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.
Why 90% here? The coin is still fair, and anthropic reasoning should still remain, since you have to take into account the probability of receiving the observation when updating on it. Otherwise you become vulnerable to filtered evidence.
Edit: I take back the sentence on filtered evidence.
Edit 2: So it looks like the 90% probability estimate is actually correct, and the error is in estimating (acausal) consequences of possible decisions.
I don't understand your comment. There is no anthropic reasoning and no filtered evidence involved. Everyone gets told their status, deciders and non-deciders alike.
Imagine I have two sacks of marbles, one containing 9 black and 1 white, the other containing 1 black and 9 white. I flip a fair coin to choose one of the sacks, and offer you to draw a marble from it. Now, if you draw a black marble, you must update to 90% credence that I picked the first sack. This is a very standard problem of probability theory that is completely analogous to the situation in the post, or am I missing something?
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.)