I think you’re confusing preferences about the world and preferences about an un-observable cause. As an altruist, Alice cares about Bob’s preferences whether to have a dollar or not. Bob has no way of having knowledge of (or a preference over) Alice’s prediction, and she knows it, so she’d be an idiot to project that onto her choice. If she thinks Bob may be right, then she updated her probability estimate, in contradiction of the story.
options 2 and 3 are twice as likely to lose than option 1. This is what it means for Alice to have the belief that there is a 2⁄3 chance of heads.
I think you’re confusing preferences about the world and preferences about an un-observable cause. As an altruist, Alice cares about Bob’s preferences whether to have a dollar or not. Bob has no way of having knowledge of (or a preference over) Alice’s prediction, and she knows it, so she’d be an idiot to project that onto her choice. If she thinks Bob may be right, then she updated her probability estimate, in contradiction of the story.
options 2 and 3 are twice as likely to lose than option 1. This is what it means for Alice to have the belief that there is a 2⁄3 chance of heads.