Ok, thanks, that makes more sense than anything I'd guessed.
There's a difference between shortcutting a calculation and not accounting for something in the first place. In the debate between all the topics mentioned in the paper (e.g. SSI/SSA, split responsibility, precommitments and so on) not one method would give a different answer if that 0 was a 5, a 9, or a -100. It's not because they're shortcutting the maths, it's because, as I said in my first comment, they assume that it's effectively not possible for the two people to vote differently anyway. Which is fine in the abstract, even if it's a little suspect in practice (since this, for once, is a quite realisable experiment).
I'll rephrase my final line then: "If a method says to vote tails, and yet would give the same answer with the 0 changed to a 9, then it is clearly suspect". Incidentally I don't know of a method which says "vote tails" and would give a different answer if you changed the 0 to a 9 either.
I think the reason I didn't get your comment originally is that the first thing I do with this problem is work with the differences - which in this case means subtracting everything from 10 and think in terms of money lost on bad votes, not absolute values. So I wouldn't be multiplying by 0. It's neither better nor worse, just explains why I didn't know what you meant.
Oh, okay. Looks like I didn't really understand your point when I commented :)
Perhaps I still don't - you say "no method gives a probability higher than 3/4 for the coin being tails," but you've in fact been given information that should cause you to update that probability. It's like someone had a bag with 10 balls in it. That person flipped a coin, and if the coin was heads the bag has 9 black balls and 1 white ball, but if the coin was tails the bag has 9 white balls and 1 black ball. They reach into the bag and hand you a ball at random, and ...
A technical report of the Future of Humanity Institute (authored by me), on why anthropic probability isn't enough to reach decisions in anthropic situations. You also have to choose your decision theory, and take into account your altruism towards your copies. And these components can co-vary while leaving your ultimate decision the same - typically, EDT agents using SSA will reach the same decisions as CDT agents using SIA, and altruistic causal agents may decide the same way as selfish evidential agents.
Anthropics: why probability isn't enough
This paper argues that the current treatment of anthropic and self-locating problems over-emphasises the importance of anthropic probabilities, and ignores other relevant and important factors, such as whether the various copies of the agents in question consider that they are acting in a linked fashion and whether they are mutually altruistic towards each other. These issues, generally irrelevant for non-anthropic problems, come to the forefront in anthropic situations and are at least as important as the anthropic probabilities: indeed they can erase the difference between different theories of anthropic probability, or increase their divergence. These help to reinterpret the decisions, rather than probabilities, as the fundamental objects of interest in anthropic problems.