Thanks for your thoughtful answer.
To achieve magic, we need the ability to merge minds, which can be easily done with programs and doesn’t require anything quantum. If we merge 21 and 1, both will be in the same red room after awakening. If awakening in the red room means getting 100 USD, and the green room means losing it, then the machine will be profitable from the subjective point of view of the mind which enters it. Or we can just turn off 21 without awakening, in which case we will get 1⁄3 and 2⁄3 chances for green and red.
The interesting question here is whether this can be replicated at the quantum level (we know there is a way to get quantum magic in MWI, and it is quantum suicide with money prizes, but I am interested in a more subtle probability shift where all variants remain). If yes, such ability may naturally evolve via quantum Darwinism because it would give an enormous fitness advantage – I will write a separate post about this.
Now the next interesting thing: If I look at the experiment from outside, I will give all three variants 1⁄3, but from inside it will be 1⁄4, 1⁄4, and 1⁄2. The probability distribution is exactly the same as in Sleeping Beauty, and likely both experiments are isomorphic. In the SB experiment, there are two different ways of “copying”: first is the coin and second is awakenings with amnesia, which complicates things.
Identity is indeed confusing. Interestingly, in the art world, path-based identity is used to define identity, that is, the provenance of artworks = history of ownership. Blockchain is also an example of path-based identity. Also, in path-based identity, the Ship of Theseus remains the same.
For the outside view: Imagine that an outside observer uses a fair coin to observe one of two rooms (assuming merging in the red room has happened). They will observe either a red room or a green room, with a copy in each. However, the observer who was copied has different chances of observing the green and red rooms. Even if the outside observer has access to the entire current state of the world (but not the character of mixing of the paths in the past), they can’t determine the copied observer’s subjective chances. This implies that subjective unmeasurable probabilities are real.
Even without merging, an outside observer will observe three rooms with equal 1⁄3 probability for each, while an insider will observe room 1 with 1⁄2 probability. In cases of multiple sequential copying events, the subjective probability for the last copy becomes extremely small, making the difference between outside and inside perspectives significant.
When I spoke about the similarity with the Sleeping Beauty problem, I meant its typical interpretation. It’s an important contribution to recognize that Monday-tails and Tuesday-tails are not independent events.
However, I have an impression that this may result in a paradoxical two-thirder solution: In it, Sleeping Beauty updates only once – recognizing that there are two more chances to be in tails. But she doesn’t update again upon knowing it’s Monday, as Monday-tails and Tuesday-tails are the same event. In that case, despite knowing it’s Monday, she maintains a 2⁄3 credence that she’s in the tails world. This is technically equivalent to the ‘future anthropic shadow’ or anti-doomsday argument – the belief that one is now in the world with the longest possible survival.