That’s all perfectly true, but compare her strategy in this experiment to, say, an ordinary bet at 2:1 odds. If Beauty bets $10 on heads, she will either win $20 or lose $20 with equal likelihood over the course of the experiment—but if she bets $10 on an ordinary one-in-three chance, she will either win $20 or lose $10, with losing $10 being twice as likely. Mere risk aversion would make these two options different.
I’ll concede that, of the two options, 1⁄3 probably makes more sense to describe her credence, but it’s not sufficient to describe the variables she must account for.
I agree but don’t think it’s necessary to talk about risk at all (except to say that we wish to ignore it) for the purpose of the hypothetical bets an agent should make given a certain credence. I also think you confused the direction of the odds; if I believe something is 2⁄3 likely, I should take the positive side if I can gain anything more than half of what I stand to lose if the negative occurs (with p=1/3). But of course that doesn’t change the interesting difference you point out (that the bet involves a $40 swing rather than a $30 one).
That’s all perfectly true, but compare her strategy in this experiment to, say, an ordinary bet at 2:1 odds. If Beauty bets $10 on heads, she will either win $20 or lose $20 with equal likelihood over the course of the experiment—but if she bets $10 on an ordinary one-in-three chance, she will either win $20 or lose $10, with losing $10 being twice as likely. Mere risk aversion would make these two options different.
I’ll concede that, of the two options, 1⁄3 probably makes more sense to describe her credence, but it’s not sufficient to describe the variables she must account for.
I agree but don’t think it’s necessary to talk about risk at all (except to say that we wish to ignore it) for the purpose of the hypothetical bets an agent should make given a certain credence. I also think you confused the direction of the odds; if I believe something is 2⁄3 likely, I should take the positive side if I can gain anything more than half of what I stand to lose if the negative occurs (with p=1/3). But of course that doesn’t change the interesting difference you point out (that the bet involves a $40 swing rather than a $30 one).
Agreed. I have indicated a change of opinion at my original comment.