The Ellsberg paradox is a way of explaining human nature's aversion to uncertainty. It is also often referred to as ambiguity aversion. It was named after Daniel Ellsberg who popularized the concept.
There are many variations on the ways of explaining the Ellsberg paradox, many of which involves marbles inside non-transparent urns.
Suppose you have two urns which you can not see inside. The first urn has 5 black marbles and 5 white marbles. The second urn also has 10 marbles, but with a mixture of black and white marbles that is unknown.
Let's say we play a game where you are asked to draw a marble from one urn and if the marble drawn is black, you win say $5. Which urn would you draw from?
Most people would draw from the first urn. But it turns out that it doesn't matter which urn you draw from. The odds of winning is the same: 50/50. And can be proven mathematically.
The reason that most people draw from the first urn is that they have more information about the first urn. In particular, they know the distribution of the marbles and can more readily see that the odds are 50/50.
The reason that most people avoid the second urn is because they do not know the distribution of the second urn. There is an extra element of ambiguity or uncertainty. People do not like uncertainty and this is an example of ambiguity adversion.
Let's say we play a second game. This time if you draw a white marble, you win. Which urn would you draw from? Again, the first urn? And most people again would draw from the first urn.
But that is paradoxical, because if in the first game they draw from the first urn believing that the first urn is better for winning black marbles. Then why they are drawing from the first urn again when they are now trying to win with white marbles in the second game?
It's because people are avoiding the second urn due to its ambiguity (even though the second urn has just as good odds of winning).
Ellsberg Paradox involving a more Complicated Example
Consider another example of Ellsberg paradox. This time we have only one urn. It has 90 marbles that are either red, black, or yellow. All I'm going to tell you is that there are exactly 30 red marbles in there. So you do not know the combination of black and yellow. It is some random combination. But you know that the sum of the black and yellow marbles is 60.
You are giving the choice of two games. In the first game, you win if you draw a red or yellow marble. In the second game, you win if you draw a yellow or black marble. Which game would you play?
Turns out that most people would play the second game. At least in the second game, they know that their odds of winning is 2 out of 3. That is 60 of the marbles out of 90 will either yellow or black.
Well, it can be mathematically proven that the odds of winning in the first game is 2 out of 3 as well. Exactly the same. But that is less apparently. Most people do not know the odds of the first game and hence will not choose that game. Just another example of ambiguity aversion.
References to Ellsberg Paradox
Economist Andrew Lo spoke at MIT about economics and physics (video here). He explains the Ellsberg Paradox by introducing to the audience a hypothetical game.
New York Public Media Thirteen has a good video example demonstrating Ellsberg Paradox.