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.
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.