This struck an emotional nerve with me, so I'm going to answer as if this were an actual real-life situation, rather than an interesting hypothetical math problem about maximizing expected utility.
IMHO, if this was a situation that occurred in real life, neither of the solutions is correct. This is basically another version of Sophie's Choice. The correct solution would be to punch you in the face for using the lives of children as pawns in your sick game and trying to shift the feelings of guilt onto me, and staying silent. Give the money or not as you see fit, but don't try to saddle me with guilt over lives that YOU chose not to save.
The correct option is refusal to play combined with retaliation for blackmail. If - as with the Nazi officer - this is impossible and one is stuck with choosing from the two options you gave THEN calculation can enter into it. However, future vengeance opportunities should be explored.
It may be better to phrase such problems in terms of game-shows or other less emotionally charged situations. Given a twisted enough game, people will look for ways to pull the rope sideways.
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.)