This problem has a neat symmetry in that the players are copies of you; so all copies finding themselves in blue rooms will assign p(tails | blue)=x, and conversely all copies finding themselves in red rooms will assign p(tails | red)=1-x.
This way (outside view), the problem reduces to finding what x gives the best sum log-score for all observers.
Running the numbers gets x=99/100, but this problem has way too much symmetry to attribute this to any particular explanation. Basically, this is faul_sname’s reasoning with more math.
This problem has a neat symmetry in that the players are copies of you; so all copies finding themselves in blue rooms will assign p(tails | blue)=x, and conversely all copies finding themselves in red rooms will assign p(tails | red)=1-x. This way (outside view), the problem reduces to finding what x gives the best sum log-score for all observers. Running the numbers gets x=99/100, but this problem has way too much symmetry to attribute this to any particular explanation. Basically, this is faul_sname’s reasoning with more math.