Published 13 September 2026

Ellsberg paradox

An urn contains thirty red balls and sixty balls that are black or yellow in an undisclosed proportion. One ball will be drawn uniformly. Would you rather win on red or on black? Now consider a separate choice: would you rather win on red or yellow, or on black or yellow?

The combination “red” and “black or yellow” chooses the option with a known winning probability in both questions: one third and two thirds. The challenge is to represent both strict preferences using one additive probability assignment and the same utility for winning and losing.

Make both choices before inspecting a composition

The hypothetical black-count control explores the possible hidden compositions. It is not evidence that was available when making the original choices. Changing it should not be confused with learning the true count.

If there are b black balls, there are 60 − b yellow balls. Red wins with probability 30/90, black with b/90, red-or-yellow with (90 − b)/90, and black-or-yellow with 60/90. The two constant probabilities are known from the setup; the others depend on the undisclosed b.

For example, b = 20 gives winning probabilities 1/3, 2/9, 7/9, and 2/3. Red beats black, but red-or-yellow then beats black-or-yellow. At b = 40 the rankings reverse together. At b = 30 both comparisons are tied.

Cancel the event shared by the two bets

Let q be the chooser’s subjective probability of black, without assuming they know a physical black count. If winning is preferable to losing, a strict preference for red over black requires q to be less than one third. A strict preference for black-or-yellow over red-or-yellow requires q to be greater than one third, because the yellow event has the same consequence in both bets and cancels.

One q cannot satisfy both inequalities. This is not fixed by choosing a more curved utility function: each bet has only the same two outcomes, so the utility difference between winning and losing multiplies both probability differences by the same positive number. The conflict concerns the representation of uncertainty and common consequences, not just risk aversion over prize amounts.

The preference controls check this common-probability constraint, including indifference. They do not score a person as rational or irrational. A chooser might reject the representation assumptions, attach significance to how a prize is obtained, or use an ambiguity-sensitive decision rule. Each response changes the decision model and should be named explicitly.

A set of probabilities is not a prior distribution

Knowing only that b lies between zero and sixty yields ranges for the four winning probabilities. It does not tell us that each composition is equally likely. Assigning a uniform prior over compositions would be an additional assumption. Under any single prior over b, ordinary averaging produces one predictive probability q for black, and the same cancellation constraint still applies.

The worst-case column instead evaluates each bet at its least favorable permitted composition. Red’s worst-case winning probability is one third and black’s is zero, so this criterion prefers red. Red-or-yellow has worst case one third and black-or-yellow has two thirds, so it prefers black-or-yellow. This illustrates how a criterion defined over a set of distributions can represent the characteristic preference pair.

Different bets attain their worst cases at different compositions. The column does not assert that the urn simultaneously contains both zero black balls and sixty black balls. Nor does it claim that worst-case evaluation is the uniquely correct response to ambiguity.

Ellsberg’s 1961 paper introduced these challenges to the Savage axioms. Continue to the Allais paradox, where all lottery probabilities are known and the tension instead appears when a shared consequence changes.