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.
Good idea. Though since it's a ratio, you do miss out on a scale factor - In my example, you don't know whether to scale the heads world by 1/3 or the tails world by 3. Or mess with both by factors of 3/7 and 9/7, who knows?
Scaling by the ratio does successfully help you correct if you want to compare options between two worlds - for example, if you know you would pay 1 in the tails world, you now know you would pay 1/3 in the heads world. But if you don't know something along those lines, that missing scale factor seems like it would become an actual problem.
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.