I've just finished a post showing that thirders actually have more problems with betting in Sleeping Beauty and its derivatives than halfers, which follow the correct (not Lewis's) model.
You can't appeal to indifference when you already have some knowledge about the events. The Beauty knows that her awakening routine is determined by a fair coin toss. That means that she is not indifferent between the three awakenings anymore. If there were no coin toss - just three possible awakenings only one of which can happen in the experiment, or at least the Beauty didn't know about the coin toss and the order between awakenings, then yes, she should follow the indifference and be a thirder - this is the No-Coin-Toss problem for which Elga's model applies.
> One way to see that Bergman’s reasoning is wrong is to imagine a scenario like the original where Beauty wakes up, looks outside, and infers from the weather that it’s a Monday. Upon finding this out, it seems she should be 50/50 by Bergman’s logic—she knew she’d be awake on a Monday so she learned nothing new. But if she should be 50/50 conditional on finding out that it’s a Monday and conditional on finding out that it’s a Tuesday she should be 100% sure that the coin came up tails
There is nothing wrong here. This is exactly how one is supposed to reason about the problem while sticking to probability theory. And in the comment below I explain why.
> then being unsure which of the three days it is she should spill her credence three ways and thus third.
What three days? You mean which of three awakenings? It would be true if the awakenings were mutually exclusive and therefore could be treated as outcomes for a sample space. But as there is order between them and the Beauty knows about it, she can't lawfully spill her credence between them.
> You might reply that she learns something
She didn't and that's the whole point. She knew that she is to be awake on Monday regardless of the outcome of the coin and thius she doesn't learn anything new when she is told that she indeed was awakened on Monday
P(Heads) = P(Heads|Monday) = 1/2
> But this means that what you should think on Monday depends on what could have happened on Tuesday, because if you would never have been born on Tuesday conditional on the coin coming up tails, then you should have a 50% credence in coin having come up tails upon finding out that it’s Monday. But surely after finding out that it’s Monday, to decide upon your credences, you don’t have to know what will happen on Tuesday conditional on the coin coming up tails! It hasn’t happened yet so what will happen conditional on the coin coming up tails can’t be relevant evidence.
Ehm... What? Sorry, I don't understand what are you talking about here. No one is being born in the experiment. Could you maybe rephrase your argument here differently?
> Here’s another way to see that Bergman’s diagnosis is wrong. Bergman claims that the relevant difference between the sleeping beauty and the experimenter case is that the experimenter would have been around the time slice is unoccupied while beauty wouldn’t be. But surely this can’t be right.
Bergman might have formulated the actual principle poorly. The relevant difference is whether the person might expect not to observe some evidence. The experimenters on a random day might not observe that the Beauty is awake, either because they see her asleep or because they are killed. And therefore observing the Beauty awake is relevant evidence that updates them in favor of Tails. The Beauty always observes herself awake so it's not relevant evidence in favor of Tails. This is just how the conservation of expected evidence works.
> Accepting SIA gets really weird results. But so does accepting SSA and every other view of anthropics.
And that's why we should accept neither SIA nor SSA, nor any other anthropic theory that produces weird results and keep looking for an approach that produce correct results in every case.
> In this case, the most specific version of the evidence that Beauty has is not “I’m awake at some point,” but “I’m awake now.”
No, the Beauty doesn't have such evidence. Probability theory doesn't allows to deal with time moments unless they are randomly sampled in some manner. Monday and Tuesday awakenings on Tails are not random, they happen in ordered manner. So the Beauty can't lawfully reason about them separately.
The Beauty observes event "I'm awaken during the experiment at least once". This is what she expected to happen so she doesn't update her probability estimate for Heads in any way. Likewise, if she is told that it's Monday she observed event "I'm awake on Monday in this experiment", which also is something she expected regardless the outcome of the coin toss, so she doesn't update.
Regarding your first argument, we can modify your betting hypothetical so that halving is optimal. Let's say you are given a bet and you are told it has the following conditions:
* If today is Monday: If the coin was Heads, you pay 6 dollars; If the coin was Tails, you receive 4 dollars.
* If today is Tuesday: There is no payoff.
This is the same as your original bet, except now the payoffs only occur on Monday. But the bet is clearly suboptimal in this case. Therefore, halving rather than thirding will be optimal in this case, since thirding favors taking the bet whereas halving favors not taking the bet.
You might say that thirding does not favor taking the bet. But it's not clear why that would be the case. After all, when one wakes up on a Monday (without knowing that it's a Monday), thirding favors believing that there is a 2/3 chance that the coin was Tails. Thus, thirding favors believing that the payoff of the bet will be (probability that today is Monday) * [(probability that coin was Tails) * 4 - (probability that the coin was Heads) * 6] = (probability that today is Monday) * [(2/3) * 4 - (1/3) * 6] which is positive. So thirding favors believing that the expected payoff is positive.
You might say that thirding does not favor believing that there is a 2/3 chance of Tails. But it's not clear why that would be the case. In my betting case, you haven't been given any information about the coin: you would have been offered this bet no matter what the coin flipped and no matter what day it is. So if thirding favors believing that there is a 2/3 chance of Tails without the bet, then thirding favors believing that there is a 2/3 chance of Tails with the bet.
You might say that this is a case where betting based on rational credences does not optimize payoffs. But then you need to explain why we should expect rational credences to optimize payoffs in your betting experiment but not in my betting experiment.
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Regarding the second argument, it just seems question-begging. The Principle of Indifference favors assigning equal probabilities to different events only when we have no reason to believe to think any particular event is more likely than the others. But we do have reason to believe that 3 events (Monday-Heads, Monday-Tails, and Tuesday-Tails) are not equally likely, at least if halving is true. If halving is true, then there is a 1/2 chance that the coin was Heads. And if the coin was Heads, then there is a 100% chance that today is the Monday-Heads event. Otherwise, there is a 1/2 chance that today is Monday-Tails and a 1/2 chance that today is Tuesday-Tails. Thus, if halving is true, then we should assign the probabilities as follows: Monday-Heads = 50%, Monday-Tails = 25%, and Tuesday-Tails = 25%.
Yeah fair enough, my actual response is: risk neutral betting odds are not identical to subjective probability. They’re almost always the same, but this is an edge case where they come apart
A clearer analogue in this case is: I flip a fair coin. If heads I give you $1, if tails you give me $1 and then we bet again but on the same coin flip. So it’s functionally identical to you giving me $2 if tails
All this is established before the coin flip, there’s no hidden info or trickery
Presumably your answer is “fuck u I don’t want to play” which like yeah obviously! It was an unfair setup on top of a fair coin flip
It's underappreciated in rationalist cicles that some problems have a "Wittgensteinian" solution , where the problem itself is criticised as being ambiguous or paradoxical.
"The Principle of Indifference at first seems to favor the halfer since we have no reason to believe that the coin is any more likely to land heads than tails or vice-versa. But, thirders argue, what we should really be indifferent over is Beauty’s three indistinguishable possible waking events (waking Monday when the coin landed heads, waking Monday when the coin landed tails, and waking Tuesday when the coin landed tails), "
...and that is clearly the case here. Halfers and thiders aren't just arguing about the solutions to the problem, they are disagreeing about what the problem is -- finding the objective probability of a coin landing, or of Beauty's subjective observations.
Hey... I couldn't follow the whole thing, but I think I got the gist of it.
Wanted to know what you thought of the inverse gamblers fallacy in fine tuning argument (The "This universe objection" against the multiverse).
Because the objection says that you cannot infer a multiverse from the fact that THIS universe is fine tuned for life. You only can infer that some universe is finely tuned. Since you have to use the most specific version of the evidence you have, which is that THIS universe is finely tuned, there is no multiverse
Okay here's a worry about this- something I plan on writing a paper on one day.
Let us suppose that there are two possibilities A- the dice lands on a 2 or 3 and B- the dice lands on any of the other four numbers on that dice. Generally we would expect a 1/3 chance of A.
Now suppose there's some hypothesis on which an infinite number of me's will roll the dice with different results. On your reasoning we should think that is infinitely more likely than the scenario where I just roll the dice once.
It follows that the probability of A is:
A*(Infinity)
And the probability of B is:
B*(Infinity)
So the probability of the outcome A is:
(Infinity)/(Infinity)+(Infinity)
Now I don't think I've done all the transfinite maths properly there, but you can see the worry I'm gesturing at- I am worried that every probability will become undefined. Thus the underlying logic of thirderism leads to all probabilities being undefined, if there is some possibility that the events will happen infinite times, with a probability >0, the thirder will infer to that, but then all possibilities will happen an infinite number of times and...
Not sure if this is a problem for thirderism specifically, or if it's just a problem with probabilities in general if something like the MWH is true. My gut says there's a problem here specifically for thirderism, but I need to think some more.
> Here’s an argument for why one should third
I've just finished a post showing that thirders actually have more problems with betting in Sleeping Beauty and its derivatives than halfers, which follow the correct (not Lewis's) model.
https://www.lesswrong.com/posts/cvCQgFFmELuyord7a/beauty-and-the-bets
> The Principle of Indifference
You can't appeal to indifference when you already have some knowledge about the events. The Beauty knows that her awakening routine is determined by a fair coin toss. That means that she is not indifferent between the three awakenings anymore. If there were no coin toss - just three possible awakenings only one of which can happen in the experiment, or at least the Beauty didn't know about the coin toss and the order between awakenings, then yes, she should follow the indifference and be a thirder - this is the No-Coin-Toss problem for which Elga's model applies.
> One way to see that Bergman’s reasoning is wrong is to imagine a scenario like the original where Beauty wakes up, looks outside, and infers from the weather that it’s a Monday. Upon finding this out, it seems she should be 50/50 by Bergman’s logic—she knew she’d be awake on a Monday so she learned nothing new. But if she should be 50/50 conditional on finding out that it’s a Monday and conditional on finding out that it’s a Tuesday she should be 100% sure that the coin came up tails
There is nothing wrong here. This is exactly how one is supposed to reason about the problem while sticking to probability theory. And in the comment below I explain why.
> then being unsure which of the three days it is she should spill her credence three ways and thus third.
What three days? You mean which of three awakenings? It would be true if the awakenings were mutually exclusive and therefore could be treated as outcomes for a sample space. But as there is order between them and the Beauty knows about it, she can't lawfully spill her credence between them.
> You might reply that she learns something
She didn't and that's the whole point. She knew that she is to be awake on Monday regardless of the outcome of the coin and thius she doesn't learn anything new when she is told that she indeed was awakened on Monday
P(Heads) = P(Heads|Monday) = 1/2
> But this means that what you should think on Monday depends on what could have happened on Tuesday, because if you would never have been born on Tuesday conditional on the coin coming up tails, then you should have a 50% credence in coin having come up tails upon finding out that it’s Monday. But surely after finding out that it’s Monday, to decide upon your credences, you don’t have to know what will happen on Tuesday conditional on the coin coming up tails! It hasn’t happened yet so what will happen conditional on the coin coming up tails can’t be relevant evidence.
Ehm... What? Sorry, I don't understand what are you talking about here. No one is being born in the experiment. Could you maybe rephrase your argument here differently?
> Here’s another way to see that Bergman’s diagnosis is wrong. Bergman claims that the relevant difference between the sleeping beauty and the experimenter case is that the experimenter would have been around the time slice is unoccupied while beauty wouldn’t be. But surely this can’t be right.
Bergman might have formulated the actual principle poorly. The relevant difference is whether the person might expect not to observe some evidence. The experimenters on a random day might not observe that the Beauty is awake, either because they see her asleep or because they are killed. And therefore observing the Beauty awake is relevant evidence that updates them in favor of Tails. The Beauty always observes herself awake so it's not relevant evidence in favor of Tails. This is just how the conservation of expected evidence works.
> Accepting SIA gets really weird results. But so does accepting SSA and every other view of anthropics.
And that's why we should accept neither SIA nor SSA, nor any other anthropic theory that produces weird results and keep looking for an approach that produce correct results in every case.
> In this case, the most specific version of the evidence that Beauty has is not “I’m awake at some point,” but “I’m awake now.”
No, the Beauty doesn't have such evidence. Probability theory doesn't allows to deal with time moments unless they are randomly sampled in some manner. Monday and Tuesday awakenings on Tails are not random, they happen in ordered manner. So the Beauty can't lawfully reason about them separately.
The Beauty observes event "I'm awaken during the experiment at least once". This is what she expected to happen so she doesn't update her probability estimate for Heads in any way. Likewise, if she is told that it's Monday she observed event "I'm awake on Monday in this experiment", which also is something she expected regardless the outcome of the coin toss, so she doesn't update.
P(Heads|Monday)=P(Heads&Monday)=P(Heads|Awake)=P(Heads&Awake)=P(Heads)=1/2
P(Monday)=P(Awake)=1
P(Tuesday)=P(Tails)=1/2
Regarding your first argument, we can modify your betting hypothetical so that halving is optimal. Let's say you are given a bet and you are told it has the following conditions:
* If today is Monday: If the coin was Heads, you pay 6 dollars; If the coin was Tails, you receive 4 dollars.
* If today is Tuesday: There is no payoff.
This is the same as your original bet, except now the payoffs only occur on Monday. But the bet is clearly suboptimal in this case. Therefore, halving rather than thirding will be optimal in this case, since thirding favors taking the bet whereas halving favors not taking the bet.
You might say that thirding does not favor taking the bet. But it's not clear why that would be the case. After all, when one wakes up on a Monday (without knowing that it's a Monday), thirding favors believing that there is a 2/3 chance that the coin was Tails. Thus, thirding favors believing that the payoff of the bet will be (probability that today is Monday) * [(probability that coin was Tails) * 4 - (probability that the coin was Heads) * 6] = (probability that today is Monday) * [(2/3) * 4 - (1/3) * 6] which is positive. So thirding favors believing that the expected payoff is positive.
You might say that thirding does not favor believing that there is a 2/3 chance of Tails. But it's not clear why that would be the case. In my betting case, you haven't been given any information about the coin: you would have been offered this bet no matter what the coin flipped and no matter what day it is. So if thirding favors believing that there is a 2/3 chance of Tails without the bet, then thirding favors believing that there is a 2/3 chance of Tails with the bet.
You might say that this is a case where betting based on rational credences does not optimize payoffs. But then you need to explain why we should expect rational credences to optimize payoffs in your betting experiment but not in my betting experiment.
------
Regarding the second argument, it just seems question-begging. The Principle of Indifference favors assigning equal probabilities to different events only when we have no reason to believe to think any particular event is more likely than the others. But we do have reason to believe that 3 events (Monday-Heads, Monday-Tails, and Tuesday-Tails) are not equally likely, at least if halving is true. If halving is true, then there is a 1/2 chance that the coin was Heads. And if the coin was Heads, then there is a 100% chance that today is the Monday-Heads event. Otherwise, there is a 1/2 chance that today is Monday-Tails and a 1/2 chance that today is Tuesday-Tails. Thus, if halving is true, then we should assign the probabilities as follows: Monday-Heads = 50%, Monday-Tails = 25%, and Tuesday-Tails = 25%.
Haven’t even finished reading but here’s the text of a tweet I sent while debating this with others a few months ago (https://x.com/aaronbergman18/status/1692575917632111002)
Yeah fair enough, my actual response is: risk neutral betting odds are not identical to subjective probability. They’re almost always the same, but this is an edge case where they come apart
A clearer analogue in this case is: I flip a fair coin. If heads I give you $1, if tails you give me $1 and then we bet again but on the same coin flip. So it’s functionally identical to you giving me $2 if tails
All this is established before the coin flip, there’s no hidden info or trickery
Presumably your answer is “fuck u I don’t want to play” which like yeah obviously! It was an unfair setup on top of a fair coin flip
It's underappreciated in rationalist cicles that some problems have a "Wittgensteinian" solution , where the problem itself is criticised as being ambiguous or paradoxical.
"The Principle of Indifference at first seems to favor the halfer since we have no reason to believe that the coin is any more likely to land heads than tails or vice-versa. But, thirders argue, what we should really be indifferent over is Beauty’s three indistinguishable possible waking events (waking Monday when the coin landed heads, waking Monday when the coin landed tails, and waking Tuesday when the coin landed tails), "
...and that is clearly the case here. Halfers and thiders aren't just arguing about the solutions to the problem, they are disagreeing about what the problem is -- finding the objective probability of a coin landing, or of Beauty's subjective observations.
Hey... I couldn't follow the whole thing, but I think I got the gist of it.
Wanted to know what you thought of the inverse gamblers fallacy in fine tuning argument (The "This universe objection" against the multiverse).
Because the objection says that you cannot infer a multiverse from the fact that THIS universe is fine tuned for life. You only can infer that some universe is finely tuned. Since you have to use the most specific version of the evidence you have, which is that THIS universe is finely tuned, there is no multiverse
Okay here's a worry about this- something I plan on writing a paper on one day.
Let us suppose that there are two possibilities A- the dice lands on a 2 or 3 and B- the dice lands on any of the other four numbers on that dice. Generally we would expect a 1/3 chance of A.
Now suppose there's some hypothesis on which an infinite number of me's will roll the dice with different results. On your reasoning we should think that is infinitely more likely than the scenario where I just roll the dice once.
It follows that the probability of A is:
A*(Infinity)
And the probability of B is:
B*(Infinity)
So the probability of the outcome A is:
(Infinity)/(Infinity)+(Infinity)
Now I don't think I've done all the transfinite maths properly there, but you can see the worry I'm gesturing at- I am worried that every probability will become undefined. Thus the underlying logic of thirderism leads to all probabilities being undefined, if there is some possibility that the events will happen infinite times, with a probability >0, the thirder will infer to that, but then all possibilities will happen an infinite number of times and...
Not sure if this is a problem for thirderism specifically, or if it's just a problem with probabilities in general if something like the MWH is true. My gut says there's a problem here specifically for thirderism, but I need to think some more.