Consider the Sleeping Beauty problem. What do we mean by fair coin? It is meant that the coin will have 50-50 probability of heads or tails. But that is fake. It will ether come up heads or tail, because the real world is deterministic. It is true that I don't know the outcome. I don't know if I am in a world of type "the coin will come up heads" or a world of type "the coin will come up tails". But in this situation I should be allowed to put what ever prior I want on the coins behavior.
Consider the Born rule of quantum mechanics. If I measure the spin of an electron, then I will entangle the large apparatus that is the my measuring equipment with the spin of the electron. We say that there are now two Everett branches, one where the apparatus measured spin up and one where the apparatus measured spin down. Before I read of the result, I don't know which Hilbert branch I am in. I could be in ether, and I should be allowed to have what ever prior I want. So why the Born rule? Why to I do I believe that the square amplitude is the correct way of assigning probability to which Hilbert branch I am in?
I believe in the Born rule because of the frequency of experimental outcomes in the past. The distribution of galaxies in the sky can be traced back to the Born rule. I don't have the gears on what is causing the Born rule, but there are something undeniably real about galaxies that trumps mere philosophical Bayesian arguments about freedom of priors.
Imagine that you are offered a bet. Should you take it or not? There are several argument about what you should do in different situations. For example, if you have finite amount of money, you should maximize the E(log(money)) for each bet, (see e.g. Kelly criterion). However, every such argument I have ever seen, is assuming that you will be confronted by a large number of similar bets. This is because probabilities only relay make sense if you sample enough times from the random distribution you are considering.
The notion of "fair coin" does not make sense if the coin is flipped only once. The right way to view the Sleeping Beauty problem is to view it in it in the context of Repeated Sleeping Beauty.