But there is no possible world with a perfect predictor, unless it has a perfect track record by chance. More obviously, there is no possible world in which we can deduce, from a finite number of observations, that a predictor is perfect. The Newcomb paradox requires the decider to know, with certainty, that Omega is a perfect predictor. That hypothesis is impossible, and thus inadmissible; so any argument in which something is deduced from that fact is invalid.
The argument goes through on probabilities of each possible world, the limit toward perfection is not singular. given the 1000:1 reward ratio, for any predictor who is substantially better than chance once ought to one-box to maximize EV. Anyway, this is an old argument where people rarely manage to convince the other side.
Take a possible world in which the predictor is perfect (meaning: they were able to make a prediction, and there was no possible extension of that world’s trajectory in which what I will actually do deviates from what they have predicted). In that world, by definition, I no longer have a choice. By definition I will do what the predictor has predicted. Whatever has caused what I will do lies in the past of the prediction, hence in the past of the current time point. There is no point in asking myself now what I should do as I have no longer causal influence on what I will do. I can simply relax and watch myself doing what I have been caused to do some time before. I can of course ask myself what might have caused my action and try to predict myself from that what I will do. If I come to believe that it was myself who decided at some earlier point in time what I will do, then I can ask myself what I should have decided at that earlier point in time. If I believe that at that earlier point in time I already knew that the predictor would act in the way it did, and if I believe that I have made the decision rationally, then I should conclude that I have decided to one-box.
The original version of Newcomb’s paradox in Nozick 1969 is not about a perfect predictor however. It begins with (1) “Suppose a being in whose power to predict your choices you have enormous confidence.… You know that this being has often correctly predicted your choices in the past (and has never, so far as you know, made an incorrect prediction about your choices), and furthermore you know that this being has often correctly predicted the choices of other people, many of whom are similar to you, in the particular situation to be described below”. So the information you are given is explicitly only about things from the past (how could it be otherwise). It goes on to say (2) “You have a choice between two actions”. Information (2) implies that what I will do has not been decided yet and I still have causal influence on what I will do. Hence the information what I will do cannot have been available to the predictor. This implies that the predictor cannot have made a perfect prediction about my behaviour. Indeed nothing in (1) implies that they have, the information given is not about my future action at all. After I will have made my decision, it might turn out, of course, that it happens to coincides with what the predictor has predicted. But that is irrelevant for my choice as it would only imply that the predictor will have been lucky this time. What should I make of information (1)? If I am confident that I still have a choice, that question is of no significance for the decision problem at hand and I should two-box. If I am confident that I don’t have a choice but have decided already, the reasoning of the previous paragraph applies and I should hope to observe that I will one-box.
What if I am unsure whether or not I still have a choice? I might have the impression that I can try to move my muscles this way or that way, without being perfectly confident that they will obey. What action should I then decide to try? I should decide to try two-boxing. Why? Because that decision is the dominant strategy: if it turns out that indeed I can decide my action now, then we’re in a world where the predictor was not perfect but merely lucky and in that world two-boxing is dominant; if it instead turns out that I was not able to override my earlier decision at this point, then we’re in a world where what I try now makes no difference. In either case, trying to two-box is undominated by any other strategy.
In reply to your first point, I agree, in a deterministic world with perfect predictors the whole question is moot. I think we agree there.
Also, yes, assuming “you have a choice between two actions”, what you will do has not been decided by you yet. Which is different from “Hence the information what I will do cannot have been available to the predictor.” If the latter statement is correct, then how can could have “often correctly predicted the choices of other people, many of whom are similar to you, in the particular situation”? Presumably some information about your decision-making process is available to the predictor in this particular situation, or else the problem setup would not be possible, would it? If you think that you are a very special case, and other people like you are not really like you, then yes, it makes sense to decide that you can get lucky and outsmart the predictor, precisely because you are special. If you think that you are not special, and other people in your situation thought the same way, two-boxed and lost, then maybe your logic is not airtight and your conclusion to two-box is flawed in some way that you cannot quite put your finger on, but the experimental evidence tells you that it is. I cannot see a third case here, though maybe I am missing something. Either you are like others, and so one-boxing gives you more money than two boxing, or you are special and not subject to the setup at all, in which case two-boxing is a reasonable approach.
I should decide to try two-boxing. Why? Because that decision is the dominant strategy: if it turns out that indeed I can decide my action now, then we’re in a world where the predictor was not perfect but merely lucky and in that world two-boxing is dominant
Right, that is, I guess, the third alternative: you are like other people who lost when two-boxing, but they were merely unlucky, the predictor did not have any predictive powers after all. Which is a possibility: maybe you were fooled by a clever con or dumb luck. Maybe you were also fooled by a clever con or dumb luck when the predictor “has never, so far as you know, made an incorrect prediction about your choices”. Maybe this all led to this moment, where you finally get to make a decision, and the right decision is to two-box and not one-box, leaving money on the table.
I guess in a world where your choice is not predetermined and you are certain that the predictor is fooling you or is just lucky, you can rely on using the dominant strategy, which is to two-box.
So, the question is, what kind of a world you think you live in, given Nozick’s setup? The setup does not say it explicitly, so it is up to you to evaluate the probabilities (which also applies to a deterministic world, only your calculation would also be predetermined).
What would a winning agent do? Look at other people like itself who won and take one box, or look at other people ostensibly like itself and who nevertheless lost and two-box still?
I know what kind of an agent I would want to be. I do not know what kind of an agent you are, but my bet is that if you are the two-boxing kind, then you will lose when push comes to shove, like all the other two-boxers before you, as far as we both know.
“Hence the information what I will do cannot have been available to the predictor.” If the latter statement is correct, then how can could have “often correctly predicted the choices of other people, many of whom are similar to you, in the particular situation”?
There’s many possible explanations for this data. Let’s say I start my analysis with the model that the predictor is guessing, and my model attaches some prior probability for them guessing right in a single case. I might also have a prior about the likelihood of being lied about the predictor’s success rate, etc. Now I make the observation that I am being told the predictor was right every single time in a row. Based on this incoming data, I can easily update my beliefs about what happened in the previous prediction excercises: I will conclude that (with some credence) the predictor was guessed right in each individual case or that (also with some credence) I am being lied to about their prediction success. This is all very simple Bayesian updating, no problem at all. As long as my prior beliefs assign nonzero credence to the possibility that the predictor guesses right (and I see not reason why that shouldn’t be a possibility), I don’t need to assign any posterior credence to the (physically impossible) assumption that they could actually foretell the actions.
Let’s say I start my analysis with the model that the predictor is guessing, and my model attaches some prior probability for them guessing right in a single case. I might also have a prior about the likelihood of being lied about the predictor’s success rate, etc. Now I make the observation that I am being told the predictor was right every single time in a row. Based on this incoming data, I can easily update my beliefs about what happened in the previous prediction excercises: I will conclude that (with some credence) the predictor was guessed right in each individual case or that (also with some credence) I am being lied to about their prediction success. This is all very simple Bayesian updating, no problem at all.
Right! If I understand your point correctly, given a strong enough prior for the predictor being lucky or deceptive, it would have to be a lot of evidence to change one’s mind, and the evidence would have to be varied. This condition is certainly not satisfied by the original setup. If your extremely confident prior is that foretelling one’s actions is physically impossible, then the lie/luck hypothesis would have to be much more likely than changing your mind about physical impossibility. That makes perfect sense to me.
I guess one would want to simplify the original setup a bit. What if you had full confidence that the predictor is not a trickster? Would you one-box or two-box? To get the physical impossibility out of the way, they do not necessarily have to predict every atom in your body and mind, just observe you (and read your LW posts, maybe) to Sherlock-like make a very accurate conclusion about what you would decide.
Another question: what kind of experiment, in addition to what is in the setup, would change your mind?
Can you please explain the “zero-probability possible world”?
There is no possible world with a perfect predictor where a two-boxer wins without breaking the condition of it being perfect.
But there is no possible world with a perfect predictor, unless it has a perfect track record by chance. More obviously, there is no possible world in which we can deduce, from a finite number of observations, that a predictor is perfect. The Newcomb paradox requires the decider to know, with certainty, that Omega is a perfect predictor. That hypothesis is impossible, and thus inadmissible; so any argument in which something is deduced from that fact is invalid.
The argument goes through on probabilities of each possible world, the limit toward perfection is not singular. given the 1000:1 reward ratio, for any predictor who is substantially better than chance once ought to one-box to maximize EV. Anyway, this is an old argument where people rarely manage to convince the other side.
Take a possible world in which the predictor is perfect (meaning: they were able to make a prediction, and there was no possible extension of that world’s trajectory in which what I will actually do deviates from what they have predicted). In that world, by definition, I no longer have a choice. By definition I will do what the predictor has predicted. Whatever has caused what I will do lies in the past of the prediction, hence in the past of the current time point. There is no point in asking myself now what I should do as I have no longer causal influence on what I will do. I can simply relax and watch myself doing what I have been caused to do some time before. I can of course ask myself what might have caused my action and try to predict myself from that what I will do. If I come to believe that it was myself who decided at some earlier point in time what I will do, then I can ask myself what I should have decided at that earlier point in time. If I believe that at that earlier point in time I already knew that the predictor would act in the way it did, and if I believe that I have made the decision rationally, then I should conclude that I have decided to one-box.
The original version of Newcomb’s paradox in Nozick 1969 is not about a perfect predictor however. It begins with (1) “Suppose a being in whose power to predict your choices you have enormous confidence.… You know that this being has often correctly predicted your choices in the past (and has never, so far as you know, made an incorrect prediction about your choices), and furthermore you know that this being has often correctly predicted the choices of other people, many of whom are similar to you, in the particular situation to be described below”. So the information you are given is explicitly only about things from the past (how could it be otherwise). It goes on to say (2) “You have a choice between two actions”. Information (2) implies that what I will do has not been decided yet and I still have causal influence on what I will do. Hence the information what I will do cannot have been available to the predictor. This implies that the predictor cannot have made a perfect prediction about my behaviour. Indeed nothing in (1) implies that they have, the information given is not about my future action at all. After I will have made my decision, it might turn out, of course, that it happens to coincides with what the predictor has predicted. But that is irrelevant for my choice as it would only imply that the predictor will have been lucky this time. What should I make of information (1)? If I am confident that I still have a choice, that question is of no significance for the decision problem at hand and I should two-box. If I am confident that I don’t have a choice but have decided already, the reasoning of the previous paragraph applies and I should hope to observe that I will one-box.
What if I am unsure whether or not I still have a choice? I might have the impression that I can try to move my muscles this way or that way, without being perfectly confident that they will obey. What action should I then decide to try? I should decide to try two-boxing. Why? Because that decision is the dominant strategy: if it turns out that indeed I can decide my action now, then we’re in a world where the predictor was not perfect but merely lucky and in that world two-boxing is dominant; if it instead turns out that I was not able to override my earlier decision at this point, then we’re in a world where what I try now makes no difference. In either case, trying to two-box is undominated by any other strategy.
Sorry, could not reply due to rate limit.
In reply to your first point, I agree, in a deterministic world with perfect predictors the whole question is moot. I think we agree there.
Also, yes, assuming “you have a choice between two actions”, what you will do has not been decided by you yet. Which is different from “Hence the information what I will do cannot have been available to the predictor.” If the latter statement is correct, then how can could have “often correctly predicted the choices of other people, many of whom are similar to you, in the particular situation”? Presumably some information about your decision-making process is available to the predictor in this particular situation, or else the problem setup would not be possible, would it? If you think that you are a very special case, and other people like you are not really like you, then yes, it makes sense to decide that you can get lucky and outsmart the predictor, precisely because you are special. If you think that you are not special, and other people in your situation thought the same way, two-boxed and lost, then maybe your logic is not airtight and your conclusion to two-box is flawed in some way that you cannot quite put your finger on, but the experimental evidence tells you that it is. I cannot see a third case here, though maybe I am missing something. Either you are like others, and so one-boxing gives you more money than two boxing, or you are special and not subject to the setup at all, in which case two-boxing is a reasonable approach.
Right, that is, I guess, the third alternative: you are like other people who lost when two-boxing, but they were merely unlucky, the predictor did not have any predictive powers after all. Which is a possibility: maybe you were fooled by a clever con or dumb luck. Maybe you were also fooled by a clever con or dumb luck when the predictor “has never, so far as you know, made an incorrect prediction about your choices”. Maybe this all led to this moment, where you finally get to make a decision, and the right decision is to two-box and not one-box, leaving money on the table.
I guess in a world where your choice is not predetermined and you are certain that the predictor is fooling you or is just lucky, you can rely on using the dominant strategy, which is to two-box.
So, the question is, what kind of a world you think you live in, given Nozick’s setup? The setup does not say it explicitly, so it is up to you to evaluate the probabilities (which also applies to a deterministic world, only your calculation would also be predetermined).
What would a winning agent do? Look at other people like itself who won and take one box, or look at other people ostensibly like itself and who nevertheless lost and two-box still?
I know what kind of an agent I would want to be. I do not know what kind of an agent you are, but my bet is that if you are the two-boxing kind, then you will lose when push comes to shove, like all the other two-boxers before you, as far as we both know.
There’s many possible explanations for this data. Let’s say I start my analysis with the model that the predictor is guessing, and my model attaches some prior probability for them guessing right in a single case. I might also have a prior about the likelihood of being lied about the predictor’s success rate, etc. Now I make the observation that I am being told the predictor was right every single time in a row. Based on this incoming data, I can easily update my beliefs about what happened in the previous prediction excercises: I will conclude that (with some credence) the predictor was guessed right in each individual case or that (also with some credence) I am being lied to about their prediction success. This is all very simple Bayesian updating, no problem at all. As long as my prior beliefs assign nonzero credence to the possibility that the predictor guesses right (and I see not reason why that shouldn’t be a possibility), I don’t need to assign any posterior credence to the (physically impossible) assumption that they could actually foretell the actions.
Right! If I understand your point correctly, given a strong enough prior for the predictor being lucky or deceptive, it would have to be a lot of evidence to change one’s mind, and the evidence would have to be varied. This condition is certainly not satisfied by the original setup. If your extremely confident prior is that foretelling one’s actions is physically impossible, then the lie/luck hypothesis would have to be much more likely than changing your mind about physical impossibility. That makes perfect sense to me.
I guess one would want to simplify the original setup a bit. What if you had full confidence that the predictor is not a trickster? Would you one-box or two-box? To get the physical impossibility out of the way, they do not necessarily have to predict every atom in your body and mind, just observe you (and read your LW posts, maybe) to Sherlock-like make a very accurate conclusion about what you would decide.
Another question: what kind of experiment, in addition to what is in the setup, would change your mind?