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saturn comments on Solve Psy-Kosh's non-anthropic problem - Less Wrong
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After being told whether they are deciders or not, 9 people will correctly infer the outcome of the coin flip, and 1 person will have been misled and will guess incorrectly. So far so good. The problem is that there is a 50% chance that the one person who is wrong is going to be put in charge of the decision. So even though I have a 90% chance of guessing the state of the coin, the structure of the game prevents me from ever having more than a 50% chance of the better payoff.
eta: Since I know my attempt to choose the better payoff will be thwarted 50% of the time, the statement "saying 'yea' gives 0.9*1000 + 0.1*100 = 910 expected donation" isn't true.
This seems to be the correct answer, but I'm still not sure how to modify my intuitions so I don't get confused by this kind of thing in the future. The key insight is that a group of fully coordinated/identical people (in this case the 9 people who guess the coin outcome) can be effectively treated as one once their situation is permanently identical, right?