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Alicorn comments on Bead Jar Guesses - Less Wrong
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I do not know how related this is to your comment, but it made me think of another response to the Dutch book objection. (Am I using that term correctly?)
If Omega asks me about a red bead I can say 100%. If he then asks about a blue bead I can adjust my original estimate so that both red and blue are the same at 50/50. Every question asked is adding more information. If Omega asks about green beads all three answers get shifted to 1/3.
This translates into an example with numbered balls just fine. The more colors or numbers Omega asks about decreases the expected probability that any particular one of them will come out of the jar simply because the known space of colors and numbers is growing. Until Omega acknowledges that there could be a bead of that color or number there is no particular reason to assume that such a bead exists.
If the example was rewritten to simply say any type of object could be in the jar, this still makes sense. If Omega asks about a red bead, we say 100%. If Omega asks about a blue chair, both become 50%. The restriction of colors and numbers is our assumed knowledge and has nothing to do with the problem at hand. We can meta-game all we want, but it has nothing to do with what could be in the jar.
The state of the initial problem is this:
After the second question:
I suppose it makes some sense to include an "other" category, but there is no knowledge of anything other than red and green beads. The question of probability implies that another may exist, but is that enough to assign it a probability?
What if Omega wants you to commit to a bet based on your probabilities at every step?
Or what if he just straight up asks you what color you want to guess the bead will be, without asking about any individual colors? (Then you'd probably be best served by switching to a language with fewer basic color words, but that aside...)
Than you are forced to bid 0 because you have to account for any further questions, which sounds similar to what Vladimir_Nesov said.
By the way, I think adding another restriction to your example to force it back into your specific response is not particularly meaningful. In the case where you do not have to commit to a bet at every step, does what I say make sense? If so, than what Vladimir_Nesov suggested seems to be on the right path with regards to your restrictions.
Switching languages is a semantic trick. If we are allowed to use any words to describe the bead we can just say "not-clear" because the space of "not-clear" covers what we generally mean by "color". We may as well say "the bead will be a colored bead." All of this breaks the assumed principle of no information.
If Omega wanted a particular color and forced us into actually answering the annoying question, we are completely off the path of probabilities and it does not matter what you answer as long as you picked a color. If Omega then asked us what the probability of that particular color coming out of the jar would be, the answer should be the same as if you picked any other color. This drops to zero unless you self-restrict by the number of colors you can personally remember.
MrHen, whatever strategy you're employing here, it doesn't sound like a strategy for arriving at the really truly correct answer, but some sort of clever set of verbal responses with a different purpose entirely. In real life, just because Omega asked if the bead is red simply does not mean there is probability 0 of it being green.
Mmm... I was not trying to employ a strategy with clever verbal responses. I thought I was arguing against that, actually, so I must be far from where I think I am.
I feel like I am trying to answer a completely different question than the one originally asked. Is the question:
I admittedly assumed the latter even though the article used words closer to the former. Perhaps this was my mistake?
I would agree. I do think that Omega asking about a red bead implies nothing about the probability of it being green. What I am currently wondering is if the question implies anything about the probability of the bead being red. If Omega acknowledges that the bead could be red, does that give red a higher probability than green?
I suppose I instinctively would answer affirmatively. The reasoning is that "red" is now included in the jar's potential outcomes while green has not been acknowledged yet. In other words, green doesn't even have a probability. Strictly speaking, this makes little sense, so I must be misstepping somewhere. My hunches are pointing toward my disallowing green into the potential outcomes.
This does not mean that I refuse to think of green as a color, but that green is not automatically included in the jar's potential outcomes just because Omega used the word "color". Is this the verbal cleverness you were referring to?
(Switching thoughts) In terms of arriving at the really truly correct answer, it seems that a strategy that gets closer as more beads is what is desired. If no beads are revealed, what sort of strategy is possible? I think the answer to this revolves around my potential confusion of the original question.
I apologize if I am mudding things up and am way off base.
Is Omega privileging the hypothesis that the bead is red?:-)