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fool comments on The Ellsberg paradox and money pumps - Less Wrong
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Comments (72)
Yes, replacing the new one. I.e. given a choice between trading the bet on green for a new randomised bet, we prefer to keep the bet on green. And no, the virtual interval is not part of any bet, it is persistent.
OK, now I understand why this is a necessary part of the framework.
I do think there is a problem with strictly choosing the lesser of the two utilities. For example, you would choose 1U with certainty over something like 10U ± 10U. You said that you would be still make the ambiguity-adverse choice if a few red balls were taken out, but what if almost all of them were removed?
On a more abstract note, your stated reasons for your decision seem to be that you actually care about what might have happened for reasons other than the possibility of it actually happening (does this make sense and accurately describe your position?). I don't think humans actually care about such things. Probability is in the mind; a difference in what might have happened is a difference in states of knowledge about states of knowledge. A sentence like "I know now that my irresponsible actions could have resulted in injuries or deaths" isn't actually true given determinism, it's about what you know believe you should have known in the past. [1] [2]
Getting back to the topic, people's desires about counterfactuals are desires about their own minds. What Irina and Joey's mother wants is to not intend to favour either of her children. [3] In reality, the coin is just as determininstic as her decision. Her preference for randomness is about her mind, not reality.
[1] True randomness like that postulated by some interpretations of QM is different and I'm not saying that people absolutely couldn't have preferences about truly random couterfactuals. Such a world would have to be pretty weird though. It would have to be timeful, for instance, since the randomness would have to be fundamentally indeterminite before it happens, rather than just not known yet, and timeful physics doesn't even make sense to me.
[2] This is itself a counterfactual, but that's irrelevent for this context.
[3] Well, my model of her prefers flipping a coin to drawing green or blue balls from an urn, but my model of her does not agree with me on a lot of things. If she were a Bayesian decision theorist, I would expect her to be indifferent between the coin and the urn, but prefer either to having to choose for herself.
If I had set P(green) = 1/3 +- 1/3, then yes. But in this case I'm not ambiguity averse to the extreme, like I mentioned. P(green) = 1/3 +- 1/9 was what I had, i.e. (1/2 +- 1/6)(2/3). The tie point would be 20 red balls, i.e. 1/4 exactly versus (1/2 +- 1/6)(3/4).
It makes sense, but I don't feel this really describes me. I'm not sure how to clarify. Maybe an analogy:
Maybe. Though I put it to you that the mother wants nothing more than what is "best for her children". Even if we did agree with her about what his best for each child separately, we might still disagree with her about what is "best for her children".
Perhaps I just want the "best chance of winning".
(ADDED:) If it helps, I don't think the fact that it is she making the decision is the issue - she would wish the same thing to happen if her children were in someone else's care.
Well utility is invariant under positive affine transformations, so you could have 30U +- 10U and shift the origin so you have 10U +- 10U. More intuitively, if you have 30U +- 10U, you can regard this as 20U + (20U,0U) and you would be willing to trade this for 21U, but you're guaranteed the first 20U and you would think it's excessive to trade (20U,0U) for just 1U.
Interesting.
What if they were in the care of her future self who already flipped the coin? Why is this different?
Bonus scenario: There are two standard Elisberg-paradox urns, each paired with a coin. You are asked to pick one to get a reward for iff ((green and heads) or (blue and tails)). At first you are indifferent, as both are identical. However, before you make your selection, one of the coins is flipped. Are you still indifferent?
What bet did you have in mind that was worth (20U,0) ? One of the simplest examples, if P(green) = 1/3 +- 1/9, would be 70U if green, -20U if not green. Does it still seem excessive to be neutral to that bet, and to trade it for a certain 1U (with the caveats mentioned)
This I don't understand. She is her future self isn't she?
Oh boy!
So there are two urns, one coin is going to be flipped. No matter what I'm offered a randomised bet on the second urn. If the coin comes up heads I'll be offered a bet on green on the first urn, if the coin comes up tails I'll be offered a bet on blue on the first urn. So looks like my options are:
A) choose urn 1 either way
B) choose urn 1 (i.e. green) if the coin comes up heads, choose urn 2 if the coin comes up tails
C) choose urn 2 if the coin comes up heads, choose urn 1 (i.e. blue) if the coin comes up tails
D) choose urn 2 either way
And to be pedantic: E) flip my own coin to randomise between options B and C.
I am indifferent between A, D, and E, which I prefer to B or C.
Generally, we seem to be really overanalysing the phrase "ought to flip a coin".
Huh, my explanations in that last post were really bad. I may have used a level of detail calibrated for simpler points, or I may have just not given enough thought to my level of detail in the first place.
What if I told you that the balls were either all green or all blue? Would you regard that as (20U,0U) (that was basically the bet I was imagining but, on reflection, it is not obvious that you would assign it that expected utility)? Would you think it equivalent to the (20U,0U) bet you mentioned and not preferrable to 1U?
So in the standard Ellisberg paradox, you wouldn't act nonbayesianally if you were told "The reason I'm asking you to choose between red and green rather than red and blue is because of a coin flip.", but you'd still prefer red if all three options were allowed? I guess that is at least consistent.
This is getting at a similar idea as the last one. What seems like the same option, like green or Irina, becomes more valuable when there is an interval due to a random event, even though the random event has already occurred and the result is now known with certainty. This seems to be going against the whole idea of probability being about mental states; even though the uncertainty has been resolved, its status as 'random' still matters.
Hmm. Well, with the interval prior I had in mind (footnote 7), this would result in very high (but not complete) ambiguity. My guess is that's a limitation of two dimensions -- it'll handle updating on draws from the urn but not "internals" like that. But I'm guessing. (1/2 +- 1/6) seems like a reasonable prior interval for a structureless event.
If I take the statement at face value, sure.
Yes, but again I could flip a coin to decide between green and blue then.
Well, okay. I don't think this method has any metaphysical consequences, so I should be able to adopt your stance on probability. I'd say (for the sake of argument) that the probability intervals are still about the mental states that I think you mean. However these mental states still leave the correct course of action underdetermined, and the virtual interval represents one degree of freedom. There is no rule for selecting the prior virtual interval. 0 is the obvious value, but any initial value is still dynamically consistent.
Would a single ball that is either green or blue work?
I agree that your decision procedure is consistent, not susceptible to Dutch books, etc.
I don't think this is true. Whether or not you flip the coin, you have the same information about the number of green balls in the urn, so, while the total information is different, the part about the green balls is the same. In order to follow your decision algorithm while believing that probability is about incomplete information, you have to always use all your knowledge in decisions, even knowledge that, like the coin flip, is 'uncorrelated', if I can use that word for something that isn't being assigned a probability, with what you are betting on. This is consistent with the letter of what I wrote, but I think that a bet that is about whether a green ball will be drawn next should use your knowledge about the number of green balls in the urn, not your entire mental state.
That still seems like a structureless event. No abstract example comes to mind, but there must be concrete cases where Bayesians disagree wildly about the prior probability of an event (<5% vs >95%). Some of these cases should be candidates for very high (but not complete) ambiguity.
I think you're really saying two things: the correct decision is a function of (present, relevant) probability, and probability is in the mind. I'd say the former is your key proposition. It would be sufficient to rule out that an agent's internal variables, like the virtual interval, could have any effect. I'd say the metaphysical status of probability is a red herring (but I would also accept a 50:50 green-blue herring).
Of course even for Bayesians there are equiprobable options, so decisions can't be entirely a function of probability. More precisely, your key proposition is that probabilistic indecision has to be transitive. Ambiguity would be an example of intransitive indecision.
Okay.
Well, once you assign probabilities to everything, you're mostly a Bayesian already. I think the best summary would be that when one must make a decision under uncertainty, preference between actions should depend on and only on one's knowledge about the possible outcomes.
Aren't you violating the axiom of independence but not the axiom of transitivity?
I'm not really sure what a lot of this means. The virtual interval seems to me to be subjectively objective in the same way probability is. Also, do you mean 'could have any effect' in the normative sense of an effect on what the right choice is?