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 it's black - what's the probability the coin was heads?
If you reward disagreement, then what you're really rewarding in this case are mixed (probabilistic) actions. The reward only pays out if the coin landed tails, so that there's someone else to disagree with. So people will give what seems to them to be the same honest answer when you change the result of disagreeing from 0 to 0+epsilon. But when the payoff from disagreeing passes the expected payoff of honesty, agents will pick mixed actions.
To be more precise: If we simplify a little and only let them choose 50/50 if they want to disagree, then we have that the expected utility of honesty is P(heads)U(choice,heads) + P(tails)U(choice,heads), while the expected utility of coin-flipping is pretty much P(heads)U(average,heads) + P(tails)*U(disagree,tails). These will pass each other at different values of U(disagree, tails) depending on that you think P(heads) and P(tails) are, and also depending on which choice you think is best.
I tried to cover what you're talking about with my statement in brackets at the end of the first paragraph. Set the value for disagreeing too high and you're rewarding it, in which case people start deliberately making randomised choices in order to disagree. Too low and they ought to be going out of their way to try and agree above all else - except there's no way to do that in practice, and no way not to do it in the abstract analysis that assumes they think the same. A value of 9 though is actually in between these two cases - it's exactly the average o...
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.