But do you also agree that there isn’t any kind of bet with any terms or resolution mechanism which supports the halfer probabilities? While you did not say it explicitly, your comment’s structure seems to imply that one of the bet structure you gave (the one I’ve quoted) supports the halfer side. My comment is an analysis showing that that’s not true (which was apriori pretty surprising to me).
I’m not sure where the error is, but I love that you’ve shown how thirders are the true halfers!
Oh, it may also be because the randomization interacts with the “if they don’t match, bets are off” stipulation, which was intended to acknowledge that the wakings on monday and tuesday are identical, to Beauty. It turns out that disfavors tails, which is the only opportunity for a mismatch. The fix is to either disallow randomization, or to say that “if there are two wagers which disagree, we’ll randomize between them as to which is binding”.
Amusingly, this means that both halfer and thirder are indifferent between heads and tails. Making it very clear that it’s an incomplete question. In fact, I don’t mean to “support the halfer side”, I mean that having a side, without specifying precisely what future experience(s) are being predicted, is incorrect.
Thank you for making it further clear that the problem is deeply rooted in intuitions of identity and the confusion between there being one entity on Sunday, one OR two on Monday, and one again on Wednesday. I do think that it’s purely a modeling choice whether to consider heads&tuesday to be 0.25 probability that just doesn’t happen to have Beauty awake in it, or whether to distribute that probability among the others.
I’m not sure where the error is in your calculations (I suspect in double-counting tuesday, or forgetting that tuesday happens even if not woken up, so it still gets it’s “matches Monday bet” payout), but I love that you’ve shown how thirders are the true halfers!
To be precise, I’ve shown that in a given betting structure (which is commonly used as an argument for the halfer side even if you didn’t use it that way now) using thirder probabilities leads to correct behaviour. In fact my belief is that in ANY kind of setup using thirder probabilities leads to correct behaviour, while using the halfer probabilities leads to worse or equivalent results. I wouldn’t characterize this as ″thirders are the true halfers!’. I disagree that there is a mistake, is the only reason you think there is a mistake that the result of the calculation disagrees with your prior belief?
I don’t mean to “support the halfer side”, I mean that having a side, without specifying precisely what future experience(s) are being predicted, is incorrect.
But if every reasonable way to specify precisely what future experiences are being predicted gives the same set of probabilities, couldn’t we say that one side is correct?
But do you also agree that there isn’t any kind of bet with any terms or resolution mechanism which supports the halfer probabilities? While you did not say it explicitly, your comment’s structure seems to imply that one of the bet structure you gave (the one I’ve quoted) supports the halfer side. My comment is an analysis showing that that’s not true (which was apriori pretty surprising to me).
I’m not sure where the error is, but I love that you’ve shown how thirders are the true halfers!
Oh, it may also be because the randomization interacts with the “if they don’t match, bets are off” stipulation, which was intended to acknowledge that the wakings on monday and tuesday are identical, to Beauty. It turns out that disfavors tails, which is the only opportunity for a mismatch. The fix is to either disallow randomization, or to say that “if there are two wagers which disagree, we’ll randomize between them as to which is binding”.
Amusingly, this means that both halfer and thirder are indifferent between heads and tails. Making it very clear that it’s an incomplete question. In fact, I don’t mean to “support the halfer side”, I mean that having a side, without specifying precisely what future experience(s) are being predicted, is incorrect.
Thank you for making it further clear that the problem is deeply rooted in intuitions of identity and the confusion between there being one entity on Sunday, one OR two on Monday, and one again on Wednesday. I do think that it’s purely a modeling choice whether to consider heads&tuesday to be 0.25 probability that just doesn’t happen to have Beauty awake in it, or whether to distribute that probability among the others.
To be precise, I’ve shown that in a given betting structure (which is commonly used as an argument for the halfer side even if you didn’t use it that way now) using thirder probabilities leads to correct behaviour. In fact my belief is that in ANY kind of setup using thirder probabilities leads to correct behaviour, while using the halfer probabilities leads to worse or equivalent results. I wouldn’t characterize this as ″thirders are the true halfers!’. I disagree that there is a mistake, is the only reason you think there is a mistake that the result of the calculation disagrees with your prior belief?
But if every reasonable way to specify precisely what future experiences are being predicted gives the same set of probabilities, couldn’t we say that one side is correct?