At any given point, you don’t know whether it’s the first or second wakening. The betting argument depends on what the wager is (and more generally, what future experience is being predicted by the probability). If it’s “on wednesday, you’ll be paid $1 if your predicion(s) were correct, and lose $1 if they were incorrect (and voided if somehow there are two wakenings and you make different predictions)”, you should be indifferent to heads or tails as your prediction. If it’s “for each wakening, you’ll win $1 if it’s correct, and lose $1 if incorrrect”, you should NOT be indifferent—you lose twice if you bet heads and are wrong, and the reverse if you bet tails.
If it’s “on wednesday, you’ll be paid $1 if your predicion(s) were correct, and lose $1 if they were incorrect (and voided if somehow there are two wakenings and you make different predictions)”, you should be indifferent to heads or tails as your prediction.
I recommend setting aside around an hour and studying this comment closely.
In particular, you will see that just because the text I quoted from you is true, that is not an argument for believing that the probability of heads is 1⁄2. Halfers are actually those who are NOT indifferent between heads and tails when they are awakened in this setup, they will change their mind about their randomized strategy!
Consider randomized strategies: before the experiment you decide that you will bet Heads with q probability and tails with 1-q. (Before the experiment, both halfers and thirders agree that all qs are equally good)
Thirder wakes up:
Expected value of betting heads: P(Heads)*1$ + P(Tails&Monday)*P(you will bet heads on Tuesday)*(-1$) + P(Tails&Tuesday)*P(you bet heads on Monday)*(-1$) = 1/3*1$ + 1/3*q*(-1$) +1/3*q*(-1$) = 1⁄3 − 2/3*q
Expected value of betting tails: P(Heads)*(-1$) + P(Tails&Monday)*P(you will bet tails on Tuesday)*1$ + P(Tails&Tuesday)*P(you bet tails on Monday)*1$ = 1/3*(-1$) + 1/3*(1-q)*1$ +1/3*(1-q)*1$ = 1⁄3 − 2/3*q
Exactly equal for all q!!!!
Halfer wakes up:
Expected value of betting heads: P(Heads)*1$ + P(Tails&Monday)*P(you will bet heads on Tuesday)*(-1$) + P(Tails&Tuesday)*P(you bet heads on Monday)*(-1$) = 1/2*1$ + 1/4*q*(-1$) +1/4*q*(-1$) = 1⁄2 − 1/2*q
Expected value of betting tails: P(Heads)*(-1$) + P(Tails&Monday)*P(you will bet tails on Tuesday)*1$ + P(Tails&Tuesday)*P(you bet tails on Monday)*1$ = 1/2*(-1$) + 1/4*(1-q)*1$ +1/4*(1-q)*1$ = −1/2*q
For all q halfers believe betting heads has higher expected value and so they are not indifferent between the two. (Because of your example’s payoffs you can’t get positive expected value even with randomized strategies, and so halfers won’t fare worse by departing from their random strategy than thirders do staying with their randomized one, but that’s just a coincidence. See the linked comment for an example where halfers’ false beliefs DO lead them to make worse decisions! (that example has the same structure as yours but with different numbers))
You are already aware of this but, for the benefits of other readers all mention it anyway.
In this post I demonstrate that the narrative of betting arguments validating thirdism is generally wrong and is just a result of the fact that the first and therefore most popular ha;fer model is wrong.
Both thirders and halfers, following the correct model, make the same bets in Sleeping Beauty, though for different reasons. The disagreement is about how to factorize the product of probability of event and utility of event.
And if we investigate a bit deeper, halfer way to do it makes more sense, because its utilities do not shift back and forth during the same iteration of the experiment.
Yes, I basically agree: My above comment is only an argument against the most popular halfer model.
However, in the interest of sparing reader’s time I have to mention that your model doesn’t have a probability for ‘today is Monday’ nor for ‘today is Tuesday’. If they want to see your reasoning for this choice, they should start with the post you linked second instead of the post you linked first.
I’m probably not going to spend an hour on this, but at first glance, it appears that both that comment and yours are making very clear betting arguments. I FULLY agree that the terms and resolution mechanism for the bets (aka the experience definition for the prediction) are the definition of probability, and control what probability Beauty should use.
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?
Not directly, but all probability is betting. Or at least the modeling part is the same, where you define what the prediction is that your probability assessment applies to.
Sleeping beauty problems are interesting because they mess with the number of agents making predictions, and this very much confuses our intuitions. The confusion is in how to aggregate the two wakings (which are framed as independent, but I haven’t seen anyone argue that they’ll ever be different).
I think we all agree that post-amnesia, on Wednesday, you should predict 50% that the experimenter will reveal heads, and you were awoken once, and 50% tails, twice. When woken and you don’t know if it’s Monday or Tuesday, you should acknowledge that on Wednesday you’ll predict 50%. If right now you bet 1⁄3, it’s because you’re predicting something different than you will on Wednesday.
Of course you’re predicting something different. In all cases you’re making a conditional prediction of a state of the world given your epistemic state at the time. Your epistemic state on Wednesday is different from that on Monday or Tuesday. On Tuesday you have a 50% chance of not being asked anything at all due to being asleep, which breaks the symmetry between heads and tails.
By Wednesday the symmetry may have been restored due to the amnesia drug—you may not know whether the awakening you remember was Monday (which would imply heads) or Tuesday (which would imply tails). However, there may be other clues such as feeling extra hungry due to sleeping 30+ hours without eating.
you’re making a conditional prediction of a state of the world given your epistemic state at the time.
I think this is a crux. IMO, you can’t predict the state of the world, since you have no access to that except via your perceptions/experiences. You’re making a prediction of a future epistemic state (aka experience), given (of course) your current epistemic state, conditional on which prediction you make (what will happen if you guess either way, and if you’re right/wrong).
It’s perfectly reasonable to bet 1⁄3 if the reveal/payout is instantaneous and multiple, and to bet 1⁄2 if the reveal/payout is post-merge and singular. Each is correct, for predicting different future experiences.
At any given point, you don’t know whether it’s the first or second wakening. The betting argument depends on what the wager is (and more generally, what future experience is being predicted by the probability). If it’s “on wednesday, you’ll be paid $1 if your predicion(s) were correct, and lose $1 if they were incorrect (and voided if somehow there are two wakenings and you make different predictions)”, you should be indifferent to heads or tails as your prediction. If it’s “for each wakening, you’ll win $1 if it’s correct, and lose $1 if incorrrect”, you should NOT be indifferent—you lose twice if you bet heads and are wrong, and the reverse if you bet tails.
I recommend setting aside around an hour and studying this comment closely.
In particular, you will see that just because the text I quoted from you is true, that is not an argument for believing that the probability of heads is 1⁄2. Halfers are actually those who are NOT indifferent between heads and tails when they are awakened in this setup, they will change their mind about their randomized strategy!
Consider randomized strategies: before the experiment you decide that you will bet Heads with q probability and tails with 1-q. (Before the experiment, both halfers and thirders agree that all qs are equally good)
Thirder wakes up:
Expected value of betting heads: P(Heads)*1$ + P(Tails&Monday)*P(you will bet heads on Tuesday)*(-1$) + P(Tails&Tuesday)*P(you bet heads on Monday)*(-1$) = 1/3*1$ + 1/3*q*(-1$) +1/3*q*(-1$) = 1⁄3 − 2/3*q
Expected value of betting tails: P(Heads)*(-1$) + P(Tails&Monday)*P(you will bet tails on Tuesday)*1$ + P(Tails&Tuesday)*P(you bet tails on Monday)*1$ = 1/3*(-1$) + 1/3*(1-q)*1$ +1/3*(1-q)*1$ = 1⁄3 − 2/3*q
Exactly equal for all q!!!!
Halfer wakes up:
Expected value of betting heads: P(Heads)*1$ + P(Tails&Monday)*P(you will bet heads on Tuesday)*(-1$) + P(Tails&Tuesday)*P(you bet heads on Monday)*(-1$) = 1/2*1$ + 1/4*q*(-1$) +1/4*q*(-1$) = 1⁄2 − 1/2*q
Expected value of betting tails: P(Heads)*(-1$) + P(Tails&Monday)*P(you will bet tails on Tuesday)*1$ + P(Tails&Tuesday)*P(you bet tails on Monday)*1$ = 1/2*(-1$) + 1/4*(1-q)*1$ +1/4*(1-q)*1$ = −1/2*q
For all q halfers believe betting heads has higher expected value and so they are not indifferent between the two. (Because of your example’s payoffs you can’t get positive expected value even with randomized strategies, and so halfers won’t fare worse by departing from their random strategy than thirders do staying with their randomized one, but that’s just a coincidence. See the linked comment for an example where halfers’ false beliefs DO lead them to make worse decisions! (that example has the same structure as yours but with different numbers))
You are already aware of this but, for the benefits of other readers all mention it anyway.
In this post I demonstrate that the narrative of betting arguments validating thirdism is generally wrong and is just a result of the fact that the first and therefore most popular ha;fer model is wrong.
Both thirders and halfers, following the correct model, make the same bets in Sleeping Beauty, though for different reasons. The disagreement is about how to factorize the product of probability of event and utility of event.
And if we investigate a bit deeper, halfer way to do it makes more sense, because its utilities do not shift back and forth during the same iteration of the experiment.
Yes, I basically agree: My above comment is only an argument against the most popular halfer model.
However, in the interest of sparing reader’s time I have to mention that your model doesn’t have a probability for ‘today is Monday’ nor for ‘today is Tuesday’. If they want to see your reasoning for this choice, they should start with the post you linked second instead of the post you linked first.
I’m probably not going to spend an hour on this, but at first glance, it appears that both that comment and yours are making very clear betting arguments. I FULLY agree that the terms and resolution mechanism for the bets (aka the experience definition for the prediction) are the definition of probability, and control what probability Beauty should use.
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?
I didn’t make a betting argument.
Not directly, but all probability is betting. Or at least the modeling part is the same, where you define what the prediction is that your probability assessment applies to.
Sleeping beauty problems are interesting because they mess with the number of agents making predictions, and this very much confuses our intuitions. The confusion is in how to aggregate the two wakings (which are framed as independent, but I haven’t seen anyone argue that they’ll ever be different).
I think we all agree that post-amnesia, on Wednesday, you should predict 50% that the experimenter will reveal heads, and you were awoken once, and 50% tails, twice. When woken and you don’t know if it’s Monday or Tuesday, you should acknowledge that on Wednesday you’ll predict 50%. If right now you bet 1⁄3, it’s because you’re predicting something different than you will on Wednesday.
Of course you’re predicting something different. In all cases you’re making a conditional prediction of a state of the world given your epistemic state at the time. Your epistemic state on Wednesday is different from that on Monday or Tuesday. On Tuesday you have a 50% chance of not being asked anything at all due to being asleep, which breaks the symmetry between heads and tails.
By Wednesday the symmetry may have been restored due to the amnesia drug—you may not know whether the awakening you remember was Monday (which would imply heads) or Tuesday (which would imply tails). However, there may be other clues such as feeling extra hungry due to sleeping 30+ hours without eating.
I think this is a crux. IMO, you can’t predict the state of the world, since you have no access to that except via your perceptions/experiences. You’re making a prediction of a future epistemic state (aka experience), given (of course) your current epistemic state, conditional on which prediction you make (what will happen if you guess either way, and if you’re right/wrong).
It’s perfectly reasonable to bet 1⁄3 if the reveal/payout is instantaneous and multiple, and to bet 1⁄2 if the reveal/payout is post-merge and singular. Each is correct, for predicting different future experiences.