Okay, so let's say you're given some weak evidence which world you're in - for example, if you're asked the question when you've been awake for 4 hours if the coin was Tails vs. awake for 3.5 hours if Heads. In the Doomsday problem, this would be like learning facts about the earth that would be different if we were about to go extinct vs. if it wasn't (we know lots of these, in fact).
So let's say that your internal chronometer is telling you that if "feels like it's been 4 hours" when you're asked the question, but you're not totally sure - let's say that the only two options are "feels like it's been 4 hours" and "feels like it's been 3.5 hours," and that your internal chronometer is correctly influenced by the world 75% of the time. So P(feels like 4 | heads) = 0.25, P(feels like 3.5 | heads) = 0.75, and vice versa for tails.
A utility-maximizing agent would then make decisions based on P(heads | feels like 4 hours) - but an ADT agent has to do something else. In order to update on the evidence, an ADT agent can just weight the different worlds by the update ratio. For example, if told that the coin is more likely to land heads than tails, an ADT agent successfully updates in favor of heads.
However, what if the update ratio also depended on the anthropic probabilities (that is, SIA vs. SSA)? That would be bad - we couldn't do the same updating thing . If our new probability is P(A|B), Bayes' rule says that's P(A)*P(B|A)/P(B), so the update ratio is P(B|A)/P(B). The numerator is easy - it's just 0.75 or 0.25. Does the denominator, on the other hand, depend on the anthropic probabilities?
If we look at the odds ratios, then P(A|B)/P(¬A|B)=P(A)/P(¬A) * P(B|A)/P(B|¬A). So as long as we have P(B|A) and P(B|¬A), it seems to work exactly as usual.
A few weeks ago at a Seattle LW meetup, we were discussing the Sleeping Beauty problem and the Doomsday argument. We talked about how framing Sleeping Beauty problem as a decision problem basically solves it and then got the idea of using same heuristic on the Doomsday problem. I think you would need to specify more about the Doomsday setup than is usually done to do this.
We didn't spend a lot of time on it, but it got me thinking: Are there papers on trying to gain insight into the Doomsday problem and other anthropic reasoning problems by framing them as decision problems? I'm surprised I haven't seen this approach talked about here before. The idea seems relatively simple, so perhaps there is some major problem that I'm not seeing.