Ellsberg Paradox

The urn experiment

In Ellsberg's (1961) best-known example, an urn contains 30 red balls and 60 balls that are black or yellow in unknown proportion. Many people prefer betting on red to betting on black, while also preferring to bet on black or yellow rather than on red or yellow. No single subjective probability can rationalize both preferences, so the choices violate Savage's (1954) sure-thing principle. Both choices reveal a tendency to pick the option whose probability is known.

Risk and ambiguity

The paradox connects to Knight's (1921) distinction between measurable risk and unmeasurable uncertainty and to Keynes's (1921) notion of the weight of evidence. Ellsberg called situations with missing probability information ambiguous and argued that people avoid them. Heath and Tversky (1991) proposed a competence hypothesis, finding that people prefer ambiguous bets in domains where they feel knowledgeable, and Fox and Tversky (1995) proposed comparative ignorance, finding that ambiguity aversion is strongest when a clearer alternative is present. Camerer and Weber (1992) surveyed the early experimental evidence.

Theories of ambiguity aversion

Gilboa and Schmeidler (1989) proposed maxmin expected utility, in which a decision maker maximizes the worst expected utility over a set of possible priors. Schmeidler (1989) developed Choquet expected utility using non-additive probabilities, and Klibanoff, Marinacci and Mukerji (2005) built a smooth model of second-order uncertainty over distributions. The variational preferences of Maccheroni, Marinacci and Rustichini (2006) and the robust control approach of Hansen and Sargent (2001) have been applied in macroeconomics and finance. Trautmann and van de Kuilen (2015) reviewed the experimental evidence on ambiguity attitudes.

FAQ

What is the Ellsberg paradox in simple terms?

Given a bet with known 50–50 odds and a bet with completely unknown odds, people tend to choose the known one even when the expected gain is the same.

How does it differ from the Allais paradox?

The Allais paradox concerns overweighting certainty under known risk, whereas the Ellsberg paradox concerns avoiding ambiguity when the probabilities themselves are unknown.

Sources reviewed: 2026-10-26

References

1. Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics, 75(4), 643–669.
2. Knight, F. H. (1921). Risk, Uncertainty and Profit. Boston: Houghton Mifflin.
3. Keynes, J. M. (1921). A Treatise on Probability. London: Macmillan.
4. Savage, L. J. (1954). The Foundations of Statistics. New York: Wiley.
5. Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with non-unique prior. Journal of Mathematical Economics, 18(2), 141–153.
6. Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57(3), 571–587.
7. Klibanoff, P., Marinacci, M., & Mukerji, S. (2005). A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892.
8. Maccheroni, F., Marinacci, M., & Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica, 74(6), 1447–1498.
9. Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of Risk and Uncertainty, 5(4), 325–370.
10. Heath, C., & Tversky, A. (1991). Preference and belief: Ambiguity and competence in choice under uncertainty. Journal of Risk and Uncertainty, 4(1), 5–28.
11. Fox, C. R., & Tversky, A. (1995). Ambiguity aversion and comparative ignorance. Quarterly Journal of Economics, 110(3), 585–603.
12. Hansen, L. P., & Sargent, T. J. (2001). Robust control and model uncertainty. American Economic Review, 91(2), 60–66.
13. Trautmann, S. T., & van de Kuilen, G. (2015). Ambiguity attitudes. In G. Keren & G. Wu (Eds.), The Wiley Blackwell Handbook of Judgment and Decision Making (pp. 89–116). Chichester: Wiley.