awesome-thought-experiments

The Allais Paradox

Category: economics
Associated thinker(s): Maurice Allais
Original publication or approximate period: “Le comportement de l’homme rationnel devant le risque” (1953)
Difficulty: Intermediate
Type: puzzle
Themes: risk; independence; expected utility
Related experiments: ellsberg-paradox; st-petersburg-paradox

1. The Scenario

Consider two separate choices with monetary outcomes measured in the same units. First choose A, a sure 1 million, or B, a 10% chance of 5 million, an 89% chance of 1 million, and a 1% chance of nothing. Then choose C, an 11% chance of 1 million and otherwise nothing, or D, a 10% chance of 5 million and otherwise nothing.

A familiar preference pattern selects A over B but D over C. Let u0, u1, and u5 be the utilities of the three amounts. The first preference requires 0.11u1 > 0.10u5 + 0.01u0; the second requires the reverse inequality. A single expected-utility representation cannot satisfy both.

2. The Central Question

Is the paired preference pattern irrational, or does it expose an overly restrictive standard of rational choice?

3. Why It Matters

The example distinguishes expected money from expected utility. Curving the utility function to represent ordinary risk aversion does not remove the displayed contradiction. The issue is how preferences should respond when the same consequence on a common event is changed in both options.

4. Major Interpretations

A defense of expected utility treats independence as a norm and regards the paired choices as a mistake to reconsider after seeing their structure. Risk-sensitive alternatives permit additional attitudes toward the distribution of outcomes. Outcome-redescription approaches include regret, disappointment, or certainty-related experiences among the valued consequences. Their challenge is to avoid making the theory untestable by redescribing any awkward choice after the fact.

5. Common Misunderstandings

The paradox is not merely that people dislike risk. Expected-utility theory already accommodates many kinds of risk aversion. Descriptive evidence that people make a choice does not by itself establish that the choice is normatively justified. The two lotteries should be considered with consistent stakes and background assumptions.

6. Think for Yourself

1. After seeing the shared-event structure, which of your two preferences would you revise, if either?
2. Is certainty itself part of the outcome you value?
3. What would count as a principled rather than ad hoc redescription of the consequences?

7. Variations and Extensions

Common-consequence and common-ratio effects explore related deviations. Ellsberg’s urn example instead emphasizes ambiguity about probabilities, so it should not be collapsed into the same psychological explanation.

8. Related Thought Experiments

• The Ellsberg Paradox — Choices involving an unknown color mixture challenge a single precise probability representation.
• The St. Petersburg Paradox — A finite payout on every completed play can nevertheless have infinite expected monetary value.

9. Further Reading

• Normative Theories of Rational Choice: Expected Utility — Stanford Encyclopedia of Philosophy, Spring 2020 archive

10. Sources and Attribution

The retrieved expected-utility entry supplies the 1953 attribution and paired-lottery structure. The monetary scale is simplified here without changing probabilities or the algebraic conflict. This is a decision-theoretic counterexample, not a formal logical paradox.

Verified on 2026-10-30. Source coverage: [1] Section 3.2.2: Allais (1953), Ellsberg (1961), numerical cases and alternative responses. Reflection questions and illustrative numerical examples are original editorial material.

This site is open source. Improve this page.