Exactly the same as what you were saying
The reason that's "exactly as I was saying" is because you adjusted a free parameter to fit the problem, after you learned the subjective probabilities. The free parameter was which world to regard as "normal" and which one to apply a correction to. If you already know that the (1/2, 1/4, 1/4) problem is the "normal" one, then you already solved the probability problem and should just maximize expected utility.
Er no - you gave me an underspecified problem. You told me the agents were selfish (good), but then just gave me anthropic probabilities, without giving me the non-anthropic probabilities. I assumed you were meaning to use SSA, and worked back from there. This may have been incorrect - were you assuming SIA? In that case the coin odds are (1/2,1/2) and (2/3,1/3), and ADT would reach different conclusions. But only because the problem was underspecified (giving anthropic probabilities without explaining the theory that goes with them is not specifying the p...
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.