Posted by: Tobias | October 4, 2008

Ambiguity aversion

Yet again, I learn something new about the wonderfully quirky machine that is the human brain (via Boing Boing)

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).

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Responses

  1. Correct me if I am wrong, but the interpretation depends pretty much on whether we know in advance that there will be a second round, in which we try to draw the opposite colour. This does not quite become clear from your description.

    Under the assumption that we do not know before that we’ll be able to draw a second marble of opposite colour afterwards, the observed behaviour appears to me just as yet another sign of risk-aversion revealed by most human beings. After all, while one knows for sure that the chance of getting a black marble in the left urn is 50-50, the right urn might offer anything from 100-0 to 0-100. So, given identical expected values for the probabilities of drawing a black marble, a risk-averse individual would seek to minimize the variance of probabilities.
    If this is correct, the same strategy holds in the second round, and it is rational to pick again the left urn.

    If we know that there’s a second draw, then we might have an incentive to chose the right urn, in order to gain additional information. If we’re successful, we might pick from the other urn in the second case. If we’re unsuccessful, we might stick to the same urn.

    I am curious to find out whether I made an error in reasoning.

  2. Your argument seems to rely on a draw actually taking place and you being told about what ball was drawn. I admit I have no idea how exactly the experiment was conducted, but as far as I understand it, there is no such draw. They simply ask you what you would prefer without ever actually drawing. In this case their argument seems completely reasonable.

    BTW: I just spent half an hour trying to figure out what the probability of drawing a white ball from the “unknown” urn is, given that you’ve already drawn one (suppose you put that white ball back into the urn afterwards). In classical statistics your estimate would be 1, but in Bayesian statistics using a non-informative prior, the answer would be different, right?


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