I’m going to try and slowly introduce indexicality into the setup, to argue that it’s really there all along and you ought to answer 0.99.
Do you have to be copied? The multiple coexisting copies are a bit of a distraction. Let’s say the setup is that you’re told you are going to be put in the first room and subsequently given a memory-erasing drug; then the second room and so on.
Now one more adjustment: the memory-erasing drug is imperfect; it merely makes your memory of each trial very fuzzy and uncertain. You can’t remember what you saw and deduced each time, but you remember that it happened. In particular, you know that you’re in the 65th room right now, although you have no idea what you saw in the previous 64.
Finally, instead of moving you from the first room to the 100th orderly, let’s say that a random room is chosen for you every time, and this is repeated a million times.
In this case, the answer 0.99 seems inevitable. The choice of the room for you was randomly independent in this particular trial, and brings you fresh legitimate evidence to bayes upon. Note that you can say that you would, with very high probability, see blue in one of the trials regardless of what the coin showed. This doesn’t faze you; if the coin were heads, it’s very unlikely that this particular 65th trial would be one to turn blue on you; and your answer reflects this unlikelyhood.
Now go back to orderly traversing the 100 rooms. This time you can say with certainty that you’d see blue in one of the trials no matter what. But this certainty is only very very slightly more than the very high probability of the previous paragraph; going from 0.99 to 0.5 on the strength of that change is hardly warranted. As before, if the coin were heads, it’s very unlikely that this particular 65th trial would be the one with the blue room; and your answer should reflect this unlikelyhood.
Now go back yet again: they improved the drug and you no longer remember how many trials you had before this one. Should this matter a lot to your response? When you knew this was the 65th trial, you knew nothing about your experience in the previous 64. And it was guaranteed that you’d eventually reach the 65th. It seems that the number of the trial isn’t giving you any real information, and your answer ought to remain unchanged without access to it.
I’m going to try and slowly introduce indexicality into the setup, to argue that it’s really there all along and you ought to answer 0.99.
Do you have to be copied? The multiple coexisting copies are a bit of a distraction. Let’s say the setup is that you’re told you are going to be put in the first room and subsequently given a memory-erasing drug; then the second room and so on.
Now one more adjustment: the memory-erasing drug is imperfect; it merely makes your memory of each trial very fuzzy and uncertain. You can’t remember what you saw and deduced each time, but you remember that it happened. In particular, you know that you’re in the 65th room right now, although you have no idea what you saw in the previous 64.
Finally, instead of moving you from the first room to the 100th orderly, let’s say that a random room is chosen for you every time, and this is repeated a million times.
In this case, the answer 0.99 seems inevitable. The choice of the room for you was randomly independent in this particular trial, and brings you fresh legitimate evidence to bayes upon. Note that you can say that you would, with very high probability, see blue in one of the trials regardless of what the coin showed. This doesn’t faze you; if the coin were heads, it’s very unlikely that this particular 65th trial would be one to turn blue on you; and your answer reflects this unlikelyhood.
Now go back to orderly traversing the 100 rooms. This time you can say with certainty that you’d see blue in one of the trials no matter what. But this certainty is only very very slightly more than the very high probability of the previous paragraph; going from 0.99 to 0.5 on the strength of that change is hardly warranted. As before, if the coin were heads, it’s very unlikely that this particular 65th trial would be the one with the blue room; and your answer should reflect this unlikelyhood.
Now go back yet again: they improved the drug and you no longer remember how many trials you had before this one. Should this matter a lot to your response? When you knew this was the 65th trial, you knew nothing about your experience in the previous 64. And it was guaranteed that you’d eventually reach the 65th. It seems that the number of the trial isn’t giving you any real information, and your answer ought to remain unchanged without access to it.