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fool comments on The Ellsberg paradox and money pumps - Less Wrong
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Comments (72)
Well then, P(green) = 1/3 +- 1/3 would be extreme ambiguity aversion (such as would match the adversary I think you are proposing), and P(green) = 1/3 exactly would be no ambiguity aversion , so something like P(green) = 1/3 +- 1/9 would be such a compromise, no? And why is that untenable?
To clarify: the aversary you have in mind, what powers does it have, exactly?
Generally speaking, an adversary would affect my behaviour, unless the loss of ambiguity aversion from the fact that all probabilities are known were exactly balanced by the gain in ambiguity aversion from the fact that said probabilities are under control of a (limited) aversary.
(Which is similar to saying that finding out the true distribution from which the urn was drawn would indeed affect your behaviour, unless you happened to find that the distribution was the prior you had in mind anyway.)
I don't get what this range signifies. There should be a data point about how ambiguous it is, which you could use or not use to influence actions. (For instance, if someone says they looekd in the urn and it seemed about even, that reduced ambiguity.) But then you want to convert that into a range, which does not refer to the actual range of frequencies (which could be 1/3 +- 1/3) and is dependent on your degree of aversion, but then you want to convert that into a decision?
Well, in terms of decisions, P(green) = 1/3 +- 1/9 means that I'd buy a bet on green for the price of a true randomised bet with probability 2/9, and sell for 4/9, with the caveats mentioned.
We might say that the price of a left boot is $15 +- $5 and the price of a right boot is $15 -+ $5.
Yes. So basically you are biting a certain bullet that most of us are unwilling to bite, of not having a procedure to determine your decisions and just kind of choosing a number in the middle of your range of choices that seems reasonable.
You're also biting a bullet where you have a certain kind of discontinuity in your preferences with very small bets, I think.
I don't understand what you mean in the first paragraph. I've given an exact procedure for my decisions.
What kind of discontinuities to you have in mind?
How do you choose the interval? I have not been able to see any method other than choosing something that sounds good (choosing the minimum and maximum conceivable would lead to silly Pascal's Wager - type things, and probably total paralysis.)
The discontinuity: Suppose you are asked to put a fair price f(N) on a bet that returns N if A occurs and 1 if it does not. The function f will have a sharp bend at 1, equivalent to a discontinuity in the derivative.
An alternative ambiguity aversion function, more complicated to define, would give a smooth bend.
Heh. I'm the one being accused of huffing priors? :-)
Okay, granted, there are methods like maximum entropy for Bayesian priors that can be applied in some situations, and the Ellsberg urn is such a situation.
Yes, you are correct about the discontinuity in the derivative.
Yes. Because you're huffing priors. Twice as much, in fact - we have to make up one number, you have to make up two.