The big problem for halfers is that in the iterated version of sleeping beauty, they agree thirding is sensible (for in the limit, 2/3 of coinflips are created by tails coinflips). So then avoiding the money pumps for thirders is everyone's problem--halfers have precisely the same problem as thirders to, so long as the scenario is iterated.
Indeed, this makes the halfer money pumps flip in favor of thirding. If halfers go the EDT route and suggest you ought to bet at twice the odds you really believe the probabilities to be, then they end up with irrational and money-pumpable betting odds of 1/5 in the iterated version of the sleeping beauty problem.
For that reason, I think the standard halfer money pump backfires. In addition, both single and double halfing have their own money pumps.
I couldn't agree more with most of this & it does get messy. I think I'd be wary of arguing that EDT halfers don't escape dutch books, because I don't think one is required to have a single principled heuristic. The EDT halfer could adjust and avoid the 1/5 scenario. (Who says they have to update to 1/3? Who says they can't throw out their heuristic once it no longer makes sense?) Regardless, calling oneself an EDT halfer is really just a veneer, they are admitting reality reflects something different from their beliefs. I don't see how they aren't SIA in everything but name. It's not like SIA people don't know that a coin flip is 50/50.
I'm afraid you are making a common mistake while discussing Sleeping Beauty problem: not distinguishing between single and double halfism. This is crucial because while single halfism is quite stupid, double halfism is, in fact, the correct answer.
I saw one of your posts on this while writing my follow-up post! It is kind of you to read this as it's really more introductory on the decision theory space and meant to focus more on getting people to think about how seemingly innocuous assumptions affect their beliefs.
Single halfism is indeed stupid. (75% chance it's Monday is obviously false.) I tend to think double halfism is also wrong. (I originally also took it to imply Tuesday does not exist. There I go, building a sample space again.)
I find your arguments about mutual exclusivity here very compelling. I think it does end up implying that Sleeping Beauty cannot have any credence at all about whether it's Monday or Tuesday, which I find to be subject to dutch books (if she is asked to bet on which day it is). You might argue that it is possible to evaluate the bet by computing expected utility across the whole experiment without assigning P(Monday), but I think this is just thirding with extra steps.
The Technicolor argument is also persuasive. However, I suspect the halfer has no principled reason to update on color if they don't update on awakening and the thirder doesn't need to update on color because we've already correctly accounted for the probability. The halfer updating on color is just exploiting a strategy that allows them to see the probability as it is. The thirder could use the same strategy to the same effect. I could be wrong here.
Also –– I don't think anything about SIA requires treating awakenings as independent/not acknowledging the tails awakenings to be sequential.
> I think it does end up implying that Sleeping Beauty cannot have any credence at all about whether it's Monday or Tuesday
Yes, completely correct. Under amnesia and repetition of the same experience the notion of "Today" becomes ill-defined as it points to both days in the trials where the coin is Tails.
It doesn't mean that P(Monday) is not defined, mind you. It means that instead of ill-defined "Monday is Today", where it's not clear what exactly we mean by "Today" it is properly defined "Monday awakening happens during this iteration of probability experiment". Probability of this event is 1. Likewise P(Tuesday) =1/2
> I find to be subject to dutch books (if she is asked to bet on which day it is)
No dutch book here. She simply bets based on probability of Monday and Tuesday awakening happening during the experiment.
P(Monday) = 1
P(Tuesday) = 1/2
Therefore the betting odds are
2:1
> I think this is just thirding with extra steps.
I agree that if you start from thirders assumptions it seems this way. But, strictly speaking, this is thirdism with *less* steps.
More specifically the step which is not required is accepting the framework of centred possible worlds which disregards the fact that sample space has to be constructed from mutually exclusive outcomes of the experiment, leading to utility shifts during the same trial and all the standard anthropic paradoxes.
> The Technicolor argument is also persuasive. However, I suspect the halfer has no principled reason to update on color if they don't update on awakening
The principled reason for update in general is the ability to distinguish between trials where a event C is realized and all the other trials of probability experiment. Initially we are uncertain between all the possible iterations of the probability experiment. And have our unconditional probabilities for every event:
P(E1), P(E2)... P(En)
Then we are uncertain between all the iterations of conditional probability experiment - *where event C is realized*. The ratio of realization of the events E1... En can be different between the unconditional and conditional probability experiments. So now we have our credences:
The reason why double halfers do not update their probabilities on awakening is because Beauty awakens in every trial of the experiment.
P(Awake) = 1
So the fact of awakening doesn't allow to distinguish between the trials where coin is Heads and where it's Tails. Conditional probability experiment equals the unconditional therefore:
P(Heads) = P(Heads|Awake)
P(Tails) = P(Tails|Awake)
With Technicolor Sleeping Beauty there are two cases: where we can distinguish the iterations where a particular color is shown and where we can't.
If we just go into the experiment unprepared and see that the room is some color - Blue or Red this indeed doesn't give us anything as in every iteration of the experiment event Blue or Red is realized:
P(Blue or Red) = 1
However, if we prepared ourself to pay attention to a specific color before the experiment started, for instance, Blue, we can distinguish between iterations. The room is Blue in only 75% of iteration - in half of the iterations where the coin is Heads and in every iteration where it's Tails.
P(Blue) = 3/4
P(Blue|Heads) = 1/2
P(Blue|Tails) = 1
Therefore
P(Heads|Blue) = 1/3
> thirder doesn't need to update on color because we've already correctly accounted for the probability.
No, that's the whole point. Thirder's think that they've already updated and so there is no difference whether the room is Blue or Red. For them Technicolor Sleeping Beauty is just isomorphic to regular Sleeping Beauty problem. As a result thirders miss the opportunity to win extra money in technicolor, that double halfers take.
Consider: you are about to be subject to multiple iterations of Technicolor Sleeping Beaty experiment. Your goal is to guess iterations of the experiment where the coin came Tails, which you can do with a button. Press it whenever you think the coin is Tails - every press of the button costs you 100 dollar, however you get 160 dollars per iteration of experiment (not awakening!) when you've pressed the button at least once, and the coin happened to be Tails.
A thirder would just be annoyed that the scoring rule was designed by a halfer and would not see any reason to press a button. Thirder P(Tails) = 2/3 means that the coin is Tails in about 2/3 of awakenings, not iterations of experiment! Of course they can't guess it better than chance if the scoring rule is *unfair*.
A double halfer, meanwhile, would on average become 50 dollars richer per Tails iteration of experiment.
> I don't think anything about SIA requires treating awakenings as independent/not acknowledging the tails awakenings to be sequential.
It all goes back to the formal definition of sample space - that it has to consist of mutually exclusives outcomes of the experiment. As thirder's sample space does not consist of mutually exclusive outcomes of the experiment it's not sound.
Some thirders may try to circumvent this by claiming that individual awakenings should themselves be considered trials of a random sample among three options: Heads&Monday, Tails&Monday, Tails&Tuesday. This, however, also doesn't work because individual trials has to be independent from each other, while in Sleeping Beauty there is strict order between awakenings.
Four volunteers, named SB1, SB2, SB3, and SB4, participate in a two day experiment. Each is assigned a day from (Mon, Tue) and a coin face from (H, T) such that each has a different combination. On Sunday night, all are put to sleep and a fair coin is flipped. On Monday, and again on Tuesday, three are awakened. The one who is left asleep is the one whose assignment matches the current day and the coin flip.
Each of the awake volunteers is asked for the [probability/credence/degree of belief/whatever word you want] that their coin assignment matches the coin.
"I think halfers’ arguments are subject to Dutch books, i.e. in the betting version, they will lose money consistently."
This is true, but so are thirders' arguments. What if I set up the bet differently? So every time you wake up you get to say "bet" or "don't bet". But the rule is that I only take into account your final answer in deciding whether to bet the $10. Now, the right odds to take are 1:1. If I instead offer you a bet, and I say it's counted as half if you bet before, then the right odds are 2:3, 40%. I can get any value between 1/3 and 1/2 by setting up the bet the right way.
Another way to see this is that if I bet with you right after flipping the coin but also every time you wake up, I can make money off you. After flipping the fair coin the odds are half, but when being woken up it's a third. So I can just Dutch book you already by exploiting the information asymmetry here.
I don't think I'm either a thirder or a halfer since I don't think every event can be meaningfully given a probability. Let S be a set with outer measure 1/2 and inner measure 1/3. If I take random element in [0,1), what's the probability it lies in S? Probability is a measure. There are no non-measurable sets in Sleeping Beauty sure, but there is a uniform measure over the universes induced by the coin (halfer), and a uniform measure over the states of consciousness you experience when you wake up (thirder).
1/2 position can't be Dutch Booked; rewarding people for each waking guess will obviously multiply the winnings for guessing in the direction of more winnings, which is something a 1/2er can easily account for.
The 1/3 position, however...
Let's say the coin is no longer a fair coin. Now it’s a magical coin that flips Heads one million times more often than it flips Tails.
If the coin lands Tails, you’ll be woken up a billion billion times.
If the coin lands Heads, you’ll be woken up only once.
Each time you wake up, you will guess the coin flip result. If you ever guess wrong, you will be killed after all the wakings are over. What should you guess?
Guessing Tails will result with you dead 99.9999% of the time.
Nice post! Incidentally, I think that money pump arguments are a bit messy but once you work through them all, they decisively favor SIA https://benthams.substack.com/p/money-pump-arguments-destroy-non?utm_source=publication-search.
The big problem for halfers is that in the iterated version of sleeping beauty, they agree thirding is sensible (for in the limit, 2/3 of coinflips are created by tails coinflips). So then avoiding the money pumps for thirders is everyone's problem--halfers have precisely the same problem as thirders to, so long as the scenario is iterated.
Indeed, this makes the halfer money pumps flip in favor of thirding. If halfers go the EDT route and suggest you ought to bet at twice the odds you really believe the probabilities to be, then they end up with irrational and money-pumpable betting odds of 1/5 in the iterated version of the sleeping beauty problem.
For that reason, I think the standard halfer money pump backfires. In addition, both single and double halfing have their own money pumps.
I couldn't agree more with most of this & it does get messy. I think I'd be wary of arguing that EDT halfers don't escape dutch books, because I don't think one is required to have a single principled heuristic. The EDT halfer could adjust and avoid the 1/5 scenario. (Who says they have to update to 1/3? Who says they can't throw out their heuristic once it no longer makes sense?) Regardless, calling oneself an EDT halfer is really just a veneer, they are admitting reality reflects something different from their beliefs. I don't see how they aren't SIA in everything but name. It's not like SIA people don't know that a coin flip is 50/50.
I'm afraid you are making a common mistake while discussing Sleeping Beauty problem: not distinguishing between single and double halfism. This is crucial because while single halfism is quite stupid, double halfism is, in fact, the correct answer.
Here you may read about the double halfer's model for sleeping beauty: https://www.lesswrong.com/posts/gwfgFwrrYnDpcF4JP/the-solution-to-sleeping-beauty
Previous and next posts are also relevant if you want to dig deeper in the topic
As a nice bonus, it isn't presumptious in cosmology and simulations while also not predicting doom in doomsday.
I saw one of your posts on this while writing my follow-up post! It is kind of you to read this as it's really more introductory on the decision theory space and meant to focus more on getting people to think about how seemingly innocuous assumptions affect their beliefs.
Single halfism is indeed stupid. (75% chance it's Monday is obviously false.) I tend to think double halfism is also wrong. (I originally also took it to imply Tuesday does not exist. There I go, building a sample space again.)
I find your arguments about mutual exclusivity here very compelling. I think it does end up implying that Sleeping Beauty cannot have any credence at all about whether it's Monday or Tuesday, which I find to be subject to dutch books (if she is asked to bet on which day it is). You might argue that it is possible to evaluate the bet by computing expected utility across the whole experiment without assigning P(Monday), but I think this is just thirding with extra steps.
The Technicolor argument is also persuasive. However, I suspect the halfer has no principled reason to update on color if they don't update on awakening and the thirder doesn't need to update on color because we've already correctly accounted for the probability. The halfer updating on color is just exploiting a strategy that allows them to see the probability as it is. The thirder could use the same strategy to the same effect. I could be wrong here.
Also –– I don't think anything about SIA requires treating awakenings as independent/not acknowledging the tails awakenings to be sequential.
> I think it does end up implying that Sleeping Beauty cannot have any credence at all about whether it's Monday or Tuesday
Yes, completely correct. Under amnesia and repetition of the same experience the notion of "Today" becomes ill-defined as it points to both days in the trials where the coin is Tails.
It doesn't mean that P(Monday) is not defined, mind you. It means that instead of ill-defined "Monday is Today", where it's not clear what exactly we mean by "Today" it is properly defined "Monday awakening happens during this iteration of probability experiment". Probability of this event is 1. Likewise P(Tuesday) =1/2
> I find to be subject to dutch books (if she is asked to bet on which day it is)
No dutch book here. She simply bets based on probability of Monday and Tuesday awakening happening during the experiment.
P(Monday) = 1
P(Tuesday) = 1/2
Therefore the betting odds are
2:1
> I think this is just thirding with extra steps.
I agree that if you start from thirders assumptions it seems this way. But, strictly speaking, this is thirdism with *less* steps.
More specifically the step which is not required is accepting the framework of centred possible worlds which disregards the fact that sample space has to be constructed from mutually exclusive outcomes of the experiment, leading to utility shifts during the same trial and all the standard anthropic paradoxes.
> The Technicolor argument is also persuasive. However, I suspect the halfer has no principled reason to update on color if they don't update on awakening
The principled reason for update in general is the ability to distinguish between trials where a event C is realized and all the other trials of probability experiment. Initially we are uncertain between all the possible iterations of the probability experiment. And have our unconditional probabilities for every event:
P(E1), P(E2)... P(En)
Then we are uncertain between all the iterations of conditional probability experiment - *where event C is realized*. The ratio of realization of the events E1... En can be different between the unconditional and conditional probability experiments. So now we have our credences:
P(E1|C), P(E2|C)... P(En|C)
If you are interested to dig deeper, I talk in details about it here: https://www.lesswrong.com/posts/6TsqoQo9nnwQS5ina/ptf-102-conditionalization-and-events
Going back to Sleeping Beauty
The reason why double halfers do not update their probabilities on awakening is because Beauty awakens in every trial of the experiment.
P(Awake) = 1
So the fact of awakening doesn't allow to distinguish between the trials where coin is Heads and where it's Tails. Conditional probability experiment equals the unconditional therefore:
P(Heads) = P(Heads|Awake)
P(Tails) = P(Tails|Awake)
With Technicolor Sleeping Beauty there are two cases: where we can distinguish the iterations where a particular color is shown and where we can't.
If we just go into the experiment unprepared and see that the room is some color - Blue or Red this indeed doesn't give us anything as in every iteration of the experiment event Blue or Red is realized:
P(Blue or Red) = 1
However, if we prepared ourself to pay attention to a specific color before the experiment started, for instance, Blue, we can distinguish between iterations. The room is Blue in only 75% of iteration - in half of the iterations where the coin is Heads and in every iteration where it's Tails.
P(Blue) = 3/4
P(Blue|Heads) = 1/2
P(Blue|Tails) = 1
Therefore
P(Heads|Blue) = 1/3
> thirder doesn't need to update on color because we've already correctly accounted for the probability.
No, that's the whole point. Thirder's think that they've already updated and so there is no difference whether the room is Blue or Red. For them Technicolor Sleeping Beauty is just isomorphic to regular Sleeping Beauty problem. As a result thirders miss the opportunity to win extra money in technicolor, that double halfers take.
Consider: you are about to be subject to multiple iterations of Technicolor Sleeping Beaty experiment. Your goal is to guess iterations of the experiment where the coin came Tails, which you can do with a button. Press it whenever you think the coin is Tails - every press of the button costs you 100 dollar, however you get 160 dollars per iteration of experiment (not awakening!) when you've pressed the button at least once, and the coin happened to be Tails.
A thirder would just be annoyed that the scoring rule was designed by a halfer and would not see any reason to press a button. Thirder P(Tails) = 2/3 means that the coin is Tails in about 2/3 of awakenings, not iterations of experiment! Of course they can't guess it better than chance if the scoring rule is *unfair*.
A double halfer, meanwhile, would on average become 50 dollars richer per Tails iteration of experiment.
> I don't think anything about SIA requires treating awakenings as independent/not acknowledging the tails awakenings to be sequential.
It all goes back to the formal definition of sample space - that it has to consist of mutually exclusives outcomes of the experiment. As thirder's sample space does not consist of mutually exclusive outcomes of the experiment it's not sound.
Some thirders may try to circumvent this by claiming that individual awakenings should themselves be considered trials of a random sample among three options: Heads&Monday, Tails&Monday, Tails&Tuesday. This, however, also doesn't work because individual trials has to be independent from each other, while in Sleeping Beauty there is strict order between awakenings.
Four volunteers, named SB1, SB2, SB3, and SB4, participate in a two day experiment. Each is assigned a day from (Mon, Tue) and a coin face from (H, T) such that each has a different combination. On Sunday night, all are put to sleep and a fair coin is flipped. On Monday, and again on Tuesday, three are awakened. The one who is left asleep is the one whose assignment matches the current day and the coin flip.
Each of the awake volunteers is asked for the [probability/credence/degree of belief/whatever word you want] that their coin assignment matches the coin.
The only possible answer is 1/3.
Great intro to Evidential decision theory got me thinking, thanks <3
"I think halfers’ arguments are subject to Dutch books, i.e. in the betting version, they will lose money consistently."
This is true, but so are thirders' arguments. What if I set up the bet differently? So every time you wake up you get to say "bet" or "don't bet". But the rule is that I only take into account your final answer in deciding whether to bet the $10. Now, the right odds to take are 1:1. If I instead offer you a bet, and I say it's counted as half if you bet before, then the right odds are 2:3, 40%. I can get any value between 1/3 and 1/2 by setting up the bet the right way.
Another way to see this is that if I bet with you right after flipping the coin but also every time you wake up, I can make money off you. After flipping the fair coin the odds are half, but when being woken up it's a third. So I can just Dutch book you already by exploiting the information asymmetry here.
I don't think I'm either a thirder or a halfer since I don't think every event can be meaningfully given a probability. Let S be a set with outer measure 1/2 and inner measure 1/3. If I take random element in [0,1), what's the probability it lies in S? Probability is a measure. There are no non-measurable sets in Sleeping Beauty sure, but there is a uniform measure over the universes induced by the coin (halfer), and a uniform measure over the states of consciousness you experience when you wake up (thirder).
1/2 position can't be Dutch Booked; rewarding people for each waking guess will obviously multiply the winnings for guessing in the direction of more winnings, which is something a 1/2er can easily account for.
The 1/3 position, however...
Let's say the coin is no longer a fair coin. Now it’s a magical coin that flips Heads one million times more often than it flips Tails.
If the coin lands Tails, you’ll be woken up a billion billion times.
If the coin lands Heads, you’ll be woken up only once.
Each time you wake up, you will guess the coin flip result. If you ever guess wrong, you will be killed after all the wakings are over. What should you guess?
Guessing Tails will result with you dead 99.9999% of the time.
I explain more here:
https://ramblingafter.substack.com/p/always-thirders-are-wrong-about-the
Furthermore, you don't need either the SSA or SIA:
https://ramblingafter.substack.com/p/a-devastating-explanation-of-sleeping