Then the self-sampling assumption (SSA) produces the doomsday argument.
I have heard this around FHI before, Nick seems unbothered by such utterances. Nevertheless, I think this false to some extent. This is true: (S)SSA ⊨ (A ⊢ DA), A being some still open claims about the reference class. SSA produces a much weaker version of DA, which might not deserve this name and using it can be misleading. On the other hand, claiming DA was refuted by (S)SSA is also misleading. Between these two misleads, Nick seems to have chosen the former. His choice made sense from an utilitarian-memetic perspective, 10 years ago, among philosophers. However, I would prefer people already concerned with x-risks to use another terminology, which avoided both misleads.
SSA has a preference for universe with smaller numbers of observers (since it's more likely that you're one-in-a-hundred than one-in-a-billion).
I didn't understand this. Why would the reference class be the universe you belong to? The reference class ought to be composed by all epistemically indistinguishable individuals(let's call these EII), modulo all beliefs with no relevance to the questions being assessed. I think we have discussed my views on this last time I was here (I'm Joao, btw). I believe there are some grounds for claiming that (S)SSA entails you are in the universe with the biggest number of copies of epistemically indistinguishable individuals like you(let's call these the class Z universes). On the other hand, if there are much more non-Z universes containing EII like you, as to counterbalance the fact there are many more of EII like you on the Z universes, then it can be the case is much more likely you are not on a Z universe. If we are talking about possible worlds in the logical space, instead of possible universes, then the Z class seems to be composed by one member. This is so because, iff a world has the maximum number EII like you, then it is full, it exhausted the logical possible permutations, there are no more degrees of freedom where one could put more of anything else. This seems to be a good argument to say (S)SSA does not entail an absurdity, for then it becomes clear the class Z (or the descending classes (Y, X..) regarding the amount of EII like you) is much smaller than the classes of worlds where the frequency of EII like you is not oddly huge. If we are talking about physical possible universes, then I'm not sure how to prevent absurdity. But it is still the case it seems much more likely you are on an universe with a lot of members of your reference class (or EII like you) than in one with few of them, provided the reference class cuts between universes - as I think it should, in most scenarios. In the end, physical universes or light-cones are much less relevant than the reference class. Rational observers live in the latter, atoms in the former. Whereas we live in an evolutionary mess.
*(S)SSA should probably mean SSSA, the one I'm sure where my arguments make sense. I think they also work on SSA, but I'm not sure.
Ideas developed with Paul Almond, who kept on flogging a dead horse until it started showing signs of life again.
Doomsday, SSA and SIA
Imagine there's a giant box filled with people, and clearly labelled (inside and out) "(year of some people's lord) 2013". There's another giant box somewhere else in space-time, labelled "2014". You happen to be currently in the 2013 box.
Then the self-sampling assumption (SSA) produces the doomsday argument. It works approximately like this: SSA has a preference for universe with smaller numbers of observers (since it's more likely that you're one-in-a-hundred than one-in-a-billion). Therefore we expect that the number of observers in 2014 is smaller than we would otherwise "objectively" believe: the likelihood of doomsday is higher than we thought.
What about the self-indication assumption (SIA) - that makes the doomsday argument go away, right? Not at all! SIA has no effect on the number of observers expected in the 2014, but increases the expected number of observers in 2013. Thus we still expect that the number of observers in 2014 to be lower than we otherwise thought. There's an SIA doomsday too!
Enter causality
What's going on? SIA was supposed to defeat the doomsday argument! What happens is that I've implicitly cheated - by naming the boxes "2013" and "2014", I've heavily implied that these "boxes" figuratively correspond two subsequent years. But then I've treated them as independent for SIA, like two literal distinct boxes.
In reality, of course, the contents of two years are not independent. Causality connects the two: it's much more likely that there are many observers in 2014, if there were many in 2013. Indeed, most of the observers will exist in both years (and there will be some subtle SSA issues of "observer moments" that we're eliding here - does a future you count as the same observer or a different one). So causality removes the independence assumption, and though there may be some interesting SIA effects on changes in growth rates, we won't see a real SIA doomsday.
Exit causality
But is causality itself a justified assumption? To some extent it is: some people who think they live in 2013/2014 will certainly be in a causal relationship with people who think they live in 2014/2013.
But many will not! What of Boltzmann brains? What of Boltzmann worlds - brief worlds that last less than a year?
What about deluded worlds: worlds where the background data coincidentally (or conspiratorially) imply that we exist on the planet Earth around the Sun, in 2013 - but where we really exist inside a star circling a space-station a few seconds after the seventh toroidal big bang, or something, and will soon wake up to this fact. What about simply deluded people - may we be alien nutjobs, dreaming we're humans? And of great relevance to arguments often presented here, what if we are short-term simulations run by some advanced species?
All these are possible, with non-zero probability (the probability of simulations may be very high indeed, under some assumptions). All of these break the causal link to 2014 or other future events. And hence all of them allow a genuine SIA doomsday argument to flourish: we should expect that seeing 2014 is less likely than is objectively implied, given that we think we are in the year 2013.
Is anthropic decision theory (ADT) subject to the same doomsday argument? In that form, no. In any situation where causality breaks down, your decisions cease to have consequences, and ADT tosses them aside. But more complicated preferences or setups could bring doomsday into ADT as well.