But if we’re talking about an ordinary Sleeping Beauty problem, there are no repeats—no multiple instances of Beauty with exactly the same memories. Whatever betting scheme may have been defined, when Beauty decides what to bet, her decision is made at a single moment in time, and applies only for that time, affecting her payoff according to whatever the rules of the betting scheme may be. She is allowed to make a different decision at a different time (though of course she may in fact make the same decision), and again that will affect her payoff (or not) according to the rules of the scheme. There is no scope for any unusual relationship between probability and betting odds.
But if we’re talking about an ordinary Sleeping Beauty problem, there are no repeats—no multiple instances of Beauty with exactly the same memories. Whatever betting scheme may have been defined, when Beauty decides what to bet, her decision is made at a single moment in time, and applies only for that time, affecting her payoff according to whatever the rules of the betting scheme may be. She is allowed to make a different decision at a different time (though of course she may in fact make the same decision), and again that will affect her payoff (or not) according to the rules of the scheme. There is no scope for any unusual relationship between probability and betting odds.