Followup to: The Savage theorem and the Ellsberg paradox In the previous post, I presented a simple version of Savage's theorem, and I introduced the Ellsberg paradox. At the end of the post, I mentioned a strong Bayesian thesis, which can be summarised: "There is always a price to pay...
Followup to: A summary of Savage's foundation for probability and utility. In 1961, Daniel Ellsberg, most famous for leaking the Pentagon Papers, published the decision-theoretic paradox which is now named after him 1. It is a cousin to the Allais paradox. They both involve violations of an independence or separability...
No, (un)fortunately it is not so.
I say this has nothing to do with ambiguity aversion, because we can replace (1/2, 1/2+-1/4, 1/10) with all sorts of things which don't involve uncertainty. We can make anyone "leave money on the table". In my previous message, using ($100, a rock, $10), I "proved" that a rock ought to be worth at least $90.
If this is still unclear, then I offer your example back to you with one minor change: the trading incentive is still 1/10, and one agent still has 1/2+-1/4, but instead the other agent has 1/4. The Bayesian agent holding 1/2+-1/4 thinks it's worth more than 1/4 plus 1/10, so it refuses to trade. Whereas the ambiguity averse agents are under no such illustion.
So, the boot's on the other foot: we trade, and you don't. If your example was correct, then mine would be too. But presumably you don't agree that you are "leaving money on the table".
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.
To quote the article you linked: "Jaynes certainly believed very firmly that probability was in the mind ... there was only one correct prior distribution to use, given your state of partial information at the start of the problem."
I have not specified how prior intervals are chosen. I could (for the sake of argument) claim that there was only one correct prior probability interval to assign to any event, given the state of partial information.
At no time is the agent... (read more)
Right, except this doesn't seem to have anything to do with ambiguity aversion.
Imagine that one agent owns $100 and the other owns a rock. A government agency wishes to promote trade, and so will offer $10 to any agents that do trade (a one-off gift). If the two agents believe that a rock is worth more than $90, they will trade; if they don't, they won't, etc etc
But it still remains that in a many circumstances (such as single draws in this setup), there exists information that a Bayesian will find useless and an ambiguity-averter will find valuable. If agents have the opportunity to sell this information, the Bayesian will get a free bonus.
How does this work, then? Can you justify that the bonus is free without circularity?
... (read more)From a more financial persepective, the ambiguity-averter gives up the opportunity to be a market-maker: a Bayesian can quote a price and be willing to either buy and sell at that price (plus a small fee), wherease the ambiguity-averter's required spread is pushed up by the ambiguity (so all other agents will
Would a single ball that is either green or blue work?
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 (95%). Some of these cases should be candidates for very high (but not complete) ambiguity.
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.
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... (read more)
What if I told you that the balls were either all green or all blue?
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.
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."
If... (read more)
Suppose in the Ellsberg paradox that the proportion of blue and green balls was determined, initially, by a coin flip (or series of coin flips). In this view, there is no ambiguity at all, just classical probabilities
Correct.
Where do you draw the line
1) I have no reason to think A is more likely than B and I have no reason to think B is more likely than A
2) I have good reason to think A is as likely as B.
These are different of course. I argue the difference matters.
The boots and the mother example can all be dealt with using standard Bayesian techniques
Correct. See last paragraph of the post.
... (read more)You would pay
If you mean repeated draws from the same urn, then they'd all have the same orientation. If you mean draws from different unrelated urns, then you'd need to add dimensions. It wouldn't converge the way I think you're suggesting.
Here's an alternate interpretation of this method:
If two events have probability intervals that don't overlap, or they overlap but they have the same orientation and neither contains the other, then I'll say that one event is unambiguously more likely than the other. If two events have the exact same probability intervals (including orientation), then I'll say that are equally likely. Otherwise they are incomparable.
Under this interpretation, I claim that I do obey rule 2 (see prev post): if A is unambiguously more likely than B, then (A but not B) is unambiguously more likely than (B but not A), and conversely. I still obey rule 3: (A or B) is either unambiguously... (read more)
Followup to: The Savage theorem and the Ellsberg paradox
In the previous post, I presented a simple version of Savage's theorem, and I introduced the Ellsberg paradox. At the end of the post, I mentioned a strong Bayesian thesis, which can be summarised: "There is always a price to pay for leaving the Bayesian Way."1 But not always, it turns out. I claimed that there was a method that is Ellsberg-paradoxical, therefore non-Bayesian, but can't be money-pumped (or "Dutch booked"). I will present the method in this post.
I'm afraid this is another long post. There's a short summary of the method at the very end, if you want to skip the jibba jabba... (read 4043 more words →)
Followup to: A summary of Savage's foundation for probability and utility.
In 1961, Daniel Ellsberg, most famous for leaking the Pentagon Papers, published the decision-theoretic paradox which is now named after him 1. It is a cousin to the Allais paradox. They both involve violations of an independence or separability principle. But they go off in different directions: one is a violation of expected utility, while the other is a violation of subjective probability. The Allais paradox has been discussed on LW before, but when I do a search it seems that the first discussion of the Ellsberg paradox on LW was my comments on the previous post 2. It seems to me... (read 3191 more words →)
If there is nothing wrong with having a state variable, then sure, I can give a rule for initialising it, and call it "objective". It is "objective" in that it looks like the sort of thing that Bayesians call "objective" priors.
Eg. you have an objective prior in mind for the Ellsberg urn, presumably uniform over the 61 configurations, perhaps based on max entropy. What if instead there had been one draw (with replacement) from the urn, and it had been green? You can't apply max entropy now. That's ok: apply max entropy "retroactively" and run the usual update process to get your initial probabilities.
So we could normally start the state variable... (read more)