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Even after reading ape's chain of articles, I find this reasoning very unconvincing. Beauty is asked, per awakening, how likely tails is. The obvious answer is 2/3, as Ape (and you) acknowledge through the betting odds. That it is possible to construe some weird betting scheme that restores the original coin toss likelihood is true, but entirely irrelevant, in my view, to the original though experiment; It just transforms it into a different (rather boring) thought experiment, namely: "you toss a coin. Some stuff happens on monday or tuesday but it doesn't matter. It's wednesday now, how likely was the coin to come up heads?". The scheme is deliberately designed so that your awakening doesn't matter anymore, the only thing that matters is that after the summations are applied on wednesday you have to arrive at the original coin toss likelihood. You can of course also construe many betting scheme for various odds once you allow for weighed summation. We can get p=1 by only summing over tuesday, for example. We can also do even more degenerate shenanigans, like explicitly summing only if the coin toss was heads, so the correct bet would become p=0. The original question was still, however, per awakening.
The technicolor problem doesn't change this, either (though I agree it's interesting, so still thanks for the link!).
That is rather the point, yeah. The goal is to show that the probabilities you use to guide your decision should be based on how that decision will be used.
Let's say Sleeping Beauty is actually a mind upload, and if the coin comes up heads I will run two copies of her and only use her answer if the two copies match (but the hardware is very good and the two copies will match 99.999% of the time), and if the coin comes up tails I will only run one copy. Halfer or thirder?
How about if, in the heads case, instead of running two independent copies of her entire mind, I run two independent copies of each neuron's computations, and at each step, if there's a mismatch, I run a third copy as a tiebreaker (but mismatches are incredibly rare). Halfer or thirder?
Actually it turns out I'm just using a less efficient algorithm if the coin came up heads which happens to use twice as much compute. Halfer or thirder?
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