Rebuttal: This confuses expected value with probability. The betting strategy is optimal due to the asymmetric nature of the payoffs (betting twice on Tails vs. once on Heads), not because Tails is more likely. The underlying probability of the coin flip remains 50⁄50, regardless of the betting structure.
(This is not a rhetorical question:) What do you mean by “probability” here? A common way of arguing for “having probabilities” is that it’s how you make consistent bets—bets that aren’t obviously leaving utility on the table (e.g. Dutch bookable). But you’re dismissing arguments of the form [I want to bet like this] → [therefore my probabilities should be such and such].
I would think that what we’re learning is that there’s some sort of equivalence principle or something, where it becomes hard to disentangle [I care about my actions in this information-set twice as much] from the allegedly more narrow [This information-set is “truly twice as likely”]. See probutilities.
An answer might be “The world happens to be the case that there pretty strongly tends to be a bunch of stuff that’s external to you, which isn’t correlated with the size of your information-sets (i.e. how many instances of you there are who you can’t distinguish yourself from). That stuff is what we call “reality” and what we have “probabilities” about.”. But that doesn’t seem like a very fundamental notion, and would break down in some cases [citation needed].
This is not a rhetorical question:) What do you mean by “probability” here?
Yeah, since posting this question:
I have updated towards thinking that it’s in a sense not obvious/not clear what exactly “probability” is supposed to be interpreted as here.
And once you pin down an unambiguous interpretation of probability the problem dissolves.
I had a firm notion in mind for what I thought probability meant. But Rafael Harth’s answer really made me unconfident that the notion I had in mind was the right notion of probability for the question.
I think the question is underdefined. Some bets are posed once per instance of you, some bets are posed once per instance of a world (whatever that means), etc.
(This is not a rhetorical question:) What do you mean by “probability” here? A common way of arguing for “having probabilities” is that it’s how you make consistent bets—bets that aren’t obviously leaving utility on the table (e.g. Dutch bookable). But you’re dismissing arguments of the form [I want to bet like this] → [therefore my probabilities should be such and such].
I would think that what we’re learning is that there’s some sort of equivalence principle or something, where it becomes hard to disentangle [I care about my actions in this information-set twice as much] from the allegedly more narrow [This information-set is “truly twice as likely”]. See probutilities.
An answer might be “The world happens to be the case that there pretty strongly tends to be a bunch of stuff that’s external to you, which isn’t correlated with the size of your information-sets (i.e. how many instances of you there are who you can’t distinguish yourself from). That stuff is what we call “reality” and what we have “probabilities” about.”. But that doesn’t seem like a very fundamental notion, and would break down in some cases [citation needed].
Yeah, since posting this question:
I had a firm notion in mind for what I thought probability meant. But Rafael Harth’s answer really made me unconfident that the notion I had in mind was the right notion of probability for the question.
I think the question is underdefined. Some bets are posed once per instance of you, some bets are posed once per instance of a world (whatever that means), etc.