Ellsberg Paradox

The Ellsberg Paradox is a paradox in decision theory and experimental economics in which people’s choices violate the expected utility hypothesis.[1] It is generally taken to be evidence for ambiguity aversion. The paradox was popularized by Daniel Ellsberg, although a version of it was noted considerably earlier by John Maynard Keynes.[2]

Ellsberg raised two problems: 1 urn problem and 2 urn problem. Here, 1 urn problem is described, which is the better known one.

There are two large urns placed in front of you. The urns are completely opaque, so you cannot see their contents. The urn on the left contains ten black marbles and ten white ones. The urn on the right contains twenty marbles, but you do not know the proportion of black to white. Now, the game is to draw a black marble from one of the urns. If you are successful, you win $100. You only have one chance, so which urn will you draw from? Keep the answer in mind.

Let’s play again. Now, the game is to draw a white marble. Again, you only have one chance, so which urn will it be?

Most people when confronted with these choices choose the urn on the left — the one with the known proportions of black and white marbles. And therein lies the paradox. If you choose the left-hand urn when trying to pull a black marble, that means you think your chances are better for that urn. But because there are only two colors in both urns, the odds of pulling a white must be complementary to the odds of pulling a black. Logically, if you thought the left-hand urn was the better choice for a black marble, the right-hand urn should be the better choice for a white marble. The fact that most people avoid the right-hand urn altogether suggests that people have an inherent fear of the unknown (also called the ambiguity aversion).

Generality of the paradox

Note that the result holds regardless of your utility function. Indeed, the amount of the payoff is likewise irrelevant. Whichever gamble you choose, the prize for winning it is the same, and the cost of losing it is the same (no cost), so ultimately, there are only two outcomes: you receive a specific amount of money, or you receive nothing. Therefore it is sufficient to assume that you prefer receiving some money to receiving nothing (and in fact, this assumption is not necessary — in the mathematical treatment above, it was assumed U($100) > U($0), but a contradiction can still be obtained for U($100) < U($0) and for U($100) = U($0)).

In addition, the result holds regardless of your risk aversion. All the gambles involve risk. By choosing Gamble D, you have a 1 in 3 chance of receiving nothing, and by choosing Gamble A, you have a 2 in 3 chance of receiving nothing. If Gamble A was less risky than Gamble B, it would follow that Gamble C was less risky than Gamble D (and vice versa), so, risk is not averted in this way.

However, because the exact chances of winning are known for Gambles A and D, and not known for Gambles B and C, this can be taken as evidence for some sort of ambiguity aversion which cannot be accounted for in expected utility theory. It has been demonstrated that this phenomenon occurs only when the choice set permits comparison of the ambiguous proposition with a less vague proposition (but not when ambiguous propositions are evaluated in isolation)