Maurice Allais presented his famous paradox in 1953, demonstrating a pattern of choices that violates the independence axiom — the most controversial axiom of expected utility theory. The paradox involves two pairs of choices where people's preferences are inconsistent with any utility function that satisfies the von Neumann-Morgenstern axioms.
The Paradox
Problem 2: C = ($1M, .11; $0, .89) vs. D = ($5M, .10; $0, .90)
Common pattern: A ≻ B and D ≻ C
This violates independence (EU requires A ≻ B ⟹ C ≻ D)
Most people choose the sure $1M (A) in Problem 1 — preferring certainty over a slightly higher expected value gamble that includes a chance of getting nothing. But in Problem 2, most choose D — preferring the chance at $5M, since both options have similar probabilities of winning nothing. This pattern is inconsistent with any expected utility function.
Theoretical Implications
The Allais paradox is explained by several non-expected utility theories. Prospect theory accounts for it through probability weighting: the certainty effect (overweighting certain outcomes relative to merely probable ones). Rank-dependent utility theory accommodates it by weighting probabilities based on their rank position. Regret theory explains it through the anticipated regret of receiving nothing when $1M was available for certain. The paradox remains the single most important empirical challenge to normative expected utility theory.