My impression is that they have in mind something more than just looking at betting odds (for some unmentioned bet) rather than probabilities. Probability as normally conceived provides a basis for betting choices, using ordinary decision theory, with any sort of betting setup. In Sleeping Beauty with real people, there are no issues that could possibly make ordinary decision theory inapplicable. But it seems the OP has something strange in mind...
My impression is that they have in mind something more than just looking at betting odds (for some unmentioned bet) rather than probabilities. Probability as normally conceived provides a basis for betting choices, using ordinary decision theory, with any sort of betting setup. In Sleeping Beauty with real people, there are no issues that could possibly make ordinary decision theory inapplicable. But it seems the OP has something strange in mind...