In the case of facing the problem for the first time, in real-time, a person can only 1 box by ignoring the concept of a “dominant” strategy. Or by not really understanding the problem (the boxes really are there with either $1 million in or not and you can’t actually change that: Omega has no time travel or reverse causation powers). Or by having a utility something other than money, which is not in itself irrational, but goes against the statement of the problem.
For instance, I think an astute rational thinker could (perhaps) argue in real-time “this looks like a sort of disguised moral problem; Omega seems to be implicitly testing my ethics i.e. testing my self-restraint versus my greed. So perhaps I should take 1”. However, at that stage the 1-boxer probably values acting ethicallly more than being $1000 richer. Or there might be other rational preferences for not 2-boxing such as getting a really strong urge to 1-box at the time, and prefering to satisfy the urge than to be $1000 richer. Or knowing that if you 2-box you’ll worry for the rest of your life whether that was the right thing, and this is just not worth $1000. I think these are well-known “solutions” which all shift the utility function and hence sidestep the problem.
I understand the argument, I just don’t understand what the novelty of the problem has to do with it. That is, it seems the same problem arises whether it’s a new problem or not.
You’re of course right that there’s no timetravel involved. If I’m the sort of person who two-boxes, Omega will put $1000 in. If I’m the sort of person who one-boxes, Omega will put $1000000 in. (If I’m the sort of person whose behavior can’t be predicted ahead of time, then Omega is lying to me.)
So, what sort of person am I? Well, geez, how should I know? Unlike Omega, I’m not a reliable predictor of my behavior. The way I find out what sort of person I am is by seeing what I do in the situation.
You seem to be insisting on there being a reason for my one-boxing beyond that (like “I think Omega is testing my ethics” or “I precommitted to one-boxing” or some such thing). I guess that’s what I don’t understand, here. Either I one-box, or I two-box. My reasons don’t matter.
You seem to be insisting on there being a reason for my one-boxing beyond that (like “I think Omega is testing my ethics” or “I precommitted to one-boxing” or some such thing). I guess that’s what I don’t understand, here. Either I one-box, or I two-box.
Indeed. “I like money” seems like a good enough reason to one box without anything more complicated!
I call it “I take free monies theory!” I don’t need a theoretical framework to do that. At this point in time there isn’t a formal decision theory that results in all the same decisions that I endorse—basically because the guys are still working out the kinks in UDT and formalization is a real bitch sometimes. They haven’t figured out a way to generalize the handling of counterfactuals the way I would see them handled.
I’m going to track what’s happened on the other threads discussing Newcomb’s paradox, since I suspect there’s quite a lot of repetition or overlap. Before signing off though, does anyone here have a view on whether it matters whether Omega is a perfect predictor, or just a very goodpredictor?
Personally, I think it does matter, and matters rather a lot. The Newcomb problem can be stated either way.
Let’s start with the “very good” predictor case, which I think is the most plausible one, since it just requires Omega to be a good judge of character.
Consider Alf, who is the “sort of person who 2-boxes”. Let’s say he has >99% chance of 2-boxing and <1% chance of 1 boxing (but he’s not totally deterministic and has occasional whims, lapses or whatever). If Omega is a good predictor based on general judge of character, then Omega won’t have put the $1 million in Alf’s boxes. So in the unlikely event that Alf actually does take just the one box then he’ll win nothing at all. This means that if Alf knows he’s basically a 2-boxer (he assigns something like 99% credence to the event that he 2-boxes) and knows that Omega is a good but imperfect predictor, Alf has a rationale for remaining a 2-boxer. This holds under both causal decision theory and evidential decision theory. The solution of being a 2-boxer is reflectively-stable; Alf can know he’s like that and stay like that.
But now consider Beth who’s the sort of person who 1-boxes. In the unlikely event that she takes both boxes, Omega will still have put the $1 million in, and so Beth will win $1001000. But now if Beth knows she’s a 1-boxer (say assigns 99% credence to taking 1 box), and again knows that Omega is good but imperfect, this puts her in an odd self-assessment position, since it seems she has a clear rationale to take both boxes (again under both evidential and causal decision thery). If she remains a 1-boxer, then she is essentially projecting of herself that she has only 1% chance of making a $-optimal choice i.e. she believes of herself either that she is not a rational utility maximiser, or her utility function is different from $. If Beth truly is a $ utility maximiser, then Beth’s position doesn’t look reflectively stable; though she could maybe have “trained” herself to act in this way in Newcomb situations and is aware of the pre-conditioning.
Finally, consider Charles, who has never heard of the Newcomb problem, and doesn’t know whether he will 1-box or 2-box. However, Charles is sure he is a $-utility maximizer. If he is causal decision theorist, he will quickly decide to 2 box, and so will model himself like Alf. If he’s an evidential decision theorist, then he will initially assign some probability to either 1 or 2 boxing, calculate that his expected utility is higher by 1 boxing, and then start to model himself like Beth. But then, he will realize this self-model is reflectively unstable, since it requires him to model himself as something other than a $ utility maximiser, and he’s sure that’s what he is. After flapping about a bit, he will realze that the only reflectively stable solution is to model himself like Alf, and this makes it better for him to 2 box. Thinking about the problem too much forces him to 2 box.
In the event that Omega is a perfect predictor, and the box-chooser knows this, then things get messier, because now the only reflectively-stable solution for the evidential decision theorist is to 1-box. (Beth thinks “I have 99% chance of 1-boxing, and in the rare event that I decide to 2-box, Omega will have predicted this, and my expected utility will be lower; so I still have a rationale to 1 box!). What about the causal decision theorist though? One difficulty is how the causal theorist can really believe in Omega as a perfect predictor without also believing in some form of retrograde causation or time travel. This seems a strange set of beliefs to hold in combination. If the causal decision theorist squares the circle by assigning some sort of pre-cognitive faculty to Omega, or at least assigning some non-trivial credence to such a pre-cog faculty, then he can reason that there is after all (with some credence) a genuine (if bizarre) causal relation between what he chooses, and what goes in the box, so he should 1-box. If he remains sure that there is no such causal relation, then he should 2 box. But we should note that the 2 box position is distinctly weaker in this case than in the “good but imperfect” case.
It is not clear to me that Alf’s position as described here is stable.
You say Alf knows Omega is a good (but imperfect) predictor. Just for specificity, let’s say Alf has (and believes he has) .95 confidence that Omega can predict Alf’s box-selection behavior with .95 accuracy. (Never mind how he arrived at such a high confidence; perhaps he’s seen several hundred trials.) And let’s say Alf values money.
Given just that belief, Alf ought to be able to reason as follows: “Suppose I open just one box. In that case, I expect with ~.9 confidence that Omega placed $1m+$1k in the box. OTOH, suppose I open both boxes. In that case, I expect with ~.9 confidence that Omega placed $1k in the box.”
For simplicity, let’s assume Alf believes Omega always puts either $1k or $1m+$1k in the boxes (as opposed to, say, putting in an angry bobcat). So if Alf has .9 confidence in there being $1k in the boxes, he has .1 confidence in ($1m+1k) in the boxes.
So, Alf ought to be able to conclude that one-boxing has an expected value of (.9 $1m + .1 $1k) and two-boxing has an expected value of (.9 $1k + .1 $1m+1k). The expected value of one-boxing is greater than that of two-boxing, so Alf ought to one-box.
So far, so good. But you also say that Alf has .99 confidence that Alf two-boxes… that is, he has .99 confidence that he will take the lower-value choice. (Again, never mind how he arrived at such high confidence… although ironically, we are now positing that Alf is a better predictor than Omega is.)
Well, this is a pickle! There do seem to be some contradictions in Alf’s position.
Perhaps I’m missing some key implications of being a causal vs. an evidential decision theorist, here. But I don’t really see why it should matter. That just affects how Alf arrived at those various confidence estimates, doesn’t it? Once we know the estimates themselves, we should no longer care.
Incidentally, if Alf believes Omega is a perfect predictor (that is, Alf has .95 confidence that Omega can predict Alf’s box-selection with 1-epsilon accuracy) the situation doesn’t really change much; the EV calculation is (.95 $1m + .05 $1k) vs (.9 $1k + .05 $1m+1k), which gets you to the same place.
OK, maybe it wasn’t totally clear. Alf is very confident that he 2-boxes, since he thinks that’s the “right” answer to Newcomb’s problem. Alf is very confident that Omega is a good predictor, because he’s a good judge of character, and will spot that Alf is a 2-boxer.
Alf believes that in the rare, fluky event that he actually 1-boxes, then Omega won’t have predicted that, since it is so out of character for Alf. Alf thinks Omega is a great predictor, but not a perfect predictor, and can’t foresee such rare, fluky, out-of-character events. So there still won’t be the $1 million in Alf’s boxes in the flukey event that he 1-boxes, and he will win nothing at all, not $1 million. Given this belief set, Alf should 2-box, even if he’s an evidential decision theorist rather than a causal decision theorist. The position is consistent and stable.
Alf believes that in the rare, fluky event that he actually 1-boxes, then Omega won’t have predicted that.
Ah! Yes, this clarifies matters.
Sure, if Alf believes that Omega has a .95 chance of predicting Alf will two-box regardless of whether or not he does, then Alf should two-box. Similarly, if Beth believes Omega has a .95 chance of predicting Beth will one-box regardless of whether or not she does, then she also should two-box. (Though if she does, she should immediately lower her earlier confidence that she’s the sort of person who one-boxes.)
This is importantly different from the standard Newcomb’s problem, though.
You seem to be operating under the principle that if a condition is unlikely (e.g., Alf 1-boxing) then it is also unpredictable. I’m not sure where you’re getting that from.
By way of analogy… my fire alarm is, generally speaking, the sort of thing that remains silent… if I observe it in six-minute intervals for a thousand observations, I’m pretty likely to find it silent in each case. However, if I’m a good predictor of fire alarm behavior, I don’t therefore assume that if there’s a fire, it will still remain silent.
Rather, as a good predictor of fire alarms, what my model of fire alarms tells me is that “when there’s no fire, I’m .99+ confident it will remain silent; when there is a fire, I’m .99+ confident it will make noise.” I can therefore test to see if there’s a fire and, if there is, predict it will make noise. Its noise is rare, but predictable (for a good enough predictor of fire alarm behavior).
Remember I have two models of how Omega could work.
1) Omega is in essence an excellent judge of character. It can reliably decide which of its candidates is “the sort of person who 1-boxes” and which is “the sort of person who 2-boxes”. However, if he chooser actually does something extremely unlikely and out of character, Omega will get its prediction wrong. This is a model for Omega that I could actually see working, so it is the most natural way for me to interpret Newcomb’s thought experiment.
If Omega behaves like this, then I think causal and evidential decision theory align. Both tell the chooser to 2-box, unless the chooser has already pre-committed to 1-boxing. Both imply the chooser should pre-commit to 1-boxing (if they can).
2) Omega is a perfect predictor, and always gets its predictions right. I can’t actually see how his model would work without reverse causation. If reverse causatiion is implied by the problem statement, or choosers can reasonably think it is implied, then both causal and evidential decision theory align and tell the chooser to 1-box.
From the sound of things, you are describing a third model in which Omega can not only judge character, but can also reliably decide whether someone will act out of character or not. When faced with “the sort of person who 1-boxes”, but then—out of character − 2 boxes after all, Omega will still with high probability guess correctly that the 2-boxing is going to happen, and so withhold the $ 1 million.
I tend to agree that in this third model causal and evidential decision theory may become decoupled, but again I’m not really sure how this model works, or whether it requires backward causation again. I think it could work if the causal factors leading the chooser to act “out of character” in the particular case are already embedded in the chooser’s brain state when scanned by Omega, so at that stage it is already highly probable that the chooser will act out of character this time. But the model won’t work if the factors causing out of character behaviour arise because of very rare, random, brain events happening after the scanning (say a few stray neurons fire which in 99% of cases wouldn’t fire after the scanned brain state, and these cause a cascade eventually leading to a different choice). Omega can’t predict that type of event without being a pre-cog.
Thanks anyway though; you’ve certainly made me think about the problem a bit further...
So, what does it mean for a brain to do one thing 99% of the time and something else 1% of the time?
If the 1% case is a genuinely random event, or the result of some mysterious sort of unpredictable free will, or otherwise something that isn’t the effect of the causes that precede it, and therefore can’t be predicted short of some mysterious acausal precognition, then I agree that it follows that if Omega is a good-but-not-perfect predictor, then Omega cannot predict the 1% case, and Newcomb’s problem in its standard form can’t be implemented even in principle, with all the consequences previously discussed.
Conversely, if brain events—even rare ones—are instead the effects of causes that precede them, then a good-but-not-perfect predictor can make good-but-not-perfect predictions of the 1% case just as readily as the 99% case, and these problems don’t arise.
Personally, I consider brain events the effects of causes that precede them. So if I’m the sort of person who one-boxes 99% of the time and two-boxes 1% of the time, and Omega has a sufficient understanding of the causes of human behavior to make 95% accurate predictions of what I do, then Omega will predict 95% of my (common) one-boxing as well as 95% of my (rare) two-boxing. Further, if I somehow come to believe that Omega has such an understanding, then I will predict that Omega will predict my (rare) two-boxing, and therefore I will predict that two-boxing loses me money, and therefore I will one-box stably.
So, what does it mean for a brain to do one thing 99% of the time and something else 1% of the time?
For the sake of the least convenient world assume that the brain is particularly sensitive to quantum noise. This applies in the actual world too albeit at a far, far lower rate than 1% (but hey… perfect). That leaves a perfect predictor perfectly predicting that in the branches with most of the quantum goo (pick a word) the brain will make one choice while in the others it will make the other.
In this case it becomes a matter of how the counterfactual is specified. The most appropriate one seems to be with Omega filling the large box with an amount of money proportional to how much of the brain will be one boxing. A brain that actively flips a quantum coin would then be granted a large box with half the million.
The only other obvious alternative specification of Omega that wouldn’t break the counterfactual given this this context are a hard cutoff and some specific degree of ‘probability’.
As you say the one boxing remains stable under this uncertainty and even imperfect predictors.
I’m not sure what the quantum-goo explanation is adding here.
If Omega can’t predict the 1% case (whether because it’s due to unpredictable quantum goo, or for whatever other reason… picking a specific explanation only subjects me to a conjunction fallacy) then Omega’s behavior will not reflect the 1% case, and that completely changes the math. Someone for whom the 1% case is two-boxing is then entirely justified in two-boxing in the 1% case, since they ought to predict that Omega cannot predict their two-boxing. (Assuming that they can recognize that they are in such a case. If not, they are best off one-boxing in all cases. Though it follows from our premises that they will two-box 1% of the time anyway, though they might not have any idea why they did that. That said, compatibilist decision theory makes my teeth ache.)
Anyway, yeah, this is assuming some kind of hard cutoff strategy, where Omega puts a million dollars in a box for someone it has > N% confidence will one-box.
If instead Omega puts N% of $1m in the box if Omega has N% confidence the subject will one-box, the result isn’t terribly different if Omega is a good predictor.
I’m completely lost by the “proportional to how much of the brain will be one boxing” strategy. Can you say more about what you mean by this? It seems likely to me that most of the brain neither one-boxes nor two-boxes (that is, is not involved in this choice at all) and most of the remainder does both (that is, performs the same operations in the two-boxing case as in the one-boxing case).
I’m not sure what the quantum-goo explanation is adding here.
A perfect predictor will predict correctly and perfectly that the brain both one boxes and two boxes in different Everett branches (with vastly different weights). This is different in nature to an imperfect predictor that isn’t able to model the behavior of the brain with complete certainty yet given preferences that add up to normal it requires that you use the same math. It means you do not have to abandon the premise “perfect predictor” for the probabilistic reasoning to be necessary.
I’m completely lost by the “proportional to how much of the brain will be one boxing” strategy.
How much weight the everett branches in which it one box have relative to the everett branches in which it two boxes.
Allow me to emphasise:
As you say the one boxing remains stable under this uncertainty and even imperfect predictors.
Omega can’t predict that type of event without being a pre-cog.
Assume that the person choosing the boxes is a whole brain emulation, and that Omega runs a perfect simulation, which guarantees formal identity of Omega’s prediction and person’s actual decision, even though the computations are performed separately.
So the chooser in this case is a fully deterministic system, not a real-live human brain with some chance of random firings screwing up Omega’s prediction?
Wow, that’s an interesting case, and I hadn’t really thought about it! One interesting point though—suppose I am the chooser in that case; how can I tell which simulation I am? Am I the one which runs after Omega made its prediction? Or am I the one which Omega runs in order to make its prediction, and which does have a genuine causal effect on what goes in the boxes? It seems I have no way of telling, and I might (in some strange sense) be both of them. So causal decision theory might advise me to 1-box after all.
So the chooser in this case is a fully deterministic system, not a real-live human brain with some chance of random firings screwing up Omega’s prediction?
This is more of a way of pointing out a special case that shares relevant considerations with TDT-like approach to decision theory (in this extreme identical-simulation case it’s just Hofstadter’s “superrationality”).
If we start from this case and gradually make the prediction model and the player less and less similar to each other (perhaps by making the model less detailed), at which point do the considerations that make you one-box in this edge case break? Clearly, if you change the prediction model just a little bit, correct answer shouldn’t immediately flip, but CDT is no longer applicable out-of-the-box (arguably, even if you “control” two identical copies, it’s also not directly applicable). Thus, a need for generalization that admits imperfect acausal “control” over sufficiently similar decision-makers (and sufficiently accurate predictions) in the same sense in which you “control” your identical copies.
That might give you the right answer when Omega is simulating you perfectly, but presumably you’d want to one-box when Omega was simulating a slightly lossy, non-sentient version of you and only predicted correctly 90% of the time. For that (i.e. for all real world Newcomblike problems), you need a more sophisticated decision theory.
Well no, not necessarily. Again, let’s take Alf’s view. (Note I edited this post recently to correct the display of the matrices)
Remember that Alf has a high probability of 2 boxing, and he knows this about himself. Whether he would actually do better by 1-boxing will depend how well Omega’s “mistaken” simulations are correlated with the (rare, freaky) event that Alf 1 boxes. Basically, Alf knows that Omega is right at least 90% of the time, but this doesn’t require a very sophisticated simulation at all, certainly not in Alf’s own case. Omega can run a very crude simulation, say “a clear” 2-boxer, and not fill box B (so Alf won’t get the $ 1 million. Basically, the game outcome would have a probability matrix like this:
Box B filled. Box B empty.
0. 0.99. Alf 2 boxes
0. 0.01. Alf 1 boxes
Notice that Omega has less than 1% chance of a mistaken prediction.
But, I’m sure you’re thinking, won’t Omega run a fuller simulation with 90% accuracy and produce a probability matrix like this?
Box B filled. Box B empty.
0.099. 0.891. Alf 2 boxes
0.009. 0.001. Alf 1 boxes
Well Omega could do that, but now its probability of error has gone up from 1% to 10%, so why would Omega bother?
Let’s compare to a more basic case: weather forecasting. Say I have a simulation model which takes in the current atmospheric state above a land surface, runs it forward a day, and tries to predict rain. It’s pretty good: if there is going to be rain, then the simulation predicts rain 90% of the time; if there is not going to be rain, then it predicts rain only 10% of the time. But now someone shows me a desert, and asks me to predict rain: I’m not going to use a simulation with a 10% error rate, I’m just going to say “no rain”.
So it seems in the case of Alf. Provided Alf’s chance of 1-boxing is low enough (i.e. lower than the underlying error rate of Omega’s simulations) then Omega can always do best by just saying “a clear 2-boxer” and not filling the B box. Omega may also say to himself “what an utter schmuck” but he can’t fault Alf’s application of decision theory. And this applies whether or not Alf is a causal decision theorist or an evidential decision theorist.
Incidentally, your fire alarm may be practically useless in the circumstances you describe. Depending on the relative probabilities (small probability that the alarm goes off when there is not a fire versus even smaller probability that there genuinely is a fire) then you may find that essentially all the alarms are false alarms. You may get fed up responding to false alarms and ignore them. When predicting very rare events, the prediction system has to be extremely accurate.
This is related to the analysis below about Omega’s simulation being only 90% accurate versus a really convinced 2-boxer (who has only a 1% chance of 1-boxing). Or of simulating rain in a desert.
In the case of facing the problem for the first time, in real-time, a person can only 1 box by ignoring the concept of a “dominant” strategy. Or by not really understanding the problem (the boxes really are there with either $1 million in or not and you can’t actually change that: Omega has no time travel or reverse causation powers). Or by having a utility something other than money, which is not in itself irrational, but goes against the statement of the problem.
For instance, I think an astute rational thinker could (perhaps) argue in real-time “this looks like a sort of disguised moral problem; Omega seems to be implicitly testing my ethics i.e. testing my self-restraint versus my greed. So perhaps I should take 1”. However, at that stage the 1-boxer probably values acting ethicallly more than being $1000 richer. Or there might be other rational preferences for not 2-boxing such as getting a really strong urge to 1-box at the time, and prefering to satisfy the urge than to be $1000 richer. Or knowing that if you 2-box you’ll worry for the rest of your life whether that was the right thing, and this is just not worth $1000. I think these are well-known “solutions” which all shift the utility function and hence sidestep the problem.
I understand the argument, I just don’t understand what the novelty of the problem has to do with it. That is, it seems the same problem arises whether it’s a new problem or not.
You’re of course right that there’s no timetravel involved. If I’m the sort of person who two-boxes, Omega will put $1000 in. If I’m the sort of person who one-boxes, Omega will put $1000000 in. (If I’m the sort of person whose behavior can’t be predicted ahead of time, then Omega is lying to me.)
So, what sort of person am I? Well, geez, how should I know? Unlike Omega, I’m not a reliable predictor of my behavior. The way I find out what sort of person I am is by seeing what I do in the situation.
You seem to be insisting on there being a reason for my one-boxing beyond that (like “I think Omega is testing my ethics” or “I precommitted to one-boxing” or some such thing). I guess that’s what I don’t understand, here. Either I one-box, or I two-box. My reasons don’t matter.
Indeed. “I like money” seems like a good enough reason to one box without anything more complicated!
That’s just evidential decision theory, right?
I call it “I take free monies theory!” I don’t need a theoretical framework to do that. At this point in time there isn’t a formal decision theory that results in all the same decisions that I endorse—basically because the guys are still working out the kinks in UDT and formalization is a real bitch sometimes. They haven’t figured out a way to generalize the handling of counterfactuals the way I would see them handled.
(ArisKatsaris nails it in the sibling).
Well, Newcomb’s problem is simple enough that evidential decision theory suffices.
I’m going to track what’s happened on the other threads discussing Newcomb’s paradox, since I suspect there’s quite a lot of repetition or overlap. Before signing off though, does anyone here have a view on whether it matters whether Omega is a perfect predictor, or just a very goodpredictor?
Personally, I think it does matter, and matters rather a lot. The Newcomb problem can be stated either way.
Let’s start with the “very good” predictor case, which I think is the most plausible one, since it just requires Omega to be a good judge of character.
Consider Alf, who is the “sort of person who 2-boxes”. Let’s say he has >99% chance of 2-boxing and <1% chance of 1 boxing (but he’s not totally deterministic and has occasional whims, lapses or whatever). If Omega is a good predictor based on general judge of character, then Omega won’t have put the $1 million in Alf’s boxes. So in the unlikely event that Alf actually does take just the one box then he’ll win nothing at all. This means that if Alf knows he’s basically a 2-boxer (he assigns something like 99% credence to the event that he 2-boxes) and knows that Omega is a good but imperfect predictor, Alf has a rationale for remaining a 2-boxer. This holds under both causal decision theory and evidential decision theory. The solution of being a 2-boxer is reflectively-stable; Alf can know he’s like that and stay like that.
But now consider Beth who’s the sort of person who 1-boxes. In the unlikely event that she takes both boxes, Omega will still have put the $1 million in, and so Beth will win $1001000. But now if Beth knows she’s a 1-boxer (say assigns 99% credence to taking 1 box), and again knows that Omega is good but imperfect, this puts her in an odd self-assessment position, since it seems she has a clear rationale to take both boxes (again under both evidential and causal decision thery). If she remains a 1-boxer, then she is essentially projecting of herself that she has only 1% chance of making a $-optimal choice i.e. she believes of herself either that she is not a rational utility maximiser, or her utility function is different from $. If Beth truly is a $ utility maximiser, then Beth’s position doesn’t look reflectively stable; though she could maybe have “trained” herself to act in this way in Newcomb situations and is aware of the pre-conditioning.
Finally, consider Charles, who has never heard of the Newcomb problem, and doesn’t know whether he will 1-box or 2-box. However, Charles is sure he is a $-utility maximizer. If he is causal decision theorist, he will quickly decide to 2 box, and so will model himself like Alf. If he’s an evidential decision theorist, then he will initially assign some probability to either 1 or 2 boxing, calculate that his expected utility is higher by 1 boxing, and then start to model himself like Beth. But then, he will realize this self-model is reflectively unstable, since it requires him to model himself as something other than a $ utility maximiser, and he’s sure that’s what he is. After flapping about a bit, he will realze that the only reflectively stable solution is to model himself like Alf, and this makes it better for him to 2 box. Thinking about the problem too much forces him to 2 box.
In the event that Omega is a perfect predictor, and the box-chooser knows this, then things get messier, because now the only reflectively-stable solution for the evidential decision theorist is to 1-box. (Beth thinks “I have 99% chance of 1-boxing, and in the rare event that I decide to 2-box, Omega will have predicted this, and my expected utility will be lower; so I still have a rationale to 1 box!). What about the causal decision theorist though? One difficulty is how the causal theorist can really believe in Omega as a perfect predictor without also believing in some form of retrograde causation or time travel. This seems a strange set of beliefs to hold in combination. If the causal decision theorist squares the circle by assigning some sort of pre-cognitive faculty to Omega, or at least assigning some non-trivial credence to such a pre-cog faculty, then he can reason that there is after all (with some credence) a genuine (if bizarre) causal relation between what he chooses, and what goes in the box, so he should 1-box. If he remains sure that there is no such causal relation, then he should 2 box. But we should note that the 2 box position is distinctly weaker in this case than in the “good but imperfect” case.
Try my analysis and Anna Salamon’s.
It is not clear to me that Alf’s position as described here is stable.
You say Alf knows Omega is a good (but imperfect) predictor. Just for specificity, let’s say Alf has (and believes he has) .95 confidence that Omega can predict Alf’s box-selection behavior with .95 accuracy. (Never mind how he arrived at such a high confidence; perhaps he’s seen several hundred trials.) And let’s say Alf values money.
Given just that belief, Alf ought to be able to reason as follows: “Suppose I open just one box. In that case, I expect with ~.9 confidence that Omega placed $1m+$1k in the box. OTOH, suppose I open both boxes. In that case, I expect with ~.9 confidence that Omega placed $1k in the box.”
For simplicity, let’s assume Alf believes Omega always puts either $1k or $1m+$1k in the boxes (as opposed to, say, putting in an angry bobcat). So if Alf has .9 confidence in there being $1k in the boxes, he has .1 confidence in ($1m+1k) in the boxes.
So, Alf ought to be able to conclude that one-boxing has an expected value of (.9 $1m + .1 $1k) and two-boxing has an expected value of (.9 $1k + .1 $1m+1k). The expected value of one-boxing is greater than that of two-boxing, so Alf ought to one-box.
So far, so good. But you also say that Alf has .99 confidence that Alf two-boxes… that is, he has .99 confidence that he will take the lower-value choice. (Again, never mind how he arrived at such high confidence… although ironically, we are now positing that Alf is a better predictor than Omega is.)
Well, this is a pickle! There do seem to be some contradictions in Alf’s position.
Perhaps I’m missing some key implications of being a causal vs. an evidential decision theorist, here. But I don’t really see why it should matter. That just affects how Alf arrived at those various confidence estimates, doesn’t it? Once we know the estimates themselves, we should no longer care.
Incidentally, if Alf believes Omega is a perfect predictor (that is, Alf has .95 confidence that Omega can predict Alf’s box-selection with 1-epsilon accuracy) the situation doesn’t really change much; the EV calculation is (.95 $1m + .05 $1k) vs (.9 $1k + .05 $1m+1k), which gets you to the same place.
OK, maybe it wasn’t totally clear. Alf is very confident that he 2-boxes, since he thinks that’s the “right” answer to Newcomb’s problem. Alf is very confident that Omega is a good predictor, because he’s a good judge of character, and will spot that Alf is a 2-boxer.
Alf believes that in the rare, fluky event that he actually 1-boxes, then Omega won’t have predicted that, since it is so out of character for Alf. Alf thinks Omega is a great predictor, but not a perfect predictor, and can’t foresee such rare, fluky, out-of-character events. So there still won’t be the $1 million in Alf’s boxes in the flukey event that he 1-boxes, and he will win nothing at all, not $1 million. Given this belief set, Alf should 2-box, even if he’s an evidential decision theorist rather than a causal decision theorist. The position is consistent and stable.
Is that clearer?
Ah! Yes, this clarifies matters.
Sure, if Alf believes that Omega has a .95 chance of predicting Alf will two-box regardless of whether or not he does, then Alf should two-box. Similarly, if Beth believes Omega has a .95 chance of predicting Beth will one-box regardless of whether or not she does, then she also should two-box. (Though if she does, she should immediately lower her earlier confidence that she’s the sort of person who one-boxes.)
This is importantly different from the standard Newcomb’s problem, though.
You seem to be operating under the principle that if a condition is unlikely (e.g., Alf 1-boxing) then it is also unpredictable. I’m not sure where you’re getting that from.
By way of analogy… my fire alarm is, generally speaking, the sort of thing that remains silent… if I observe it in six-minute intervals for a thousand observations, I’m pretty likely to find it silent in each case. However, if I’m a good predictor of fire alarm behavior, I don’t therefore assume that if there’s a fire, it will still remain silent.
Rather, as a good predictor of fire alarms, what my model of fire alarms tells me is that “when there’s no fire, I’m .99+ confident it will remain silent; when there is a fire, I’m .99+ confident it will make noise.” I can therefore test to see if there’s a fire and, if there is, predict it will make noise. Its noise is rare, but predictable (for a good enough predictor of fire alarm behavior).
Remember I have two models of how Omega could work.
1) Omega is in essence an excellent judge of character. It can reliably decide which of its candidates is “the sort of person who 1-boxes” and which is “the sort of person who 2-boxes”. However, if he chooser actually does something extremely unlikely and out of character, Omega will get its prediction wrong. This is a model for Omega that I could actually see working, so it is the most natural way for me to interpret Newcomb’s thought experiment.
If Omega behaves like this, then I think causal and evidential decision theory align. Both tell the chooser to 2-box, unless the chooser has already pre-committed to 1-boxing. Both imply the chooser should pre-commit to 1-boxing (if they can).
2) Omega is a perfect predictor, and always gets its predictions right. I can’t actually see how his model would work without reverse causation. If reverse causatiion is implied by the problem statement, or choosers can reasonably think it is implied, then both causal and evidential decision theory align and tell the chooser to 1-box.
From the sound of things, you are describing a third model in which Omega can not only judge character, but can also reliably decide whether someone will act out of character or not. When faced with “the sort of person who 1-boxes”, but then—out of character − 2 boxes after all, Omega will still with high probability guess correctly that the 2-boxing is going to happen, and so withhold the $ 1 million.
I tend to agree that in this third model causal and evidential decision theory may become decoupled, but again I’m not really sure how this model works, or whether it requires backward causation again. I think it could work if the causal factors leading the chooser to act “out of character” in the particular case are already embedded in the chooser’s brain state when scanned by Omega, so at that stage it is already highly probable that the chooser will act out of character this time. But the model won’t work if the factors causing out of character behaviour arise because of very rare, random, brain events happening after the scanning (say a few stray neurons fire which in 99% of cases wouldn’t fire after the scanned brain state, and these cause a cascade eventually leading to a different choice). Omega can’t predict that type of event without being a pre-cog.
Thanks anyway though; you’ve certainly made me think about the problem a bit further...
So, what does it mean for a brain to do one thing 99% of the time and something else 1% of the time?
If the 1% case is a genuinely random event, or the result of some mysterious sort of unpredictable free will, or otherwise something that isn’t the effect of the causes that precede it, and therefore can’t be predicted short of some mysterious acausal precognition, then I agree that it follows that if Omega is a good-but-not-perfect predictor, then Omega cannot predict the 1% case, and Newcomb’s problem in its standard form can’t be implemented even in principle, with all the consequences previously discussed.
Conversely, if brain events—even rare ones—are instead the effects of causes that precede them, then a good-but-not-perfect predictor can make good-but-not-perfect predictions of the 1% case just as readily as the 99% case, and these problems don’t arise.
Personally, I consider brain events the effects of causes that precede them. So if I’m the sort of person who one-boxes 99% of the time and two-boxes 1% of the time, and Omega has a sufficient understanding of the causes of human behavior to make 95% accurate predictions of what I do, then Omega will predict 95% of my (common) one-boxing as well as 95% of my (rare) two-boxing. Further, if I somehow come to believe that Omega has such an understanding, then I will predict that Omega will predict my (rare) two-boxing, and therefore I will predict that two-boxing loses me money, and therefore I will one-box stably.
For the sake of the least convenient world assume that the brain is particularly sensitive to quantum noise. This applies in the actual world too albeit at a far, far lower rate than 1% (but hey… perfect). That leaves a perfect predictor perfectly predicting that in the branches with most of the quantum goo (pick a word) the brain will make one choice while in the others it will make the other.
In this case it becomes a matter of how the counterfactual is specified. The most appropriate one seems to be with Omega filling the large box with an amount of money proportional to how much of the brain will be one boxing. A brain that actively flips a quantum coin would then be granted a large box with half the million.
The only other obvious alternative specification of Omega that wouldn’t break the counterfactual given this this context are a hard cutoff and some specific degree of ‘probability’.
As you say the one boxing remains stable under this uncertainty and even imperfect predictors.
I’m not sure what the quantum-goo explanation is adding here.
If Omega can’t predict the 1% case (whether because it’s due to unpredictable quantum goo, or for whatever other reason… picking a specific explanation only subjects me to a conjunction fallacy) then Omega’s behavior will not reflect the 1% case, and that completely changes the math. Someone for whom the 1% case is two-boxing is then entirely justified in two-boxing in the 1% case, since they ought to predict that Omega cannot predict their two-boxing. (Assuming that they can recognize that they are in such a case. If not, they are best off one-boxing in all cases. Though it follows from our premises that they will two-box 1% of the time anyway, though they might not have any idea why they did that. That said, compatibilist decision theory makes my teeth ache.)
Anyway, yeah, this is assuming some kind of hard cutoff strategy, where Omega puts a million dollars in a box for someone it has > N% confidence will one-box.
If instead Omega puts N% of $1m in the box if Omega has N% confidence the subject will one-box, the result isn’t terribly different if Omega is a good predictor.
I’m completely lost by the “proportional to how much of the brain will be one boxing” strategy. Can you say more about what you mean by this? It seems likely to me that most of the brain neither one-boxes nor two-boxes (that is, is not involved in this choice at all) and most of the remainder does both (that is, performs the same operations in the two-boxing case as in the one-boxing case).
A perfect predictor will predict correctly and perfectly that the brain both one boxes and two boxes in different Everett branches (with vastly different weights). This is different in nature to an imperfect predictor that isn’t able to model the behavior of the brain with complete certainty yet given preferences that add up to normal it requires that you use the same math. It means you do not have to abandon the premise “perfect predictor” for the probabilistic reasoning to be necessary.
How much weight the everett branches in which it one box have relative to the everett branches in which it two boxes.
Allow me to emphasise:
(I think we agree?)
Ah, I see what you mean.
Yes, I think we agree. (I had previously been unsure.)
Assume that the person choosing the boxes is a whole brain emulation, and that Omega runs a perfect simulation, which guarantees formal identity of Omega’s prediction and person’s actual decision, even though the computations are performed separately.
So the chooser in this case is a fully deterministic system, not a real-live human brain with some chance of random firings screwing up Omega’s prediction?
Wow, that’s an interesting case, and I hadn’t really thought about it! One interesting point though—suppose I am the chooser in that case; how can I tell which simulation I am? Am I the one which runs after Omega made its prediction? Or am I the one which Omega runs in order to make its prediction, and which does have a genuine causal effect on what goes in the boxes? It seems I have no way of telling, and I might (in some strange sense) be both of them. So causal decision theory might advise me to 1-box after all.
This is more of a way of pointing out a special case that shares relevant considerations with TDT-like approach to decision theory (in this extreme identical-simulation case it’s just Hofstadter’s “superrationality”).
If we start from this case and gradually make the prediction model and the player less and less similar to each other (perhaps by making the model less detailed), at which point do the considerations that make you one-box in this edge case break? Clearly, if you change the prediction model just a little bit, correct answer shouldn’t immediately flip, but CDT is no longer applicable out-of-the-box (arguably, even if you “control” two identical copies, it’s also not directly applicable). Thus, a need for generalization that admits imperfect acausal “control” over sufficiently similar decision-makers (and sufficiently accurate predictions) in the same sense in which you “control” your identical copies.
That might give you the right answer when Omega is simulating you perfectly, but presumably you’d want to one-box when Omega was simulating a slightly lossy, non-sentient version of you and only predicted correctly 90% of the time. For that (i.e. for all real world Newcomblike problems), you need a more sophisticated decision theory.
Well no, not necessarily. Again, let’s take Alf’s view. (Note I edited this post recently to correct the display of the matrices)
Remember that Alf has a high probability of 2 boxing, and he knows this about himself. Whether he would actually do better by 1-boxing will depend how well Omega’s “mistaken” simulations are correlated with the (rare, freaky) event that Alf 1 boxes. Basically, Alf knows that Omega is right at least 90% of the time, but this doesn’t require a very sophisticated simulation at all, certainly not in Alf’s own case. Omega can run a very crude simulation, say “a clear” 2-boxer, and not fill box B (so Alf won’t get the $ 1 million. Basically, the game outcome would have a probability matrix like this:
Notice that Omega has less than 1% chance of a mistaken prediction.
But, I’m sure you’re thinking, won’t Omega run a fuller simulation with 90% accuracy and produce a probability matrix like this?
Well Omega could do that, but now its probability of error has gone up from 1% to 10%, so why would Omega bother?
Let’s compare to a more basic case: weather forecasting. Say I have a simulation model which takes in the current atmospheric state above a land surface, runs it forward a day, and tries to predict rain. It’s pretty good: if there is going to be rain, then the simulation predicts rain 90% of the time; if there is not going to be rain, then it predicts rain only 10% of the time. But now someone shows me a desert, and asks me to predict rain: I’m not going to use a simulation with a 10% error rate, I’m just going to say “no rain”.
So it seems in the case of Alf. Provided Alf’s chance of 1-boxing is low enough (i.e. lower than the underlying error rate of Omega’s simulations) then Omega can always do best by just saying “a clear 2-boxer” and not filling the B box. Omega may also say to himself “what an utter schmuck” but he can’t fault Alf’s application of decision theory. And this applies whether or not Alf is a causal decision theorist or an evidential decision theorist.
Incidentally, your fire alarm may be practically useless in the circumstances you describe. Depending on the relative probabilities (small probability that the alarm goes off when there is not a fire versus even smaller probability that there genuinely is a fire) then you may find that essentially all the alarms are false alarms. You may get fed up responding to false alarms and ignore them. When predicting very rare events, the prediction system has to be extremely accurate.
This is related to the analysis below about Omega’s simulation being only 90% accurate versus a really convinced 2-boxer (who has only a 1% chance of 1-boxing). Or of simulating rain in a desert.