Initially either 9 or 1 of the 10 people will have been chosen with equal likelihood, meaning I had a 50% chance of being chosen. If being chosen means I should find 90% likelihood that the coin came up tails, then not being chosen should mean I find 90% likelihood that the coin came up heads (which it does). If that were the case, I'd want nay to be what the others choose (0.9 100 + 0.1 1000 = 190 < 700). Since both branches are equally likely (initially), and my decision of what to choose in the branch in which I choose corresponds (presumably) to the others' decision of what to choose in the branch in which I don't choose, I should therefore say nay. You can't win with mere straightforward optimization when your action in 1 branch affects other (no longer reachable) branches.
The source is here. I'll restate the problem in simpler terms:
You are one of a group of 10 people who care about saving African kids. You will all be put in separate rooms, then I will flip a coin. If the coin comes up heads, a random one of you will be designated as the "decider". If it comes up tails, nine of you will be designated as "deciders". Next, I will tell everyone their status, without telling the status of others. Each decider will be asked to say "yea" or "nay". If the coin came up tails and all nine deciders say "yea", I donate $1000 to VillageReach. If the coin came up heads and the sole decider says "yea", I donate only $100. If all deciders say "nay", I donate $700 regardless of the result of the coin toss. If the deciders disagree, I don't donate anything.
First let's work out what joint strategy you should coordinate on beforehand. If everyone pledges to answer "yea" in case they end up as deciders, you get 0.5*1000 + 0.5*100 = 550 expected donation. Pledging to say "nay" gives 700 for sure, so it's the better strategy.
But consider what happens when you're already in your room, and I tell you that you're a decider, and you don't know how many other deciders there are. This gives you new information you didn't know before - no anthropic funny business, just your regular kind of information - so you should do a Bayesian update: the coin is 90% likely to have come up tails. So saying "yea" gives 0.9*1000 + 0.1*100 = 910 expected donation. This looks more attractive than the 700 for "nay", so you decide to go with "yea" after all.
Only one answer can be correct. Which is it and why?
(No points for saying that UDT or reflective consistency forces the first solution. If that's your answer, you must also find the error in the second one.)