She responds "I'm sorry, but while I am a highly skilled mathematician, I'm actually from an alternate universe which is identical to yours except that in mine 'subjective probability' is the name of a particularly delicious ice cream flavor. Please precisely define what you mean by 'subjective probability', preferably by describing in detail a payoff structure such that my winnings will be maximized by selecting the correct answer to your query."
I agree with that response to the sleeping beauty problem, and the way you set up the payoff structure will probably make this problem equivalent to the St. Petersburg Paradox.
I got into a heated debate a couple days ago with some of my (math grad student) colleagues about the Sleeping Beauty Problem. Out of this discussion came the following thought experiment:
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: She will be put to sleep. During the experiment, Beauty will be wakened, interviewed, and put back to sleep with an amnesia-inducing anti-aging drug that makes her forget that awakening. A fair coin will be tossed until it comes up heads to determine which experimental procedure to undertake: if the coin takes n flips to come up heads, Beauty will be wakened and interviewed exactly 3^n times. Any time Sleeping Beauty is wakened and interviewed, she is asked, "What is your subjective probability now that the coin was flipped an even number of times?"
I will defer my analysis to the comments.