Category: economics
Associated thinker(s): Daniel Ellsberg
Original publication or approximate period: “Risk, Ambiguity, and the Savage Axioms” (1961)
Difficulty: Intermediate
Type: puzzle
Themes: ambiguity; subjective probability; sure-thing principle
Related experiments: allais-paradox; two-envelopes
An urn contains 90 balls: 30 red and 60 that are black or yellow in an unknown proportion. In the first choice, a winning bet pays the same prize if the ball is red or, alternatively, if it is black. Many favor red, whose probability is known. In the second choice, compare winning on red-or-yellow with winning on black-or-yellow. Many favor black-or-yellow, whose probability is known to be 2/3.
For a precise probability distribution, preferring red to black implies P(red) > P(black). Adding the same P(yellow) to both sides implies preference for red-or-yellow. The paired preference pattern reverses that implication.
Can aversion to unknown probabilities be rational even when it violates a standard subjective expected-utility representation?
The urn distinguishes risk with specified probabilities from ambiguity about probabilities. Treating all uncertainty as one precise probability distribution can hide attitudes toward the reliability or completeness of information. The challenge is to explain those attitudes without abandoning coherent comparison of options.
The orthodox response defends the sure-thing principle and asks the chooser to revise the paired preferences. Ambiguity-sensitive approaches use sets of probabilities or alternative evaluation rules that can represent cautious attitudes toward poorly known distributions. Richer outcome descriptions consider whether uncertainty itself matters to the agent. These are different theories of rationality, not evidence that the urn has inconsistent physical probabilities.
Unknown does not mean equally split. Saying that the red bet feels safer is not yet a proof of rational superiority. The paradox concerns the combination of choices: either choice alone can fit a precise probability distribution. It also differs from ordinary risk aversion over monetary gains.
Two-urn versions contrast a known mixture with an unknown one. Allais’s example uses fully known probabilities and therefore tests a related axiom without the same ambiguity feature.
The retrieved scholarly entry supports the 1961 origin, urn structure, and normative debate. Color names and win/lose presentation are simplified consistently. No empirical frequency of a preference is asserted without a specific study.
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.