As I mentioned earlier, it's not an argument against halfers in general; it's against halfers with a specific kind of utility function, which sounds like this: "In any possible world I value only my own current and future subjective happiness, averaged over all of the subjectively indistinguishable people who could equally be "me" right now."

In the above scenario, there is a 1/2 chance that both Jack and Roger will be created, a 1/4 chance of only Jack, and a 1/4 chance of only Roger.

Before finding out who you are, averaging would lead to a 1:1 odds ratio, and so (as you've agreed) this would lead to a cutoff of 1/2.

After finding out whether you are, in fact, Jack or Roger, you have only one possible self in the TAILS world, and one possible self in the relevant HEADS+Jack/HEADS+Roger world, which leads to a 2:1 odds ratio and a cutoff of 2/3.

Ultimately, I guess the essence here is that this kind of utility function is equivalent to a failure to properly conditionalise, and thus even though you're not using probabilities you're still "Dutch-bookable" with respect to your own utility function.

I guess it could be argued that this result is somewhat trivial, but the utility function mentioned above is at least intuitively reasonable, so I don't think it's meaningless to show that having that kind of utility function is going to put you in trouble.

I agree with this. In probability terms, this is saying that P(there are 9 copies of me) is not necessarily meaningful because the event is not necessarily well defined.

My first response is / was that the event "the internet says it's Monday" seems a lot better-defined than "there are 9 of me," and should therefore still have a meaningful probability, even in anthropic situations. But an example may be necessary here.

I think you'd agree that a good example of "certain weird situations" is the divisible brain. Suppose we ran a mind on transistors and wires of macroscopic size. That is, we could make them half as big and they'd still run the same program. Then one can imagine splitting this mind down the middle into two half-sized copies. If this single amount of material counts as two people when split, does it also count as two people when it's together?

Whether it does or doesn't is, to some extent, mere semantics. If we set up a Sleeping Beauty problem except that there's the same amount of total width on both sides, it then becomes semantics whether there is equal anthropic probability on both sides, or unequal. So the "anthropic probabilities are meaningless" argument is looking pretty good. And if it's okay to define amount of personhood based on thickness, why not define it however you like and make probability pointless?

But I don't think it's quite as bad as all that, because of the restriction that your definition of personhood is part of how you view the world, not a free parameter. You don't try to change your mind about the gravitational constant so that you can jump higher. So agents can have this highly arbitrary factor in what they expect to see, but still behave somewhat reasonably. (Of course, any time an agent has some arbitrary-seeming information, I'd like to ask "how do you know what you think you know?" Exploring the possibilities better in this case would be a bit of a rabbit hole, though.)

Then, if I'm pretending to be Stuart Armstrong, I note that there's an equivalence in the aforementioned equal-total-width sleeping beauty problem between e.g. agents who think that anthropic probability is proportional to total width but have the same payoffs in both worlds ("width-selfish agents"), and agents who ignore anthropic probability, but weight the payoffs to agents by their total widths, per total width ("width-average-utilitarian outside perspective [UDT] predictors").

Sure, these two different agents have different information/probabilities and different internal experience, but to the extent that we only care about the actions in this game, they're the same.

Even if an agent starts in multiple identical copies that then diverge into non-identical versions, a selfish agent will want to self-modify to be an average utilitarian between non-identical versions. But this is a bit different from the typical usage of "average utilitarianism" in population ethics. A population-ethics average utilitarian would feed one of their copies to hungry alligators if it paid of for the other copies. But a reflectively-selfish average utilitarian would expect some chance of

beingthe one fed to the alligators, and wouldn't like that plan at all.Actually, I think the cause of this departure from average utilitarianism over copies is the starting state. When you start already defined as one of multiple copies, like in the divisible brain case, the UDT agent that naive selfish agents want to self-modify to be no longer looks just like an average utilitarian.

So that's one caveat about this equivalence - that it might not apply to all problems, and to get these other problems right, the proper thing to do is to go back and derive the best strategy in terms of selfish preferences.

Which is sort of the general closing thought I have: your arguments make a lot more sense to me than they did before, but as long as you have some preferences that are indexically selfish, there will be cases where you need to do anthropic reasoning just to go from the selfish preferences to the "outside perspective" payoffs that generate the same behavior. And it doesn't particularly matter if you have some contrived state of information that tells you you're one person on Mondays and ten people on Tuesdays.

Man, I haven't had a journey like this since DWFTTW. I was so sure that thing couldn't be going downwind faster than the wind.

P.S. So I have this written down somewhere, the causal buzzword important for an abstract description of the game with the marbles is "factorizable probability distribution." I may check out a causality textbook and try and figure the application of this out with less handwaving, then write a post on it.

That would be interesting.