Mitchell, you are on to an important point: Observers must be well-defined.

Worlds are not well-defined, and there is no definite number of worlds (given standard physics).

You may be interested in my proposed Many Computations Interpretation, in which observers are identified not with so-called 'worlds' but with implementations of computations: http://arxiv.org/abs/0709.0544

See my blog for further discussion: http://onqm.blogspot.com/

I wasn't sneaky about it.

I don't think I got visibly hurt or angry. In fact, when I did it, I was feeling more tempted than angry. I was in the middle of a conversation with another guy, and her rear appeared nearby, and I couldn't resist.

It made me seem like a jerk, which is bad, but not necessarily low status. Acting without apparent fear of the consequences, even stupidly, is often respected as long as you get away with it.

Another factor is that this was a 'high status' woman. I'm not sure but she might be related to a celebrity. (I didn't know that at the time.) Hence, any story linking me and her may be 'bad publicity' for me but there is the old saying 'there's no such thing as bad publicity'.

It was a single swat to the buttocks, done in full sight of everyone. There was other ass-spanking going on, between people who knew each other - done as a joke - so in context it was not so unusual. I would not have done it outside of that context, nor would I have done it if my inhibitions had not been lowered by alcohol; nor would I do it again even if they are.

Yes, she deserved it!

It was a mistake. Why? It exposed me to more risk than was worthwhile, and while I might have hoped that (aside from simple punishment) it would teach her the lesson that she ought to follow the Golden Rule, or at least should not pull the same tricks on guys, in retrospect it was unlikely to do so.

Other people (that I have talked to) seem to be divided on whether it was a good thing to do or not.

Women seem to have a strong urge to check out what shoes a man has on, and judge their quality. Even they can't explain it. Perhaps at some unconscious level, they are guarding against men who 'cheat' by wearing high heels.

I can confirm that this does happen at least sometimes (USA). I was at a bar, and I approached a woman who is probably considered attractive by many (skinny, bottle blonde) and started talking to her. She soon asked me to buy her a drink. Being not well versed in such matters, I agreed, and asked her what she wanted. She named an expensive wine, which I agreed to get her a glass of. She largely ignored me thereafter, and didn't even bother taking the drink!

(I did obtain some measure of revenge later that night by spanking her rear end hard, though I do not advise doing such things. She was not amused and her brother threatened me, though as I had apologized, that was the end of it. She did tell some other lies so I don't know if she is neurotypical; my impression was that she was well below average in morality, being a spoiled brat.)

But Stuart_Armstrong's description is asking us to condition on the camera showing 'you' surviving.

That condition imposes post-selection.

I guess it doesn't matter much if we agree on what the probabilities are for the pre-selection v. the post-selection case.

Wrong - it matters a lot because you are using the wrong probabilities for the survivor (in practice this affects things like belief in the Doomsday argument).

I believe the strong law of large numbers implies that the relative frequency converges almost surely to p as the number of Bernoulli trials becomes arbitrarily large. As p represents the 'one-shot probability,' this justifies interpreting the relative frequency in the infinite limit as the 'one-shot probability.'

You have things backwards. The "relative frequency in the infinite limit" can be defined that way (sort of, as the infinite limit is not actually doable) and is then equal to the pre-defined probability p for each shot if they are independent trials. You can't go the other way; we don't have any infinite sequences to examine, so we can't get p from them, we have to start out with it. It's true that if we have a large but finite sequence, we can guess that p is "probably" close to our ratio of finite outcomes, but that's just Bayesian updating given our prior distribution on likely values of p. Also, in the 1-shot case at hand, it is crucial that there is only the 1 shot.

It is only possible to fairly "test" beliefs when a related objective probability is agreed upon

That's wrong; behavioral tests (properly set up) can reveal what people really believe, bypassing talk of probabilities.

Would you really guess "red", or do we agree?

Under the strict conditions above and the other conditions I have outlined (long-time-after, no other observers in the multiverse besides the prisoners), then sure, I'd be a fool not to guess red.

But I wouldn't recommend it to others, because if there are more people, that would only happen in the blue case. This is a case in which the number of observers depends on the unknown, so maximizing expected average utility (which is appropriate for decision theory for a given observer) is not the same as maximizing expected total utility (appropriate for a class of observers).

More tellingly, once I find out the result (and obviously the result becomes known when I get paid or punished), if it is red, I would not be surprised. (Could be either, 50% chance.)

Not that I've answered your question, it's time for you to answer mine: What would you vote, given that the majority of votes determines what SB gets? If you really believe you are probably in a blue room, it seems to me that you should vote blue; and it seems obvious that would be irrational.

Then if you find out it was red, would you be surprised?

The way you set up the decision is not a fair test of belief, because the stakes are more like $1.50 to $99.

To fix that, we need to make 2 changes:

1) Let us give any reward/punishment to a third party we care about, e.g. SB.

2) The total reward/punishment she gets won't depend on the number of people who make the decision. Instead, we will poll all of the survivors from all trials and pool the results (or we can pick 1 survivor at random, but let's do it the first way).

The majority decides what guess to use, on the principle of one man, one vote. That is surely what we want from our theory - for the majority of observers to guess optimally.

Under these rules, if I know it's the 1-shot case, I should guess red, since the chance is 50% and the payoff to SB is larger. Surely you see that SB would prefer us to guess red in this case.

OTOH if I know it's the multi-shot case, the majority will be probably be blue, so I should guess blue.

In practice, of course, it will be the multi-shot case. The universe (and even the population of Earth) is large; besides, I believe in the MWI of QM.

The practical significance of the distinction has nothing to do with casino-style gambling. It is more that 1) it shows that the MWI can give different predictions from a single-world theory, and 2) it disproves the SIA.

If that were the case, the camera might show the person being killed; indeed, that is 50% likely.

Pre-selection is not the same as our case of post-selection. My calculation shows the difference it makes.

Now, if the fraction of observers of each type that are killed is the same, the difference between the two selections cancels out. That is what tends to happen in the many-shot case, and we can then replace probabilities with relative frequencies. One-shot probability is not relative frequency.