If you look at all 100 copies, the problem disappears.
Before seeing rooms: All have probability estimate of 0.5 that coin was heads.
If the coin lands heads:
All 100 give a probability of 0.5 that the coin landed heads.
If the coin lands tails:
All 100 give a probability of 0.5 that the coin landed heads.
100 of 200 predictions with 0.5 confidence were correct. This is perfect calibration.
If the coin lands heads:
99 of the 100 will see a red room, and give a probability of 0.99 that the coin landed on heads.
1 of the 100 will see a blue room, and give a probability of 0.99 that the coin landed tails.
If the coin lands tails:
99 of the 100 will see a blue room, and give a probability of 0.99 that the coin landed on tails.
1 of the 100 will see a red room, and give a probability of 0.99 that the coin landed heads.
198 of 200 predictions with 0.99 confidence were correct. Once again, this is perfect calibration.
If you are in the 100 rooms situation, you should guess in such a way as to maximize the total accuracy of all the yous.
If you look at all 100 copies, the problem disappears.
Before seeing rooms: All have probability estimate of 0.5 that coin was heads.
If the coin lands heads: All 100 give a probability of 0.5 that the coin landed heads.
If the coin lands tails: All 100 give a probability of 0.5 that the coin landed heads.
100 of 200 predictions with 0.5 confidence were correct. This is perfect calibration.
If the coin lands heads: 99 of the 100 will see a red room, and give a probability of 0.99 that the coin landed on heads. 1 of the 100 will see a blue room, and give a probability of 0.99 that the coin landed tails.
If the coin lands tails: 99 of the 100 will see a blue room, and give a probability of 0.99 that the coin landed on tails. 1 of the 100 will see a red room, and give a probability of 0.99 that the coin landed heads.
198 of 200 predictions with 0.99 confidence were correct. Once again, this is perfect calibration.
If you are in the 100 rooms situation, you should guess in such a way as to maximize the total accuracy of all the yous.
I knew there was an elegant solution to be had, my own reply became pointless. Thank you.