In 1961, Daniel Ellsberg published the results of a hypothetical experiment he had conducted, which, to many, constitutes an even worse violation of the expected utility axioms than the Allais Paradox. Ellsberg's subjects in his thought experiment seemed to run the gamut of noted economists of the time, from Gerard Debreu to Paul Samuelson and Howard Raiffa.

Subjects are presented with 2 urns. Urn I contains 100 red and black balls, but in an unknown ratio. Urn II has exactly 50 red and 50 black balls. Subjects must choose an urn to draw from, and bet on the color that will be drawn - they will receive a $100 payoff if that color is drawn, and $0 if the other color is drawn. Subjects must decide which they would rather bet on:

1. A red draw from Urn I, or a black draw from Urn I
2. A red draw from Urn II, or a black draw from Urn II
3. A red draw from Urn I, or a red draw from Urn II
4. A black draw from Urn I, or a black draw from Urn II

One would expect subjects to be indifferent in the first two cases, and they are. However, people uniformly prefer a draw from Urn I in cases 3 and 4. It is impossible to infer judgements about probabilities from these choices - do people regard a draw of a particular color from Urn I as more likely? Certainly not, because otherwise they would not choose Urn I in both cases 3 and 4.

Why is this? Well, if they choose Urn I in case 3, this implies that they believe (rightly or wrongly) that Urn II has more black balls than red. However, if that is their belief, then they ought to choose Urn II in case 4 - but they don't. Both the completeness and monotonicity axioms are violated.

The primary conclusion one draws from this experiment is that people always prefer definite information to indefinite - Urn II may have more black balls than red, but it may also have more red balls than black. People tend to "prefer the devil they know".

Subjects are presented an urn containing 30 red balls and 60 black and yellow balls, the latter in an unknown proportion. Once again, subjects must bet on the color that will be drawn - they will receive a $100 payoff if that color is drawn, and $0 if either of the other colors is drawn. Let action I be a bet on red; action II be a bet on black.

OR:

Subjects are offered $100 if either black or yellow, call this action III; or either red or yellow is drawn, call this action IV.

It is very common for subjects to prefer I to II, and IV to III. Again, the axioms require the ordering of I preferred to II to be preserved in the preferences, and III should be preferred to IV.

Ellsberg Paradox experiment software from Econport cataloged resources

Ellsberg, Daniel, Risk, Ambiguity, and the Savage Axioms, The Quarterly Journal of Economics (Nov., 1961)