Good overall, but you are making a serious mistake: confusing single halfism, with double halfism.
SSA is a generalized single halfism reasoning. It's very obviously wrong in Sleeping Beauty as it implies that if the Beauty knows that she is awakened on Monday, she expects that there is 3/2 chance that the coin is Heads. Generally if you actually do the math, SSA can't produce correct betting odds for Sleeping Beauty problem. It's, in a sense, even worse than SIA for SB, but ultimately both of them are based on the flawed and unjustified framework of "centred possible worlds".
Double halfism is the correct approach and has no problem with correct betting odds. All your argument in favor of halfism are, in fact, double halfer reasoning. However you mistakenly credit them to SSA.
Took me a while to get back to you because I wanted to read your whole sequence (linked to me by someone else) before responding.
Anyway, it was far better than anything I wrote, and the thinking was much clearer.
I’m also a bit surprised that I hadn’t seen more discussion about double-halferism. It’s a single, barely comprehensible paragraph on the Sleeping Beauty wikipedia page, and you can barely find any links to it when googling “Sleeping Beauty” or “anthropics.” All this, despite the fact that it’s clearly the correct solution!
Anyway, thank you for the good read. I’m hoping to take a second attempt at this problem that should be a bit more thorough. I think I had gotten most of the way there, reasoning wise, but there were pieces I was clearly missing.
maximize “number of people who make the correct decision.”
Isn't that it - if you maximize the number of people who make the correct decision, then its 50/50 and you can choose either. If you maximize the number of correct decisions, then you choose tails (2 vs 1). The problem doesn't specify in enough detail what you are actually trying to maximize.
Yes good point, I hadn't thought about it exactly that way before. The observers are created/possible after the universe is created so you can't average across universes. How do you decide when one universe starts and another ends? Light cone? Part of the same inflation bubble even if up to infinitely far away?
Math is math, and at the end of the day the SB problem is just a math problem.
No, it's also an identity/assumption problem. Probability is subjective - it's an agent's estimate of future experience. In the sleeping beauty case, there is an undefined and out-of-domain intuition about "will it be one or two individuals having this future experience?" We just don't have any identity quantification experience in the case of split/merge from this memory-wipe setup.
the unstated disagreement is whether it's one or two experiences that resolve the probability. This ambiguity is clear by the fact that simplifications into clearly-distinct people don't trigger the same confusions. The memory-wipe is the defining element of this problem.
And to tie this to the universe question - how will the probability be resolved? What future experience are you predicting with either interpretation?
That comment about math was just intended as a rhetorical flourish. My apologies if it was over stated. In context, though, hopefully it was clear that I meant once you’ve established what question you’re actually asking, the math is straightforward.
Would you be willing to expand a little on your last sentence and explain a bit more about what you’re asking? I’m not sure I followed.
I think it goes to our main point of agreement: there is ambiguity in what question is being asked. For sleeping beauty, the ambiguity is "probability of WHAT future experience for WHOM" is she calculating a probability for. I was curious if you can answer that for your universe question: whose future experience will be used to resolve the truth of the matter for what probability was appropriate to use for the prediction?
I think we may be approaching these questions too differently, and I'm having trouble appreciating your question. I want to make sure I actually answer it.
The way I'm modelling the situation is this, the implication being that this is the closest to the way we would want to understand our universe:
1. A universe is created
2. Observers are "placed" in this universe as a result of the universe's inherent processes
3. You are "assigned" to one of these observers at random
In this framework, you don't necessarily get to verify anything. It's merely the case that if you were modelling the universe that way, then you would find that the probability of being in any given universe was determined only by step 1, unaffected by step 3.
I don't think I agree with #3 (and I'd frame #2 as "localities of space-time gain the ability to to sense and model things", but I'm not sure if that's important to our miscommunication). I think each of the observers happens to exist, and observes what it can independently of the others. Each of them experiences "you-ness", and none are privileged over the others, as far as any 3rd observer can tell.
So I think I'd say
I don't think active verbs are justified here - not necessarily "created", "placed", or "assigned".
I don't know for sure whether there is a god's eye view or "outside" observation point, but I suspect not, or at least I suspect that I can never get access to it or any effects of it, and can't think of what evidence I could find one way or the other.
You hit one point on the head: "it’s really just a math problem." But what that means is that it is not a philosophy problem. It was originally posed as a problem in philosophy, but the only thing such considerations do is obfuscate the math problem. Without justification. Just the need to put it in the realm of philosophy to match the original intent.
Consider a variation of the problem, that isn't a variation at all, but I'm sure will get a response as to why it doesn't fit in with a desired philosophical solution. I know this, because it has happened before. Use the same details with four volunteers, but one coin, and different schedules. One volunteer will be left asleep on Tuesday, after Heads, as in the popular[1] version. Each of the other three will be left asleep on a different one of the other three possible combinations of the day (Monday or Tuesday) and the coin result (Heads or Tails).
So on each day in the experiment, three of the volunteers will awakened. Each will be asked, individually, for the probability[2] that this is her only waking[3]. Then, the three will be brought together and asked to discuss the same question. The only taboo topic is the combinations where each would be left asleep.
In the individual interview, each is solving an problem that is identical to the popular version. In the combined one, each knows that she has a 1/3 chance to have the same assigned coin result as the missing volunteer. The only possible issue, is why should the answer change between these interviews.
--
[1] It isn't the original, or even the version posed in the paper that created the popular version.
[2] "Odds" means a ratio, not a probability. Odds of X:Y mean a probability of X/(X+Y). Halfers say the odds are 1:1, or a probability of 1/2. Thirders say the odds are 1:2, or a probability or 1/3.
[3] This is the question as it was in the original version. It is equivalent to "What is the probability that the coin result means you will be wakened only once?"
Here is a similar problem based on the same Mathematics. I call it Camp Sleeping Beauty. The same sleep and Amnesia details will be used, but at a week-long camp. On the arrival day, Sunday, you will be told all these details.
Like I said, this is a Math problem. The probability for each die roll is the number of times today's activity appears in that row, divided by the number of times it appears on the board. That is, if the activity appears only once on the board - or if every time it appears it is in the same row - there is a 100% chance that that was the die roll. If it appears twice on different rows, there is a 1/2 chance for each of those two rolls. Etc.
The important part here is that the arrangement of the other activities, or the actual activities, is completely irrelevant to this answer. "Today" is not an enigmatic value that you need a "universe model" to access, it is a random sampling of this six-by-six array.
Now, change the "Exercise" activity to "Extra sleep," where you get to sleep all day. That is, no interview. Nothing changes on the days where you do wake up. The answers do not change to the "Halfer" solution of 1/6 for each just because you know you would not wake up those days. The information you have about them is exactly the same - they are inconsistent with what happened today.
The popular Sleeping Beauty Problem is just a two-by-two version of this, with three of the day's activities being "wake up" and one "sleep." The answer, when you do wake up, is the number of times "wake" appears in the Heads row (one) divided by the number of times it appears on the board (three).
I have thought a lot about anthropics.
In an infinite universe, there are an infinite number of identical observers. You cannot define a probability distribution with an infinite sample space, and infinite cardinalities cannot help you. You cannot ask for the probability that it is Monday or Tuesday upon flipping tails because there are an infinite number of observers in both cases.
Do you agree that anthropic questions like these are meaningless if we live in an infinite universe?
[Since this piece no longer represents any positions I hold, I have decided to remove it from my LessWrong page. It can still be read on OneManyNone.substack.com.
If you are interested in my actual thoughts on anthropics, please see here.]