I'm not trying to be mysterious. As far as I can see, there is a distinction. The expected value of switching to Yea from your point of view is affected by whether or not you care about the kids in the branches you are not yourself in.
After being told your status, you're split:
1/20 are U=Decider, U=Heads. Yea is very bad here.
9/20 are U=Decider, U=Tails. Yea is good here.
9/20 are U=Passenger, U=Heads. Yea is very bad here.
1/20 are U=Passenger, U=Tails. Yea is good here.
After being told your status, the new information changes the expected values across the set of branches you could now be in, because that set has changed. It is now only the first 2 lines, above, and is heavily weighted towards Yea = good, so for the kids in your own branches, Yea wins.
But the other branches still exist. If all deciders must come to the same decision (see above), then the expected value of Yea is lower than Nay as long as you care about the kids in branches you're not in yourself - Nay wins. If fact, this expected value is exactly what it was before you had the new information about which branches you can now be in yourself.
Okay. You're bringing up quantum mechanics needlessly, though. This is exactly the same reasoning as cousin it went through in the post, and leads to exactly the same problem, since everyone can be expected to reason like you. If yea is only said because it generates better results, and you always switch to yea, then QED always saying yea should have better results. But it doesn't!
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.)