Sleeping Beauty and other anthropic problems considered in terms of bets illustrate how most ways of assigning anthropic probabilities are not about beliefs of fact in a general sense, their use is more of appeal to consequences. At the very least the betting setup should remain a salient companion to these probabilities whenever they are produced. Anthropic probabilities make no more sense on their own, without the utilities, than whatever arbitrary numbers you get after applying Bolker-Jeffrey rotation. The main difference is in how the utilities of anthropics are not as arbitrary, so failing to carefully discuss what they are in a given setup makes the whole construction ill-justified.
Sleeping Beauty and other anthropic problems considered in terms of bets illustrate how most ways of assigning anthropic probabilities are not about beliefs of fact in a general sense, their use is more of appeal to consequences. At the very least the betting setup should remain a salient companion to these probabilities whenever they are produced. Anthropic probabilities make no more sense on their own, without the utilities, than whatever arbitrary numbers you get after applying Bolker-Jeffrey rotation. The main difference is in how the utilities of anthropics are not as arbitrary, so failing to carefully discuss what they are in a given setup makes the whole construction ill-justified.