I find this idea very interesting, especially since it seems to me that it gives different probabilities than most other version of halfing. I wonder if you agree with me about how it would answer this scenario (due to Conitzer):
Two coins are tossed on Sunday. The possibilities are
HH: wake wake
HT: wake sleep
TH: sleep wake
TT: sleep sleep
When you wake up (which is not guaranteed now), what probability should you give that the coins come up differently?
According to most versions of halfing, it would be 2⁄3. You could say that when you wake up you learn that you wake up at least once, eliminating TT. Alternatively, you could say that when you wake up the day is selected randomly from all the days you wake up in reality. Either way you get 2⁄3.
However, what if we say that “today” is not selected at all from our perspective? If “today” wasn’t selected at all, it can’t possibly tell us anything about the other day. So it would be 1⁄2 probability that the coins are different.
The weird thing about this is that if we change the situation into:
HH: wake wake
HT: wake sleep
TH: sleep wake
TT: wake wake
Now it seems like we are back to the original sleeping beauty problem, where again we would say 1⁄2 for the probability that the coins are different. How can the probability not change despite TT: sleep sleep turning into TT: wake wake?
And yet, from my own perspective, I could still say that “today” was not selected. So it still gives me no information about whether the other coin is different, and the probability has to stay at 1⁄2.
There is new information in the first scenario, but how does it allow you to update the probability that the coins are different without thinking of today as randomly selected?
Imagine you are woken up every day, but the color of the room may be different. You are asked the probability that the coins are different.
HH: blue blue
HT: blue red
TH: red blue
TT: red red
Now you wake up and see “blue.” That is new information. You now know that there is at least one “blue”, and you can eliminate TT.
However, I think everyone would agree that the probability is still 1⁄2. It was 1⁄2 to begin with, and seeing “blue” as opposed to “red”, while it is new information, is not relevant to deciding the coins are different.
Back to scenario 1:
HH: wake wake
HT: wake sleep
TH: sleep wake
TT: sleep sleep
Now you wake up. That is new information, and you can eliminate TT. But the question is, how is that relevant to the coins being different? If you are treating “today” as randomly selected from all days that exist in reality, then that would allow you to update. But if you are not treating “today” as randomly selected at all, then by what mechanism can you update?
Just going by intuition, I personally don’t think you should update. In this scenario the coin doesn’t need to be tossed until the morning. Heads they wake you up, tails they don’t. So when you wake up, you do get new information just like in the blue/red example. But since the coins are independent of each other, how can learning about that morning’s coin tell you something about the other coin you don’t see? Unless you are using a random selection process in which “today” not primitive.