All this problem illustrates is that one needs to either (1) really pay attention to information contained in an observation, or (2) be really clear about how you're sampling.

If you have a big pile of universes with their full timeline available to you, and you teleport the soul of Sleeping Beauty into a just-awakening body, uniformly at random, then the chance of heads is 1/3 and the chance of tails is 2/3, because you pick the "tails" branch twice as often by construction. The Bayesian reasoning implicitly imports this view by treating each waking-up instance as equivalent. But it's got nothing to do with Bayesian logic vs frequentist: I just gave the equivalent frequentist interpretation above.

If, in contrast, you take your big pile of universes and you teleport the soul of Sleeping Beauty in at the beginning of the experiment, then the chance of being in the heads branch is 1/2 and the tails branch is 1/2. SB souls who uniformly pick "heads" end up emitting the correct answer (once) half the time, and the wrong answer (but twice) half the time. SB souls who uniformly pick "tails" again are 50/50 but in the branch where they're right, they give the right answer twice, and when they're wrong they only give it once.

If SB gets a two-point payoff for saying the right thing twice, her expected winnings are 1 if she consistently chooses tails but only 1/2 if she consistently chooses heads. If that's the payoff matrix, sure, go for tails! On the other hand, if she gets one point if she's never wrong, and zero if she ever is, then her payoff is equal (1/2 point expectation) with both consistent strategies. (A uniformly random strategy gives her only a 3/8 point expectation and is strictly worse.) Alternatively, if she gets a point if she's ever right, the two strategies also give a 1/2 point expectation (but the random strategy gives her a 5/8 point expectation and is strictly better.)

I've done this from the perspective of sampling many worlds, but you can also use the straightforward observational approach as a Bayesian.

Let Ht be the hypothesis that I'm in a Tails World. Let Hh be the hypothesis that I'm in a Heads World. I wake up! Then P(wake up|Ht) = 1, because in a tails world of course we do wake up. P(wake up|Hh) = 1 also, because in a heads world we also wake up. We don't observe Monday or Tuesday; if we wanted to conclude something about that, that'd have to be part of our hypothesis. So, using p(H|e) = p(e|H)p(H)/p(e) where p(e) is the chance of the evidence and p(H) is the prior chance of the hypothesis, we get p(Ht|wake up) = 1*0.5/1 = 0.5 = p(Ht), because you always wake up. So observing that we wake up doesn't actually add any information, and doesn't change any probabilities! And "was it heads or tails" is strictly a matter of asking about whether we have more evidence to believe Heads World or Tails World hypotheses.

You see, it was your uniform prior on the hypotheses {Heads World Monday, Tails World Monday, Tails World Tuesday} the whole time. You just did a little math to read out your prior, but it was your prior that gave you the answer.

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This is a good exercise for a thoughtful student of statistics, but it is an exercise in assumptions about sampling and/or carefully structuring the statement of your hypotheses.