the sleeping beauty must decide beforehand
You've changed the problem to suit your answer.
Yes, but to be clear, my 'answer' is that theres' no universal right answer: whenever the question asked is about the single anthropic position, thirders are correct, when it's about the global structure of the branch they're in, halfers' is the correct answer.
The argument would have carried through even if I had not destroyed the symmetry between the branches, because in that case it would have been that both positions would have won on average, and so there was no 'obviously' correct answer, but I think this way is clearer, because in the Monty Hall version one of the branch gets almost all probability mass.
A friend referred me to another paper on the Sleeping Beauty problem. It comes down on the side of the halfers.
I didn't have the patience to finish it, because I think SB is a pointless argument about what "belief" means. If, instead of asking Sleeping Beauty about her "subjective probability", you asked her to place a bet, or take some action, everyone could agree what the best answer was. That it perplexes people is a sign that they're talking non-sense, using words without agreeing on their meanings.
But, we can make it more obvious what the argument is about by using a trick that works with the Monty Hall problem: Add more doors. By doors I mean days.
The Monty Hall Sleeping Beauty Problem is then:
The halfer position implies that she should still say 1/2 in this scenario.
Does stating it this way make it clearer what the argument is about?