. Interpretation: certainty and common consequence effects
3. Development of non-expected utility theories
4. FAQ
5. References

Structure of the paradox

Allais (1953) posed two pairs of choices. In the first, A pays 1 million for certain, while B pays 5 million with probability 10%, 1 million with probability 89% and nothing with probability 1%. In the second, C pays 1 million with probability 11% and D pays 5 million with probability 10%. Many people choose A and D, yet the two pairs differ only in a common 89% consequence, so expected utility theory requires anyone choosing A to choose C and anyone choosing B to choose D. The inconsistency violates the independence axiom, Savage's (1954) sure-thing principle.

Interpretation: certainty and common consequence effects

Savage reported that he had intuitively chosen A and D himself but revised his choices on reflection, defending the axiom as a normative standard. MacCrimmon and Larsson (1979) showed that violations persisted across many variants. Kahneman and Tversky (1979) explained the pattern by a certainty effect, the overweighting of outcomes that are certain, and later work labeled the reversal produced by changing a common outcome the common consequence effect. Conlisk (1989) reported that presenting the problem in sequential stages sharply reduced violations.

Development of non-expected utility theories

The paradox inspired several generalizations of expected utility. Machina (1982) showed how to analyze preferences locally like expected utility without the independence axiom, and Quiggin (1982) developed rank-dependent utility, which weights probabilities. Loomes and Sugden (1982) proposed regret theory, and Tversky and Kahneman (1992) proposed cumulative prospect theory. Starmer (2000) surveyed these theories, and Huck and Müller (2012) found the paradox replicated in a large representative sample, with its strength varying by education and presentation.

FAQ

What is the Allais paradox in simple terms?: People avoid risk when a large prize is certain but chase the bigger prize when both options are uncertain, a combination that breaks the consistency conditions of expected utility theory.
Does the paradox show people are irrational?: Interpretations differ. Those who treat the independence axiom as normative see an error, while Allais and others argued that a preference for certainty can itself be rational.

Sources reviewed: 2026-10-26

References

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.
von Neumann, J., & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton: Princeton University Press.
Savage, L. J. (1954). The Foundations of Statistics. New York: Wiley.
MacCrimmon, K. R., & Larsson, S. (1979). Utility theory: Axioms versus 'paradoxes'. In M. Allais & O. Hagen (Eds.), Expected Utility Hypotheses and the Allais Paradox (pp. 333–409). Dordrecht: Reidel.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–291.
Machina, M. J. (1982). "Expected utility" analysis without the independence axiom. Econometrica, 50(2), 277–323.
Machina, M. J. (1987). Choice under uncertainty: Problems solved and unsolved. Journal of Economic Perspectives, 1(1), 121–154.
Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behavior & Organization, 3(4), 323–343.
Loomes, G., & Sugden, R. (1982). Regret theory: An alternative theory of rational choice under uncertainty. Economic Journal, 92(368), 805–824.
Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297–323.
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.
Conlisk, J. (1989). Three variants on the Allais example. American Economic Review, 79(3), 392–407.
Huck, S., & Müller, W. (2012). Allais for all: Revisiting the paradox in a large representative sample. Journal of Risk and Uncertainty, 44(3), 261–293.

Reports and materials that use this theory (Korean)

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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.