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The values of A, C and P are all equivalent. You insist on making CDT determine C in a model where it does not know these are correlated. This is a problem with your model.
This only shows that the model is no good, because the model does not respect the assumptions of the decision theory.
Decision theories do not compute what the world will be like. Decision theories select the best choice, given a model with this information included. How the world works is not something a decision theory figures out, it is not a physicist and it has no means to perform experiments outside of its current model. You need take care of that yourself, and build it into your model.
If a decision theory had the weakness that certain, possible scenarios could not be modeled, that would be a problem. Any decision theory will have the feature that they work with the model they are given, not with the model they should have been given.
You are applying a decision theory to the node C, which means you are implicitly stating: there are multiple possible choices to be made at this point, and this decision can be made independent of nodes not in front of this one. This means that your model does not model the Newcomb's problem we have been discussing - it models another problem, where C can have values independent of P, which is indeed solved by two-boxing.
It is not the decision theory's responsibility to know that the values of node C is somehow supposed to retrospectively alter the state of the branch the decision theory is working in. This is, however,a consequence of the modelling you do. You are on purpose applying CDT too late in your network, such that P and thus the cost of being a two-boxer has gone over the horizon and such that the node C must affect P backwards, not because the problem actually contains backwards causality, but because you want to fix the value of nodes in the wrong order.
If you do not want to make the assumption of free choice at C, then you can just not promote it to an action node. If the decision at C is casually determined from A, then you can apply a decision theory at node A and follow the causal inference. Then you will, once again, get a correct answer from CDT, this time for the version of Newcomb's problem where A and C are fully correlated.
If you refuse to reevaluate your model, then we might as well leave it at this. I do agree that if you insist on applying CDT at C in your model, then it will two-box. I do not agree that this is a problem.
Could you try to maybe give a straight answer to, what is your problem with my model above? It accurately models the situation. It allows CDT to give a correct answer. It does not superficially resemble the word for word statement of Newcomb's problem.
Therefore, even if the CDT algorithm knows that its choice is predetermined, it cannot make use of that in its decision, because it cannot update contrary to the direction of causality.
You are trying to use a decision theory to determine which choice an agent should make, after the agent has already had its algorithm fixed, which causally determines which choice the agent must make. Do you honestly blame that on CDT?
If you apply CDT at T=4 with a model which builds in the knowledge that the choice C and the prediction P are perfectly correlated, it will one-box. The model is exceedingly simple:
- T'=0: Choose either C1 or C2
- T'=1: If C1, then gain 1000. If C2, then gain 1.
This excludes the two other impossibilities, C1P2 and C2P1, since these violate the correlation constraint. CDT makes a wrong choice when these two are included, because then you have removed the information of the correlation constraint from the model, changing the problem to one in which Omega is not a predictor.
What is your problem with this model?
If you take a careful look at the model, you will realize that the agent has to be precommited, in the sense that what he is going to do is already fixed. Otherwise, the step at T=1 is impossible. I do not mean that he has precommited himself consciously to win at Newcomb's problem, but trivially, a deterministic agent must be precommited.
It is meaningless to apply any sort of decision theory to a deterministic system. You might as well try to apply decision theory to the balls in a game of billiards, which assign high utility to remaining on the table but have no free choices to make. For decision theory to have a function, there needs to be a choice to be made between multiple, legal options.
As far as I have understood, your problem is that, if you apply CDT with an action node at T=4, it gives the wrong answer. At T=4, there is only one option to choose, so the choice of decision theory is not exactly critical. If you want to analyse Newcomb's problem, you have to insert an action node at T<1, while there is still a choice to be made, and CDT will do this admirably.
Playing prisoner's dilemma against a copy of yourself is mostly the same problem as Newcomb's. Instead of Omega's prediction being perfectly correlated with your choice, you have an identical agent whose choice will be perfectly correlated with yours - or, possibly, randomly distributed in the same manner. If you can also assume that both copies know this with certainty, then you can do the exact same analysis as for Newcomb's problem.
Whether you have a prediction made by an Omega or a decision made by a copy really does not matter, as long as they both are automatically going to be the same as your own choice, by assumption in the problem statement.
Excellent.
I think laughably stupid is a bit too harsh. As I understand thing, confusion regarding Newcomb's leads to new decision theories, which in turn makes the smoking lesion problem interesting because the new decision theories introduce new, critical weaknesses in order to solve Newcomb's problem. I do, agree, however, that the smoking lesion problem is trivial if you stick to a sensible, CDT model.
We do, by and large, agree. I just thought, and still think, the terminology is somewhat misleading. This is probably not a point I should press, because I have no mandate to dictate how words should be used, and I think we understand each other, but maybe it is worth a shot.
I fully agree that some values in the past and future can be correlated. This is more or less the basis of my analysis of Newcomb's problem, and I think it is also what you mean by imposing constraints on the past light cone. I just prefer to use different words for backwards correlation and forwards causation.
I would say that the robot getting the extra pack necessitates that it had already been charged and did not need the extra pack, while not having been charged earlier would cause it to fail to recharge itself. I think there is a significant difference between how not being charged causes the robot to run out of power, versus how running out of power necessitates that is has not been charged.
You may of course argue that the future and the past are the same from the viewpoint of physics, and that either can said to cause another. However, as long as people consider the future and the past to be conceptually completely different, I do not see the hurry to erode these differences in the language we use. It probably would not be a good idea to make tomorrow refer to both the day before and the day after today, either.
I guess I will repeat: This is probably not a point I should press, because I have no mandate to dictate how words should be used.
I agree with the content, though I am not sure if I approve of a terminology where causation traverses time like a two-way street.
I tend to agree with mwengler - value is not a property of physical objects or world states, but a property of an observer having unequal preferences for different possible futures.
There is a risk we might be disagreeing because we are working with different interpretations of emotion.
Imagine a work of fiction involving no sentient beings, not even metaphorically - can you possibly write a happy or tragic ending? Is it not first when you introduce some form of intelligence with preferences that destruction becomes bad and serenity good? And are not preferences for this over that the same as emotion?
I do not want to make estimates on how and with what accuracy Omega can predict. There is not nearly enough context available for this. Wikipedia's version has no detail whatsoever on the nature of Omega. There seems to be enough discussion to be had, even with the perhaps impossible assumption that Omega can predict perfectly, always, and that this can be known by the subject with absolute certainty.
I do not think the standard usage is well defined, and avoiding these terms altogether is not possible, seeing as they are in the definition of the problem we are discussing.
Interpretations of the words and arguments for the claim are the whole content of the ancestor post. Maybe you should start there instead of quoting snippets out of context and linking unrelated fallacies? Perhaps, by specifically stating the better and more standard interpretations?
Then I guess I will try to leave it to you to come up with a satisfactory example. The challenge is to include Newcomblike predictive power for Omega, but not without substantiating how Omega achieves this, while still passing your own standards of subject makes choice from own point of view. It is very easy to accidentally create paradoxes in mathematics, by assuming mutually exclusive properties for an object, and the best way to discover these is generally to see if it is possible construct or find an instance of the object described.
I don't think it is, actually. It just seems so because it presupposes that your own choice is predetermined, which is kind of hard to reason with when you're right in the process of making the choice. But that's a problem with your reasoning, not with the scenario. In particular, the CDT agent has a problem with conceiving of his own choice as predetermined, and therefore has trouble formulating Newcomb's problem in a way that he can use - he has to choose between getting two-boxing as the solution or assuming backward causation, neither of which is attractive.
This is not a failure of CDT, but one of your imagination. Here is a simple, five minute model which has no problems conceiving Newcomb's problem without any backwards causation:
- T=0: Subject is initiated in a deterministic state which can be predicted by Omega.
- T=1: Omega makes an accurate prediction for the subject's decision in Newcomb's problem by magic / simulation / reading code / infallible heuristics. Denote the possible predictions P1 (one-box) and P2.
- T=2: Omega sets up Newcomb's problem with appropriate box contents.
- T=3: Omega explains the setup to the subject and disappears.
- T=4: Subject deliberates.
- T=5: Subject chooses either C1 (one-box) or C2.
- T=6: Subject opens box(es) and receives payoff dependent on P and C.
You can pretend to enter this situation at T=4 as suggested by the standard Newcomb's problem. Then you can use the dominance principle and you will lose. But this just using a terrible model. You entered at T=0, because you were needed at T=1 for Omega's inspection. If you did not enter the situation at T=0, then you can freely make a choice C at T=5 without any correlation to P, but that is not Newcomb's problem.
Instead, at T=4 you become aware of the situation, and your decision making algorithm must return a value for C. If you consider this only from T=4 and onward, this is completely uninteresting, because C is already determined. At T=1, P was determined to either P1 or P2, and the value of C follow directly from this. Obviously, healthy one-boxing code wins and unhealthy two-boxing code loses, but there is no choice being made here, just different code with different return values being rewarded differently, and that is not Newcomb's problem either.
Finally, we will work under illusion of choice with Omega as a perfect predictor. We realize that T=0 is the critical moment, seeing as all subsequent T follows directly from this. We work backwards as follows:
- T=6: My preferences are P1C2 > P1C1 > P2C2 > P2C1.
- T=5: I should choose either C2 or C1 depending on the current value of P.
- T=4: this is when all this introspection is happening
- T=3: this is why
- T=2: I would really like there to be a million dollars present.
- T=1: I want Omega to make prediction P1.
- T=0: Whew, I'm glad I could do all this introspection which made me realize that I want P1 and the way to achieve this is C1. It would have been terrible if my decision making just worked by the dominance principle. Luckily, the epiphany I just had, C1, was already predetermined at T=0, Omega would have been aware of this at T=1 and made the prediction P1, so (...) and P1 C1 = a million dollars is mine.
Shorthand version of all the above; if the decision is necessarily predetermined before T=4, then you should not pretend you make the decision at T=4. Insert a decision making step at T=0.5, which causally determines the value of P and C. Apply your CDT to this step.
This is the only way of doing CDT honestly, and it is the slightest bit messy, but that is exactly what happens when you create a reference to the decision the decision theory is going to make in the future in the problem itself with perfect correlation to the decision before the decision has overtly been made. This sort of self reference creates impossibilities out of the thin air every day of week, such as when Pinocchio says my nose will grow now. The good news is that this way of doing it is a lot less messy than inventing a new, superfluous decision theory, and it also allows you to deal with problems like the psychopath button without any trouble whatsoever.
The post scav made more or less represents my opinion here. Compatibilism, choice, free will and determinism are too many vague definitions for me to discuss with. For compatibilism to make any sort of sense to me, I would need a new definition of free will. It is already difficult to discuss how stuff is, without simultaneously having to discuss how to use and interpret words.
Trying to leave the problematic words out of this, my claim is that the only reason CDT ever gives a wrong answer in a Newcomb's problem is that you are feeding it the wrong model. http://lesswrong.com/lw/gu1/decision_theory_faq/8kef elaborates on this without muddying the waters too much with the vaguely defined terms.
I think the barbering example is excellent - it illustrates that, while controlled experiments more or less is physics, and while physics is great, it is probably not going to bring a paradigm shift to barbering any time soon. One should not expect all domains to be equally well suited to a cut and dried scientific approach.
Where medicine lies on this continuum of suitedness is an open question - it is probably even a misleading question, with medicine being a collection of vastly different problems. However, it is not at all obvious that simply turning up the scientificness dial is going to make things better. It is for instance conceivable that there are already people treating medicine as a hard science, and that the current balance of intuition and evidence in medicine reflects how effective these two approaches are.
I am not trying to argue whether astrology is evidence-based or not. I am saying that the very inclusive definition of evidence-based which encompasses barbering is, (a) nearly useless because it includes every possible way of doing medicine and (b) probably not the one intended by the others using the term.
If you interpret evidence-based in the widest sense possible, the phrase sort of loses its meaning. Note that the very post you quote explains the intended contrast between systematic and statistical use of evidence versus intuition and traditional experience based human learning.
Besides, would you not say that astrologers figure out both how to be optimally vague, avoiding being wrong while exciting their readers, much the same way musicians figure out what sounds good?
Ironically, this whole exchange might have been a bit more constructive with less taking of offense.
I think I agree, by and large, despite the length of this post.
Whether choice and predictability are mutually exclusive depends on what choice is supposed to mean. The word is not exactly well defined in this context. In some sense, if variable > threshold then A, else B is a choice.
I am not sure where you think I am conflating. As far as I can see, perfect prediction is obviously impossible unless the system in question is deterministic. On the other hand, determinism does not guarantee that perfect prediction is practical or feasible. The computational complexity might be arbitrarily large, even if you have complete knowledge of an algorithm and its input. I can not really see the relevance to my above post.
Finally, I am myself confused as to why you want two different decision theories (CDT and EDT) instead of two different models for the two different problems conflated into the single identifier Newcomb's paradox. If you assume a perfect predictor, and thus full correlation between prediction and choice, then you have to make sure your model actually reflects that.
Let's start out with a simple matrix, P/C/1/2 are shorthands for prediction, choice, one-box, two-box.
- P1 C1: 1000
- P1 C2: 1001
- P2 C1: 0
- P2 C2: 1
If the value of P is unknown, but independent of C: Dominance principle, C=2, entirely straightforward CDT.
If, however, the value of P is completely correlated with C, then the matrix above is misleading, P and C can not be different and are really only a single variable, which should be wrapped in a single identifier. The matrix you are actually applying CDT to is the following one:
- (P&C)1: 1000
- (P&C)2: 1
The best choice is (P&C)=1, again by straightforward CDT.
The only failure of CDT is that it gives different, correct solutions to different, problems with a properly defined correlation of prediction and choice. The only advantage of EDT is that it is easier to cheat in this information without noticing it - even when it would be incorrect to do so. It is entirely possible to have a situation where prediction and choice are correlated, but the decision theory is not allowed to know this and must assume that they are uncorrelated. The decision theory should give the wrong answer in this case.
Only in the sense that the term "pro-life" implies than there exist people opposed to life.
Maybe he means something along the lines of same cause, same effect is just a placeholder for as long as all the things which matter stay the same, you get the same effect. After all, some things, such as time since the man invented fire and position relative to Neptune and so on and so forth cannot possibly be the same for two different events. And this in turn sort of means things which matter -> same effect is a circular definition. Maybe he means to say that the law of causality is not the actually useful principle for making predictions, while there are indeed repeatable experiments and useful predictions to be made.
(Thanks for discussing!)
I will address your last paragraph first. The only significant difference between my original example and the proper Newcomb's paradox is that, in Newcomb's paradox, Omega is made a predictor by fiat and without explanation. This allows perfect prediction and choice to sneak into the same paragraph without obvious contradiction. It seems, if I try to make the mode of prediction transparent, you protest there is no choice being made.
From Omega's point of view, its Newcomb subjects are not making choices in any substantial sense, they are just predictably acting out their own personality. That is what allows Omega its predictive power. Choice is not something inherent to a system, but a feature of an outsider's model of a system, in much the same sense as random is not something inherent to a Eeny, meeny, miny, moe however much it might seem that way to children.
As for the rest of our disagreement, I am not sure why you insist that CDT must work with a misleading model. The standard formulation of Newcomb's paradox is inconsistent or underspecified. Here are some messy explanations for why, in list form:
- Omega predicts accurately, then you get to choose is a false model, because Omega has predicted you will two-box, then you get to choose does not actually let you choose; one-boxing is an illegal choice, and two-boxing the only legal choice (In Soviet Russia joke goes here)
- You get to choose, then Omega retroactively fixes the contents of the boxes is fine and CDT solves it by one-boxing
- Omega tries to predict but is just blindly guessing, then you really get to choose is fine and CDT solves it by two-boxing
- You know that Omega has perfect predictive power and are free to be committed to either one- or two-boxing as you prefer is nowhere near similar to the original Newcomb's formulation, but is obviously solved by one-boxing
- You are not sure about Omega's predictive power and are torn between trying to 'game' it and cooperating with it is not Newcomb's problem
- Your choice has to be determined by a deterministic algorithm, but you are not allowed to know this when designing the algorithm, so you must instead work in ignorance and design it by a false dominance principle is just cheating
I am not sure where our disagreement lies at the moment.
Are you using choice to signify strongly free will? Because that means the hypothetical Omega is impossible without backwards causation, leaving us at (b) but not (a) and the whole of Newcomb's paradox moot. Whereas, if you include in Newcomb's paradox, the choice of two-boxing will actually cause the big box to be empty, whereas the choice of one-boxing will actually cause the big box to contain a million dollars by a mechanism of backwards causation, then any CDT model will solve the problem.
Perhaps we can narrow down our disagreement by taking the following variation of my example, where there is at least a bit more of choice involved:
Imagine John, who never understood why he gets thirsty. Despite there being a regularity in when he chooses to drink, this is for him a mystery. Every hour, Omega must predict whether John will choose to drink within the next hour. Omega's prediction is made secret to John until after the time interval has passed. Omega and John play this game every hour for a month, and it turns out that while far from perfect, Omega's predictions are a bit better than random. Afterwards, Omega explains that it beats blind guesses by knowing that John will very rarely wake up in the middle of the night to drink, and that his daily water consumption follows a normal distribution with a mean and standard deviation that Omega has estimated.
Thanks for the link.
I like how he just brute forces the problem with (simple) mathematics, but I am not sure if it is a good thing to deal with a paradox without properly investigating why it seems to be a paradox in the first place. It is sort of like saying that this super convincing card trick you have seen, there is actually no real magic involved without taking time to address what seems to require magic and how it is done mundanely.
I do not agree that a CDT must conclude that P(A)+P(B) = 1. The argument only holds if you assume the agent's decision is perfectly unpredictable, i.e. that there can be no correlation between the prediction and the decision. This contradicts one of the premises of Newcomb's Paradox, which assumes an entity with exactly the power to predict the agent's choice. Incidentally, this reduces to the (b) but not (a) from above.
By adopting my (a) but not (b) from above, i.e. Omega as a programmer and the agent as predictable code, you can easily see that P(A)+P(B) = 2, which means one-boxing code will perform the best.
Further elaboration of the above:
Imagine John, who never understood how the days of the week succeed each other. Rather, each morning, a cab arrives to take him to work if it is a work day, else he just stays at home. Omega must predict if he will go to work or not the before the cab would normally arrive. Omega knows that weekdays are generally workdays, while weekends are not, but Omega does not know the ins and outs of particular holidays such as fourth of July. Omega and John play this game each day of the week for a year.
Tallying the results, John finds that the score is as follows: P( Omega is right | I go to work) = 1.00, P( Omega is right | I do not go to work) = 0.85, which sums to 1.85. John, seeing that the sum is larger than 1.00, concludes that Omega seems to have rather good predictive power about whether he will go to work, but is somewhat short of perfect accuracy. He realizes that this has a certain significance for what bets he should take with Omega, regarding whether he will go to work tomorrow or not.
I don't really think Newcomb's problem or any of its variations belong in here. Newcomb's problem is not a decision theory problem, the real difficulty is translating the underspecified English into a payoff matrix.
The ambiguity comes from the the combination of the two claims, (a) Omega being a perfect predictor and (b) the subject being allowed to choose after Omega has made its prediction. Either these two are inconsistent, or they necessitate further unstated assumptions such as backwards causality.
First, let us assume (a) but not (b), which can be formulated as follows: Omega, a computer engineer, can read your code and test run it as many times as he would like in advance. You must submit (simple, unobfuscated) code which either chooses to one- or two-box. The contents of the boxes will depend on Omega's prediction of your code's choice. Do you submit one- or two-boxing code?
Second, let us assume (b) but not (a), which can be formulated as follows: Omega has subjected you to the Newcomb's setup, but because of a bug in its code, its prediction is based on someone else's choice than yours, which has no correlation with your choice whatsoever. Do you one- or two-box?
Both of these formulations translate straightforwardly into payoff matrices and any sort of sensible decision theory you throw at them give the correct solution. The paradox disappears when the ambiguity between the two above possibilities are removed. As far as I can see, all disagreement between one-boxers and two-boxers are simply a matter of one-boxers choosing the first and two-boxers choosing the second interpretation. If so, Newcomb's paradox is not as much interesting as poorly specified. The supposed superiority of TDT over CDT either relies on the paradox not reducing to either of the above or by fiat forcing CDT to work with the wrong payoff matrices.
I would be interested to see an unambiguous and nontrivial formulation of the paradox.
Some quick and messy addenda:
- Allowing Omega to do its prediction by time travel directly contradicts box B contains either $0 or $1,000,000 before the game begins, and once the game begins even the Predictor is powerless to change the contents of the boxes. Also, this obviously make one-boxing the correct choice.
- Allowing Omega to accurately simulate the subject reduces to problem to submit code for Omega to evaluate; this is not exactly paradoxical, but then the player is called upon to choose which boxes to take actually means the code then runs and returns its expected value, which clearly reduces to one-boxing.
- Making Omega an imperfect predictor, with an accuracy of p<1.0 simply creates a superposition of the first and second case above, which still allows for straightforward analysis.
- Allowing unpredictable, probabilistic strategies violates the supposed predictive power of Omega, but again cleanly reduces to payoff matrices.
- Finally, the number of variations such as the psychopath button are completely transparent, once you decide between choice is magical and free will and stuff which leads to pressing the button, and the supposed choice is deterministic and there is no choice to make, but code which does not press the button is clearly the most healthy.