However, in the A,B-Game we assume that a specific gene makes people presented with two options choose the worse one—please note that I have not mentioned Omega in this sentence yet! So the claim is not that Omega is able to predict something, but that the gene can determine something, even in absence of the Omega. It’s no longer about Omega’s superior human-predicting powers; the Omega is there merely to explain the powers of the gene.
I think there might be a misunderstanding. Although I don’t believe it to be impossible that a gene causes you to think in specific ways, in the setting of the game such a mechanism is not required. You can also imagine a game where Omega predicts that those who pick a carrot out of a basket of vegetables are the ones that will die shortly of a heart attack. As long as we believe in Omega’s forecasting power, its statements are relevant even if we cannot point at any underlying causal mechanisms. As long as the predicted situation is logically possible (here, all agents that pick the carrot die), we don’t need to reject Omega’s prediction just because such a compilation of events would be unlikely. Though we might call Omega’s predictions into question. Still, as long as we believe in its forecasting power (after such a update), we have to take the prediction into account.
Hence, the A,B-Game holds even if you don’t know of any causal connection between the genes and the behaviour, we only need a credible Omega.
Although I don’t believe it to be impossible that a gene causes you to think in specific ways, in the setting of the game such a mechanism is not required.
It is required. If Omega is making true statements, they are (leaving aside those cases where someone is made aware of the prediction before choosing) true independently of Omega making them. That means that everyone with gene A makes choice A and everyone with gene B makes choice B. This strong entanglement implies the existence of some sort of causal connection, whether or not Omega exists.
More generally, I think that every one of those problems would be made clear by exhibiting the causal relationships that are being presumed to hold. Here is my attempt.
For the School Mark problem, the causal diagram I obtain from the description is one of these:
pupil's character ----> teacher's prediction ----> final mark
|
|
V
studying ----> exam performance
or
pupil's character ----> teacher's prediction
|
|
V
studying ----> exam performance ----> final mark
For the first of these, the teacher has waived the requirement of actually sitting the exam, and the student needn’t bother. In the second, the pupil will not get the marks except by studying for and taking the exam. See also the decision problem I describe at the end of this comment.
For Newcomb we have:
person's qualities --> Omega's prediction --> contents of boxes
| |
| |
V V
person's decision --------------------------> payoff
(ETA: the second down arrow should go from “contents of boxes” to “payoff”. Apparently Markdown’s code mode isn’t as code-modey as I expected.)
The hypotheses prevent us from performing surgery on this graph to model do(person’s decision). The do() operator requires deleting all in-edges to the node operated on, making it causally independent of all of its non-descendants in the graph. The hypotheses of Newcomb stipulate that this cannot be done: every consideration you could possibly employ in making a decision is assumed to be already present in the personal qualities that Omega’s prediction is based on.
A-B:
Unknown factors ---> Gene ---> Lifespan
|
|
V
Choice
or:
Gene ---> Lifespan
|
|
V
Choice
or both together.
Here, it may be unfortunate to discover oneself making choice B, but by the hypotheses of this problem, you have no choice. As with Newcomb, causal surgery is excluded by the problem. To the extent that your choice is causally independent of the given arrow, to that extent you can ignore lifespan in making your choice—indeed, it is to that extent that you have a choice.
For Solomon’s Problem (which, despite the great length of the article, you didn’t set out) the diagram is:
charisma ----> overthrow
|
|
V
commit adultery
This implies that while it may be unfortunate for Solomon to discover adulterous desires, he will not make himself worse off by acting on them. This differs from A-B because we are given some causal mechanisms, and know that they are not deterministic: an uncharismatic leader still has a choice to make about adultery, and to the extent that it is causally independent of the lack of charisma, it can be made, without regard to the likelihood of overthrow.
(ETA: the arrow from “chew gun” to “throat abcesses” didn’t come out very well.)
in which chewing gum is protective against throat abscesses, and positively to be recommended.
Newcomb’s Soda:
soda assignment ---> $1M
|
|
V
choice of ice cream ---> $1K
Here, your inclination to choose a flavour of ice-cream is informative about the $1M prize, but the causal mechanism is limited to experiencing a preference. If you would prefer $1K to a chocolate ice-cream then you can safely choose vanilla.
Finally, here’s another decision problem I thought of. Unlike all of the above, it requires no sci-fi hypotheses, real-world examples exist everywhere, and correctly solving them is an important practical skill.
I want to catch a train in half an hour. I judge that this is enough time to get to the station, buy a ticket, and board the train. Based on a large number of similar experiences in the past, I can confidently predict that I will catch the train. Since I know I will catch the train, should I actually do anything to catch the train?
The general form of this problem can be applied to many others. I predict that I’m going to ace an upcoming exam. Should I study? I predict I’ll win an upcoming tennis match. Should I train for it? I predict I’ll complete a piece of contract work on time. Should I work on it? I predict that I will post this. Should I click the “Comment” button?
For the School Mark problem, the causal diagram I obtain from the description is one of these:
diagram
or
diagram
For the first of these, the teacher has waived the requirement of actually sitting the exam, and the student needn’t > bother. In the second, the pupil will not get the marks except by studying for and taking the exam. See also the
decision problem I describe at the end of this comment.
I think it’s clear that Pallas had the first diagram in mind, and his point was exactly that the rational thing to do is to study despite the fact that the mark has already been written down. I agree with this.
Think of the following three scenarios:
A: No prediction is made and the final grade is determined by the exam performance.
B: A perfect prediction is made and the final grade is determined by the exam performance.
C: A perfect prediction is made and the final grade is based on the prediction.
Clearly, in scenario A the student should study. You are saying that in scenario C, the rational thing to do is not studying. Therefore, you think that the rational decision differs between either A and B, or between B and C. Going from A to B, why should the existence of someone who predicts your decision (without you knowing the prediction!) affect which decision the rational one is? That the final mark is the same in B and C follows from the very definition of a “perfect prediction”. Since each possible decision gives the same final mark in B and C, why should the rational decision differ?
In all three scenarios, the mapping from the set of possible decisions to the set of possible outcomes is identical—and this mapping is arguably all you need to know in order to make the correct decision. ETA: “Possible” here means “subjectively seen as possible”.
By deciding whether or not to learn, you can, from your subjective point of view, “choose” wheter you were determined to learn or not.
My first diagram is scenario C and my second is scenario B. In the first diagram there is no (ETA: causal) dependence of the final mark on exam performance. I think pallas’ intended scenario was more likely to be B: the mark does (ETA: causally) depend on exam performance and has been predicted. Since in B the mark depends on final performance it is necessary to study and take the exam.
In the real world, where teachers do not possess Omega’s magic powers, teachers may very well be able to predict pretty much how their students will do. For that matter, the students themselves can predict how they will do, which transforms the problem into the very ordinary, non-magical one I gave at the end of my comment. If you know how well you will do on the exam, and want to do well on it, should you (i.e. is it the correct decision to) put in the work? Or for another example of topical interest, consider the effects of genes on character.
Unless you draw out the causal diagrams, Omega is just magic: an imaginary phenomenon with no moving parts. As has been observed by someone before on LessWrong, any decision theory can be defeated by suitably crafted magic: Omega fills the boxes, or whatever, in the opposite way to whatever your decision theory will conclude. Problems of that sort offer little insight into decision theory.
You can also imagine a game where Omega predicts that those who pick a carrot out of a basket of vegetables are the ones that will die shortly of a heart attack.
Those who pick a carrot after hearing Omega’s prediction, or without hearing the prediction? Those are two very different situations, and I am not sure which one you meant.
If some people even after hearing the Omega’s prediction pick the carrot and then die of a heart attack, there must be something very special about them. They are suicidal, or strongly believe that Omega is wrong and want to prove it, or some other confusion.
If people who without hearing the Omega’s prediction pick the carrot and die, that does not mean they would have also picked the carrot if they were warned in advance. So saying “we should also press A here” provides no actionable advice about how people should behave, because it only works for people who don’t know it.
Those who pick a carrot after hearing Omega’s prediction, or without hearing the prediction? Those are two very different situations, and I am not sure which one you meant.
That’s a good point. I agree with you that it is crucial to keep apart those two situations. This is exactly what I was trying to address considering Newcomb’s Problem and Newcomb’s Soda. What do the agents (previous study-subjects) know? It seems to me that the games aren’t defined precise enough. Once we specify a game in a way that all the agents hear Omega’s prediction (like in Newcomb’s Problem), the prediction provides actionable advice as all the agents belong to the same reference class. If we, and we alone, know about a prediction (whereas other agents don’t) the situation is different and the actionable advice is not provided anymore, at least not to the same extent. When I propose a game where Omega predicts whether people pick carrots or not and I don’t specify that this only applies to those who don’t know about the prediction then I would not assume prima facie that the prediction only applies to those who don’t know about the prediction. Without further specification, I would assume that it applies to “people” which is a superset of “people who know of the prediction”.
We believe in the forecasting power, but we are uncertain as to what mechanism that forecasting power is taking advantage of to predict the world.
analogously, I know Omega will defeat me at Chess, but I do not know which opening move he will play.
In this case, the TDT decision depends critically on which causal mechanism underlies that forecasting power. Since we do not know, we will have to apply some principles for decision under uncertainty, which will depend on the payoffs, and on other features of the situation. The EDT decision does not. My intuitions and, I believe, the intuitions of many other commenters here, are much closer to the TDT approach than the EDT approach. Thus your examples are not very helpful to us—they lump things we would rather split, because our decisions in the sort of situation you described would depend in a fine-grained way on what causal explanations we found most plausible.
Suppose it is well-known that the wealthy in your country are more likely to adopt a certain distinctive manner of speaking due to the mysterious HavingRichParents gene. If you desire money, could you choose to have this gene by training yourself to speak in this way?
I agree that it is challenging to assign forecasting power to a study, as we’re uncertain about lots of background conditions. There is forecasting power to the degree that the set A of all variables involved with previous subjects allow for predictions about the set A’ of variables involved in our case. Though when we deal with Omega who is defined to make true predictions, then we need to take this forecasting power into account, no matter what the underlying mechanism is. I mean, what if Omega in Newcomb’s Problem was defined to make true predictions and you don’t know anything about the underlying mechanism? Wouldn’t you one-box after all?
Let’s call Omega’s prediction P and the future event F. Once Omega’s prediction are defined to be true, we can denote the following logical equivalences:
P(1 boxing) <--> F(1 boxing) and P(2 boxing) <--> P(2 boxing). Given this conditions, it impossible to 2-box when box B is filled with a million dollars (you could also formulate it in terms of probabilities where such an impossible event would have the probability of 0).
I admit that we have to be cautious when we deal with instances that are not defined to make true predictions.
Suppose it is well-known that the wealthy in your country are more likely to adopt a certain distinctive manner of speaking due to the mysterious HavingRichParents gene. If you desire money, could you choose to have this gene by training yourself to speak in this way?
My answer depends on the specific set-up. What exactly do we mean with “It is well-known”? It doesn’t seem to be a study that would describe the set A of all factors involved which we then could use to derive A’ that applied to our own case. Unless we define “It is well-known” as a instance that allows for predictions in the direction A --> A’, I see little reason to assume a forecasting power. Without forecasting power, screening off applies and it would be foolish to train the distinctive manner of speaking.
If we specified the game in a way that there is forecasting power at work (or at least we had reason to believe so), depending on your definition of choice (I prefer one that is devoid of free will) you can or cannot choose the gene. These kind of thoughts are listed here or in the section “Newcomb’s Problem’s Problem of Free Will” in the post.
Suppose I am deciding now whether to one-box or two-box on the problem. That’s a reasonable supposition, because I am deciding now whether to one-box or two-box. There are a couple possibilities for what Omega could be doing:
Omega observes my brain, and predicts what I am going to do accurately.
Omega makes an inaccurate prediction, probabilistically independent from my behavior.
Omega modifies my brain to a being it knows will one-box or will two-box, then makes the corresponding prediction.
If Omega uses predictive methods that aren’t 100% effective, I’ll treat it as combination of case 1 and 2. If Omega uses very powerful mind-influencing technology that isn’t 100% effective, I’ll treat it as a combination of case 2 and 3.
In case 1 , I should decide now to one-box. In case 2, I should decide now to two-box. In case 3, it doesn’t matter what I decide now.
If Omega is 100% accurate, I know for certain I am in case 1 or case 3. So I should definitely one-box. This is true even if case 1 is vanishingly unlikely.
If Omega is even 99.9% accurate, then I am in some combination of case 1, case 2, or case 3. Whether I should decide now to one-box or two-box depends on the relative probability of case 1 and case 2, ignoring case 3. So even if Omega is very accurate, ensuring that the probability of case 2 is small, if the probability of case 1 is even smaller, I should decide now to two-box.
I mean, I am describing a very specific forecasting technique that you can use to make forecasts right now. Perhaps a more precise version is, you observer children in one of two different preschools, and observe which school they are in. You observe that almost 100% of the children in one preschool end up richer than the children in the other preschool. You are then able to forecast that future children observed in preschool A will grow up to be rich, and future children observed in preschool B will grow up to be poor. You then have a child. Should you bring them to preschool A? (Here I don’t mean have them attend the school. They can simply go to the building at whatever time of day the study was conducted, then leave. That is sufficient to make highly accurate predictions, after all!)
I don’t really know what you mean by “the set A of all factors involved”
If the scenario you describe is coherent, there has to be a causal mechanism, even if you don’t know what it is. If Omega is a perfect predictor, he can’t predict that carrot-choosers have heart attacks unless carrot-choosers have heart attacks.
I think I agree. But I would formulate it otherwise:
i) Omega’s prediction are true.
ii) Omega predicts that carrot-choosers have heart attacks.
c) Therefore, carrot-choosers have heart attacks.
As soon as you accept i), c) follows if we add ii). I don’t know how you define “causal mechanism”. But I can imagine a possible world where no biological mechanism connects carrot-choosing with heart attacks but where “accidentally” all the carrot-choosers have heart-attacks (Let’s imagine running worlds on a computer countless times. One day we might observe such a freak world). Then c) would be true without there being some sort of “causal mechanism” (as you might define it I suppose?). If you say that in such a freak world carrot-choosing and heart attacks are causally connected, then I would agree that c) can only be true if there is a underlying causal mechanism.
But in that case, when Omega tells me that if I choose carrots I’ll have a heart attack, then almost certainly I’m not in a freak world, and actually Omega is wrong.
Assuming i), I would rather say that when Omega tells me that if I choose carrots I’ll have a heart attack, then almost certainly I’m not in a freak world, but in a “normal” world where there is a causal mechanism (as common sense would call it). But the point stands that there is no necessity for a causal mechanism so that c) can be true and the game can be coherent. (Again, this point only stands as long as one’s definition of causal mechanism excludes the freak case.)
Seems like there are two possible cases. Either:
a) There is a causal mechanism
b) None of the reasoning you might sensibly make actually works.
Since the reasoning only works in the causal mechanism case, the existence of the freak world case doesn’t actually make any difference, so we’re back to the case where we have a causal mechanism and where RichardKennaway has explained everything far better than I have.
Thanks for the comment!
I think there might be a misunderstanding. Although I don’t believe it to be impossible that a gene causes you to think in specific ways, in the setting of the game such a mechanism is not required. You can also imagine a game where Omega predicts that those who pick a carrot out of a basket of vegetables are the ones that will die shortly of a heart attack. As long as we believe in Omega’s forecasting power, its statements are relevant even if we cannot point at any underlying causal mechanisms. As long as the predicted situation is logically possible (here, all agents that pick the carrot die), we don’t need to reject Omega’s prediction just because such a compilation of events would be unlikely. Though we might call Omega’s predictions into question. Still, as long as we believe in its forecasting power (after such a update), we have to take the prediction into account. Hence, the A,B-Game holds even if you don’t know of any causal connection between the genes and the behaviour, we only need a credible Omega.
It is required. If Omega is making true statements, they are (leaving aside those cases where someone is made aware of the prediction before choosing) true independently of Omega making them. That means that everyone with gene A makes choice A and everyone with gene B makes choice B. This strong entanglement implies the existence of some sort of causal connection, whether or not Omega exists.
More generally, I think that every one of those problems would be made clear by exhibiting the causal relationships that are being presumed to hold. Here is my attempt.
For the School Mark problem, the causal diagram I obtain from the description is one of these:
or
For the first of these, the teacher has waived the requirement of actually sitting the exam, and the student needn’t bother. In the second, the pupil will not get the marks except by studying for and taking the exam. See also the decision problem I describe at the end of this comment.
For Newcomb we have:
(ETA: the second down arrow should go from “contents of boxes” to “payoff”. Apparently Markdown’s code mode isn’t as code-modey as I expected.)
The hypotheses prevent us from performing surgery on this graph to model do(person’s decision). The do() operator requires deleting all in-edges to the node operated on, making it causally independent of all of its non-descendants in the graph. The hypotheses of Newcomb stipulate that this cannot be done: every consideration you could possibly employ in making a decision is assumed to be already present in the personal qualities that Omega’s prediction is based on.
A-B:
or:
or both together.
Here, it may be unfortunate to discover oneself making choice B, but by the hypotheses of this problem, you have no choice. As with Newcomb, causal surgery is excluded by the problem. To the extent that your choice is causally independent of the given arrow, to that extent you can ignore lifespan in making your choice—indeed, it is to that extent that you have a choice.
For Solomon’s Problem (which, despite the great length of the article, you didn’t set out) the diagram is:
This implies that while it may be unfortunate for Solomon to discover adulterous desires, he will not make himself worse off by acting on them. This differs from A-B because we are given some causal mechanisms, and know that they are not deterministic: an uncharismatic leader still has a choice to make about adultery, and to the extent that it is causally independent of the lack of charisma, it can be made, without regard to the likelihood of overthrow.
Similarly CGTA:
and the variant:
(ETA: the arrow from “chew gun” to “throat abcesses” didn’t come out very well.)
in which chewing gum is protective against throat abscesses, and positively to be recommended.
Newcomb’s Soda:
Here, your inclination to choose a flavour of ice-cream is informative about the $1M prize, but the causal mechanism is limited to experiencing a preference. If you would prefer $1K to a chocolate ice-cream then you can safely choose vanilla.
Finally, here’s another decision problem I thought of. Unlike all of the above, it requires no sci-fi hypotheses, real-world examples exist everywhere, and correctly solving them is an important practical skill.
I want to catch a train in half an hour. I judge that this is enough time to get to the station, buy a ticket, and board the train. Based on a large number of similar experiences in the past, I can confidently predict that I will catch the train. Since I know I will catch the train, should I actually do anything to catch the train?
The general form of this problem can be applied to many others. I predict that I’m going to ace an upcoming exam. Should I study? I predict I’ll win an upcoming tennis match. Should I train for it? I predict I’ll complete a piece of contract work on time. Should I work on it? I predict that I will post this. Should I click the “Comment” button?
I think it’s clear that Pallas had the first diagram in mind, and his point was exactly that the rational thing to do is to study despite the fact that the mark has already been written down. I agree with this.
Think of the following three scenarios:
A: No prediction is made and the final grade is determined by the exam performance.
B: A perfect prediction is made and the final grade is determined by the exam performance.
C: A perfect prediction is made and the final grade is based on the prediction.
Clearly, in scenario A the student should study. You are saying that in scenario C, the rational thing to do is not studying. Therefore, you think that the rational decision differs between either A and B, or between B and C. Going from A to B, why should the existence of someone who predicts your decision (without you knowing the prediction!) affect which decision the rational one is? That the final mark is the same in B and C follows from the very definition of a “perfect prediction”. Since each possible decision gives the same final mark in B and C, why should the rational decision differ?
In all three scenarios, the mapping from the set of possible decisions to the set of possible outcomes is identical—and this mapping is arguably all you need to know in order to make the correct decision. ETA: “Possible” here means “subjectively seen as possible”.
By deciding whether or not to learn, you can, from your subjective point of view, “choose” wheter you were determined to learn or not.
My first diagram is scenario C and my second is scenario B. In the first diagram there is no (ETA: causal) dependence of the final mark on exam performance. I think pallas’ intended scenario was more likely to be B: the mark does (ETA: causally) depend on exam performance and has been predicted. Since in B the mark depends on final performance it is necessary to study and take the exam.
In the real world, where teachers do not possess Omega’s magic powers, teachers may very well be able to predict pretty much how their students will do. For that matter, the students themselves can predict how they will do, which transforms the problem into the very ordinary, non-magical one I gave at the end of my comment. If you know how well you will do on the exam, and want to do well on it, should you (i.e. is it the correct decision to) put in the work? Or for another example of topical interest, consider the effects of genes on character.
Unless you draw out the causal diagrams, Omega is just magic: an imaginary phenomenon with no moving parts. As has been observed by someone before on LessWrong, any decision theory can be defeated by suitably crafted magic: Omega fills the boxes, or whatever, in the opposite way to whatever your decision theory will conclude. Problems of that sort offer little insight into decision theory.
Those who pick a carrot after hearing Omega’s prediction, or without hearing the prediction? Those are two very different situations, and I am not sure which one you meant.
If some people even after hearing the Omega’s prediction pick the carrot and then die of a heart attack, there must be something very special about them. They are suicidal, or strongly believe that Omega is wrong and want to prove it, or some other confusion.
If people who without hearing the Omega’s prediction pick the carrot and die, that does not mean they would have also picked the carrot if they were warned in advance. So saying “we should also press A here” provides no actionable advice about how people should behave, because it only works for people who don’t know it.
That’s a good point. I agree with you that it is crucial to keep apart those two situations. This is exactly what I was trying to address considering Newcomb’s Problem and Newcomb’s Soda. What do the agents (previous study-subjects) know? It seems to me that the games aren’t defined precise enough.
Once we specify a game in a way that all the agents hear Omega’s prediction (like in Newcomb’s Problem), the prediction provides actionable advice as all the agents belong to the same reference class. If we, and we alone, know about a prediction (whereas other agents don’t) the situation is different and the actionable advice is not provided anymore, at least not to the same extent.
When I propose a game where Omega predicts whether people pick carrots or not and I don’t specify that this only applies to those who don’t know about the prediction then I would not assume prima facie that the prediction only applies to those who don’t know about the prediction. Without further specification, I would assume that it applies to “people” which is a superset of “people who know of the prediction”.
We believe in the forecasting power, but we are uncertain as to what mechanism that forecasting power is taking advantage of to predict the world.
analogously, I know Omega will defeat me at Chess, but I do not know which opening move he will play.
In this case, the TDT decision depends critically on which causal mechanism underlies that forecasting power. Since we do not know, we will have to apply some principles for decision under uncertainty, which will depend on the payoffs, and on other features of the situation. The EDT decision does not. My intuitions and, I believe, the intuitions of many other commenters here, are much closer to the TDT approach than the EDT approach. Thus your examples are not very helpful to us—they lump things we would rather split, because our decisions in the sort of situation you described would depend in a fine-grained way on what causal explanations we found most plausible.
Suppose it is well-known that the wealthy in your country are more likely to adopt a certain distinctive manner of speaking due to the mysterious HavingRichParents gene. If you desire money, could you choose to have this gene by training yourself to speak in this way?
I agree that it is challenging to assign forecasting power to a study, as we’re uncertain about lots of background conditions. There is forecasting power to the degree that the set A of all variables involved with previous subjects allow for predictions about the set A’ of variables involved in our case. Though when we deal with Omega who is defined to make true predictions, then we need to take this forecasting power into account, no matter what the underlying mechanism is. I mean, what if Omega in Newcomb’s Problem was defined to make true predictions and you don’t know anything about the underlying mechanism? Wouldn’t you one-box after all? Let’s call Omega’s prediction P and the future event F. Once Omega’s prediction are defined to be true, we can denote the following logical equivalences: P(1 boxing) <--> F(1 boxing) and P(2 boxing) <--> P(2 boxing). Given this conditions, it impossible to 2-box when box B is filled with a million dollars (you could also formulate it in terms of probabilities where such an impossible event would have the probability of 0). I admit that we have to be cautious when we deal with instances that are not defined to make true predictions.
My answer depends on the specific set-up. What exactly do we mean with “It is well-known”? It doesn’t seem to be a study that would describe the set A of all factors involved which we then could use to derive A’ that applied to our own case. Unless we define “It is well-known” as a instance that allows for predictions in the direction A --> A’, I see little reason to assume a forecasting power. Without forecasting power, screening off applies and it would be foolish to train the distinctive manner of speaking. If we specified the game in a way that there is forecasting power at work (or at least we had reason to believe so), depending on your definition of choice (I prefer one that is devoid of free will) you can or cannot choose the gene. These kind of thoughts are listed here or in the section “Newcomb’s Problem’s Problem of Free Will” in the post.
Suppose I am deciding now whether to one-box or two-box on the problem. That’s a reasonable supposition, because I am deciding now whether to one-box or two-box. There are a couple possibilities for what Omega could be doing:
Omega observes my brain, and predicts what I am going to do accurately.
Omega makes an inaccurate prediction, probabilistically independent from my behavior.
Omega modifies my brain to a being it knows will one-box or will two-box, then makes the corresponding prediction.
If Omega uses predictive methods that aren’t 100% effective, I’ll treat it as combination of case 1 and 2. If Omega uses very powerful mind-influencing technology that isn’t 100% effective, I’ll treat it as a combination of case 2 and 3.
In case 1 , I should decide now to one-box. In case 2, I should decide now to two-box. In case 3, it doesn’t matter what I decide now.
If Omega is 100% accurate, I know for certain I am in case 1 or case 3. So I should definitely one-box. This is true even if case 1 is vanishingly unlikely.
If Omega is even 99.9% accurate, then I am in some combination of case 1, case 2, or case 3. Whether I should decide now to one-box or two-box depends on the relative probability of case 1 and case 2, ignoring case 3. So even if Omega is very accurate, ensuring that the probability of case 2 is small, if the probability of case 1 is even smaller, I should decide now to two-box.
I mean, I am describing a very specific forecasting technique that you can use to make forecasts right now. Perhaps a more precise version is, you observer children in one of two different preschools, and observe which school they are in. You observe that almost 100% of the children in one preschool end up richer than the children in the other preschool. You are then able to forecast that future children observed in preschool A will grow up to be rich, and future children observed in preschool B will grow up to be poor. You then have a child. Should you bring them to preschool A? (Here I don’t mean have them attend the school. They can simply go to the building at whatever time of day the study was conducted, then leave. That is sufficient to make highly accurate predictions, after all!)
I don’t really know what you mean by “the set A of all factors involved”
If the scenario you describe is coherent, there has to be a causal mechanism, even if you don’t know what it is. If Omega is a perfect predictor, he can’t predict that carrot-choosers have heart attacks unless carrot-choosers have heart attacks.
I think I agree. But I would formulate it otherwise:
i) Omega’s prediction are true. ii) Omega predicts that carrot-choosers have heart attacks.
c) Therefore, carrot-choosers have heart attacks.
As soon as you accept i), c) follows if we add ii). I don’t know how you define “causal mechanism”. But I can imagine a possible world where no biological mechanism connects carrot-choosing with heart attacks but where “accidentally” all the carrot-choosers have heart-attacks (Let’s imagine running worlds on a computer countless times. One day we might observe such a freak world). Then c) would be true without there being some sort of “causal mechanism” (as you might define it I suppose?). If you say that in such a freak world carrot-choosing and heart attacks are causally connected, then I would agree that c) can only be true if there is a underlying causal mechanism.
But in that case, when Omega tells me that if I choose carrots I’ll have a heart attack, then almost certainly I’m not in a freak world, and actually Omega is wrong.
Assuming i), I would rather say that when Omega tells me that if I choose carrots I’ll have a heart attack, then almost certainly I’m not in a freak world, but in a “normal” world where there is a causal mechanism (as common sense would call it). But the point stands that there is no necessity for a causal mechanism so that c) can be true and the game can be coherent. (Again, this point only stands as long as one’s definition of causal mechanism excludes the freak case.)
Seems like there are two possible cases. Either: a) There is a causal mechanism b) None of the reasoning you might sensibly make actually works.
Since the reasoning only works in the causal mechanism case, the existence of the freak world case doesn’t actually make any difference, so we’re back to the case where we have a causal mechanism and where RichardKennaway has explained everything far better than I have.