You’ve swapped Monday and Tuesday compared to the usual description of the problem, but other than that, your description is what I am working with. You just have a mistaken intuition regarding how the probabilities relate to decisions—it’s slightly non-obvious (but maybe not obvious that it’s non-obvious). Note that this is all using completely standard probability and decision theory—I’m not doing anything strange here.
In this situation, as explained in detail in my reply above, Beauty gets the right answer regarding how to bet only if she gives probability 1⁄3 to Heads whenever she is woken, in which case she is indifferent to guessing Heads versus Tails (as she should be—as you say, it’s just a coin flip), whereas if she gives probability 1⁄2 to Heads, she will have a definite preference for guessing Heads. If we give guessing Heads a small penalty (say on Monday only, to resolve how this works if her guesses differ on the two days), in order to tip the scales away from indifference, the Thirder Beauty correctly guesses Tails, which does indeed maximizes her expected reward, whereas the Halfer Beauty does the wrong thing by still guessing Heads.
You’ve swapped Monday and Tuesday compared to the usual description of the problem, but other than that, your description is what I am working with. You just have a mistaken intuition regarding how the probabilities relate to decisions—it’s slightly non-obvious (but maybe not obvious that it’s non-obvious). Note that this is all using completely standard probability and decision theory—I’m not doing anything strange here.
In this situation, as explained in detail in my reply above, Beauty gets the right answer regarding how to bet only if she gives probability 1⁄3 to Heads whenever she is woken, in which case she is indifferent to guessing Heads versus Tails (as she should be—as you say, it’s just a coin flip), whereas if she gives probability 1⁄2 to Heads, she will have a definite preference for guessing Heads. If we give guessing Heads a small penalty (say on Monday only, to resolve how this works if her guesses differ on the two days), in order to tip the scales away from indifference, the Thirder Beauty correctly guesses Tails, which does indeed maximizes her expected reward, whereas the Halfer Beauty does the wrong thing by still guessing Heads.