This would be more to my taste (I can’t speak for anyone else’s) if we were told more about the space of possible unfairnesses. In particular, it wasn’t clear to me whether
the coin flips are allowed to depend on our predictions (and, if so, which predictions)
nor whether
we were actually looking for a genuinely probabilistic rule (in which case it would have to be a very simple one for there to be any chance of guessing it) or for a deterministic one, possibly depending on the predictions (in which case it might be more complicated).
Again, I don’t claim that my taste is anyone else’s; but my reaction to the extreme open-endedness is along the lines of “this could be practically anything, and some varieties of thing-it-could-be are obviously not deducible with any confidence from 50 bits of information, and this has probably been designed so that it’s solvable if you correctly guess what space of possibilities the creator had in mind but I don’t feel like trying to read his mind”.
The spoiler-blocks above aren’t very spoilery since they are just asking questions. But for the benefit of anyone who feels the same way as I do, here are what I now believe to be the answers to those questions:
The coin does not know anything about your predictions; you are trying to model a thing that autonomously emits coin flips, not something that actively responds to your probing. Unsurprisingly-given-that, the coin is (at least potentially) probabilistic.
This would be more to my taste (I can’t speak for anyone else’s) if we were told more about the space of possible unfairnesses. In particular, it wasn’t clear to me whether
the coin flips are allowed to depend on our predictions (and, if so, which predictions)
nor whether
we were actually looking for a genuinely probabilistic rule (in which case it would have to be a very simple one for there to be any chance of guessing it) or for a deterministic one, possibly depending on the predictions (in which case it might be more complicated).
Again, I don’t claim that my taste is anyone else’s; but my reaction to the extreme open-endedness is along the lines of “this could be practically anything, and some varieties of thing-it-could-be are obviously not deducible with any confidence from 50 bits of information, and this has probably been designed so that it’s solvable if you correctly guess what space of possibilities the creator had in mind but I don’t feel like trying to read his mind”.
The spoiler-blocks above aren’t very spoilery since they are just asking questions. But for the benefit of anyone who feels the same way as I do, here are what I now believe to be the answers to those questions:
The coin does not know anything about your predictions; you are trying to model a thing that autonomously emits coin flips, not something that actively responds to your probing. Unsurprisingly-given-that, the coin is (at least potentially) probabilistic.