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Vladimir_Nesov comments on Sleeping Beauty gets counterfactually mugged - Less Wrong
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It's a combination of Sleeping Beauty and Counterfactual Mugging, with the decision depending on the resolution of both problems. It doesn't look like the problems interact, but if you are a 1/3-er, you don't give away the money, and if you don't care about the counterfactual, you don't give it away either. You factored out the Sleeping Beauty in your example, and equivalently the Counterfactual Mugging can be factored out by asking the question before the coin toss.
I think it's not quite the Sleeping Beauty problem. That's about the semantics of belief; this is about the semantics of what a "decision" is.
Making a decision to give or not to give means making the decision for both days, and you're aware of that in the scenario. Since the problem requires that Omega can simulate you and predict your answer, you can't be a being that can say yes on one day and no on another day. It would be the same problem if there were no amnesia and he asked you to give him 200 pounds once.
In other words, you don't get to make 2 independent decisions on the two days, so it is incorrect to say you are making decisions on those days. The scenario is incoherent.