Curses on this problem; I spent the whole day worrying about it, and am now so much of a wreck that the following may or may not make sense. For better or worse, I came to a similar conclusion of Psy-Kosh: that this could work in less anthropic problems. Here's the equivalent I was using:

Imagine Omega has a coin biased so that it comes up the same way nine out of ten times. You know this, but you don't know which way it's biased. Omega allows you to flip the coin once, and asks for your probability that it's biased in favor of heads. The coin comes up heads. You give your probability as 9/10.

Now Omega takes 20 people and puts them in the same situation as in the original problem. It lets each of them flip their coins. Then it goes to each of the people who got tails, and offers $1 to charity for each coin that came up tails, but threatens to steal $3 from charity for each coin that came up heads.

This nonanthropic problem works the same way as the original anthropic problem. If the coin is really biased heads, 18 people will get heads and 2 people will get tails. In this case,the correct subjective probability to assign is definitely 9/10 in favor of whatever result you got; after all, this is the correct probability when you're the only person in the experiment, and just knowing that 19 other people are also participating in the experiment shouldn't change matters.

I don't have a formal answer for why this happens, but I can think of one more example that might throw a little light on it. In another thread, someone mentioned that lottery winners have excellent evidence that they are brains-in-a-vat and that the rest of the world is an illusion being put on by the Dark Lord of the Matrix for their entertainment. After all, if this was true, it wouldn't be too unlikely for them to win the lottery, so for a sufficiently large lottery, the chance of winning it this way exceeds the chance of winning it through luck.

Suppose Bob has won the lottery and so believes himself to be a brain in a vat. And suppose that the evidence for the simulation argument is poor enough that there is no other good reason to believe yourself to be a brain in a vat. Omega goes up to Bob and asks him to take a bet on whether he is a brain in a vat. Bob says he is, he loses, and Omega laughs at him. What did he do wrong? Nothing. Omega was just being mean by specifically asking the one person whom ve knew would get the answer wrong.

Omega's little prank would still work if ve announced ver intention to perform it beforehand. Ve would say "When one of you wins the lottery, I will be asking this person to take a bet whether they are a brain in a vat or not!" Everyone would say "That lottery winner shouldn't accept Omega's bet. We know we're not brains in vats." Then someone wins the lottery, Omega asks if they're a brain in a vat, and they say yes, and Omega laughs at them (note that this also works if we consider a coin with a bias such that it lands the same way 999999 out of a million times, let a million people flip it once, and ask people what they think the coin's bias is, asking the people who get the counter-to-expectations result more often than chance.)

Omega's being equally mean in the original problem. There's a 50% chance ve will go and ask the two out of twenty people who are specifically most likely to be wrong and can't do anything about it. The best course I can think of would be for everyone to swear an oath not to take the offer before they got assigned into rooms.