To me adding a bet fundamentally changes the question. Canonically, the question is phrased based on credence, maximizing a bet isn't the same as being rational about reality. Asking for a bet (odds) is not the same as asking for probability of the coin flip(credence).
For the same reason that I know that I may die in the next 5 years but simultaneously be unwilling to take that as a nontransferable bet at even the most generous odds, Sleeping Beauty should believe that the coin flip was 50/50 but be unwilling take a bet of the same odds.
So, using deductive reasoning to answer the question: A fair coin has a 50/50 value, a fair coin was flipped, Sleeping Beauty gains no new information, therefore Sleeping Beauty should respond that she believes there is a 50% chance of Heads. A bet on each day changes to formulation, there are stakes that need to maximized and so now we are considering the odds: Using the credence that a fair coin is fair and given the set of outcomes from the initial condition of the problem, we can apply statistics and say that the odds are 1/3 that the coin flip resulted in heads.
A more rigorous argument can be found in this paper:
When Betting Odds and Credences Come Apart: More Worries for Dutch Book Arguments Darren Bradley and Hannes Leitgeb
To me adding a bet fundamentally changes the question. Canonically, the question is phrased based on credence, maximizing a bet isn't the same as being rational about reality. Asking for a bet (odds) is not the same as asking for probability of the coin flip(credence).
For the same reason that I know that I may die in the next 5 years but simultaneously be unwilling to take that as a nontransferable bet at even the most generous odds, Sleeping Beauty should believe that the coin flip was 50/50 but be unwilling take a bet of the same odds.
So, using deductive reasoning to answer the question: A fair coin has a 50/50 value, a fair coin was flipped, Sleeping Beauty gains no new information, therefore Sleeping Beauty should respond that she believes there is a 50% chance of Heads. A bet on each day changes to formulation, there are stakes that need to maximized and so now we are considering the odds: Using the credence that a fair coin is fair and given the set of outcomes from the initial condition of the problem, we can apply statistics and say that the odds are 1/3 that the coin flip resulted in heads.
A more rigorous argument can be found in this paper:
Darren Bradley and Hannes Leitgeb