“Ellsberg offered several groups of people a chance to bet on drawing either a red ball or a black ball from two different urns, each holding 100 balls. Urn 1 held 50 balls of each color; the breakdown in Urn 2 was unknown. Probability theory would suggest that Urn 2 was split 50-50, for there was no basis for any other distribution.”

From page 280 of Against the Gods: The Remarkable Story of Risk.

Tags: books

“Probability theory would suggest that Urn 2 was split 50-50, for there was no basis for any other distribution.”

I agree this is problematic, but I don’t think it’s terribly wrong. In the absence of prior information on the distribution of marble colors in the second urn, we assume that there is a 50-50 split because we possess no more informative prior information, and a decision-theoretic approach to whatever problem followed would suggest that unless the game was explicitly engineered to exploit assumptions of uniformity in the distrubution, assuming uniformity in the distribution is going to be the most accurate (and highest payoff) approach.

The problem with the statement is the use of the phrase “probability theory suggests…” Without any information, probability theory suggests that the probability the marbles are split 50-50 is the same is any other possible distribution or marble colors. But that’s just poor word choice on the author’s part. If he had said that Decision Theory suggests we should assume that the second urn was uniformly distributed in order to attain an optimal outcome in the absence of perfect information, he would have been closer to correct.

Unless I missed something completely?

To Tony and Matt: why problematic?

To Kate: insert persnickety comment that uniform priors do not technically reflect the most uninformed state in all situations (e.g., Jeffrey’s prior) here.

It’s problematic because it creates a fundamental misunderstanding of the Ellsberg “paradox.” The *point* of the Ellsberg experiment is that it doesn’t actually *matter* what the person believes about the second urn, because if they take the second urn, they also get to *pick* the color on which to bet. So, the person gets to choose between a 50/50 gamble on one hand, or an *at worst* 50/50 gamble on the other. The Ellsberg *paradox* is that almost everyone picks the first urn even though we would *expect* that no matter *what* their beliefs about the second, they would weakly prefer it if it was just a question of maximizing expected winnings. If probability theory really said that everyone should assume 50/50 in the second urn, then the Ellsberg experiment would simply indicate that people don’t get that concept. Fortunately, probability theory doesn’t prescribe what you should assume when you don’t know anything, which makes the Ellsberg result so fascinating, and gives us this beautiful rich concept of “ambiguity aversion” to mull over.

I don’t think it’s problamatic just not very well explained. Like Tony said, if you don’t have any information the uniform prior makes the most sense. I guess you could say that you can’t make any assumptions about the underlying distrubution. To do that you would have to understand the process used by the people to pick the balls.

Beyond any awkwardness of wording, there is a clear notion being expressed that probability theory tells you what distribution to assume in the absence of any knowledge. But probability theory, in no way, shape or form, can turn an unknown (the mix of balls in the urn) into a known (“[we should assume that] Urn 2 was split 50-50”). Assuming the urn is split is such a way might be the most conservative thing to do for some calculations, but not for others, and the most conservative assumption isn’t always the best.

Also, while I wouldn’t go so far as Bryon Wall (see “The Lure of the Fundamental Probability Set of Equally Likely Events”), there is an argument to be made that explaining probability in terms of equally likely events has been detrimental to public understanding. At the very least you have to be extremely careful when picking a model of ignorance. For example, a different uniform prior might be to assume that all distributions (from 0 black and 100 red to the opposite) are equally likely.

Oh no, well, I would recommend you read books from types other than Bernstein and Co.

A rule of thumb : every economist who has had something to do with the efficient market hypothesis, is not someone doing real research in economics, in the sense of actually trying to understand things. Is just someone doing politics in a very elaborate way. Period.

And if you really want to understand something about the new developments in economics, search for the masters

http://www.imbs.uci.edu/personnel/luce/luce.html

http://kuznets.harvard.edu/~aroth/alroth.html